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.
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.
- 2008: Bernhard H. Korte and Jens Vygen: Combinatorial Optimization: Theory and Algorithms 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/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/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/Matroid satisfies Rank Axioms/Necessary Condition
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms
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
- 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
