open subset 开集
A subset belonging to the specified topology of a space.
Terminology & Translations
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operator norm 线性算子范数
For a bounded linear map T between normed spaces, its operator norm is sup{norm(Tx):norm(x)<=1}, equivalently the least constant in its global linear bound.
opposite category 反范畴
The category with the same objects as C, all arrows reversed, and composition performed in the reverse order.
orbit of a dynamical system 轨道
For a self-map $T$, the forward orbit of $x$ is $\{T^nx:n\ge0\}$; when $T$ is invertible, its full orbit is $\{T^nx:n\in\mathbb Z\}$.
orbit of a group action 群作用的轨道
For $x$ in a $G$-set, the subset $Gx=\{gx:g\in G\}$.
ordinary generating function 普通生成函数
For a sequence (a_n) indexed by nonnegative integers in a commutative ring R, its ordinary generating function is the formal series ∑ a_n x^n in R[[x]].
orientation of a smooth manifold 光滑流形的定向
On a smooth real n-manifold, an equivalence class of nowhere-vanishing smooth n-forms, where two forms are equivalent when one is a positive smooth function times the other; such a class exists exactly when the manifold…
p-adic additive valuation p 进加性赋值
For prime $p$ and nonzero $x\in\mathbb Q$, write $x=p^e a/b$ with $p\nmid ab$ and define $v_p(x)=e$; set $v_p(0)=+\infty$.
p-adic number field p 进数域
For prime $p$, $\mathbb Q_p$ is the completion of $\mathbb Q$ for the absolute value $|x|_p=p^{-v_p(x)}$.
pairwise independent events 两两独立事件
A family of events in one probability space for which $P(A_i\cap A_j)=P(A_i)P(A_j)$ whenever $i\ne j$.
parallel transport 平行移动
Given a connection on a smooth vector bundle and a smooth path in its base, the linear map between endpoint fibers that sends an initial vector to the endpoint value of the unique section along the path with zero…
Parseval identity for Fourier series Parseval等式
For a 2 pi-periodic function f in L2 and coefficients c_k=(2 pi)^(-1) integral f(x)e^{-ikx} dx, Parseval's identity states (2 pi)^(-1) integral_{-pi}^pi |f|^2=sum_{k in Z}|c_k|^2.
partial derivative 偏导数
For f on an open subset of R^n, its jth partial derivative at a is the finite limit [f(a+h e_j)-f(a)]/h as nonzero h tends to zero, when that limit exists.
partial order 偏序
A reflexive, antisymmetric and transitive binary relation on a set.
path component 道路分支
An equivalence class under the relation that two points can be joined by a path in the space.
path in a graph 路
In a simple graph, a path is a finite sequence of distinct vertices whose consecutive pairs are edges; its length is the number of consecutive pairs.
path-connected space 道路连通
A space in which every pair of points is joined by a continuous map from the unit interval.
Pell equation 佩尔方程
For a positive nonsquare integer $d$, the equation $x^2-dy^2=1$ with integer unknowns $x,y$.
perfect matching 完美匹配
A perfect matching in a finite simple graph is a matching incident with every vertex.
periodic point 周期点
A point $x$ for which $T^rx=x$ for some integer $r\ge1$; its least period is the smallest such positive integer, and the orbit is called periodic.
permutation of a finite set 排列
A permutation of a finite set X is an ordering of all its elements, equivalently a bijection from {1,…,|X|} to X. An n-element set has n! such orderings, including one empty ordering when n=0.
pigeonhole principle 鸽巢原理
If n objects are assigned to m ≥ 1 categories, at least one category contains at least the ceiling of n/m objects; in particular n>m forces a repeated category.
Poincaré duality Poincaré 对偶
For a closed oriented n-manifold with chosen integral fundamental class, cap product with that class is an isomorphism from H^k(M;Z) to H_{n-k}(M;Z).
Poincaré recurrence theorem 庞加莱回复定理
For a probability-preserving transformation $T$ and measurable $A$, almost every $x\in A$ returns to $A$ infinitely often under positive iterates.
Poincaré return map 首次返回映射
For a flow and a transverse section, the map taking a point to its next positive-time intersection with that section, on the domain where the first return exists.