singular cohomology 奇异上同调群
For an abelian coefficient group A, the cohomology of Hom(C_*(X;Z),A), with coboundary given by precomposing a cochain with the singular boundary.
Terminology & Translations
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singular homology 奇异同调
The homology of the free chain complex generated by continuous standard simplices in X, using alternating face boundaries; coefficients are integers unless stated otherwise.
singular simplex 奇异单形
A continuous map from the standard n-simplex with ordered vertices into a topological space, for some nonnegative integer n.
small category 小范畴
A category whose objects and all morphisms form sets in the chosen size convention.
smooth immersion 浸入
A smooth map between smooth manifolds whose differential is injective at every point of its domain.
smooth manifold 微分流形
A Hausdorff, second countable space locally homeomorphic to open subsets of a fixed Euclidean space, together with a smooth structure represented by charts whose transition maps are infinitely differentiable; boundary…
smooth submersion 淹没
A smooth map between smooth manifolds whose differential is surjective at every point of the domain; surjectivity of the map itself is not required.
smooth vector bundle 向量丛
A smooth projection over a smooth manifold with finite-dimensional real vector spaces of fixed rank as fibers and local product trivializations that are linear on each fiber; the changes between trivializations are…
smooth vector field 光滑向量场
A smooth section of the tangent bundle of a smooth manifold: it assigns to each point a tangent vector at that point, with smooth coefficient functions in every coordinate chart.
smooth wavefront set 波前集
For a distribution u, its smooth wavefront set consists of pairs (x,xi) with xi nonzero for which no smooth compactly supported cutoff equal to one near x has a Fourier transform of its product with u decaying faster…
Sobolev space Sobolev空间
For open Omega in R^n, an integer k>=0, and 1<=p<=infinity, W^{k,p}(Omega) consists of Lp equivalence classes whose distributional derivatives of every order at most k are represented by Lp functions.
Sobolev trace Sobolev函数的迹
For a bounded smooth domain Omega and 1 <= p < infinity, the first-order Sobolev trace is the bounded linear map from W^{1,p}(Omega) to L^p(partial Omega, dS) extending boundary restriction of smooth functions on the…
spanning tree 支撑树
A spanning tree of a nonempty finite graph G is a subgraph with the entire vertex set of G that is a tree.
spectral sequence 谱序列
In an abelian category, a sequence of differential objects $(E_r,d_r)$ with $d_r^2=0$ and specified isomorphisms $H(E_r,d_r)\cong E_{r+1}$; the cohomological bigraded version has differential of bidegree $(r,1-r)$.
spectral theorem for compact self-adjoint operators 紧自伴算子的谱定理
For a compact self-adjoint operator on a real or complex separable Hilbert space, an orthonormal basis of eigenvectors exists; nonzero eigenvalues are real with finite multiplicity and, if infinitely many, tend to zero…
splitting field 分裂域
For $f\in K[x]$, an extension generated over $K$ by all roots of $f$ in which $f$ splits into linear factors.
stabilizer of a group action 群作用的稳定化子
The subgroup $G_x=\{g\in G:gx=x\}$ fixing a specified point.
stable homotopy group 稳定同伦群
For a based CW complex X, the colimit of pi_{k+r}(Sigma^r X) under suspension homomorphisms, with r large enough to use group-valued degrees.
standard Brownian motion 布朗运动
A process $B_t$ starting at zero, with almost surely continuous paths and independent increments on disjoint intervals, such that $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$.
stochastic differential equation 随机微分方程
In the Brownian Itô setting, an equation $dX_t=b(X_t)dt+\sigma(X_t)dB_t$ means an adapted continuous process satisfies $X_t=X_0+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB_s$, with the integrals well defined.
stochastic process 随机过程
An indexed family $(X_t)_{t\in T}$ of random variables or random elements on one probability space, with a specified common state space.
Stokes theorem on manifolds Stokes 定理
For an oriented smooth n-manifold with boundary, n at least one, and a smooth compactly supported (n−1)-form ω, the integral of dω over the manifold equals the integral of the pullback of ω to its boundary, with…
stopping time 停时
For a filtration $(\mathcal F_t)$, a random time $\tau$ valued in $[0,\infty]$ is a stopping time when $\{\tau\le t\}\in\mathcal F_t$ for every $t$.
strong induction 强数学归纳法
The principle that proving $P(n)$ from all $P(k)$ with $k<n$, for each $n\in\mathbb N$, proves $P$ everywhere.
strong mixing 强混合
A probability-preserving map is strongly mixing if $\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)$ for every fixed pair of measurable events $A,B$.