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Wikijunior:Ancient Civilizations/Sumerians

What country did they live in?
The Sumerians lived in Mesopotamia around 4000 BC. Mesopotamia – or the Fertile Crescent – was a region in the Middle East between the Tigris and Euphrates rivers, in the territory of modern Iraq, Iran, and Syria. Ancient Sumeria was very small; it occupied only a narrow stretch of this territory, where the Tigris and Euphrates rivers reach the Persian Gulf. Their territory was mostly inside the modern-day borders of Iraq, with small stretches of land in Kuwait, Iran and Arabia. Sumerian traders are known to have sailed down the Gulf as far as modern-day Qatar and Bahrain, and to have traveled upstream to Haran, in the neighborhood of the modern-day cities of Mosul and Tikirit.

Since the land between these two rivers were very fertile, people first started practicing agriculture (farming), there. Sumer became home to the first civilizations and cities because of the development of farming. In the beginning these cities were independent, each city being a state or nation of its own. Each city had its own king, laws, gods, and calendar. The biggest cities were Eridu, Kish, Lagash, Uruk, Ur, and Nippur. The rulers of these cities led their people into constant wars.

What did their buildings look like?
The Sumerians built houses, palaces, and temples using mud bricks. Good stone is not found in the Euphrates delta, so it had to be transported at great expense over long distances. Small quantities of precious stone would be used to cover the brick in places, but most Sumerian buildings were brick.

Sumerians made bricks by scooping wet mud into a wooden frame. The bricks would be all of the same size, but slightly rounded on top. The Sumerians preferred low, wide bricks to the standard shape we use today. Hot sunshine would evaporate most of the water from the brick, which hardened. Builders only rarely fired their bricks to make them totally hard. This meant that heavy rains and floods would usually destroy Sumerian buildings. Fired bricks would be decorated with paints or glazes, and assembled to form pictures on walls. Sometimes the Sumerians covered walls of brick with an added layer of mud, into which they would push cones of colored clay to decorate walls.

The most famous type of Sumerian architecture was the  Ziggurat . Ziggurats were many-storied structures like Egyptian pyramids, or wedding cakes. Each layer was square, with a smaller layer built on top. Stairs on the outside of the ziggurat led up the sides to the top of the building. This put the top platform of the ziggurat high in the air. There the Sumerian priests put temples to their gods. It is possible they also watched the movements of the stars from observatories there,  built very high on a platform of mud brick. On these large platforms were built smaller and smaller platforms, creating a pyramid shape. Originally they were made out of mud brick but Ziggurats from later times were made out of fired bricks, which were glazed and adorned with pictures or faced with stone carvings.

Most people's houses did not look like this. They were simple mud brick huts, similar to those found all over the Middle and Near East. Many of them had courtyards at the center, with chambers for eating and working on the main floor, and rooms for sleeping either at the back or upstairs. We do not know what the roofs of these houses were like, but we can guess. They were built of heavy sticks set into the wall in one direction, and lighter sticks laid over them in another direction, and covered with a layer of mud.

What did they eat?
As Sumerians were mostly agricultural people, they ate what they farmed: wheat, barley, lentils, beans, garlic, onions, milk ,and milk products. They made bread and beer from grains. Meat, usually from goats or sheep and occasionally cattle, was probably rare in their diets; the animals were too expensive to kill. They ate fruits as well such as apple, banana and oranges.

Records of ancient Sumerian laws show that oxen — cattle bred, de-horned and neutered for work — were valuable but potentially dangerous animals. When they escaped, they did damage to fields and sometimes injured or killed people. Sheep and goats were probably more common. The wool and leather from the animals was also useful, because it could be turned into clothing, armor, or tools.

What did they wear?
Their clothing was mostly made out of wool and flax (which is a plant with blue flowers). The stem of flax was used in making the clothing. Clothing was also made out of other products they could grow themselves.

The men of Sumer often had long hair with a parting in the middle. Many had long beards as well, although some men preferred to be clean-shaven. The men's attire consisted of long skirts and no shirts. Women wore floor-length gowns that covered the left shoulder. The right shoulder and breast were left bare. They also wore their hair long, and most would braid their locks into one long braid which they then wrapped around the top of their heads. When they had company, they would simply wear headdresses.

Princesses wore bright coloured clothes and expensive jewelry made out of gold, silver, lapis lazuli, and carnelian.

Heavier clothing was worn in the winter and lighter clothing was worn in the summer.

