Direct Proof 直接证明
A direct proof establishes a conclusion by justified deductions from the stated assumptions.
Mathematics Encyclopedia
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Dirichlet Character 狄利克雷特征
A Dirichlet character is a multiplicative character of the unit group modulo an integer, extended by zero to nonunits.
Dirichlet L-Function 狄利克雷 L 函数
A Dirichlet L-function weights integers by a Dirichlet character, refining zeta's prime product to detect residue classes and encode abelian arithmetic over the rationals.
Dirichlet's unit theorem 狄利克雷单位定理
The signature of a number field determines the rank of its unit group. Compare real and imaginary quadratic fields, then follow the logarithmic embedding used to prove finite generation.
Distribution in Analysis 分布（分析）
A distribution is a continuous linear functional on smooth compactly supported test functions.
Dominated Convergence Theorem 控制收敛定理
An integrable bound independent of the index upgrades almost-everywhere convergence to convergence in L1.
Donsker's Invariance Principle
For independent identically distributed increments with mean zero and variance one, rescaled interpolated partial-sum paths converge in distribution to Brownian motion. The path-space statement also controls continuous functionals such as maxima.
Dynamical Flow 流
A flow is a continuous real-time action whose maps compose by addition of time and are invertible.
Equivalence of Categories 范畴等价
Equivalent categories have the same objects up to isomorphism and the same morphism structure, without requiring literal inverse functors.
Equivalence Relation 等价关系
An equivalence relation partitions a set into classes of mutually equivalent elements.
Ergodic Transformation 遍历变换
An ergodic probability-preserving transformation has no invariant measurable event of intermediate probability.
Étale Cohomology
Étale cohomology applies sheaf cohomology to coverings by étale morphisms, providing invariants suited to algebraic geometry over general fields. Begin with the topology and its sheaf condition, then use Mayer–Vietoris as a first structural calculation.
Étale Fundamental Group
The étale fundamental group encodes finite étale covers through automorphisms of a geometric fibre functor. Read the classification theorem alongside the spectrum-of-a-field example to connect geometric covering theory with Galois theory.
Euclidean Algorithm for Integers 整数辗转相除法
Repeated division with remainder computes a greatest common divisor without prime factorization.
Euler's Totient Function 欧拉函数
Euler's totient counts invertible residue classes and connects factorization with modular exponentiation.
Ext Group of Modules 模的 Ext 群
Ext groups are derived Hom groups whose first degree classifies extensions of modules with fixed end terms.
Extremal Graph Theory 极值图论
Extremal graph theory studies the largest or smallest graph parameters compatible with specified structural restrictions.
Fermat's Little Theorem 费马小定理
For a prime modulus, every nonzero residue raised to the power one less than the prime equals one.
Finite Correspondence
Finite correspondences enlarge the maps between smooth schemes to include suitable algebraic cycles in their products. Read the graph example and the composition construction as preparation for transfers and motives.
Finite Simple Graph 简单图
A finite simple graph records an undirected adjacency relation without loops or repeated edges.
Formal Completion along a Closed Subset
Formal completion isolates the infinitesimal geometry supported on a chosen closed subset. Read its description by a functor of points and compare it with the affine construction from quotients of the coordinate ring.
Fourier Series Fourier级数
A Fourier series represents a periodic function through its coefficients against exponential frequencies.
Fourier Transform 傅立叶变换
The Fourier transform represents a function by continuous frequencies and converts differentiation into multiplication.
Fpqc Descent for Quasi-Coherent Sheaves
Faithfully flat coverings let quasi-coherent sheaves be reconstructed from compatible local data. Follow the affine case first, then the passage to an arbitrary fpqc covering, keeping track of both gluing objects and descending their maps.
Fractional Ideal of a Number Field 数域的分式理想
Nonzero fractional ideals extend integral ideals by allowing bounded denominators, forming a group whose quotient by principal ideals is the ideal class group.