cyclotomic polynomial 分圆多项式
For $n\ge1$, $\Phi_n(X)$ is the monic polynomial whose roots are the complex roots of unity of exact order $n$.
Terminology & Translations
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de Rham cohomology de Rham 上同调
For a smooth manifold, the real vector spaces obtained in each degree by dividing the closed smooth differential forms by the exact ones, using the exterior derivative; degree zero has no incoming differential.
deck transformation 复迭变换
A homeomorphism of a covering's total space E satisfying p composed with that homeomorphism equals p.
Dedekind zeta function 戴德金 ζ 函数
For a number field $K$, $\zeta_K(s)=\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}|\mathcal O_K/\mathfrak a|^{-s}$ for $\operatorname{Re}s>1$, with its meromorphic continuation.
deformation retract 强形变收缩
A subspace A of X for which a homotopy of the identity of X to a retraction onto A fixes every point of A at every time.
dense subset 稠密
A subset whose closure equals the entire ambient topological space.
derangement 错排
A derangement of {1,…,n} is a bijection σ of that set with σ(i) ≠ i for every i. The empty permutation is a derangement by the vacuous convention.
derivative of a real function 导数
For a real function defined near a, its derivative at a is the finite limit of (f(a+h)-f(a))/h as nonzero h tends to zero.
Devaney chaos Devaney混沌
For a continuous self-map of an infinite compact metric space without isolated points, topological transitivity together with density of periodic points; these hypotheses imply sensitive dependence.
diagonalizable linear map 可对角化线性映射
A finite-dimensional endomorphism possessing a basis of eigenvectors over the specified scalar field.
differential form 外微分形式
For a smooth manifold and a nonnegative integer k, a smooth section of the kth exterior power of its cotangent bundle; pointwise it is an alternating real multilinear function of k tangent vectors.
differential graded category dg-范畴
A category enriched in cochain complexes over a commutative ring, with associative unital degree-zero composition that is a map of complexes.
differential of a real function 微分
For a function differentiable at a, its differential is the linear map h mapped to f'(a)h in f(a+h)=f(a)+f'(a)h+o(|h|).
dimension of a vector space 向量空间的维数
The cardinality of any basis of a vector space over its specified field.
direct proof 直接证明
An argument deriving the conclusion from the given hypotheses by justified steps, without adopting its negation as the main temporary assumption.
direct sum of modules 模的直和
The module of tuples from a family $(M_i)$ having only finitely many nonzero coordinates, with coordinatewise operations.
directional derivative 方向导数
For f on an open subset of R^n and a specified unit vector v, the directional derivative at a is the finite two-sided limit [f(a+tv)-f(a)]/t as t tends to zero, when it exists.
Dirichlet character 狄利克雷特征
For $N\ge1$, a homomorphism $(\mathbb Z/N\mathbb Z)^\times\to\mathbb C^\times$, extended periodically to integers and assigned $0$ on integers not coprime to $N$.
Dirichlet L-function 狄利克雷 L 函数
For a Dirichlet character $\chi$, $L(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s}$ for $\operatorname{Re}s>1$, with its meromorphic continuation.
Dirichlet's unit theorem 狄利克雷单位定理
If a number field has $r_1$ real embeddings and $r_2$ conjugate pairs of complex embeddings, then $\mathcal O_K^\times\cong\mu_K\times\mathbb Z^{r_1+r_2-1}$, where $\mu_K$ is its group of roots of unity.
discrete topology 离散拓扑
The topology on a set in which every subset is open.
discriminant of a number field 数域的判别式
For an integral basis $\omega_1,\ldots,\omega_n$ of $K$, $D_K=\det(\operatorname{Tr}_{K/\mathbb Q}(\omega_i\omega_j))$, independent of that basis.
disjoint union 无交并
For indexed sets, the set of pairs (i,x) with x in the i-th set, retaining the component index.
disjunction 析取
The inclusive connective producing a statement true when at least one component is true.
distribution in analysis 广义函数
A distribution on an open set Omega in R^n is a continuous linear functional on C_c^infinity(Omega); locally, its action is bounded by finitely many derivative suprema of tests with support in a fixed compact set.