Social status was determined by the quantity and quality of jewelry and clothing a person owned but they did not wear a lot of clothing because of where they lived (in a warm, dry climate).

What did their writing look like?

Sumerians were the first to invent writing. They wrote with sharpened sticks made of  reed  – a plant that grows on the banks of rivers. One of these sticks is called a  stylus . When you push this kind of pen into a tablet of raw clay, you’ll get lines that look like wedges: thicker on one side, thinner on the other. That is why their writing is called  Cuneiform , which means "wedge-shaped".

Writing consisted of small pictures. At first, these pictures were drawn into the clay with the stylus. There was a different picture for each word. That meant that a person had to know thousands of these pictures to read and write. Only very educated people, called scribes, were able to read and write. The scribes found drawing pictures to be very hard even in smooth wet clay, so they invented standard ways to combine wedges to form the pictures. After many centuries, they created very quick combinations of wedges that formed the pictures.

Children went to schools to learn how to be scribes. They began by reciting prayers to their gods and promises to their teachers to study hard and learn well. Then they learned to shape clay into many kinds of tablets — rounded squares for legal contracts, and large oblongs for poetry, and cylinders or prisms for royal decrees. They also copied out word lists — rows of symbols that belonged to specific categories, like the names of kings, and the words for different kinds of jobs, and words for different products available in the markets of the city. The teachers used sticks to beat slow learners, but also gave special tutoring to the students of parents who paid extra. After learning many words, scribes practiced making whole documents by copying out famous laws, poems and songs. This made them fast and accurate scribes.

Scribes wrote numbers with the back end of the stylus, and used different symbols for 1, 5, 10, 20, and 60. So the number "157" would be written with two 60s, a 20, a 10, a 5 and two 1s (60+60= 120 + 20 = 140 + 10 = 150 + 5 = 155 + 1 +1 = 157). Scribes could add and subtract. Some seem to have known how to multiply and divide, but most did not. Highly trained scribes appear to have known some geometry and astronomy, but the Sumerians did not have an accurate calendar.

What did they believe?
Sumerians had many gods. An was the god of the sky, Enki the god of water, Enlil the god of air, Ki the goddess of the earth, Inanna the goddess of fertility and many more. Plus, they had lots of local deities; each town had its own.

The sky god An was very important, but not always very smart. In one story, he decided to destroy all humans one day, because their noise was tremendous and it kept him awake. An ordered Enki to send a flood to destroy everyone, and Enki did so. But An had forgotten that people give the gods sacrifices of food, drink, and incense. Because of this, the gods nearly starved because An was upset about a little noise.

Fortunately, a Sumerian named Utnapishtim had gotten warning of the flood from Enlil, who is the god of air. Enlil told Utnapishtim to build a boat, and sail away on the flood with useful plants and animals in the boat. The old Sumerian man had done so, and when the flood waters receded, he sacrificed to the gods. The food revived An and the others, and they recognized their folly at last. Never again, they promised, would they try to wipe out humans.

A version of this story is recorded in the Hebrew Bible, which is repeated in the Christian Bible.

What is their history?
The Sumerians began as a series of powerful city-states. Each city had its own god and looked after itself and ruled the surrounding countryside. They fought each other as well as made alliances. This culture lasted from about 2900 - 2400 BC. They had a very distinct culture and their style of architecture, writing and religion spread throughout the Middle East.

The Akkadians settled North of Sumer along the Tigris and Euphrates rivers and copied many of the Sumerians' ideas. Later on, one of the Akkadian kings, Sargon I conquered all the city states of Sumer and Akkad and created the world's first empire. His reign lasted about 60 years (2371 - 2316 BC) and his empire stretched from the coast of Lebanon to Iran. When Sargon died, his empire broke up and the Sumerians and Akkadians were invaded by other groups of tribesmen from the mountains.

After this, the dominant forces in the Middle East were the Babylonians and Assyrians. These two civilizations were descendents of the Akkadians who had ruled over the whole area, but they split into separate empires. The Assyrians were in the North and the Babylonians were in the South. The Sumerians as a people no longer existed, though their writing, architecture and religion influenced the civilizations in this area for centuries after.They had over 100 gods.

Are some of them famous even today?

The most famous of Sumerians is Gilgamesh, legendary king of Uruk. He is known due to the Epic of Gilgamesh, a collection of Sumerian legends, later translated and written down in Akkadian. It tells about the adventure of Gilgamesh, his friendship with Enkidu, a wild man raised by animals, and his journey to obtain the secret of immortality.

