heat equation 热方程
The constant-coefficient heat equation is partial_t u=kappa Delta_x u with kappa>0, for a space-time function u that is once time differentiable and twice space differentiable in the classical formulation.
Terminology & Translations
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Hecke operator on classical modular forms 经典模形式上的 Hecke 算子
On full-level weight-$k$ modular forms, for prime $p$ define $(T_pf)(\tau)=p^{k-1}f(p\tau)+p^{-1}\sum_{b=0}^{p-1}f((\tau+b)/p)$. Set $T_1=I$, $T_{mn}=T_mT_n$ for coprime $m,n$, and…
Hensel's lemma 亨泽尔引理
For prime $p$ and $f\in\mathbb Z_p[X]$, a simple root $a$ of $f\bmod p$ lifts to a unique root $\alpha\in\mathbb Z_p$ with $\alpha\equiv a\pmod p$.
higher category 高阶范畴
A categorical structure with higher morphisms and coherent composition; the scoped (∞,1) model here is a simplicial set admitting fillers for every inner horn.
higher homotopy group 同伦群
For n at least two, the abelian group of based maps from S^n to a based space modulo based homotopy, with addition from pinching the sphere into a wedge of two spheres.
Hilbert class field 希尔伯特类域
For a number field $K$, its maximal abelian extension unramified at every finite place and split at every real place.
Hilbert space Hilbert空间
A Hilbert space is a real or complex inner-product space complete for the norm $\sqrt{\langle x,x\rangle}$ induced by that inner product.
holomorphic cusp form 尖点形式
A holomorphic modular form on a congruence subgroup whose Fourier expansion at each cusp has zero constant term.
holomorphic function 全纯函数
A complex-valued function on an open subset of C is holomorphic when its complex difference quotient has a finite limit at every point, with the increment approaching zero through all complex directions.
holomorphic modular form 模形式
For integer $k$ and a congruence subgroup $\Gamma\subseteq\operatorname{SL}_2(\mathbb Z)$, a holomorphic function on the upper half-plane satisfying $f((a\tau+b)/(c\tau+d))=(c\tau+d)^k f(\tau)$ for every matrix in…
homeomorphism 同胚
A bijective continuous map of topological spaces whose inverse is also continuous.
homology of a module complex 模复形的同调
For a chain complex $C$, the module $H_n(C)=\ker d_n/\operatorname{im}d_{n+1}$.
homotopy 同伦
A continuous map H from X times the unit interval to Y whose endpoint slices are the specified continuous maps X to Y.
homotopy equivalence 同伦等价
A continuous map f:X to Y admitting a continuous g:Y to X such that both composites are homotopic to the corresponding identities.
ideal class group of a number field 数域的理想类群
The quotient of the multiplicative group of nonzero fractional $\mathcal O_K$-ideals by the subgroup of principal fractional ideals.
ideal of a commutative ring 交换环的理想
An additive subgroup $I$ of a commutative ring $R$ with $rI\subseteq I$ for every $r\in R$.
idele group of a number field 数域的伊代尔群
For a number field $K$, the restricted product of $K_v^\times$ relative to $\mathcal O_v^\times$ at finite places; algebraically it is $\mathbb A_K^\times$.
identity element of a group 群的单位元
The unique element $e$ of a group $G$ satisfying $eg=ge=g$ for all $g\in G$.
image of a subset under a function 像
For f:A→B and S⊆A, the set of values f(s) for s∈S.
implicit function theorem 隐函数定理
If F is a C^1 map near (a,b) in R^n times R^m, F(a,b)=0, and its derivative with respect to the second variable is invertible there, then nearby solutions F(x,y)=0 uniquely have y=g(x) for a C^1 map g with g(a)=b.
inclusion–exclusion principle 容斥原理
For finite sets A_1,…,A_m, the size of their union is the sum over nonempty index sets S of (−1)^(|S|+1) times the size of the intersection indexed by S.
independent and identically distributed 独立同分布
A family of random variables that is mutually independent and whose members all have the same probability law.
independent events 独立事件
Events $A_i$ are mutually independent if every finite subfamily satisfies $P(\bigcap_{i\in J}A_i)=\prod_{i\in J}P(A_i)$.
independent random variables 独立随机变量
Variables $X_i$ whose every finite subfamily satisfies $P(X_{i_j}\in B_j,\ 1\le j\le r)=\prod_{j=1}^rP(X_{i_j}\in B_j)$ for all Borel sets $B_j$.
independent set in a graph 独立集
An independent set in a simple graph is a vertex subset containing no adjacent pair of distinct vertices.