Merge branch 'content'

vault/Astronomy/Horizontal Coordinate System.md

@@ -5,9 +5,68 @@ tags:
     - astronomy
 ---
 
+# The Celestial Horizon
+
+Due to its size, the Earth always obstructs about half of the celestial sphere, irrespective of what the location of an observer is. 
+
+>[!DEFINITION] Definition: The Celestial Horizon
+>
+>The plane tangent to the Earth at the location of an observer intersects the celestial sphere in a great circle known as the **celestial horizon** of the observer.
+>
+>![](res/The%20Celestial%20Horizon.svg)
+>
+
+The celestial horizon of an observer divides the celestial sphere into two hemispheres. The Earth obstructs the light coming from all celestial bodies in the lower hemisphere and so these objects cannot be seen by the observer.
+
+>[!NOTE]- Note: Astronomical Horizon vs True Horizon
+>
+>The aforementioned celestial horizon is also known as the observer's **astronomical horizon**. It is a simplification because it assumes that the observer has no size and is just a point on the surface of the Earth.
+>
+>In practice, however, all observers have a size. For example, your eyes are some 180 cm above Earth's surface. As height increases, so does the part of the lower hemisphere which an observer can see. The new plane below which all celestial bodies are obstructed by Earth and are thus invisible to the observer is called the **true horizon**.
+>
+>Usually, the observer's distance from the surface of the Earth is negligibly small and the astronomical horizon aligns more-or-less perfectly with the true horizon. Therefore, most of the time, we ignore the distinction between the two.
+>
+
+## Zenith and Nadir
+
+Imagine the line joining Earth's center and an observer. This line is perpendicular to the observer's celestial horizon and intersects the celestial sphere in two points.
+
+![](res/Zenith%20and%20Nadir.svg)
+
+>[!DEFINITION] Definition: Zenith
+>
+>The **zenith** of an observer is the point on the celestial sphere which is directly above them.
+>
+>>[!NOTATION]
+>>
+>>$$
+>>Z
+>>$$
+>>
+
+>[!DEFINITION] Definition: Nadir
+>
+>The **nadir** of an observer is the point on the celestial sphere which is directly below them.
+>
+>>[!NOTATION]
+>>
+>>$$
+>>Z'
+>>$$
+>>
+>
+
+## Drawing Celestial Spheres
+
+No single way to draw the celestial sphere for a given observer is more correct than another. However, there are certain conventions which have been established so as to ease the deciphering of the information on diagrams with celestial spheres.
+
+Most commonly, the celestial sphere is drawn such that the zenith of the observer is right at the top and their nadir is right at the bottom.
+
+![](res/Celestial%20Sphere%20Drawing%20Convention.svg)
+
 # Horizontal Coordinate System
 
-The **horizontal coordinate system** uses an observer's [celestial horizon](The%20Celestial%20Sphere.md#the%20celestial%20horizon) to assign coordinates to each celestial body on the [celestial sphere](The%20Celestial%20Sphere.md). 
+The **horizontal coordinate system** uses an observer's celestial horizon to assign coordinates to each celestial body on the [celestial sphere](The%20Celestial%20Sphere.md). 
 
 On the celestial horizon, we plot the points $N$, $S$, $E$ and $W$ such that when the observer is looking at $N$, they are looking north, when they are looking at $S$, they are looking south, when they are looking at $E$, they are looking east, and when they are looking at $W$, they are looking west.
 
@@ -61,7 +120,7 @@ Consider a celestial body $X$ and imagine the arc which goes through $X$ and the
 
 Most commonly, the altitude $a$ lies in the range $[-90 \degree; +90 \degree]$. Positive values are assigned to celestial bodies above the horizon and negative values are used for objects below it. The zenith has altitude $+90 \degree$ and the nadir has altitude $-90 \degree$.
 
