Localization of a Commutative Ring 交换环的局部化
Localization universally makes a chosen multiplicative set of ring elements invertible.
Mathematics Encyclopedia
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Löwenheim–Skolem Theorem (Countable Model Form)
In a countable first-order language, a satisfiable theory has a model with at most countably many elements. Skolem's paradox distinguishes an external enumeration from a bijection available inside that model.
Markov Chain 马尔可夫链
A finite-state Markov chain evolves by transition probabilities determined by its current state rather than its full observed history.
Martingale 鞅
A martingale is an integrable adapted process whose future conditional expectation equals its present value.
Mathematical Induction 数学归纳法
Mathematical induction proves a property of every natural number from a base case and a successor step.
Model Category
Weak equivalences, cofibrations, and fibrations provide an abstract setting for homotopy theory. Riehl expresses the model axioms through two weak factorization systems; unpack their lifting and factorization requirements before using replacements.
Modular Curve 模曲线
Modular curves connect quotients of the upper half-plane with moduli of elliptic curves and arithmetic models over number fields.
Modular Form 模形式
A holomorphic modular form obeys a weighted symmetry law on the upper half-plane and a regularity condition at every cusp.
Morse Function Morse函数
A Morse function is a smooth real-valued function whose critical points all have nonsingular Hessians.
Motivic Cohomology
Motivic cohomology places cycle-theoretic information in groups indexed by cohomological degree and weight. Begin with motivic complexes and their hypercohomology definition, using the book's dependency chart to plan the route toward motives.
Natural Transformation 自然变换
A natural transformation compares two functors by components commuting with every source morphism.
Noetherian Commutative Ring Noether 环
A commutative ring is Noetherian when all its ideals are finitely generated, equivalently when every ascending ideal chain stabilizes.
Number Field 数域
A number field is a finite extension of the rational numbers; its ring of integers carries arithmetic beyond ordinary integer factorization.
Orbit of a Dynamical System 轨道
An orbit is the set of states reached by iterating a map or following a flow, with forward and full time distinguished.
Ordinary Generating Function 普通生成函数
An ordinary generating function encodes a sequence as coefficients of a formal power series.
p-adic number field p 进数域
Completing the rationals for the p-adic absolute value produces a field with nonarchimedean convergence. Compute digit expansions before studying its valuation ring and local-field extensions.
Partial Order 偏序
A partial order is a reflexive, antisymmetric and transitive relation that may leave some pairs incomparable.
Perfectoid Field
Perfectoid fields provide the valued-field setting for tilting. Read the completeness, nondiscreteness, and Frobenius conditions together, then follow the inverse-limit construction of the tilt.
Perfectoid Space
Perfectoid spaces glue affinoid charts built from perfectoid algebras. Start with their local definition and use the preceding affinoid results to understand why these charts support a sheaf.
Poincaré Duality Poincaré 对偶
On a closed oriented manifold, cap product with its fundamental class identifies cohomology with homology in complementary dimensions.
Poincaré Recurrence Theorem 庞加莱回复定理
In a probability-preserving system, almost every point of each measurable set returns to that set infinitely often.
Poisson Process 泊松过程
A homogeneous Poisson process counts arrivals in continuous time with stationary independent increments and a fixed positive rate.
Prime Spectrum of a Commutative Ring 交换环的素谱空间
The prime spectrum turns a commutative ring into a topological space whose points are prime ideals.
Prism in p-adic Geometry
A prism combines a delta-ring with a Cartier-divisor ideal and a completeness condition. Read the formal definition and its perfect, bounded, and crystalline variants before approaching prismatic cohomology.
Prismatic Cohomology
Prismatic cohomology studies a smooth p-adic formal scheme using a site of prisms over a bounded base prism. Begin with the site and structure sheaf, then read the complex used to formulate Hodge–Tate comparison.