OpenLLM-France/wikimedia huggingface dataset:
# The Universe Made Simple/The Earth/The Moon

The Universe Made Simple/The Solar System/Navigation

## The Moon

The Earth itself is very interesting; however, so is its moon. The Moon or Luna is 384,000 kilometers from the Earth. It has a radius of 1,738 kilometer and weighs 7.35 × 10^22 kilograms. The moon is 1/80 the mass of the earth and 1/4 its diameter. An interesting thing about the moon is that it slows the Earth’s rotation by two milliseconds every century. This means that 900 million years ago, there were 481 days and 18 hours in an Earth year. The moon’s orbit is very nearly circular and one lunar day and night is almost fifteen Earth days. The temperature on the moon changes drastically depending on whether it is day or night. At night, the temperature can drop to almost −113° Celsius. During the day, temperatures can reach 100° Celsius. This is all because the moon does not have surface water or an atmosphere.
# The Universe Made Simple/The Solar System

| ← The Solar System | The Universe Made Simple - The Galaxy | The Universe → |

## The Solar System

Everyone learns about our solar system in elementary school, but what is the solar system?
The eight planets and the sun make up the Solar System. It also includes moon, comets, asteroids, minor planets, dust, and gas. In addition, some of these things are labeled like the asteroid belt, Kuiper belt, and the Oort cloud.
Our solar system is 4.5 billion years old. In general, objects orbiting a star constitute a solar system. The prevailing theory, called "nebular hypothesis" seems to indicate that solar systems evolve from giant clouds of dust and gas. That this cloud coalesces due to gravity's effect, which forces the cloud to collapse under its own weight. This causes the entire cloud to start circling as if it was running down a drain in the middle where a start is formed. The dust continue to aggregate until small clumps of the dust began to form. These clumps became the planets and moons and asteroids are not only leftover material but the result of collisions that ultimately result in stable orbits.
# The Universe Made Simple

The Universe Made Simple is a book that attempts to describe the universe to someone who knows little or nothing about it to start with. It aims to explain the basic structure of the universe, our place in it, and some of its core, basic workings.
This book aims to be as simple as possible, but not at the risk of inaccuracy or dumbing-down. Some things will be omitted or phrased in looser terms. The book is not Wikijunior, but it also isn't a detailed or advanced book; it is for new learners wishing for a sound grasp of the most important points.

## Contents

Contents are defined, including the titles of the pages, at The Universe Made Simple/Contents Data.
* Introduction
* The Earth
** The Moon
** Artificial Satellites
* The Solar System
** The Sun
** Mercury
** Venus
** Mars
** Jupiter
** Saturn
** Uranus
** Neptune
** Dwarf Planet - Pluto
** Asteroid - Ceres
** Comets, Asteroids, Clouds and Belts
* The Galaxy
** The Sun is a Star
** Nature of Galaxies
** Light Years
** Movement of the Galaxies
* The Universe
** Origins (Big Bang)
** Formation of Stars, Galaxies and Planets
** Expansion
** The End of the Universe
* Laws
** Newton's 3 Laws of Motion
** Gravity
** Basic Laws of Electricity
* Matter
** Atoms
** Elements
** Subatomics
* And beyond
** Closure with metaphysical speculation
# The Universe Made Simple/Introduction

| ← Title Page | The Universe Made Simple - Introduction | The Earth → |
You're reading this book because you want to know more about the universe that you live in. But what is the universe?

## What is the universe?

The universe is everything that there is. Some people like to think of the universe as a vast container, in which everything that exists resides. However, no one knows whether or not this container is infinite in size, or if it has an "outer limit." No one knows for sure, as this question of "Is the universe finite or infinite?", has been asked for ages. Many scientists speculate that it expands outward as a balloon and has no end, continuing forever in all directions without any meaning at all. Still, some believe there is a limit and that all the galaxies and stellar objects are contained within a limited universe. This moves out of the realm of science and into the realm of metaphysics and philosophy, because there is no empirical data to suggest any conclusion as of yet.
When you look at the night sky, and see the stars, you are looking out into a very close part of the universe. The stars you see are such an incredible distance away that it makes you stand awestruck by it all. The blackness you see is the empty space of the universe; the parts that are not filled with light. However, there may be objects moving in the blackness that have no light shining on them, and we can only wonder what else is moving through space.