-The azimuth $A$ lies in the range $[0; 360 \degree)$. The point $N$ is assigned an azimuth of $0$ and the coordinate increases clockwise, i.e. in the direction $N \to E \to S \to W$. Here is a table summarizing the horizontal coordinates for key points on the celestial sphere.
+The azimuth $A$ lies in the range $[0 \degree; 360 \degree)$. The point $N$ is assigned an azimuth of $0$ and the coordinate increases clockwise, i.e. in the direction $N \to E \to S \to W$. Here is a table summarizing the horizontal coordinates for key points on the celestial sphere.
 
 |Point|Altitude|Azimuth|
 |:--:|:--:|:--:|
@@ -72,3 +131,8 @@ The azimuth $A$ lies in the range $[0; 360 \degree)$. The point $N$ is assigned
 |$S$|$0 \degree$|$180 \degree$|
 |$W$|$0 \degree$|$270 \degree$|
 
+## Pros and Cons
+
+The horizontal coordinate system is the most natural choice of a coordinate system for each observer because it allows them to easily determine the positions of celestial bodies relative to them as they are in the current moment. It is particularly useful for quickly finding and observing objects as it requires little more than a compass to determine where the cardinal directions for the observer are. For example, if you want to look at Venus tonight, then you will be using its altitude and azimuth.
+
+The main disadvantage of this system is that the coordinates of celestial bodies depend on the observer's location on Earth and are always changing due to Earth's rotation.

vault/Astronomy/The Celestial Sphere.md

@@ -20,62 +20,3 @@ To achieve this, we introduce the concept of the **celestial sphere**.
 >
 
 We imagine that all celestial bodies are positioned on the surface of this sphere and then use coordinates to describe their location, ignoring the distance between us and them. Obviously, the same celestial body will have a different location, depending on where on Earth you are. In fact, two observers might not even be able to see the same object.
-
-## The Celestial Horizon
-
-Due to its size, the Earth always obstructs about half of the celestial sphere, irrespective of what the location of an observer is. 
-
->[!DEFINITION] Definition: The Celestial Horizon
->
->The plane tangent to the Earth at the location of an observer intersects the celestial sphere in a great circle known as the **celestial horizon** of the observer.
->
->![](res/The%20Celestial%20Horizon.svg)
->
-
-The celestial horizon of an observer divides the celestial sphere into two hemispheres. The Earth obstructs the light coming from all celestial bodies in the lower hemisphere and so these objects cannot be seen by the observer.
-
->[!NOTE]- Note: Astronomical Horizon vs True Horizon
->
->The aforementioned celestial horizon is also known as the observer's **astronomical horizon**. It is a simplification because it assumes that the observer has no size and is just a point on the surface of the Earth.
->
->In practice, however, all observers have a size. For example, your eyes are some 180 cm above Earth's surface. As height increases, so does the part of the lower hemisphere which an observer can see. The new plane below which all celestial bodies are obstructed by Earth and are thus invisible to the observer is called the **true horizon**.
->
->Usually, the observer's distance from the surface of the Earth is negligibly small and the astronomical horizon aligns more-or-less perfectly with the true horizon. Therefore, most of the time, we ignore the distinction between the two.
->
-
-## Zenith and Nadir
-
-Imagine the line joining Earth's center and an observer. This line is perpendicular to the observer's celestial horizon and intersects the celestial sphere in two points.
-
-![](res/Zenith%20and%20Nadir.svg)
-
->[!DEFINITION] Definition: Zenith
->
->The **zenith** of an observer is the point on the celestial sphere which is directly above them.
->
->>[!NOTATION]
->>
->>$$
->>Z
->>$$
->>
-
->[!DEFINITION] Definition: Nadir
->
->The **nadir** of an observer is the point on the celestial sphere which is directly below them.
->
->>[!NOTATION]
->>
->>$$
->>Z'
->>$$
->>
->
-
-## Drawing Celestial Spheres
-
-No single way to draw the celestial sphere for a given observer is more correct than another. However, there are certain conventions which have been established so as to ease the deciphering of the information on diagrams with celestial spheres.
-
-Most commonly, the celestial sphere is drawn such that the zenith of the observer is right at the top and their nadir is right at the bottom.
-
-![](res/Celestial%20Sphere%20Drawing%20Convention.svg)

vault/Astronomy/Units of Distance.md

@@ -0,0 +1,15 @@
+---
+title: Units of Distance
+tags:
+    - astronomy
+---
+
+
+>[!UNIT] Unit of Measurement: Astronomical Unit
+>
+>Astronomical units are units of distance defined as
+>
+>$$
+>1\, \mathrm{AU} = 149\,597\,870\,700\, \mathrm{m}
+>$$
+>

vault/Computer Science/Networking/Protocols/Address Resolution Protocol (ARP)/ARP Caching.md

