manifold with boundary 带边流形
A Hausdorff, second countable space locally homeomorphic to relatively open subsets of the closed Euclidean half-space of a fixed positive dimension; the boundary consists of points corresponding to the bounding…
Terminology & Translations
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Markov chain 马尔可夫链
In the finite-state time-homogeneous discrete-time setting, a process $X_n$ with initial law $\alpha$ and row-stochastic matrix $P=(p_{ij})$ satisfies $P(X_{n+1}=j\mid\sigma(X_0,\ldots,X_n))=p_{X_nj}$ almost surely for…
martingale 鞅
An adapted discrete-time process $M_n$ with $E|M_n|<\infty$ and $E[M_{n+1}\mid\mathcal F_n]=M_n$ almost surely for each $n$, relative to a specified filtration.
matching in a graph 匹配
A matching in a finite simple graph is a subset of edges such that no two chosen edges share an endpoint.
material implication 蕴涵
The classical connective $P\to Q$, false exactly when $P$ is true and $Q$ is false.
mathematical induction 数学归纳法
For $P$ on $\mathbb N$, the principle that $P(0)$ and every implication $P(n)\to P(n+1)$ together yield $\forall n\,P(n)$.
maximal ideal 极大理想
A proper ideal not strictly contained in any other proper ideal of the ring.
maximal matching 极大匹配
A maximal matching is a matching not properly contained in another matching of the same graph.
maximum matching 最大匹配
A maximum matching of a finite graph is a matching with at least as many edges as every other matching of that graph.
mean value theorem for real functions 微分中值定理
If a real function f is continuous on [a,b], differentiable on (a,b), and a<b, some c in (a,b) satisfies f'(c)=(f(b)-f(a))/(b-a).
measurable real function 可测函数
A real function f on a measurable space (X,A) is measurable when the inverse image of every Borel subset of R belongs to A; equivalently each set {x:f(x)>t} is measurable.
measure space 测度空间
A measure space (X,A,mu) consists of a set X, a sigma-algebra A of its subsets, and a nonnegative extended-real function mu on A with mu(empty)=0 and countable additivity on disjoint measurable sets.
measure-preserving transformation 保测变换
On a measure space $(X,\mathcal B,\mu)$, a measurable self-map $T$ that satisfies $\mu(T^{-1}A)=\mu(A)$ for all $A\in\mathcal B$.
meromorphic function 亚纯函数
A meromorphic function on a complex domain is holomorphic except at a discrete set of poles, with the identically infinite function excluded here.
metrizable space 可度量化
A topological space whose topology is induced by some metric.
minimal dynamical system 极小系统
A continuous self-map of a nonempty compact space is minimal when every forward orbit is dense in the whole space.
minimal polynomial of an algebraic element 代数元的极小多项式
The unique monic polynomial of smallest positive degree in $K[x]$ annihilating an algebraic element $\alpha$.
Möbius function on positive integers 默比乌斯函数
$\mu(1)=1$; $\mu(n)=(-1)^r$ if $n$ is a product of $r$ distinct primes, and $\mu(n)=0$ if a prime square divides $n$.
Möbius inversion for divisor sums 默比乌斯反演
For functions on positive integers, $F(n)=\sum_{d\mid n}f(d)$ implies $f(n)=\sum_{d\mid n}\mu(n/d)F(d)$.
modular function 模函数
For a congruence subgroup $\Gamma$, a meromorphic function on the upper half-plane invariant under $\Gamma$ and meromorphic at all cusps, equivalently a meromorphic weight-zero modular form.
module homomorphism 模同态
An additive map $f:M\to N$ between left $R$-modules satisfying $f(rm)=rf(m)$.
monoid 幺半群
A set with an associative binary operation and a two-sided identity element.
monotone convergence theorem 非负单调收敛定理
If measurable functions 0<=f_n increase pointwise to f on a measure space, then their integrals increase to the integral of f, allowing the value infinity.
morphism in a category 态射
An arrow with a specified source and target in a category, composable with other arrows when their endpoints match.
Morse function Morse函数
A smooth real-valued function on a smooth manifold whose Hessian at each critical point is a nonsingular symmetric bilinear form; functions with no critical points are included.