division algorithm for integers 整数带余数除法
For $a\in\mathbb Z$ and $b>0$, there are unique integers $q,r$ with $a=bq+r$ and $0\le r<b$.
Terminology & Translations
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dominated convergence theorem 控制收敛定理
If measurable f_n converge almost everywhere to a measurable f and |f_n|<=g almost everywhere for one nonnegative integrable g, then f is integrable and the integrals of f_n converge to that of f.
eigenvalue of a linear map 线性映射的特征值
A scalar $\lambda\in F$ for which an endomorphism $T$ has a nonzero vector $v$ with $Tv=\lambda v$.
empty set 空集
The unique set with no members.
entire function 整函数
An entire function is a complex-valued function holomorphic at every finite point of C.
equinumerosity 等势
The relation between sets A and B that holds precisely when a bijection A→B exists.
equivalence class 等价类
For an equivalence relation on X, the class [x] consists of all elements related to x.
equivalence of categories 范畴等价
A functor F:C→D with a functor G:D→C and natural isomorphisms GF≅1_C and FG≅1_D.
equivalence relation 等价关系
A binary relation on a specified set that is reflexive, symmetric and transitive.
ergodic transformation 遍历变换
A probability-preserving map $T$ such that $\mu(T^{-1}A\triangle A)=0$ implies $\mu(A)=0$ or $1$ for every measurable $A$.
essential isolated singularity 本性奇点
An isolated singularity of a holomorphic function is essential if it is neither removable nor a pole; equivalently its Laurent expansion has infinitely many nonzero negative-power coefficients.
Euclidean algorithm for integers 整数辗转相除法
Repeatedly replace a nonnegative integer pair $(a,b)$ with $(b,a\bmod b)$ while $b>0$; when $b=0$, the terminal first coordinate $a$ is the gcd of the original pair, including $\gcd(0,0)=0$ by convention.
Euclidean domain Euclid 环
A domain with a function $\delta:R\setminus\{0\}\to\mathbb Z_{\ge0}$ such that $a=bq+r$ with $r=0$ or $\delta(r)<\delta(b)$ whenever $b\ne0$.
Euler characteristic of a finite CW complex Euler 数
For a finite CW complex, the integer obtained by summing the number of cells in each dimension with alternating signs, starting positive in dimension zero; it is independent of the chosen finite cell structure.
Euler totient function 欧拉函数
For $n\ge1$, $\varphi(n)$ counts the integers $1\le a\le n$ satisfying $\gcd(a,n)=1$.
Euler's theorem on congruences 欧拉定理
If $n\ge2$ and $\gcd(a,n)=1$, then $a^{\varphi(n)}\equiv1\pmod n$.
event 随机事件
A member $A$ of the sigma-algebra $\mathcal F$ in a probability space, so that $P(A)$ is defined.
existential quantifier 存在量词
$\exists x\,\varphi$ holds when some domain value for $x$ makes $\varphi$ true under the remaining assignment.
expectation 数学期望
For an integrable real random variable $X$, its expectation is $E[X]=\int_\Omega X\,dP$; for nonnegative $X$, an extended value $+\infty$ is allowed.
exponential distribution 指数分布
For rate $\lambda>0$, the nonnegative law with survival probability $P(X>t)=e^{-\lambda t}$ for $t\ge0$.
exponential generating function 指数生成函数
For a sequence (a_n) in a commutative algebra over the rational numbers, its exponential generating function is the formal series ∑ a_n x^n/n!.
Ext group of modules 模的 Ext 群
For left $R$-modules $M,N$, $\operatorname{Ext}^n_R(M,N)$ is degree-$n$ cohomology of $\operatorname{Hom}_R(P_\bullet,N)$ for a projective resolution $P_\bullet\to M$.
exterior derivative 外微分运算
The degree-one real linear operator on smooth differential forms that agrees with the differential on functions, squares to zero, and satisfies the graded product rule; in local coordinates it differentiates each…
extremal graph theory 极值图论
Extremal graph theory studies optimal values of graph parameters under structural restrictions, such as the maximum edge count of an n-vertex graph avoiding a fixed subgraph.
faithful functor 忠实函子
A functor inducing an injective map on each hom-set for a fixed pair of source objects.