## How did people discover all this?

We're constantly hearing about research, and how some scientist here has discovered some new thing. But how is it actually all done? The answer is that in many cases it is extremely complicated, and that the method changes depending upon what you want to find out, and what you're looking at. Sometimes scientists build huge, special devices just to help them inspect something.
We discover things about the universe, however, by simply looking at it. We use huge telescopes that can see millions and millions of miles away. We have sent special robots called probes out into space to go and take pictures of distant things, and transmit the pictures back to Earth. We've even sent people in space-ships out into space. But space-travel is extremely expensive, difficult and dangerous. Most of the time, scientists work by taking observations, looking at how things move, and then using mathematics to work out what they must be like if they're moving like that or looking like that.

## What will this book teach you?

This book aims to be simple but informative. It will try not to miss out any important information, but it will also not confuse you with intricate details or physics.
The book will be like a journey. We will begin on Earth, then set off into the Solar System, out into the Galaxy, beyond it into the further Universe, and then even further. At each stage we hope to give you an appreciation of scale, and how everything fits together.
# The Universe Made Simple/The Moon

| ← The Earth | The Universe Made Simple - The Solar System | The Galaxy → |

## The Moon

The Earth itself is very interesting, however, so is its moon. The Moon or Luna is 384,000 kilometers from the Earth. It has a radius of 1,738 kilometer and weighs 7.35 x1022 kilograms. The moon is 1/80th the mass of the earth and 1/4th its diameter. An interesting thing about the moon is that it slows the Earth’s rotation by two milliseconds every century. This means that 900 million years ago, there were 481 days and 18 hours in an Earth year. The moon’s orbit is very nearly circular and one lunar day and night is almost fifteen Earth days. The temperature on the moon changes drastically depending on whether it is day or night. At night, the temperature can drop to almost -113° Celsius. During the day, temperatures can reach 100° Celsius. This is all because the moon does not have surface water or an atmosphere.
# The Universe Made Simple/The Solar System/Eight Planets

There are literally millions of objects orbiting the Sun - asteroids, comets, meteors and Kuiper Belt objects - but eight of these bodies are very large and seem to stand out from the rest. These eight enormous spheres are called the planets and are widely recognised to be the most interesting things in the solar system. In order from the Sun, they are:
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune.
You can use a mnemonic like "Men Very Easily Make Jugs Serve Useful Needs" to help remember the order of the planets from the Sun.
Some of you may be wondering what happened to the ninth planet, Pluto. Pluto's status has now been officially changed from a planet to a "dwarf planet". Astronomers realised that they didn't have an actual definition of the word "planet," even though they'd been using the word for centuries! By 2006 chaos had emerged and nobody could agree whether there were eight, nine or ten planets.
On August 24th of that year the International Astronomical Union (the official astronomy naming community) held a vote to determine what a planet actually is. The first definition they came up with was that a planet is something that orbits the Sun and is massive enough for gravity to pull it into a round shape. This included the nine planets known at the time (Mercury through to Pluto), but would also include several asteroids and Kuiper Belt objects as well. At the time twelve bodies were officially known to be round and orbited the Sun, but that number could have risen to hundreds as more round bodies were found. So the IAU tried again, and decided a planet should have enough mass to "clear the neighbourhood" around its orbit, and this definition won the vote.
Pluto is not a planet, because it orbits in the Kuiper Belt - its orbit is littered with thousands of other small icy bodies. However, the eight other planets have mainly empty orbits except for their moons and a few asteroids or wandering KBOs (known as centaurs.) Astronomers felt that to just get rid of Pluto would be to ignore nearly 80 years of tradition and recognition of Pluto's status as the ninth planet, and so a new category, "dwarf planet" was created. Pluto is a dwarf planet. Dwarf planets are objects not quite big enough to clear their orbits and become planets, but big enough to be pulled into a round shape.
So now our solar system contains eight planets and three dwarfs - Ceres in the asteroid belt, Pluto in the Kuiper belt and Eris in the scattered disk. This page is about them.
# Signals and Systems/Table of Fourier Transforms
## Fourier Transform

