Smooth Vector Bundle 向量丛
A smooth vector bundle is a locally trivial smooth family of vector spaces whose changes of fiber coordinates are linear.
Mathematics Encyclopedia
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Sobolev Space Sobolev空间
A Sobolev space controls a function and its weak derivatives in specified Lebesgue norms.
Sobolev Trace Sobolev函数的迹
A Sobolev trace assigns boundary values by continuously extending restriction from smooth functions.
Spectral Sequence 谱序列
A spectral sequence is a sequence of differential objects whose successive pages are their cohomology, often computing associated graded pieces of a filtered cohomology group.
Spectral Theorem for Compact Self-Adjoint Operators 紧自伴算子的谱定理
A compact self-adjoint operator on a separable Hilbert space has an orthonormal eigenbasis, with nonzero eigenvalues of finite multiplicity accumulating only at zero.
Stable Homotopy Group 稳定同伦群
Stable homotopy groups retain information that survives repeated suspension.
Stable Infinity-Category
Stability requires a zero object, fibers and cofibers of every map, and agreement of fiber and cofiber sequences. Begin with spectra, then follow how stable infinity-categories retain the homotopy information behind triangulated categories.
Stiefel–Whitney Classes
These mod-two cohomology classes detect features of real vector bundles that survive pullback. Start with the axioms and Whitney sum formula, then relate the classes to obstructions to independent sections.
Stochastic Differential Equation 随机微分方程
A stochastic differential equation specifies an integral evolution driven by noise, with solution and integration conventions forming part of the problem.
Stokes' Theorem on Manifolds Stokes 定理
Stokes' theorem equates the integral of an exterior derivative over an oriented manifold with the integral of the original form over its oriented boundary.
Stopping Time 停时
A stopping time is a random time for which deciding whether stopping has occurred uses only information already available.
Strong Mixing in Measure-Preserving Dynamics 强混合
Strong mixing means probabilities of two fixed events separated by long evolution factor in the limit.
Symplectic Manifold 辛流形
A symplectic manifold carries a closed nondegenerate differential two-form.
Szemerédi's Graph Regularity Lemma
A bounded partition makes most vertex pairs behave regularly with respect to edge density. The energy-increment proof explains why repeated refinements terminate, despite the rapidly growing number of parts.
Tangent Space 切空间
The tangent space at a manifold point captures first-order directions in local smooth coordinates.
Tensor Product of Modules 模的张量积
The tensor product represents balanced bilinear maps from a right module and a left module.
Thom Isomorphism
A Thom class shifts cohomology from a vector bundle's base into the relative cohomology of its disk and sphere bundles. Track the orientation hypothesis when moving from mod-two to integer coefficients.
Tilting of Perfectoid Spaces
Tilting relates perfectoid spaces over a perfectoid field to spaces over its characteristic-p tilt. Follow the functorial definition and the gluing theorem, noting the identification of underlying topological spaces.
Topological Compactness 紧致性
Compactness requires every open cover to admit a finite subcover.
topological conjugacy 拓扑共轭
A homeomorphism intertwining two evolution maps transports their orbit structure. Use the defining commuting equation to track periodic points and transitivity, and distinguish this equivalence from a noninvertible factor map.
topological entropy 拓扑熵
Topological entropy measures the exponential growth of orbit segments distinguishable at a fixed resolution. Compare spanning and separated sets before using the variational principle to relate this growth to invariant measures.
Topological Space 拓扑空间
A topology specifies open subsets and thereby continuity without choosing distances.
Topological Transitivity 拓扑传递
Topological transitivity means some iterate connects any prescribed pair of nonempty open regions.
Tree in Graph Theory 树
A finite tree is a nonempty connected simple graph without cycles, equivalently one with a unique path between every pair of vertices.
Uniform Continuity 一致连续
Uniform continuity uses one input tolerance for every pair of points in a domain.