@@ -1,6 +1,6 @@
 # Introduction
 
-Since [ARP](index.md) is a dynamic resolution protocol, every address resolution requires the exchange of messages on the network. this consumes bandwidth and inhibits the overall performance of the network. Whilst ARP messages are not big, sending them too often will inevitably take its toll.
+Since [ARP](./index.md) is a dynamic resolution protocol, every address resolution requires the exchange of messages on the network. this consumes bandwidth and inhibits the overall performance of the network. Whilst ARP messages are not big, sending them too often will inevitably take its toll.
 
 The solution to this is **caching**.
 

vault/Mathematics/Algebra/Algebraic Structures/Algebraic Structure.md

@@ -1,6 +1,13 @@
+---
+title: Algebraic Structures
+tags:
+    - abstract-algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Algebraic Structure
 >
->An **algebraic structure** $(S, O, R)$ consists of a [non-empty](../../Set%20Theory/The%20Empty%20Set.md) [set](../../Set%20Theory/Set.md) $S$, a set $O$ of [operations](Operations/Operation.md) on $S$ and a finite set $R$ of rules which these operations must obey.
+>An **algebraic structure** $(S, O, R)$ consists of a [non-empty](../../Set%20Theory/The%20Empty%20Set.md) [set](../../Set%20Theory/index.md) $S$, a set $O$ of [operations](Operations/Operation.md) on $S$ and a finite set $R$ of rules which these operations must obey.
 >
 >>[!NOTE]
 >>

vault/Mathematics/Algebra/Algebraic Structures/Operations/Binary Operation.md

@@ -1,6 +1,13 @@
+---
+title: Binary Operations
+tags:
+    - abstract-algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Binary Operation
 >
->A **binary operation** on a [set](../../../Set%20Theory/Set.md) $S$ is an [operation](Operation.md) $f: S \times S \to S$.
+>A **binary operation** on a [set](../../../Set%20Theory/index.md) $S$ is an [operation](Operation.md) $f: S \times S \to S$.
 >
 >>[!NOTATION]-
 >>

vault/Mathematics/Algebra/Algebraic Structures/Operations/Closure.md

@@ -1,8 +1,15 @@
+---
+title: Closure
+tags:
+    - abstract-algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Closure
 >
->Let $S$ be a [set](../../../Set%20Theory/Set.md) with an $n$-[ary operation](Operation.md) $o: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S$.
+>Let $S$ be a [set](../../../Set%20Theory/index.md) with an $n$-[ary operation](Operation.md) $o: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S$.
 >
->We say that a [subset](../../../Set%20Theory/Subset.md) $C \subseteq S$ is **closed** under $o$ if 
+>We say that a [subset](../../../Set%20Theory/index.md) $C \subseteq S$ is **closed** under $o$ if 
 >
 >$$
 >o(c_1, \cdots, c_n) \in C \qquad \forall c_1, \cdots, c_n \in C

vault/Mathematics/Algebra/Algebraic Structures/Operations/Operation.md

@@ -1,6 +1,13 @@
+---
+title: Operations
+tags:
+    - abstract-algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Operation
 >
->An $n$**-ary operation** on a [set](../../../Set%20Theory/Set.md) $S$ is a [function](../../../Analysis/Functions/index.md) from the $n$-ary [Cartesian power](../../../Set%20Theory/Operations%20with%20Sets/Cartesian%20Product.md) of $S$ to $S$.
+>An $n$**-ary operation** on a [set](../../../Set%20Theory/index.md) $S$ is a [function](../../../Analysis/Functions/index.md) from the $n$-ary [Cartesian power](../../../Set%20Theory/Set%20Operations.md) of $S$ to $S$.
 >
 >$$
 >f: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S

