Probabilistic Method 概率方法
The probabilistic method proves existence by placing candidate combinatorial objects in a probability space and showing that a desired property has positive probability.
Mathematics Encyclopedia
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Probability Space 概率空间
A probability space specifies outcomes, measurable events, and a countably additive law of total mass one.
Proof by Contradiction 反证法
Classical proof by contradiction establishes a statement by deriving an impossibility from its negation.
Pseudodifferential Operator
Pseudodifferential operators extend differential operators through kernels conormal to the diagonal. Follow the bundle-valued kernel construction into ellipticity to connect symbol information with approximate inverses.
Quadratic Reciprocity Law
Quadratic reciprocity relates whether one odd prime is a square modulo another to the reversed question. The Legendre symbol and Gauss's lemma turn the sign rule into a method for reducing computations.
Quotient Stack
A quotient stack records the orbit data of a group action together with the isomorphisms between representatives. Follow the action groupoid into its stackification and track the natural map from the original space to the quotient.
Quotient Topology 商拓扑
A quotient topology describes a space formed by identifying points.
Radon–Nikodym Theorem
Absolute continuity lets one sigma-finite measure be expressed by integrating a density against another. Read the accompanying Lebesgue decomposition to distinguish this density from the mutually singular part.
Ramsey Number 拉姆齐数
A two-color Ramsey number is the smallest complete-graph size forcing a red clique of one prescribed size or a blue clique of another.
Random Variable 随机变量
A random variable is a measurable numerical observation of an outcome, distinct from its value and distribution.
Rellich–Kondrachov Compactness
On a bounded open domain, bounded families in the zero-boundary-value Sobolev space gain strong subsequential convergence below the critical exponent. Follow the translation estimates that turn derivative control into compactness.
Representable Functor 可表函子
A representable set-valued functor describes maps to or from one object through natural bijections.
Residue at an Isolated Complex Singularity 留数
The residue is the coefficient of the inverse first power in a local Laurent expansion.
Riemann Surface 黎曼曲面
A Riemann surface is a connected complex manifold of complex dimension one, specified by holomorphic changes of local coordinates.
Riemann Zeta Function 黎曼 ζ 函数
The Riemann zeta function connects a Dirichlet series over integers with an Euler product over primes and a meromorphic continuation governed by a functional equation.
Riemannian Geodesic 测地线
A Riemannian geodesic is an affinely parametrized curve with zero covariant acceleration for the Levi–Civita connection.
Riemannian Metric Riemann度量
A Riemannian metric equips each tangent space with an inner product varying smoothly from point to point.
Scheme 概形
A scheme is a locally ringed space locally isomorphic to the prime spectrum of a commutative ring equipped with its structure sheaf.
Serre Fibration
Lifting homotopies from disks relates the homotopy groups of a fiber, total space, and base. The long exact sequence extends the lifting viewpoint already familiar from covering spaces.
Set 集合
A set is an extensional object determined by its members, with legitimate constructions controlled by the background axioms.
Sheaf 层
A sheaf assigns data to open sets with restriction maps so that compatible local sections glue uniquely.
Short Exact Sequence of Modules 模的短正合列
A short exact sequence identifies one module as a submodule and another as its quotient.
Simplicial Set
A simplicial set is a contravariant functor from the simplex category to sets. Read the order-preserving maps as operations on simplices, keeping degenerate simplices as part of the data.
Singular Homology 奇异同调
Singular homology associates abelian groups to a space by comparing cycles with boundaries.
Smooth Manifold 微分流形
A smooth manifold is a space on which multivariable calculus works through compatible local coordinate systems.