In the Epic, Gilgamesh is portrayed as a young king full of arrogance and pride. Gilgamesh is a bully, and has no friends. The gods are worried because Gilgamesh is so unkind to his people, so they send Enkidu, a wild man every bit Gilgamesh's equal, to be his friend. When they meet each other, Gilgamesh starts a fight, but he discovers that Enkidu is so strong that he cannot win; but Enkidu discovers the same thing about Gilgamesh. As soon as they have tired each other out, without one or the other winning, they become fast friends. Enkidu teaches Gilgamesh kindness, while Gilgamesh teaches Enkidu to be civilized. Both grow up to be very handsome.

Gilgamesh decides he needs a tree for the roof of his palace, so he proposes a quest to the cedar forest to cut down a tree. Enkidu is horrified and warns Gilgamesh of the terrifying monster Humbaba who guards the forest. Gilgamesh manages to persuade his friend to go anyway, and they go on the quest in search of the tree. Together they meet and kill the monster, and chop down the tree.

But the death of Humbaba makes the gods angry. Further, the goddess Inanna sees Gilgamesh and wants him for her husband. But Gilgamesh tells her that she has already had too many husbands, and she has used them all up — she even kills them when she tires of them! Gilgamesh prefers being a bachelor, thank you. Angered, Inanna persuades her father to release the Bull of Heaven in Uruk, to destroy Gilgamesh and his people. The Bull runs wild in the streets, for to touch it is death. Many are slain by the touch of the Bull, its horns or its flanks or its tail.

Enkidu and Gilgamesh go out to destroy the Bull, and they succeed in killing it. But the Bull manages to touch Enkidu, who sickens and dies. Gilgamesh is heartbroken, and refuses to let anyone care for the body or bury his friend. He delays so long that maggots begin falling out of Enkidu's eye sockets and nostrils. Finally, Gilgamesh consents to the burial of his friend, but he cannot bear the thought of dying himself. His mother tries to console him and tell him that death is normal, but the king of Uruk will not hear it. He resolves to go to the ends of the earth in search of a cure for dying — a way to become immortal.

He faces many perils on his trip. First he travels through the tunnel that the Sun uses when traveling underneath the earth. He enters it at sunrise, and races to the other end, hoping to get out before the hot sun comes in at sunset, and burns him alive. He makes it, but then he must confront the scorpion monster. However, the scorpion monster's wife tells her husband to leave the human alone, he is so tired. The scorpion monster allows Gilgamesh to live, and guides him to Sudri, a woman who keeps a tavern for the gods at the end of the earth. Sudri refuses to give Gilgamesh the secret of eternal life, but the boatman offers to guide Gilgamesh to Utnapishtim, who lives forever. Together, Gilgamesh and the boatman cross the Waters of Death, though not without some trouble.

The next chapters of this epic form the story of Utnapishtim, the king who saved humankind of the Great Flood. His tale is very similar to the Biblical story of Noah’s Ark. Utnapishtim tells Gilgamesh that he cannot handle living forever, because he cannot even stay awake seven days. Despite his weariness, Gilgamesh agrees to stay awake for seven days. Mrs. Utnapishtim promises to bring Gilgamesh a loaf of bread every day so that he can eat. No sooner than she goes away to prepare the loaf for the first day, but Gilgamesh falls asleep! He sleeps all through all seven days. When he wakes, he insists to Utnapishtim that he only dozed a moment — but the seven stale and moldy loaves of bread beside him prove otherwise. Utnapishtim denies Gilgamesh the secret, and takes him back to the boatman.

The boatman is sympathetic to Gilgamesh, though, and tells the king of Uruk that Utnapishtim gets his eternal life from a plant under the water. If Gilgamesh only dives into the water and brings up the plant, he will have eternal life. Gilgamesh, now thinking of home, decides to get the plant and bring it home to his royal gardens in Uruk. There, it will make everyone immortal, and the whole city will thank him. He dives into the water, retrieves the plant, and returns across the water. There, while he sleeps, a snake comes along and devours the plant. Not until he is within sight of the walls of his home city, Uruk, does Gilgamesh realize that his journey was for nothing. He weeps, but consoles himself with the knowledge that he has built a strong city which will endure long after he is dead and gone.