vault/Mathematics/Algebra/Algebraic Structures/Operations/Unary Operation.md

@@ -1,4 +1,11 @@
+---
+title: Unary Operations
+tags:
+    - abstract-algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Unary Operation
 >
->A **unary operation** on a [set](../../../Set%20Theory/Set.md) $S$ is an [operation](Operation.md) $f: S \to S$.
+>A **unary operation** on a [set](../../../Set%20Theory/index.md) $S$ is an [operation](Operation.md) $f: S \to S$.
 >

vault/Mathematics/Algebra/Equations/Differential Equations/Ordinary Differential Equations/index.md

@@ -1,5 +1,11 @@
 ---
 title: Ordinary Differential Equations (ODEs)
+tags:
+    - ordinary-differential-equations
+    - differential-equations
+    - equations
+    - algebra
+    - mathematics
 ---
 
 # Ordinary Differential Equations
@@ -16,7 +22,7 @@ title: Ordinary Differential Equations (ODEs)
 >
 >>[!DEFINITION] Definition: Solution to an ODE
 >>
->>A **solution** to the ODE on the [subset](../../../../Set%20Theory/Subset.md) $\mathcal{D} \subseteq \mathbb{R}$ is an $n$-times [differentiable](../../../../Analysis/Real%20Analysis/Real%20Functions/Differentiation/Differentiability%20of%20Real%20Functions.md) [real function](../../../../Analysis/Real%20Analysis/Real%20Functions/Real%20Function.md) $\phi: \mathcal{D} \to \mathbb{R}$ such that
+>>A **solution** to the ODE on the [subset](../../../../Set%20Theory/index.md) $\mathcal{D} \subseteq \mathbb{R}$ is an $n$-times [differentiable](../../../../Analysis/Real%20Analysis/Real%20Functions/Differentiation/index.md) [real function](../../../../Analysis/Real%20Analysis/Real%20Functions/index.md) $\phi: \mathcal{D} \to \mathbb{R}$ such that
 >>
 >>$$
 >>F\left(x, \phi(x), \phi'(x), \phi''(x), \dotsc, \phi^{(n)}(x)\right) = 0 \qquad \forall x \in \mathcal{D}
@@ -28,7 +34,7 @@ title: Ordinary Differential Equations (ODEs)
 
 >[!DEFINITION] Definition: Ordinary Initial Value Problem
 >
->An **ordinary initial value problem** of order $n$ consists of an $n$-th order [ODE](index.md)
+>An **ordinary initial value problem** of order $n$ consists of an $n$-th order [ODE](./index.md)
 >
 >$$
 >F\left(x, y, y', y'', \dotsc, y^{(n)}\right) = 0

vault/Mathematics/Algebra/Equations/Differential Equations/index.md

@@ -1,4 +1,9 @@
 ---
 title: Differential Equations
+tags:
+    - differential-equations
+    - equations
+    - algebra
+    - mathematics
 ---
 

vault/Mathematics/Algebra/Equations/Equation.md

@@ -1,8 +1,16 @@
+---
+title: Equations
+tags:
+    - equations
+    - algebra
+    - mathematics
+---
+
 >[!DEFINITION] Definition: Equation
 >
->Let $X$ and $Y$ be two not necessarily unique [sets](../../Set%20Theory/Set.md) such that $Y$ has an [equivalence relation](../../Set%20Theory/Relations/Equivalence%20Relation.md) $E_Y$ defined on it.
+>Let $X$ and $Y$ be two not necessarily unique [sets](../../Set%20Theory/index.md) such that $Y$ has an [equivalence relation](../../Set%20Theory/Relations/Equivalence%20Relation.md) $E_Y$ defined on it.
 >
->An **equation** over $X$ is an [expression](../../Logic/Formal%20Languages/Expression.md) of the form
+>An **equation** over $X$ is an [expression](../../Formal%20Logic/Formal%20Languages.md) of the form
 >
 >$$
 >f(x) = g(x),