hcju/proofwikips huggingface dataset:
Center is Element of Closed Ball/Normed Division Ring
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$. Then: :$a \in \map { {B_\epsilon}^-} a$
Center is Element of Closed Ball/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$. Then: :$a \in \map { {B_\epsilon}^-} a$
Length of Arc of Small Circle
Let $S$ be a sphere. Let $\bigcirc FCD$ be a small circle on $S$. Let $C$ and $D$ be the points on $\bigcirc FCD$ such that $CD$ is the arc of $\bigcirc FCD$ whose length is to be determined.
Definition:Geographical Coordinates
Let $J$ be a point on Earth's surface. The '''geographical coordinates''' of $J$ are the definition of the position of $J$ with respect to the equator and the principal meridian.
Two Parallel Lines lie in Same Plane
Let $L_1$ and $L_2$ be two lines which are parallel. Then $L_1$ and $L_2$ both lie in the same plane.
Center is Element of Open Ball
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $M$. Then: :$a \in \map {B_\epsilon} a$
Center is Element of Open Ball/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$. Then: :$a \in \map {B_\epsilon} a$
Center is Element of Open Ball/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$. Then: :$a \in \map {B_\epsilon} a$
Equivalence of Definitions of Non-Archimedean Division Ring Norm
Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.  === Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}} === Definition 2 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 2}}
Sine of Half Angle for Spherical Triangles
:$\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$ where $s = \dfrac {a + b + c} 2$.
Open Ball in P-adic Numbers is Closed Ball
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$:  :Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ :Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$.  Then: :$\forall n \in Z : \map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$
Tangent of Half Angle for Spherical Triangles
:$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$ where $s = \dfrac {a + b + c} 2$.
Countable Basis for P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.   Then: :$\BB_p = \set {\map {B_{p^{-n} } } q : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.
P-adic Numbers is Second Countable Topological Space
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is second-countable.
P-adic Numbers is Totally Disconnected Topological Space
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is totally disconnected.
P-adic Numbers is Hausdorff Topological Space
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff.
P-adic Numbers is Locally Compact Topological Space
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is locally compact.
Open and Closed Balls in P-adic Numbers are Compact Subspaces
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n}}} a$ are compact. {{disambiguate|Definition:Compact}}
Ambiguous Case
Let $\triangle ABC$ be a triangle. Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively. Let the sides $a$ and $b$ be known. Let the angle $\angle B$ also be known. Then it may not be possible to know the value of $\angle A$. This is known as the '''ambiguous case'''.
Analogue Formula for Spherical Law of Cosines
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: {{begin-eqn}} {{eqn | l = \sin a \cos B       | r = \cos b \sin c - \sin b \cos c \cos A }} {{eqn | l = \sin a \cos C       | r = \cos c \sin b - \sin c \cos b \cos A }} {{end-eqn}}
Four-Parts Formula
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. We have: :$\cos a \cos C = \sin a \cot b - \sin C \cot B$ That is: :$\map \cos {\text {inner side} } \cdot \map \cos {\text {inner angle} } = \map \sin {\text {inner side} } \cdot \map \cot {\text {other side} } - \map \sin {\text {inner angle} } \cdot \map \cot {\text {other angle} }$ This is known as the '''four-parts formula''', as it defines the relationship between each of four consecutive parts of $\triangle ABC$.
Countable Basis for P-adic Numbers/Closed Balls
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.   Then: :$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.
Countable Basis for P-adic Numbers/Cosets
Let $\Z_p$ be the $p$-adic integers. Then: :$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.
Napier's Tangent Rule for Right Spherical Triangles
Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the angle $\sphericalangle C$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''interior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''.
Napier's Cosine Rule for Quadrantal Triangles
Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the cosine of the '''opposite parts'''.
Napier's Rules for Right Spherical Triangles
'''Napier's Rules for Right Spherical Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles is a right angle. :410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let angle $\angle C$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as the '''interior''' of the above, where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''.
Napier's Rules for Quadrantal Triangles
'''Napier's Rules for Quadrantal Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose sides is a right angle. :410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as the '''exterior''' of the above, where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''.
Spherical Triangle is Polar Triangle of its Polar Triangle
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$. Then $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.
Side of Spherical Triangle is Supplement of Angle of Polar Triangle
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$. Then $A'$ is the supplement of $a$. That is: :$A' = \pi - a$ and it follows by symmetry that: :$B' = \pi - b$ :$C' = \pi - c$
Integers are Dense in P-adic Integers/Unit Ball
The integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.