    $F(j\omega )={\mathcal {F}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

## Inverse Fourier Transform

    ${\mathcal {F}}^{-1}\left\{F(j\omega )\right\}=f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

## Table of Fourier Transforms

This table contains some of the most commonly encountered Fourier transforms.
| | Time Domain | Frequency Domain |
| | $x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}$ | $X(\omega )={\mathcal {F}}\left\{x(t)\right\}$ |
| 1 | $X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$ | $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$ |
| 2 | $1\,$ | $2\pi \delta (\omega )\,$ |
| 3 | $-0.5+u(t)\,$ | ${\frac {1}{j\omega }}\,$ |
| 4 | $\delta (t)\,$ | $1\,$ |
| 5 | $\delta (t-c)\,$ | $e^{-j\omega c}\,$ |
| 6 | $u(t)\,$ | $\pi \delta (\omega )+{\frac {1}{j\omega }}\,$ |
| 7 | $e^{-bt}u(t)\,(b>0)$ | ${\frac {1}{j\omega +b}}\,$ |
| 8 | $\cos \omega _{0}t\,$ | $\pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,$ |
| 9 | $\cos(\omega _{0}t+\theta )\,$ | $\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 10 | $\sin \omega _{0}t\,$ | $j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,$ |
| 11 | $\sin(\omega _{0}t+\theta )\,$ | $j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,$ |
| 12 | ${\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | $\tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,$ |
| 13 | $\tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,$ | $2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 14 | $\left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,$ | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,$ |
| 15 | ${\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,$ | $2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,$ |
| 16 | $e^{-a|t|},\Re \{a\}>0\,$ | ${\frac {2a}{a^{2}+\omega ^{2}}}\,$ |
| Notes: | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function |
| This box: view • talk • |||
# Signals and Systems/Table of Trigonometric Identities
## Useful Mathematical Identities

| $\sin ^{2}\theta +\cos ^{2}\theta =1$ | $1+\tan ^{2}\theta =\sec ^{2}\theta$ |
| $\sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta$ | $\cos \left({\frac {\pi }{2}}-\theta \right)=\sin \theta$ |
| $\sec \left({\frac {\pi }{2}}-\theta \right)=\csc \theta$ | $\csc \left({\frac {\pi }{2}}-\theta \right)=\sec \theta$ |
| $\sin \left(-\theta \right)=-\sin \theta$ | $\cos \left(-\theta \right)=\cos \theta$ |
| $\sin 2\theta =2\sin \theta \cos \theta$ | $\cos 2\theta =\cos ^{2}-\sin ^{2}=2\cos ^{2}\theta -1=1-2\sin ^{2}\theta$ |
| $\sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}$ | $\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}$ |
| $\sin \alpha +\sin \beta =2\sin \left({\frac {\alpha +\beta }{2}}\right)\cos \left({\frac {\alpha -\beta }{2}}\right)$ | $\sin \alpha -\sin \beta =2\cos \left({\frac {\alpha +\beta }{2}}\right)\sin \left({\frac {\alpha -\beta }{2}}\right)$ |
| $\cos \alpha +\cos \beta =2\cos \left({\frac {\alpha +\beta }{2}}\right)\cos \left({\frac {\alpha -\beta }{2}}\right)$ | $\cos \alpha -\cos \beta =-2\sin \left({\frac {\alpha +\beta }{2}}\right)\sin \left({\frac {\alpha -\beta }{2}}\right)$ |
| $\sin \alpha \sin \beta ={\frac {1}{2}}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right]$ | $\cos \alpha \cos \beta ={\frac {1}{2}}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right]$ |
| $\sin \alpha \cos \beta ={\frac {1}{2}}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]$ | $1+\cot ^{2}=\csc ^{2}$ |
| $e^{j\theta }=\cos \theta +j\sin \theta$ | $\cos \theta ={\frac {e^{j\theta }+e^{-j\theta }}{2}}$ |
| $e^{-j\theta }=\cos \theta -j\sin \theta$ | $\sin \theta ={\frac {e^{j\theta }-e^{-j\theta }}{2j}}$ |
| $\tan \left({\frac {\pi }{2}}-\theta \right)=\cot \theta$ | $\cot \left({\frac {\pi }{2}}-\theta \right)=\tan \theta$ |
| $\tan(-\theta )=-\tan \theta$ | |
| $\tan ^{2}\theta ={\frac {1-\cos 2\theta }{1+\cos 2\theta }}$ | $\tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}$ |
# Signals and Systems/Resources
## Wikimedia Resources
### Wikibooks

