Ideal of a Commutative Ring 交换环的理想
An ideal is an additive subgroup closed under multiplication by arbitrary ring elements, making it suitable as the kernel of a quotient map.
Mathematics Encyclopedia
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Inclusion–Exclusion Principle 容斥原理
Inclusion–exclusion counts a finite union by alternating sums of intersection sizes.
Independent Events 独立事件
Mutual independence requires probability factorization for every finite collection, rather than only pairs.
Inertia Stack
The inertia stack organizes objects equipped with an automorphism, making stabilizer information visible across a family. Read the construction of relative and absolute inertia, then the criterion describing when inertia contributes no extra objects up to equivalence.
Infinity-Topos
An infinity-topos is an accessible left exact localization of a category of space-valued presheaves on a small infinity-category. Track both accessibility and preservation of finite limits when extending the ordinary topos analogy.
Inner Fibration
Inner fibrations impose the inner-horn lifting condition relative to a map of simplicial sets. Compare left, right, and inner fibrations, then specialize the base to a point to recover the quasicategory condition.
Integer Congruence 整数同余
Congruence identifies integers whose difference is divisible by a fixed modulus, allowing arithmetic on finitely many residue classes.
Integer Divisibility 整数整除
Divisibility asks whether an integer is an exact integer multiple of another, and organizes arithmetic without using fractions.
Integral Domain 整环
An integral domain is a nonzero commutative ring with identity and no nonzero zero divisors.
Intersection Product of Algebraic Cycles
Intersection products combine cycles with multiplicities that retain local algebraic information. Read the construction for properly intersecting cycles on a nonsingular variety before studying compatibility with rational equivalence in the projective case.
Invariant Measure 不变测度
An invariant measure assigns unchanged weights to events when pulled back by a measurable transformation.
Itô's Formula
Applying a smooth function to Brownian motion introduces a second-derivative correction absent from ordinary calculus. Work through the square-function example to connect that correction with quadratic variation.
Kan Complex
A Kan complex is a simplicial set in which every horn admits a filler. Check outer as well as inner horns to distinguish this condition from the inner-horn condition for quasicategories.
Lagrangian Submanifold 拉格朗日子流形
A Lagrangian submanifold is a half-dimensional submanifold on which the ambient symplectic form vanishes.
Law of Large Numbers 大数定律
For iid integrable observations, the sample average converges to the common expectation, almost surely and therefore in probability.
Lax–Milgram Theorem
A bounded coercive bilinear form gives a unique solution for each continuous linear functional. The elliptic application shows how adding a sufficiently large zeroth-order term can restore coercivity.
Lebesgue Integral Lebesgue积分
The Lebesgue integral accumulates measurable values by approximation with simple functions.
Lebesgue Lp Space L^p空间
An Lp space consists of measurable functions modulo almost-everywhere equality with finite p-integrability.
Lebesgue Measure Lebesgue测度
Lebesgue measure extends Euclidean volume to a complete measure on a sigma-algebra of measurable sets.
Left Module 左模
A left module replaces the scalar field of a vector space by a unital ring.
Levi–Civita Connection Riemann联络
The Levi–Civita connection is the unique rule for differentiating tangent vector fields that both preserves a Riemannian metric and has zero torsion.
Lie Group 李群
A Lie group is a finite-dimensional smooth manifold with a group structure whose multiplication and inversion are smooth.
Linear Map 线性映射
A linear map preserves addition and scalar multiplication between vector spaces over the same field.
Linear Representation of a Finite Group 有限群的线性表示
A linear representation realizes a finite group as invertible linear transformations, possibly with a nontrivial kernel.
Littlewood–Paley Decomposition Littlewood-Paley分解
Littlewood–Paley decomposition separates a distribution into smooth frequency bands to measure scale-dependent regularity.