Leigh.Samphier/Sandbox

From ProofWiki
Jump to navigation Jump to search

Common

RED

Help:LaTeX Editing

Bold

$a = \begin{cases} 1 & : i = j \\ 0 & : i \ne j \end{cases}$

$p$-adic Numbers

  • Definitions related to P-adic Number Theory can be found here.
  • Results about P-adic Number Theory can be found here.

Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with zero $0_R$ and unity $1_R$.

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-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

  • Definitions related to Matroid Theory can be found here.
  • Results about Matroid Theory can be found here.


Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.

Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.


Properties of Independent Sets and Bases

All Bases of Matroid have same Cardinality

Not Needed! Leigh.Samphier/Sandbox/Set Difference of Distinct Equal Cardinality Sets is Not Empty

Leigh.Samphier/Sandbox/Set Difference of Matroid Circuit with Element is Independent

Leigh.Samphier/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton


Leigh.Samphier/Sandbox/Matroid Base Substitution From Fundamental Circuit


Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 2

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 3

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 4

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 5

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6

Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7


Leigh.Samphier/Sandbox/Matroid Base Axiom Implies Sets Have Same Cardinality


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Necessary Condition

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 2

Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3

References:

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 4

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5/Lemma

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 3 Iff Definition 7


Properties of the Rank Function

Leigh.Samphier/Sandbox/Independent Subset is Contained in Maximal Independent Subset/Corollary

Leigh.Samphier/Sandbox/Cardinality of Maximal Independent Subset Equals Rank of Set

Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms

Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Necessary Condition

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 2

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1

The Closure Operator

Closed Sets = Flats = Subspaces

Circuits

Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality

Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit

Leigh.Samphier/Sandbox/Matroid with No Circuits Has Single Base

Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent


Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)

Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1

Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2

Leigh.Samphier/Sandbox/Matroid Satisfies Circuit Axioms

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms


The Cycle Matroid of a Graph

The Greedy Algorithm

Maximization 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

  • Definitions related to Stone Spaces can be found here.
  • Results about Stone Spaces can be found here.
Join Semilattice Ordered Subset Not Always Subsemilattice


Further Ideas

Special:WantedPages

Nagata-Smirnov Metrization Theorem, Stephen Willard - General Topology

Smirnov Metrization Theorem

Stone-Weierstrass Theorem, Stephen Willard - General Topology

Stone-Cech Compactification, Stephen Willard - General Topology

Definition:Stone Space Stone's Representation Theorem for Boolean Algebras

Definition:Frames & Locales

Gelfand-Naimark Theorem

Jordan Curve Theorem

Gelfand-Mazur Theorem

Books to Add