Leigh.Samphier/Sandbox

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

$p$-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:P-adic Number

P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number

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

Leigh.Samphier/Sandbox/Representative of P-adic Number is Representative of Equivalence Class

Leigh.Samphier/Sandbox/P-adic Numbers form Non-Archimedean Valued Field

Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers

Characterization of Closed Ball in P-adic Numbers

Characterization of Open Ball in P-adic Numbers

P-adic Norm satisfies Non-Archimedean Norm Axioms

Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric

P-adic Open Ball is Instance of Open Ball of a Norm

P-adic Closed Ball is Instance of Closed Ball of a Norm

P-adic Sphere is Instance of Sphere of a Norm

Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm

Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers

Local Basis of P-adic Number

Equivalence of Definitions of Convergent P-adic Sequence

P-adic Norm satisfies Non-Archimedean Norm Axioms

P-adic Numbers is Totally Disconnected Topological Space

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

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

P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number

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

Integers are Arbitrarily Close to P-adic Integers

P-adic Norm of p-adic Number is Power of p

Integers are Arbitrarily Close to P-adic Integers

Countable Basis for P-adic Numbers

Leigh.Samphier/Sandbox/P-adic Norm on P-adic Numbers Extends Norm on Rational Numbers

Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers

P-adic Valuation Extends to P-adic Numbers

P-adic Metric on P-adic Numbers is Non-Archimedean Metric

Valuation Ring of P-adic Norm is Subring of P-adic Integers

P-adic Norm of p-adic Number is Power of p

Other

Open and Closed Balls in P-adic Numbers are Compact Subspaces

Open and Closed Balls in P-adic Numbers are Totally Bounded

**Leigh.Samphier/Sandbox/Definition:P-adic Norm

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

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

**Leigh.Samphier/Sandbox/Definition:P-adic Norm/Rational Numbers/Definition 1

**Leigh.Samphier/Sandbox/Definition:P-adic Norm/Rational Numbers/Definition 2

**Leigh.Samphier/Sandbox/Equivalence of Definitions of P-adic Norms

**Leigh.Samphier/Sandbox/Equivalence of Definitions of P-adic Norms/Lemma 1

**Leigh.Samphier/Sandbox/P-adic Norm Characterisation of Divisibility by Power of p

**Leigh.Samphier/Sandbox/Definition:P-adic Norm on P-adic Numbers

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

**Leigh.Samphier/Sandbox/Definition:P-adic Norm/P-adic Numbers/Notation

**Leigh.Samphier/Sandbox/Definition:P-adic Metric

**Leigh.Samphier/Sandbox/Definition:P-adic Metric/Rational Numbers

**Leigh.Samphier/Sandbox/Definition:P-adic Metric on Rational Numbers

**Leigh.Samphier/Sandbox/Definition:P-adic Metric/P-adic Numbers

Leigh.Samphier/Sandbox/Metric on P-adic Numbers Extends Metric on Rationals

**Leigh.Samphier/Sandbox/Integers with Metric Induced by P-adic Valuation

**Leigh.Samphier/Sandbox/Restricted P-adic Metric is Metric

Delete Definition:P-adic Metric/Restricted

Delete Definition:Restricted P-adic Metric/Definition 1

Delete Definition:Restricted P-adic Metric/Definition 2

Continuing Svetlana Katok Book

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous): $\S 1.4$ The field of $p$-adic numbers $\Q_p$

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

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$

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

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

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:

• 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)

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

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 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 $1.3$

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

Gelfand-Naimark Theorem

Jordan Curve Theorem

Gelfand-Mazur Theorem

Books to Add

• 2018: Bernhard H. Korte and Jens Vygen: Combinatorial Optimization: Theory and Algorithms (6th ed.)

Macros

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