* Communication Systems
* Digital Signal Processing
* Control Systems
* Engineering Analysis
* Serial Communications Bookshelf
* Signal Processing
Books on Statistics and Probability
* Probability
* Statistics
* Cryptography
* Engineering Analysis

### Wikipedia

* Wikipedia:Fourier transform
* Wikipedia:Laplace transform

## Books

* Haykin, Simon and Van Veen, Barry. Signals and Systems.Second Edition, Wiley, 2003. ISBN 0471378518
* Conrad, Bruce P. Differential Equations with Boundary Value Problems: A Systems Approach. Prentice Hall, 2003. ISBN 0130934194
* Lathi, B. P. Signal Processing and Linear Systems, Oxford University Press, 1998. ISBN 0195219171
* Miller, Scott L and Childers, Donald G, Probability and Random Processes: With Applications to Signal Processing and Communications, Elsevier, 2004. ISBN 0121726517
* Oppenheim, Alan V. and Willsky, Alan S. Signals and Systems. Second Edition, Prentice-Hall, 1997. ISBN 0138147574

## Further Reading

The following books list Signals and Systems as a prerequisite:
# Signals and Systems/Table of Laplace Transforms
## Laplace Transform

    $F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

## Inverse Laplace Transform

    ${\mathcal {L}}^{-1}\left\{F(s)\right\}={1 \over {2\pi i}}\int _{c-i\infty }^{c+i\infty }e^{ft}F(s)\,ds=f(t)$

## Laplace Transform Properties

| Property | Definition |
| Linearity | ${\mathcal {L}}\left\{af(t)+bg(t)\right\}=aF(s)+bG(s)$ |
| Differentiation | ${\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0^{-})$ ${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})$ ${\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$ |
| Frequency Division | ${\mathcal {L}}\{tf(t)\}=-F'(s)$ ${\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)$ |
| Frequency Integration | ${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma$ |
| Time Integration | ${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{u(t)*f(t)\right\}={1 \over s}F(s)$ |
| Scaling | ${\mathcal {L}}\left\{f(at)\right\}={1 \over a}F\left({s \over a}\right)$ |
| Initial value theorem | $f(0^{+})=\lim _{s\to \infty }{sF(s)}$ |
| Final value theorem | $f(\infty )=\lim _{s\to 0}{sF(s)}$ |
| Frequency Shifts | ${\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)$ ${\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)$ |
| Time Shifts | ${\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)$ ${\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)$ |
| Convolution Theorem | ${\mathcal {L}}\{f(t)*g(t)\}=F(s)G(s)$ |
Where:
    $f(t)={\mathcal {L}}^{-1}\{F(s)\}$
    $g(t)={\mathcal {L}}^{-1}\{G(s)\}$
    $s=\sigma +j\omega$

