pointwise convergence 点点收敛
A sequence of real functions f_n on E converges pointwise to f if for each fixed x in E the scalar sequence f_n(x) converges to f(x).
Terminology & Translations
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Poisson distribution 泊松分布
For $\lambda>0$, a law on nonnegative integers with $P(X=k)=e^{-\lambda}\lambda^k/k!$.
Poisson equation Poisson方程
The Poisson equation is an inhomogeneous Laplace equation -Delta u=f on an open set, with prescribed source f; using Delta u=f instead changes the sign convention for the source.
Poisson process 泊松过程
For rate $\lambda>0$, a right-continuous counting process $N_t$ starting at zero with stationary independent increments and $N_t-N_s\sim\operatorname{Pois}(\lambda(t-s))$ for $s<t$.
pole of a complex function 极点
An isolated singularity a of f is a pole of order m>=1 if f(z)=g(z)/(z-a)^m near a with g holomorphic and g(a) nonzero.
polynomial ring 多项式环
For a commutative ring $R$, the ring $R[x]$ of finite formal sums $\sum a_ix^i$ with coefficientwise addition and convolution multiplication.
power series in analysis 幂级数
A power series centered at a is a series sum_{n>=0} c_n(z-a)^n with real or complex coefficients, studied as a function on the points where its partial sums converge.
power set 幂集
The set of every subset of a given set A.
preimage of a subset under a function 原像
For f:A→B and T⊆B, the members a of A such that f(a) belongs to T.
prime element 素元
A nonzero nonunit $p$ in a domain such that $p\mid ab$ implies $p\mid a$ or $p\mid b$.
prime ideal 素理想
A proper ideal $\mathfrak p$ in a commutative ring with $ab\in\mathfrak p$ implying $a\in\mathfrak p$ or $b\in\mathfrak p$.
prime number 素数
A positive integer $p>1$ whose only positive divisors are $1$ and $p$.
prime spectrum of a commutative ring 交换环的素谱空间
The set of prime ideals of $R$ with closed sets $V(I)=\{\mathfrak p:I\subseteq\mathfrak p\}$.
primitive Dirichlet character 本原狄利克雷特征
A Dirichlet character modulo $N$ that is not induced from a character modulo any proper divisor of $N$.
principal ideal domain 主理想环
A domain whose every ideal is generated by a single element.
probabilistic method 概率方法
The probabilistic method proves existence of a combinatorial object by defining a distribution on candidates and showing that the desired property has positive probability, sometimes after a controlled alteration.
probability density function 概率密度
A nonnegative measurable function $f$ with $P(X\in B)=\int_B f(x)\,dx$ for every Borel set $B$, when the law is absolutely continuous with respect to Lebesgue measure.
probability distribution 概率分布
The probability measure induced by a random variable $X$, given by $\mu_X(B)=P(X\in B)$ on Borel sets.
probability mass function 概率质量函数
For a discrete random variable with countable support $S$, the function $p(x)=P(X=x)$ on $S$, whose nonnegative values sum to one.
probability space 概率空间
A triple $(\Omega,\mathcal F,P)$ in which $\mathcal F$ is a sigma-algebra on $\Omega$ and $P$ is a countably additive nonnegative measure with $P(\Omega)=1$.
product topology 乘积空间
A Cartesian product equipped with the coarsest topology making every coordinate projection continuous; basic opens restrict only finitely many coordinates.
projective module 投射模
A module $P$ such that every module map $P\to C$ lifts through every surjective module map $B\to C$.
proof by contradiction 反证法
In classical logic, a proof of $P$ obtained by deriving $\bot$ from the premises together with $\neg P$.
proof by contraposition 逆否法证明
A classical proof of $P\to Q$ by deriving $\neg P$ from $\neg Q$.
pseudoholomorphic curve 伪全纯曲线
A smooth map from a Riemann surface with complex structure j to a smooth manifold with almost complex structure J whose differential intertwines those structures, so du composed with j equals J composed with du.