indiscrete topology 平凡拓扑
The topology on a set whose only open subsets are the empty set and the whole set.
Terminology & Translations
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induced subgraph 导出子图
For a simple graph G=(V,E) and W ⊆ V, the induced subgraph G[W] has vertex set W and exactly the edges of E whose two endpoints belong to W.
infimum of a real set 下确界
For a nonempty real set A bounded below, its infimum is the greatest real lower bound: it is no larger than any member of A and dominates every other lower bound.
initial object 始对象
An object I having exactly one morphism I→X to every object X of its category.
injective function 单射
A function $f:A\to B$ for which $f(x)=f(y)$ implies $x=y$ for all $x,y\in A$.
integer congruence 整数同余
For $m\ge1$, integers $a,b$ satisfy $a\equiv b\pmod m$ exactly when $m\mid(a-b)$.
integer divisibility 整数整除
For integers $a,b$, $a\mid b$ means that $b=ak$ for an integer $k$.
integral basis of a number field 数域的整基
A $\mathbb Z$-basis of the additive group $\mathcal O_K$ for a number field $K$.
integral domain 整环
A commutative unital ring with $1\ne0$ in which $ab=0$ implies $a=0$ or $b=0$.
interior of a subset 内部
The union of all open subsets of the ambient space contained in the given subset.
intersection of sets 交
The intersection of A and B contains exactly the objects belonging to both.
invariant measure 不变测度
For a measurable $T:X\to X$, a measure $\mu$ is invariant if $\mu(T^{-1}A)=\mu(A)$ for every measurable set $A$.
invariant set in topological dynamics 不变集
For a self-map $T$, a subset $A$ satisfying $T(A)=A$; this record uses equality rather than only forward inclusion.
inverse element in a group 群的逆元
For $g$ in a group with identity $e$, the unique element $g^{-1}$ with $gg^{-1}=g^{-1}g=e$.
inverse Fourier transform 逆傅立叶变换
For g in Schwartz space on R^n, the inverse Fourier transform in the nonunitary convention is (2 pi)^(-n) integral e^{i x dot xi}g(xi) dxi; it inverts the forward transform with the negative exponential.
irreducible element 不可约元
A nonzero nonunit $r$ in a domain such that $r=ab$ forces $a$ or $b$ to be a unit.
isomorphism in a category 同构
A morphism $f:X\to Y$ admitting $g:Y\to X$ with $gf=1_X$ and $fg=1_Y$.
jump discontinuity 跳跃间断点
A real function has a jump at a when both finite one-sided limits exist and are unequal.
kernel of a group homomorphism 群同态的核
For $f:G\to H$, the subgroup $\{g:f(g)=e_H\}$.
Lagrangian submanifold 拉格朗日子流形
An embedded smooth submanifold of a symplectic manifold of dimension 2n whose dimension is n and on whose tangent spaces the symplectic form vanishes. The submanifold need not be a closed subset under this entry's…
Laplace equation Laplace方程
The Laplace equation on an open set in R^n is the homogeneous second-order PDE Delta u=sum_{j=1}^n partial_j^2 u=0, classically for u in C^2 or with derivatives interpreted weakly.
Laurent series 洛朗展式
A Laurent series centered at a is a two-sided power expansion sum_{k in Z} c_k(z-a)^k, with its negative- and nonnegative-power parts converging on a specified annulus.
law of large numbers 大数定律
In the iid integrable setting, the sample averages $n^{-1}\sum_{j=1}^n X_j$ converge to $E[X_1]$ almost surely (strong law), and therefore in probability (weak law), assuming real iid $X_j$ and $E|X_1|<\infty$.
Lebesgue integral 勒贝格积分
For nonnegative measurable f on a measure space, its Lebesgue integral is the supremum of integrals of nonnegative simple functions below f. For signed f, subtract the negative-part integral from the positive-part…
Lebesgue measurable set 勒贝格可测集
A subset E of R^n is Lebesgue measurable if m*(A)=m*(A intersect E)+m*(A minus E) for every subset A of R^n, where m* is Lebesgue outer measure.