Analogue Formula for Spherical Law of Cosines/Corollary
{{begin-eqn}} {{eqn | l = \sin A \cos b       | r = \cos B \sin C + \sin B \cos C \cos a }} {{eqn | l = \sin A \cos c       | r = \cos C \sin B + \sin C \cos B \cos a }} {{end-eqn}}
Four-Parts Formula/Corollary
:$\cos A \cos c = \sin A \cot B - \sin c \cot b$
Cosine in Terms of Haversine
:$\cos \theta = 1 - 2 \hav \theta$ where $\cos$ denotes cosine and $\hav$ denotes haversine.
Spherical Law of Haversines
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\hav a = \map \hav {b - c} + \sin b \sin c \hav A$ where $\hav$ denotes haversine.
Sign of Haversine
The haversine is non-negative for all $\theta \in \R$.
Haversine Function is Even
The haversine is an even function: :$\forall \theta \in \R: \map \hav {-\theta} = \hav \theta$
Delambre's Analogies
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Napier's Analogies
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Half Angle Formulas for Spherical Triangles
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Half Side Formulas for Spherical Triangles
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Tangent of Half Side for Spherical Triangles
:$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$ where $S = \dfrac {a + b + c} 2$.
Zenith Distance is Complement of Celestial Altitude
Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$: :$z = 90 \degrees - a$
P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1
Then: :$\forall x, y, z \in R: \norm {x - y}_p \le \max \set {\norm {x - z}_p, \norm {y - z}_p}$
Zenith Distance of North Celestial Pole equals Colatitude of Observer
Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $z$ denote the zenith distance of $P$. Let $\psi$ denote the (terrestrial) colatitude of $O$. Then: :$z = \psi$
Altitude of North Celestial Pole equals Latitude of Observer
Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $a$ denote the altitude of $P$. Let $\phi$ denote the (terrestrial) latitude of $O$. Then: :$a = \phi$
Celestial Body moves along Parallel of Declination
Consider the celestial sphere $C$ whose observer is $O$. Let $B$ be a celestial body on $C$. Let $\delta$ be the declination of $B$. Let $P_\delta$ be the parallel of declination whose declination is $\delta$. Then $B$ appears to move along the path of $P_\delta$, in the direction from the north to east to south to west. It takes $1$ sidereal day for $B$ to travel around $P_\delta$.
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  Let $n, m \in Z$, such that $n < m$. Then: :$(1) \quad \map {B^-_{p^{-n}}} a = \displaystyle \bigcup_{i \mathop = 0}^{p^\paren {m - n} - 1} \map {B^-_{p^{-m} } } {a + i p^n}$ :$(2) \quad \set {\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dots, p^\paren {m - n} - 1}$ is a set of pairwise disjoint closed balls
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {{B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  Let $n, m \in Z$, such that $n < m$. Then: :$\map {B^-_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 0}^{p^\paren {m - n} - 1} \map {B^-_{p^{-m} } } {a + i p^\paren {m - n} }$
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.  Then: :$\forall n \in Z : \set{\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dotsc, p^\paren {m - n} - 1}$ is a set of pairwise disjoint open balls.
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1
:$\forall y \in \Q_p: \norm y_p \le p^{-n}$  there exists $i \in \Z$ such that: ::$(1)\quad 0 \le i \le p^\paren {m - n} - 1$ ::$(2)\quad \norm {y - i p^n}_p \le p^{-m}$
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $n, m \in Z$, such that $n < m$. Let $y \in \Q_p$. Let $\norm{y}_p \le p^{-n}$. Then there exists $i \in \Z$ such that: ::$(1) \quad 0 \le i \le p^\paren {m - n} - 1$ ::$(2) \quad \norm {y - i p^n}_p \le p^{-m}$
Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $y \in \Q_p$ Let $n, m \in Z$, such that $n < m$. Let there exist $i \in \Z$: :$(1): \quad 0 \le i \le p^\paren {m - n} - 1$ :$(2): \quad \norm {y - i p^n}_p \le p^{-m}$ Then: :$\norm y_p \le p^{-n}$
Open and Closed Balls in P-adic Numbers are Totally Bounded
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n} } } a$ are totally bounded subspaces.
Cofactor Sum Identity
Let $J_n$ be the $n \times n$ matrix of all ones. Let $A$ be an $n \times n$ matrix. Let $A_{ij}$ denote the cofactor of element $\paren {i,j}$ in $\det\paren A$, $1\le i,j \le n$.                  Then: :$\displaystyle \det \paren {A -J_n} = \det \paren A - \sum_{i \mathop  = 1}^n \sum_{j \mathop = 1}^n A_{ij} $
Sum of Elements of Invertible Matrix
Let $\mathbf J_n$ be the $n \times n$ square ones matrix. Let $\mathbf B$ be an $n\times n$ invertible matrix with entries $b_{i j}$, $1 \le i, j \le n$. Then: :$\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 -  \map \det {\mathbf B} \map \det {\mathbf B^{-1} - \mathbf J_n}$
Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n}}} a$ and closed ball $\map {B^-_{p^{-n}}} a$ are clopen in the $p$-adic metric.
