Leigh.Samphier/Sandbox
RED
Contents
$p$-adic Numbers
Continuing Svetlana Katok Book
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous): $\S 1.4$ The field of $p$-adic numbers $\Q_p$
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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
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
- 1976: Dominic Welsh: Matroid Theory Chapter $1.$ $\S 4.$ Loops and parallel elements
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
- 1976: Dominic Welsh: Matroid Theory Chapter $1.$ $\S 5.$ Properties of independent sets and bases
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
- 1976: Dominic Welsh: Matroid Theory ... (previous) Chapter $19.$ $\S 1.$ 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
- 1982: Peter T. Johnstone: Stone spaces: Chapter $\text I$: Preliminaries, Definition $1.3$