What is left of them today?
Though Sumerians as an ethnic group do not exist anymore, they left many technological inventions, like the wheel (which made transport easier and quicker), the potter’s wheel, baked clay, the cuneiform script, and lots of agricultural inventions: irrigation systems, agricultural tools and even beer-brewing. They were the first people to use natural forces to do some of their own work by fitting sails to boats. They were also the first to make astronomical observations, and to measure time in 60-minute hours and 60-second minutes. Their priests invented clocks too, some were just like kinds of sundials and others were a type of water clock, with water dripping slowly from a pottery jar. The peasants in Sumeria were the first ones to find out how to harness animals to plows, instead of cultivating the land with hand hoes, and this allowed people to grow much more grain.

Recent collaboration between archaeologists and DNA analysts tell that there is a strong connection between the marsh Arabs of Iraq and Iran and the Sumerian culture.

Wikijunior Alte Zivilisationen/ Sumerer


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Wikijunior talk:Ancient Civilizations/Sumerians

Internal

 * Sumer
 * Mesopotamia
 * Ziggurat
 * Cuneiform script
 * Epic of Gilgamesh

External

 * Encyclopedia Britannica
 * Middle East section of EMuseum

ImageObserver 17:08, 3 November 2005 (UTC)

New Map
The current map we have has far to much information, much of which doesn't relate to the Sumerians. I have created a map which is accurate enough. I do, however, recognize that it is in its rough form and needs more work. Feal free to work on it as you see fit. http://en.wikibooks.org/wiki/Image:Sumer.JPG

N1person 02:29, 2 March 2006 (UTC)

Typo?
Is "ethnical" a typo? Thanks, Andjam 14:58, 28 April 2006 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Embedded Systems/Terminate and Stay Resident

In the original DOS operating system, there was no capability for multi-threading, or multi-process mechanisms. However, it was found to be very beneficial to leave certain components in memory even after the process that created it ended. These program fragments were known as Terminate and Stay Resident modules, and were the precursors to the dynamic library infrastructure of current Windows operating systems. TSR routines are often used to implement device drivers, or common library functions.


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Signals and Systems/Common Distributions

There are a number of different random distributions in existence, many of which have been studied quite extensively, and many of which map very well to natural phenomena. This book will attempt to cover some of the most basic and most common distributions. This chapter will also introduce the idea of a distribution transformation, which can be used to turn a simple distribution into a more exotic distribution.

Uniform Distribution
One of the most simple distributions is a Uniform Distribution. Uniform Distributions are also very easy to model on a computer, and then they can be converted to other distribution types by a series of transforms.

A uniform distribution has a PDF that is a rectangle. This rectangle is centered about the mean, <μx, has a width of A, and a height of 1/A. This definition ensures that the total area under the PDF is 1.

Gaussian Distribution
The Gaussian (or normal) distribution is simultaneously one of the most common distributions, and also one of the most difficult distributions to work with. The problem with the Gaussian distribution is that its pdf equation is non-integratable, and therefore there is no way to find a general equation for the cdf (although some approximations are available), and there is little or no way to directly calculate certain probabilities. However, there are ways to approximate these probabilities from the Gaussian pdf, and many of the common results have been tabulated in table-format. The function that finds the area under a part of the Gaussian curve (and therefore the probability of an event under that portion of the curve) is known as the Q function, and the results are tabulated in a Q table.

PDF and CDF
The PDF of a Gaussian random variable is defined as such:

 * $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$

The CDF of the Gaussian function is the integral of this, which any mathematician will tell you is impossible to express in terms of regular functions.

The Functions Φ and Q
The normal distribution with parameters μ = 0 and σ = 1, the so-called standard normal distribution, plays an important role, because all other normal distributions may be derived from it. The CDF of the standard normal distribution is often indicated by Φ:

 * $$\Phi(x) = \frac{1}{2\pi}\int_{-\infty}^x e^{-\frac{t^2}{2}}dt$$.

It gives the probability for a standard normal distributed random variable to attain values less than x.

The Q function is the area under the right tail of the Gaussian curve and hence nothing more than 1 - Φ. The Q function is hence defined as:

 * $$Q(x) = 1 - \Phi(x)= \frac{1}{2\pi}\int_x^\infty e^{-\frac{t^2}{2}}dt$$

Mathematical texts might prefer to use the erf(x) and erfc(x) functions, which are similar. However this book (and engineering texts in general) will utilize the Q and Phi functions.

