Riemann zeta function 黎曼 ζ 函数
The function $\zeta(s)=\sum_{n\ge1}n^{-s}$ for $\operatorname{Re}s>1$, continued meromorphically to $\mathbb C$ with a simple pole at $1$.
Terminology & Translations
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Riemannian geodesic 测地线
A smooth parametrized curve on a Riemannian manifold whose velocity is parallel along the curve with respect to the Levi–Civita connection, so its covariant acceleration is zero; the parameter is affine.
Riemannian metric 黎曼度量
A smooth symmetric covariant two-tensor on a smooth real manifold whose value at every point is a positive definite bilinear form on that point's tangent space.
right-continuous real function 右连续函数
A real function is right-continuous at a if its limit along domain points decreasing to a equals f(a).
ring of integers of a number field 数域的整数环
For a number field $K$, $\mathcal O_K$ is the subring of elements integral over $\mathbb Z$.
ring of p-adic integers p 进整数环
For prime $p$, $\mathbb Z_p=\{x\in\mathbb Q_p:v_p(x)\ge0\}$, equivalently the inverse limit of $\mathbb Z/p^n\mathbb Z$ under reduction.
Rouché's theorem Rouché定理
If f and g are holomorphic near the closure of a bounded domain with piecewise smooth boundary and |g-f|<|f| on that boundary, then f and g have the same number of interior zeros, with multiplicity.
sample space 样本空间
The set $\Omega$ of possible outcomes in a specified probability model.
satisfiability 可满足
A set of first-order sentences is satisfiable if one structure satisfies all of them together.
scheme 概形
A locally ringed space in which every point has an open neighborhood isomorphic, as a locally ringed space, to the prime spectrum of a commutative unital ring equipped with its structure sheaf.
Schwartz function 速降函数
A Schwartz function on R^n is a smooth complex-valued function phi such that sup_x |x^alpha partial^beta phi(x)| is finite for every pair of multi-indices alpha,beta.
second derivative 二阶导数
If f is differentiable near a and f' is differentiable at a, the second derivative f''(a) is the derivative of f' there.
second-countable space C2 可数公理
A topological space whose topology has a countable basis.
semiflow 半动力系统
A jointly continuous action $\phi:[0,\infty)\times X\to X$ satisfying $\phi_0=\operatorname{id}$ and $\phi_{t+s}=\phi_t\circ\phi_s$ for $s,t\ge0$.
sensitive dependence on initial conditions 敏感依赖于初始条件
For a map of a metric space, some $\delta>0$ has the property that for every $x$ and every neighborhood $U$ of $x$, there exist $y\in U$ and $n\ge0$ with $d(T^nx,T^ny)>\delta$.
separable algebraic extension 可分扩张
An algebraic extension $L/K$ in which every element's minimal polynomial has distinct roots in an algebraic closure.
separable space 可分
A topological space admitting a countable dense subset.
sequential compactness 列紧
The property that every sequence in a space has a subsequence converging to a point of that space.
set 集合
A primitive collection in ZF set theory, whose identity is determined by its members.
set difference 差
The difference A∖B consists of members of A that do not belong to B.
sheaf 层
A presheaf of sets on a topological space such that every compatible family of sections on an open cover, including the empty cover, is the restriction of a unique section on the union.
short exact sequence of modules 模的短正合列
A sequence $0\to A\xrightarrow{i}B\xrightarrow{p}C\to0$ with $i$ injective, $p$ surjective and $\operatorname{im}i=\ker p$.
sigma-finite measure σ有限测度
A measure mu on X is sigma-finite if X is the union of countably many measurable sets each having finite measure.
simple measurable function 简单函数
A simple function on a measurable space is a real measurable function taking finitely many values, equivalently a finite linear combination of indicators of measurable sets.
simply connected space 单连通
A nonempty path connected space whose fundamental group at one, hence every, basepoint is trivial.