Leigh.Samphier/Sandbox

From ProofWiki
Jump to navigation Jump to search

RED

$p$-adic Numbers

Continuing Svetlana Katok Book

P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient


Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism

Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring

Leigh.Samphier/Sandbox/Distance-Preserving Homomorphism Preserves Norm

Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring

Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring/Corollary

Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion

Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary


Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Proof 1

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Proof 2

Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique


Leigh.Samphier/Sandbox/Representative of P-adic Sum

Leigh.Samphier/Sandbox/Representative of P-adic Product

Leigh.Samphier/Sandbox/Cauchy Sequence Represents P-adic Number

Leigh.Samphier/Sandbox/Constant Sequence Represents Rational in P-adic Numbers

Continuing Fernando Q. Gouvea Book

P-adic Number is Limit of Unique P-adic Expansion

Every P-adic Number is Limit of P-adic Expansion

Leigh.Samphier/Sandbox/Sequence Converges in Completion iff Sequence Represents Limit

Leigh.Samphier/Sandbox/Distinct P-adic Expansions Converge to Distinct P-adic Numbers

Leigh.Samphier/Sandbox/P-adic Number is Limit of Unique P-adic Expansion - Complete the uniqueness

Characterisation of P-adic Units

Leigh.Samphier/Sandbox/Set of P-adic Units is Unit Sphere

Leigh.Samphier/Sandbox/Characterization of Rational P-adic Units

Leigh.Samphier/Sandbox/P-adic Expansion of P-adic Units

Create theorem P-adic Numbers are Uncountable

Matroids

Loops and Parallel Elements

Element is Member of Base iff Not Loop


Leigh.Samphier/Sandbox/Doubleton of Elements is Subset

Leigh.Samphier/Sandbox/Power Set of Doubleton

Leigh.Samphier/Sandbox/Distinct Elements are Parallel iff Pair forms Circuit

Leigh.Samphier/Sandbox/Parallel Relationship is Transitive

Leigh.Samphier/Sandbox/Distinct Elements are Parallel iff Each is in Closure of Other

Leigh.Samphier/Sandbox/Closure of Subset Contains Parallel Elements

Leigh.Samphier/Sandbox/Set with Two Parallel Elements is Dependent


Leigh.Samphier/Sandbox/Definition:Simple Matroid


Leigh.Samphier/Sandbox/Loop Belongs to Every Flat

Leigh.Samphier/Sandbox/Parallel Elements Depend on Same Subsets

Leigh.Samphier/Sandbox/Matroid Contains No Loops iff Empty Set is Flat

Properties of Independent Sets and Bases

All Bases of Matroid have same Cardinality

Leigh.Samphier/Sandbox/Matroid satisfies Base Axiom

Leigh.Samphier/Sandbox/Alternative Axiomatization of Matroid


Properties of the Rank Function

Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms

The Closure Operator

Closed Sets = Flats = Subspaces

Circuits

The Cycle Matroid of a Graph

The Greedy Algorithm

Maximum Weight Problem (Greedy Algorithm)

Complete Greedy Algorithm yields Maximal Set

Complete Greedy Algorithm may not yield Maximum Weight

Complete Greedy Algorithm guarantees Maximum Weight iff Matroid

Stone Spaces

Join Semilattice Ordered Subset Not Always Subsemilattice