Poisson Distribution
The Poisson Distribution is different from the Gaussian and uniform distributions in that the Poisson Distribution only describes discrete data sets. For instance, if we wanted to model the number of telephone calls that are traveling through a given switch at one time, we cannot possibly count fractions of a phone call; phone calls come only in integer numbers. Also, you can't have a negative number of phone calls. It turns out that such situations can be easily modeled by a Poisson Distribution. Some general examples of Poisson Distribution random events are:

 * 1) The telephone calls arriving at a switch
 * 2) The internet data packets traveling through a given network
 * 3) The number of cars traveling through a given intersection

Transformations
If we have a random variable that follows a particular distribution, we would frequently like to transform that random process to use a different distribution. For instance, if we write a computer program that generates a uniform distribution of random numbers, and we would like to write one that generates a Gaussian distribution instead, we can feed the uniform numbers into a transform, and the output will be random numbers following a Gaussian distribution. Conversely, if we have a random variable in a strange, exotic distribution, and we would like to examine it using some of the easy, tabulated Gaussian distribution tools, we can transform it.


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Soil Mechanics

No engineering structure can be built on the ground or within it without the influence of geology being experienced by the engineer. Yet geology must remain an ancillary subject to students of engineering and it is therefore essential that their training is supported by a concise, reliable and usable text on the subject of geology and its relationship to engineering. In this book all the fundamental aspects of geology are described and explained, but within the limits thought suitable for engineers.

The structure of the Earth and the operation of its internal processes are described, together with the geological processes that shape and fashion the Earth and produce its rocks and soils. The commonly occurring types of rock and soil, and many types of geological structure and geological maps, are described. Throughout this book care has been taken to focus on the relationship between geology and geomechanics. For this reason emphasis has been placed on the geological processes that bear directly upon the composition, structure and mechanics of soil and rocks, and on the movement of groundwater.

1. The Earth: Surface, Structure, Age

 * /1.1 Introduction/
 * /1.2 Surface/
 * /1.3 Structure/
 * /1.4 Age/
 * /1.5 Principals of Geology/

2. Introduction to Soil Mechanics

 * /2.1 Soil phases, relationships and inter relationships/
 * /Soil classification/
 * /Engineering properties/
 * /Stress, Strain and Elastic Deformation/
 * /Consolidation/
 * /Compaction/
 * /Shear strength of soils/
 * /Bearing capacity of soils/
 * /Groundwater/
 * /Slope Stability/

3. Soil Testing

 * /Field Testing/
 * /Laboratory Testing/

4. Foundations

 * /Types of Foundations/
 * /Settlement of Foundations/
 * /Design of Foundations/

title: Soil Mechanics


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Wikijunior:Big Cats/Glossary

A glossary of terms used in this book:
 * Canine teeth: Sometimes called fangs, they are teeth used as weapons and to grasp and hold prey.
 * Captivity: When animals are kept by humans.
 * Carnivore: An animal that eats meat.
 * Carrion: The flesh of a dead animal, which is eaten by carnivores or scavengers.
 * Classification: The way scientists give animals names that everyone around the world will understand, in a way that reflects our scientific understanding of how closely different animals are related to one another.
 * Conservation: Ways of reducing the amount of natural resources used, including reusing and recycling.
 * Diurnal: Active during the day and asleep at night.
 * Domestication: Taking animals from their natural habitat into the human habitat, then changing them in ways that make them more suitable to living with humans.
 * Endangered: Animals that are in danger of becoming extinct.
 * Extinction: When all animals of a kind are gone, that kind of animal is extinct. They can never come back.
 * Family: A medium sized group used in classification, it is made up of very closely related animals.
 * Fertile: Able to have offspring.
 * Flank: The side part of an animal's body.
 * Habitat: An area able to support a type of animal.
 * Herbivore: An animal that eats plants such as leaves and berries.
 * Hybrid: The offspring of two animals of different but similar kinds, such as a lion and a tiger.
 * Hybridization: When two animals of different kinds have offspring together.
 * Instinct: An animal's inherited knowledge; knowledge that did not have to be learned.
 * Jungle: A large forest in warm countries with many different kinds of animals and plants. Also called a rainforest.
 * Life span: How long an animal usually lives.
 * Litter: A group of baby mammals with the same parents.
 * Mammal: A warm-blooded animal that has hair on its skin and gives its young milk to drink.
 * Mane: The furry crown around a male Lion's head.
 * Muzzle: The part of an animal's face that includes the nose and jaws.
 * National Park: Land that is kept in a natural state for a variety of reasons, all of them important.
 * Nature preserve: Land that is kept in a natural state for conservation reasons where the welfare of plants and animals is more important than any other uses.
 * New World: North, Central and South America.
 * Nocturnal: Active at night and asleep during the day.
 * Offspring: The children of an animal.
 * Old World: Europe, Asia, Australia and Africa.
 * Order: A very large group used in classification, it contains a number of closely related families.
 * Poaching: Illegally killing an animal.
 * Prairie: A large, flat plain with a lot of grass and almost no trees, such as the Great Plains of North America.
 * Predator: An animal that hunts other animals and eats them.
 * Prey: An animal hunted by another animal for food.
 * Rainforest: A large forest that stays warm and wet year round with many different kinds of animals and plants. Also called a jungle.
 * Rosette: A mark in an animal's fur where a number of closely placed spots form an unbroken circle.
 * Retract (claw): The hiding of claws inside a paw.
 * Savannah: A large, flat plain with a lot of grass and scattered groups of trees, such as in East Central Africa.
 * Scavenge: To look for meat for food that was killed by a different animal.
 * Scientific name: The name given to a species by scientists when they classify it.
 * Siblings: All the offspring from one litter, or sometimes offspring with the same parents from two different litters.
 * Species: A group of animals that under normal conditions can have offspring together.
 * Steppes: A large, flat plain with a lot of grass and almost no trees.
 * Sterile: Not able to have children.
 * Subtropical: An area that never gets very cold in the winter, but which has different seasons.
 * Tropical: An area that never gets cold and does not have different seasons. Tropical areas are found close to the equator, and on a world map the area between the Tropic of Capricorn and Tropic of Cancer is considered the tropics.
 * Wildlife refuge: Land that is used to protect of wild animals. Sometimes parts of a wildlife refuge are not kept in a natural state to give certain species of animals a little extra help.  Planting rice in a water bird area is one example.
 * Zoo: A type of living museum where animals are kept for people to learn about them and have fun.

