# Leigh.Samphier/Sandbox

## Common

RED

**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.

### Cleanup/Refactor

Leigh.Samphier/Sandbox/Definition:Ideal of Null Sequences

Leigh.Samphier/Sandbox/Definition:Quotient Ring of Cauchy Sequences

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Division Ring

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1

Leigh.Samphier/Sandbox/Definition:Induced Norm on Quotient of Cauchy Sequences

Leigh.Samphier/Sandbox/Definition:Normed Quotient of Cauchy Sequences

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring/Corollary 1

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3

Leigh.Samphier/Sandbox/Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4

Leigh.Samphier/Sandbox/Quotient of Cauchy Sequences is Metric Completion

Leigh.Samphier/Sandbox/Quotient of Cauchy Sequences is Metric Completion/Lemma 1

Leigh.Samphier/Sandbox/Quotient of Cauchy Sequences is Metric Completion/Lemma 2

Leigh.Samphier/Sandbox/Embedding Division Ring into Quotient Ring of Cauchy Sequences

Leigh.Samphier/Sandbox/Completion of Normed Division Ring

Leigh.Samphier/Sandbox/Definition:P-adic Number

Leigh.Samphier/Sandbox/Completion of Rational Numbers with P-adic Norm

Leigh.Samphier/Sandbox/Rational Numbers form Dense Subfield of P-adic Numbers

Leigh.Samphier/Sandbox/Definition:Rational Numbers with P-adic Norm

Leigh.Samphier/Sandbox/Rational Numbers with P-adic Norm is Non-Archimedean Valued Field

Leigh.Samphier/Sandbox/Definition:P-adic Numbers as Quotient of Cauchy Sequences replace Definition:P-adic Numbers as Quotient of Cauchy Sequences and Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm

Leigh.Samphier/Sandbox/Quotient of Cauchy Sequences Constructs P-adic Numbers

Delete Definition:P-adic Number/P-adic Norm Completion of Rational Numbers Delete Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm

Definition:P-adic Metric/P-adic Numbers

Definition:P-adic Norm/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

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

- 2018: Bernhard H. Korte and Jens Vygen:
*Combinatorial Optimization: Theory and Algorithms*(6th ed.) Chapter $13$ Matroids $\S 13.1$ Independence Systems and Matroids, Definition $13.1$

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

- 1976: Dominic Welsh:
*Matroid Theory*Chapter $1.$ $\S 5.$ Properties of independent sets and bases

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

References:

- h94 (https://math.stackexchange.com/users/305691/h94), Given bases $A$, $B$ of a matroid there is a one-to-one mapping $\omega$ from $A$ to $B$ such that $(A - {a}) \cup {\omega(a)}$ is independent, URL (version: 2018-02-09): https://math.stackexchange.com/q/2642822
- 1966: David S. Asche:
*Minimal dependent sets*(*Journal of the Australian Mathematical Society***Vol. 6**,*no. 3*: pp. 259 – 262) - 1969: Richard A. Brualdi:
*Comments on bases in dependence structures*(*Bulletin of the Australian Mathematical Society***Vol. 1**,*no. 2*: pp. 161 – 167)

### 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

### 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/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

- 1976: Dominic Welsh:
*Matroid Theory*... (previous) Chapter $19.$ $\S 1.$ 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.

- 1982: Peter T. Johnstone:
*Stone Spaces*: Chapter $\text I$: Preliminaries, Definition $1.3$

## Further Ideas

### Special:WantedPages

Nagata-Smirnov Metrization Theorem, Stephen Willard - General Topology

Stone-Weierstrass Theorem, Stephen Willard - General Topology

Stone-Cech Compactification, Stephen Willard - General Topology

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