Mathematical Theorems Flashcards
Mergelyanโs Theorem
Let ๐พ โแถแตแตแตแตแถแต โ such that โ โ ๐พ is connected. Every continuous function ๐ : ๐พ โ โ whose restriction to the interior of ๐พ is holomorphic can be uniformly approximated by polynomials.
Shell Theorem
A thin spherical shell exerts no gravitational influence on internal objects and attracts external objects as though its mass were concentrated at its center point.
Rank-Nullity Theorem
Let ๐ and ๐ be vector spaces over a field ๐ฝ, with ๐ finite-dimensional, and let ๐ : ๐ โ ๐ be a linear transformation. Then dim im ๐ + dim ker ๐ = dim ๐.
Hyperplane Separation Theorem
If ๐ด and ๐ต are two disjoint convex subsets of โโฟ, then there exist ๐ฏ โ โโฟ and ๐ โ โ such that ๐ฑแต๐ฏ โฅ ๐ for all ๐ฑ โ ๐ด and ๐ฒแต๐ฏ โค ๐ for all ๐ฒ โ ๐ต.
RobertsonโSeymour Theorem
The set of (isomorphism classes of) finite undirected graphs is well-partial-ordered by the graph minor relation.
CookโLevin Theorem
The Boolean satisfiability problem is ๐ญ๐ฏ-complete.
Max-Flow Min-Cut Theorem
Let ๐บ be a finite nonnegative-edge-weighted directed graph, and let ๐ , ๐ก โ ๐(๐บ) be distinct. The maximum value of an ๐ -๐ก flow in ๐บ equals the minimum weight of an ๐ -๐ก edge cut in ๐บ.
Brouwerโs Invariance of Domain Theorem
Let ๐ be a continuous injection from an open subset of โโฟ to โโฟ. The image of ๐ is open, and ๐ is a homeomorphism onto its image.
Brouwerโs Invariance of Dimension Theorem
If ๐ โแตแตแตโฟ โแต is homeomorphic to ๐ โแตแตแตโฟ โโฟ, then ๐ = ๐.
NovikovโBoone Theorem
There exists a finitely presented group with algorithmically undecidable word problem.
AdianโRabin Theorem
All Markov properties of finitely presented groups are algorithmically undecidable. In particular, it is undecidable whether a given finite presentation defines the trivial group.
Apรฉryโs Theorem
๐(3) is irrational.
BanachโSchrรถderโBernstein Theorem
Let ๐บ โท ๐ and ๐ด, ๐ต โ ๐บ. If ๐ด is ๐บ-equidecomposable with a subset of ๐ต and ๐ต is ๐บ-equidecomposable with a subset of ๐ด, then ๐ด and ๐ต are ๐บ-equidecomposable.
Uniformization Theorem
Every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere.
GelfandโNaimark Theorem
Every C*-algebra is *-isometric to an algebra of bounded operators on a complex Hilbert space.
Wedderburnโs Little Theorem
Every nontrivial finite ring without zero divisors is a field.
Frobeniusโ Theorem
Every finite-dimensional associative division algebra over โ is isomorphic to โ, โ, or โ.
BottโMilnorโKervaire Theorem
Every finite-dimensional division algebra over โ is isomorphic to โ, โ, โ, or ๐.
LindemannโWeierstrass Theorem
If ฮฑโ, โฆ, ฮฑโ are algebraic numbers linearly independent over โ, then exp(ฮฑโ), โฆ, exp(ฮฑโ) are algebraically independent over โ.
Rosserโs Theorem
๐โ > ๐ log ๐; improved by Dusart in 1999 to ๐โ > ๐ log ๐ + ๐ log log ๐ - ๐.
Szemerรฉdiโs Theorem
Any subset of the natural numbers with positive upper density contains arbitrarily long arithmetic progressions.
Central Limit Theorem
Suppose (๐โ, ๐โ, โฆ) is a sequence of IID random variables with finite mean ๐ and variance ๐ยฒ. As ๐โโ, the scaled sample averages (๐โ โ ๐)โ๐ converge in distribution to ๐(0, ๐ยฒ), i.e., their CDFs converge pointwise.
ChurchโRosser Theorem
ฮป-calculus is confluent under ฮฑ-conversion, ฮฒ-reduction, and ฮท-conversion. That is, if a ฮป-expression ๐ฅ can be reduced in two ways to ๐ฆโ and ๐ฆโ, then there exists a ฮป-expression ๐ง to which both ๐ฆโ and ๐ฆโ can be reduced.