Wikijunior Großkatzen/ Glossar

title: Wikijunior:Big Cats/Glossary


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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 fx, the expected value is defined as:

 * $$\mathbb{E}[X] = \int_{-\infty}^\infty x f_X(x)dx$$.

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 σx2, 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 is defined as such:

 * $$\langle f(t) \rangle = \frac{1}{T} \int_0^T f(t) dt$$


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Embedded Systems/Threading and Synchronization

One of the most useful developments in the history of computing is multitasking and multithreading. This technique isn't always available to an embedded system engineer, but some embedded systems and RTOS have multithreading (MT) capability. The chapters in this section will talk about some of the uses of MT, and will discuss some of the common pitfalls associated with MT programming. This page is only going to serve as a brief reference to multi-threaded programming.

Preemptive Multithreading
When the first multi-tasking systems were established, they did not have a central controller. Multi-tasking was established by having programs voluntarily give up control to the system, and the system would then give control to another process. This system worked reasonably well, except that any program that was misbehaving would slow down the entire system. For instance, if a program got stuck in an infinite loop, it would never give up control, and the system would freeze.

The solution to this problem is preemptive multithreading. In a preemptive environment, control could be moved from one process to another process at any given time. The process that was "preempted" would not even know that anything had happened, except maybe there would be a larger than average delay between 2 instructions. Preemptive multithreading allows for programs that do not voluntarily give up control, and it also allows a computer to continue functioning when a single process hangs.

There are a number of problems associated with preemptive multithreading that all stem from the fact that control is taken away from one process when it is not necessarily prepared to give up control. For instance, if one process were writing to a memory location, and was preempted, the next process would see half-written data, or even corrupted data in that memory location. Or, if a task was reading in data from an input port, and it was preempted, the timing would be wrong, and data would be missed from the line. Clearly, this is unacceptable.

The solution to this new problem then is the idea of synchronization. Synchronization is a series of tools provided by the preemptive multithreaded operating system to ensure that these problems are avoided. Synchronization tools can include timers, "critical sections," and locks. Timers can ensure that a given process may be preempted, but only for a certain time. Critical sections are commands in the code that prevent the system from switching control for a certain time. Locks are commands that prevent interference in atomic operations. These topics will all be discussed in the following chapters.

Mutexes
The term Mutex is short for "Mutual Exclusion", and is a type of mechanism used in a preemptive environment that can prevent unauthorized access to resources that are currently in use. Mutexes follow several rules:

 * 1) Mutexes are system wide objects, that are maintained by the kernel.
 * 2) Mutexes can only be owned by one process at a time
 * 3) Mutexes can be acquired by asking the kernel to allocate that mutex to the current task
 * 4) If a Mutex is already allocated, the request function will block until the mutex is available.

In general, it is considered good programming practice to release mutexes as quickly as possible. Some problems with mutexes will be discussed in the chapter on deadlocks.

