Leigh.Samphier/Sandbox

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

Induction example


$p$-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.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $\Z_p$ be the $p$-adic integers for some prime $p$.


FIX

Valuation Ideal is Maximal Ideal of Induced Valuation Ring

Continuing Svetlana Katok Book

Equivalence of Definitions of P-adic Integer


Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 1

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 2

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 1

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 2

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 3

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 4

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 5

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 6

Leigh.Samphier/Sandbox/P-adic Unit has Non-Zero First P-adic Digit

Definition:P-adic Unit
P-adic Unit has Norm Equal to One
P-adic Number times P-adic Norm is P-adic Unit

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic to the Left

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

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


Continuing Fernando Q. Gouvea Book

P-adic Unit has Norm Equal to One
Leigh.Samphier/Sandbox/Characterization of Rational P-adic Unit Lemma 3.3.4 (i) link between Lemma 3.3.4 and Lemma 3.3.4 (ii)

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

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

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

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence


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 $M$.


FIX

Definition:Closure Axioms (Matroid)


Properties of Independent Sets and Bases

All Bases of Matroid have same Cardinality

NOT NEEDED!! 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/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 1 Implies Condition 3/Lemma 1

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 2

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 1 Implies Condition 3/Lemma 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/Proper Subset of Matroid Circuit is Independent

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/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/Definition:Circuit Axioms (Matroid)/Definition 3


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

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

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

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 3 Implies Condition 4

Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 4 Implies Condition 1


Leigh.Samphier/Sandbox/Matroid Unique Circuit Property

Leigh.Samphier/Sandbox/Matroid Unique Circuit Property/Proof 1

Leigh.Samphier/Sandbox/Matroid Unique Circuit Property/Proof 2

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

Macros

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