## Table of Laplace Transforms

| No. | Time Domain $x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$ | Laplace Domain $X(s)={\mathcal {L}}\left\{x(t)\right\}$ |
| 1 | ${\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds$ | $\int _{-\infty }^{\infty }x(t)e^{-st}dt$ |
| 2 | $\delta (t)\,$ | $1\,$ |
| 3 | $\delta (t-a)\,$ | $e^{-as}\,$ |
| 4 | $u(t)\,$ | ${\frac {1}{s}}$ |
| 5 | $u(t-a)\,$ | ${\frac {e^{-as}}{s}}$ |
| 6 | $tu(t)\,$ | ${\frac {1}{s^{2}}}$ |
| 7 | $t^{n}u(t)\,$ | ${\frac {n!}{s^{n+1}}}$ |
| 8 | ${\frac {1}{\sqrt {\pi t}}}u(t)$ | ${\frac {1}{\sqrt {s}}}$ |
| 9 | $e^{at}u(t)\,$ | ${\frac {1}{s-a}}$ |
| 10 | $t^{n}e^{at}u(t)\,$ | ${\frac {n!}{(s-a)^{n+1}}}$ |
| 11 | $\cos(\omega t)u(t)\,$ | ${\frac {s}{s^{2}+\omega ^{2}}}$ |
| 12 | $\sin(\omega t)u(t)\,$ | ${\frac {\omega }{s^{2}+\omega ^{2}}}$ |
| 13 | $\cosh(\omega t)u(t)\,$ | ${\frac {s}{s^{2}-\omega ^{2}}}$ |
| 14 | $\sinh(\omega t)u(t)\,$ | ${\frac {\omega }{s^{2}-\omega ^{2}}}$ |
| 15 | $e^{at}\cos(\omega t)u(t)\,$ | ${\frac {s-a}{(s-a)^{2}+\omega ^{2}}}$ |
| 16 | $e^{at}\sin(\omega t)u(t)\,$ | ${\frac {\omega }{(s-a)^{2}+\omega ^{2}}}$ |
| 17 | ${\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)$ | ${\frac {1}{(s^{2}+\omega ^{2})^{2}}}$ |
| 18 | ${\frac {t}{2\omega }}\sin \omega t$ | ${\frac {s}{(s^{2}+\omega ^{2})^{2}}}$ |
| 19 | ${\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)$ | ${\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}$ |
# Signals and Systems/Probability Operations
## Expected Value Operator

The Expected value operator is a linear operator that provides a mathematical way to determine a number of different parameters of a random distribution. The downside of course is that the expected value operator is in the form of an integral, which can be difficult to calculate.
The expected value operator will be denoted by the symbol:
    $\mathbb {E} [\cdot ]$
For a random variable X with probability density f$_x$, the expected value is defined as:
    $\mathbb {E} [X]=\int _{-\infty }^{\infty }xf_{X}(x)dx$.
provided the integral exists.
The Expectation of a signal is the result of applying the expected value operator to that signal. The expectation is another word for the mean of a signal:
    $\mu _{x}=\mathbb {E} [X]$

## Moments

The expected value of the N-th power of X is called the N-th moment of X or of its distribution:
    $\mathbb {E} [X^{N}]=\int _{-\infty }^{\infty }x^{N}f_{X}(x)dx$.
Some moments have special names, and each one describes a certain aspect of the distribution.