Spin Locks
Spin locks is a quick form of synchronization methods. It is named after its behavior - spin in the loop while the condition is false. To implement spin lock system should support test-and-set idiom or give exclusive access to a locking thread by any means (masking interrupts, locking bus).

An advantage of spin locks is that they are very simple. A disadvantage is that they waste CPU cycles in loop waiting. Most common use of spin locks is to synchronize quick access to objects. It is not advisable to do a long computations while spin locked a section.

Critical Sections
A critical section is a sequence of computer instructions that may malfunction if interrupted. An atomic operation is a sequence of computer instructions that cannot be interrupted and function correctly. In practice, these two subtly different definitions are often combined. Operating systems provide synchronization objects to meet these requirements, and some actually call these objects as "critical sections," "atomic operations" or "monitors."

An example of a critical section is code that removes data from a queue that is filled by an interrupt. If the critical section is not protected, the interrupt can occur while the dequeuing function is changing pointers, and corrupt the queue pointers. An example of an atomic operation is an I/O read where the process must read all the data at a particular rate, and cannot be preempted while reading.

A generally good programming practice is to have programs exit their critical sections as quickly as possible, because holding a critical section for too long will cause other processes on the system not to get any time, and will cause a major performance decrease. Critical sections should be used with care.

Priority Scheduling
Many RTOS have a mechanism to distinguish the relative priorities of different tasks. High-priority tasks are executed more frequently than the low priority tasks. Each implementation of priority scheduling will be a little different, however.

Deadlock
A deadlock occurs when a series of synchronization objects are held in a preemptive MT system in such a way that no process can move forward. Let us take a look at an example:

Let's say that we have 2 threads: T1 and T2. We also have 2 mutexes, M1 and M2.
 * 1) T1 asks for and acquires mutex M1.
 * 2) T2 acquires M2
 * 3) T1 asks for M2, and the system transfers control to T2 until T2 releases M2.
 * 4) T2 asks for M1, and the system is in deadlock (neither thread can continue until the other releases it's mutex).

This is a very difficult problem to diagnose, and an even more difficult problem to fix. This chapter will provide some general guide-lines for preventing deadlocks.

Watchdog Timer
See ../Watchdog Timer.

read twice and compare
Perhaps the most common concurrency algorithm used in embedded systems is the "read the counter twice and compare" optimistic concurrency control pattern.

Many hardware timers and hardware counters are connected to the CPU over an 8-bit bus. If the low byte of the timer happens to be 0xFF when we start to read the timer, if the timer increments between two byte reads, a simple read of each byte separately will get a corrupted value. We get a slightly different corrupted value if we read the low byte first vs. the low byte last, but it's corrupted either way.

One simple solution is to read the counter twice and compare:

or

Because no locks are used, the double-read algorithm avoids deadlocks, avoids priority inversion, and avoids other problems with locks.

There are many other algorithms based on this read-twice-and-compare algorithm.

For example, the seqlocks used in Linux use this read-twice-and-compare algorithm.

For example, single-reader single-writer ring buffers are also common in embedded systems, and both the reader and the writer can be implemented with a similar lock-free algorithm.

incrementing counter
When the read-twice-and-compare algorithm is used to synchronize several processes (rather than, as above, a hardware counter and a process that reads the counter), the process that changes the counter needs to make that change in a way that appears atomic to the other process.

Many architectures have a single instruction that can atomically increment a variable. Alas, the exact details vary from one architecture to another, and from one C compiler to another. Some relatively portable ways to tell the C compiler to atomically increment a variable involve the std::atomic template class from the standard Atomic operations library, part of the C11 standard for the C programming language; or the tbb::atomic template class; or the boost::atomic template class; etc.

The main problem with the double-read algorithm is the ABA problem. To avoid the ABA problem, we use counters that only count up, and we give the counter enough bits that, for any reasonable amount of time a thread could possibly be delayed between reading the first byte of the first read and the last byte of the second read (say, maximum 1 second), the counter is long enough that at the worst-case (highest-frequency) count rate it takes much longer than that—say, the minimum time to overflow is once every 10 seconds.