Closed Balls Centered on P-adic Number is Countable
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Then the set of all closed balls centered on $a$ is the countable set: :$\BB^- = \set {\map {B^-_{p^{-n} } } a: n \in \Z}$
Closed Balls Centered on P-adic Number is Countable/Open Balls
Then the set of all open balls centered on $a$ is the countable set: :$\BB = \set {\map {B_{p^{-n} } } a : n \in \Z}$
Closed Balls Centered on P-adic Number is Countable/Lemma
:$\exists n \in \Z : p^{-n} \le \epsilon < p^{-\paren {n - 1} }$
Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma
:$\exists n \in \Z : p^{-\paren {n + 1} } < \epsilon \le p^{-n}$
Open Ball contains Smaller Open Ball
Let $M = \struct{A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon \le \delta$. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$. Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$. Then: :$\map {B_\epsilon} a \subseteq \map {B_\delta} a$
Closed Ball contains Smaller Closed Ball
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$. Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$. Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$. Then: :$\map {B^-_\epsilon} a \subseteq \map {B^-_\delta} a$
Open Ball contains Strictly Smaller Closed Ball
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon < \delta$. Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$. Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$. Then: :$\map {B^-_\epsilon} a \subseteq \map {B_\delta} a$
Closed Ball contains Smaller Open Ball
Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$. Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$. Then: :$\map {B_\epsilon} a \subseteq \map {B^-_\delta} a$
Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers
Then the set of $p$-adic integers $\Z_p$ is compact.
Open Balls form Local Basis for Point of Metric Space
Let $M = \struct {A, d}$ be a metric space. Let $x \in A$. Let $\BB_x$ be the set of all open balls of $M$ centered on $x$. That is: :$\BB_x = \set {\map {B_\epsilon} x : \epsilon \in \R_{>0}}$ Then $\BB$ is a local basis of $x$.
Local Basis of P-adic Number/Cosets
Let $\Z_p$ be the $p$-adic integers. Then the set $\set {a + p^n \Z_p: n \in Z}$ is a local basis of $a$ consisting of clopen sets.
Local Basis of P-adic Number/Closed Balls
Then the set of closed balls $\set {\map {B^-_{p^{-n} } } a: n \in Z}$ is a local basis of $a$ consisting of clopen sets.
Measurement of Terrestrial Longitude
To measure the (terrestrial) longitude of a position: :$(1): \quad$ Identify the exact moment of noon at the location in question :$(2): \quad$ Work out the precise time of day $t$ that this happens. Let $N$ be the time of noon at the principal meridian on the day in question. Let $x$ be the number of hours before $N$ at the principal meridian. If $t$ is after $N$, then $x$ is treated as negative. Then the (terrestrial) longitude is $15 x \degrees$. If $x$ is positive, the (terrestrial) longitude is east. If $x$ is negative, the (terrestrial) longitude is west.
Coherent Sequence is Partial Sum of P-adic Expansion
Let $p$ be a prime number. Let $\sequence{\alpha_n}$ be a coherent sequence. Then there exists a unique $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$ such that: :$\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$
Difference of Consecutive terms of Coherent Sequence
Let $p$ be a prime number. Let $\sequence {\alpha_n}$ be a coherent sequence. Then: :for all $n \in \N_{>0}$ there exists $c_n \in \N$ such that: ::$0 \le c_n < p$ ::$\alpha_n - \alpha_{n - 1} = c_n p^n$
Vandermonde Matrix Identity for Cauchy Matrix
Assume values $\set { x_1,\ldots,x_n,y_1,\ldots,y_n }$ are distinct in matrix {{begin-eqn}} {{eqn   | l = C    | r = \paren {\begin{smallmatrix}         \dfrac {1} {x_1 - y_1} & \dfrac {1} {x_1 - y_2} & \cdots & \dfrac {1} {x_1 - y_n} \\         \dfrac {1} {x_2 - y_1} & \dfrac 1 {x_2 - y_2}   & \cdots & \dfrac {1} {x_2 - y_n} \\         \vdots                 & \vdots                 & \cdots & \vdots \\         \dfrac {1} {x_n - y_1} & \dfrac {1} {x_n - y_2} & \cdots & \dfrac {1} {x_n - y_n} \\ \end{smallmatrix} }   | c = Cauchy matrix of order $n$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn   | l = C   | r = -P V_x^{-1} V_y Q^{-1}   | c = Vandermonde matrix identity for a Cauchy matrix  }} {{end-eqn}}