Fatou's lemma Fatou引理
For a sequence of nonnegative measurable functions on a measure space, the integral of the pointwise lower limit is at most the lower limit of their integrals.
Terminology & Translations
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Fermat's little theorem 费尔马小定理
If $p$ is prime and $p\nmid a$, then $a^{p-1}\equiv1\pmod p$; equivalently $a^p\equiv a$ for every integer $a$.
field 域
A commutative unital ring with $1\ne0$ in which every nonzero element is a unit.
field extension 域扩张
A field $L$ together with an injective unital field homomorphism $K\to L$, usually identified with a subfield inclusion.
filtration in probability σ代数流
An increasing family $(\mathcal F_t)$ of sub-sigma-algebras of a probability space's event sigma-algebra, indexed by ordered times: $s\le t$ implies $\mathcal F_s\subseteq\mathcal F_t$.
finite field 有限域
A field with finitely many elements; its cardinality is $p^n$ for a prime $p$ and positive integer $n$.
finite simple graph 简单图
A finite simple graph is a pair (V,E) with V finite and E a set of two-element subsets of V; edges are undirected, with no loops or repetitions.
finitely generated module 有限生成模
An $R$-module spanned by a finite list of elements under finite $R$-linear combinations.
first-countable space C1 可数公理
A space in which every point has a countable basis of open neighborhoods: every open neighborhood contains a member of that basis.
first-order compactness theorem 紧致性定理
A set of classical first-order sentences has a model if and only if every finite subset has a model.
first-order completeness theorem 完全性定理
For a sound complete classical first-order calculus, every semantic consequence of a sentence set has a finite derivation from it.
first-order formula 合式公式
A finite expression of a fixed first-order language built from atomic formulas using its permitted connectives and quantifiers.
first-order sentence 语句
A formula of a first-order language with no free variable occurrences.
first-order structure 结构
A nonempty domain equipped with interpretations of a first-order language’s constants, functions and relations of the prescribed arities.
first-order term 项
An expression naming a domain value, built from variables, constants and correctly applied function symbols in a first-order language.
fixed point in dynamics 不动点
For a map $T:X\to X$, a point $x$ with $T(x)=x$; for a flow, a point fixed by every time map.
flat module 平坦模
A module over a commutative ring for which tensoring preserves every short exact sequence.
flow 流
A jointly continuous action $\phi:\mathbb R\times X\to X$ with $\phi_0=\operatorname{id}$ and $\phi_{t+s}=\phi_t\circ\phi_s$.
formal power series 形式幂级数
Over a commutative ring R, a formal power series is a coefficient sequence written ∑ a_n x^n; equality is coefficientwise, addition is termwise, and multiplication uses finite convolution at each degree.
Fourier series Fourier级数
For a 2 pi-periodic square-integrable function f, its Fourier series is sum_{k in Z} c_k e^{ikx}, where c_k=(2 pi)^(-1) integral_{-pi}^pi f(x)e^{-ikx} dx. The series converges in L2, and stronger modes of convergence…
Fourier transform 傅立叶变换
For f in L1(R^n), the Fourier transform in the convention used here is the function f-hat(xi)=integral f(x)e^{-i x dot xi} dx.
fractional ideal of a number field 数域的分式理想
For a number field $K$, a nonzero finitely generated $\mathcal O_K$-submodule of $K$; equivalently it becomes an integral ideal after multiplication by a nonzero denominator in $\mathcal O_K$.
free module 自由模
An $R$-module with a basis, equivalently a direct sum of copies of $R$ indexed by a set.
free variable 自由变元
A variable is free in a formula if at least one occurrence is outside any quantifier binding that occurrence.
Fubini's theorem Fubini定理
For sigma-finite measure spaces X and Y, an integrable real or complex function on their product has integrable sections almost everywhere, and its product integral equals either iterated integral.