## Central Moments

Once we know the expected value of a distribution, we know its location. We may consider all other moments relative to this location and calculate the Nth moment of the random variable X - E[X]; the result is called the Nth central moment of the distribution. Each central moment has a different meaning, and describes a different facet of the random distribution. The N-th central moment of X is:
    $\mathbb {E} [(X-\mathbb {E} [X])^{N}]=\int _{-\infty }^{\infty }(x-\mathbb {E} [X])^{N}f_{X}(x)dx$.
For sake of simplicity in the notation, the first moment, the expected value is named:
    $\mathbb {E} [X]=\mu _{X}$,
The formula for the N-th central moment of X becomes then:
    $\mathbb {E} [(X-\mu _{X})^{N}]=\int _{-\infty }^{\infty }(x-\mu _{X})^{N}f_{X}(x)dx$.
It is obvious that the first central moment is zero:
    $\mathbb {E} [(X-\mu _{X})]=0$
The second central moment is the variance,
    $\mathbb {E} [(X-\mu _{X})^{2}]=\sigma ^{2}$

### Variance

The variance, the second central moment, is denoted using the symbol σ$_x$$^2$, and is defined as:
    $\sigma _{x}^{2}=\mathrm {var} (X)=\mathbb {E} [X-\mathbb {E} [X]]^{2}=\mathbb {E} [X^{2}]-\mathbb {E} [X]^{2}$

### Standard Deviation

The standard deviation of a random distribution is the square root of the variance, and is given as such:
    $\sigma _{x}={\sqrt {\sigma _{x}^{2}}}$

## Time-Average Operator

The time-average operator provides a mathematical way to determine the average value of a function over a given time range. The time average operator can provide the mean value of a given signal, but most importantly it can be used to find the average value of a small sample of a given signal. The operator also allows us a useful shorthand way for taking the average, which is used in many equations.
The time average operator is denoted by angle brackets (< and >) and is defined as such:
    $\langle f(t)\rangle ={\frac {1}{T}}\int _{0}^{T}f(t)dt$
# Solutions to Algebra (2nd Edition), M. Artin

Chapter 1
Chapter 2
Chapter 3
Chapter 4
# Algebra by Gelfand, I.M.
## Contents

Exercises 1-25
Exercises 26-50
# Solutions To Computer Engineering Textbooks

Following the example of Solutions To Mathematics Textbooks, this wikibook aims to provide solutions to any textbook belonging to the Computer Engineering field. The textbook industry of today has made it very difficult for students without access to a University course to accurately assess their progress and understanding of the material.

## Convention

We will follow the same convention of creating a sub-page for each textbook used by Solutions To Mathematics Textbooks, with the following nomenclature for the article name:
/Solutions_To_Computer_Engineering_Textbooks/BookTitle_(Edition)_(ISBN)

## Contents

* Computer Organization and Design: The Hardware-Software Interface (5th Edition) (9780124077263)
# Solutions To Computer Engineering Textbooks/Computer Organization and Design: The Hardware-Software Interface (5th Edition) (9780124077263)
## Contents

* Solutions to exercises in chapter 1
* Solutions to exercises in chapter 2
* Solutions to exercises in chapter 3
* Solutions to exercises in chapter 4
* Solutions to exercises in chapter 5
* Solutions to exercises in chapter 6
# Sport Innovation/Skins Sportswear/References

[1] SKINS Official Website
[2] SKINS Products
[3] Johnathan Thurston
[4] USA Cycling Inc.
[5] Australian Rugby Union
[6] Li-Ning Official Website
[7] SKINS Distributors
[8] Duffield, R. (2007). Comparison of three types of full-body compression garments on throwing and repeat-sprint performance in cricket players. Sports Med, 4, 409-414.
[9] Compression Garments for enhanced Performance
[10] Sporty's Warehouse
[11] Sports Pro
[12] 2XU Official Website
[13] http://www.combo.com.au/clients/case-studies/case-study-2xu
[14] Body Science Official Website
[15] Review on Body Science Compression Pants
[16] Review on Compression Gear
[17] SKINS Technolgies
[18] SKINS ICE Apparel Review
[19] New Hygienic World
[20] SKINS Atheletes
[21] SKINS on Twitter
[22] SKINS on Facebook
[23] SKINS on YouTube
[24] Starbury Shoe Phenomenon on YouTube