For example, up-down counters can be synthesized from 2 counters, one that only counts up, and the other that only counts down.


text: Template:Wikijunior Big Cats

title: Template:Wikijunior Big Cats


text:
User talk:Lord Voldemort~enwikibooks

Wikijunior Big Cats

WJ Dinosaurs

Note from John Burkitt
Thanks for the kind words. Wikipedia and Wikibooks are wonderful resources and they capture all the best of what makes the internet great. You may not know this, but of the titles published by traditional houses, two out of every three are reissues of classics that are guaranteed to sell...either as texts for literature classes or as must haves to gather dust on a bookshelf. Of the remaining third, much of it is by the If you write it they will come rich and famous with juicy secrets about what it's like to make out with Donald Trump or order a low level operative in the CIA to take the heat. That leaves a number of by the author of books that people will buy because he/she also wrote Galaxy of Blood. Maybe one out of ten books is really an innovative title by a new author. The odds are stacked against new talent, or at least they were until the Internet. Wikipedia is unique in that it capitalizes on the idealism, fire, and social conscience of a wide spectrum of our society. People like you who make this resource work are my heroes. Thank you, thank you, thank you! John Burkitt 02:04, 18 November 2005 (UTC)

Don't Hold Your Breath
I know you would like to have Jimbo make some kind of conciliatory remark about his actions, but my experience in dealing with him is that he seldom gets involved in day-to-day matters of projects like Wikibooks directly. It is interesting that he gave himself admin privileges for Wikibooks at the same time he put up the notice of speedy delete on Staff Lounge. I don't know if he intended to use it or if that was merely a threat.

I have made a comment about this whole thing on Foundation-l and the response has been very interesting to say the very least. At least this is bring the issue up to the larger Wikimedia community, including the other board members... although I'm not sure if they are going to speak out against Jimbo here. This is going to be rather interesting to say the least about how this is going to be resolved.

The main point I want to make is that we as a community need to decide if we accept or reject Jimbo's "ruling" on this matter, and more importantly on how to change policy to fit this new reality. Thank you for pointing out that Wikibooks has been sent into a tailspin over this issue. --Rob Horning 02:55, 19 November 2005 (UTC)

Game Manuals
I certainly understand what you have been saying about the game manuals. However, they have not previously been considered to be against policy, they are not explicitly against policy, jimbo did not specifically say they were against policy (although admittedly he got close). I think that at the moment it is a waste of time to fight this fight. Until we can get an enforced policy, in writing, the community will never justify the mass deletion of all the game manuals. If you want to start targeting manuals, then don't go for something so big as the pokedex: It's a high profile book with a large following. Pick the smaller books that are in a state of disrepair.

We should also probably post a prominent link in the community portal to let people know that there is currently a debate on the game manual topic, and we can call an official vote on the issue soon. -- 19:25, 1 December 2005 (UTC)

Question about a block
I prefer to block an IP listed on SORBS whatever he contributes. If he really wants to help Wikibooks, he will log from normal IP or complain that he is not an open proxy. --Derbeth talk 20:59, 2 December 2005 (UTC)

Thanks for the message
I think our only real problem is that we're 2 very opinionated people who love the project, and have different ideas of whats best for it. Given that, I can't promise we won't butt heads again in the future, but I think we can put this recent argument behind us. --Gabe Sechan 17:19, 5 December 2005 (UTC)

Oops, I edited from a history
Really sorry about that. I was backpaging through the history of the page (thats how I normally read those megapages, so I can keep track of what has changed- they're too long to read every time I go to the site). I must have hit the edit from the history text tab rather than the current text tab. I didn't actually mean to delete those, just to add my comment and vote.--Gabe Sechan 18:12, 5 December 2005 (UTC)

Notice that I even accidently deleted one of my own votes there :/ --Gabe Sechan 18:13, 5 December 2005 (UTC)

Vandalism
Thanks a lot for fighting vandalism. Unfortunately, there are so few people who partol recent changes here and I really appreciate your help.

BTW, do you check anons to be open proxies? If you find any, report them to WB:VIP and I will block them. --Derbeth talk 09:48, 6 December 2005 (UTC)

Big Cats Wikijunior
Where is the discussion surrounding the December 10th deadline for Big Cats to be up-to-code? Thanks. -- LV (Dark Mark) 18:47, 6 December 2005 (UTC)
 * On the original proposal to the Beck Foundation (http://meta.wikimedia.org/wiki/Wikijunior/proposal_to_Beck_Foundation), we said it would be printed by December 2005. All good printers likely have a backlog of jobs, because they're so in demand. Plus, I have to preflight the file to make sure it will print. So, sooner is better than later. --  user:zanimum

 * Well, thanks for the response. So what is going to happen to the Solar System book? I thought that that book was going to be the first to print. Maybe I read that wrong somewhere. Anyway, do we even know who is going to be doing the printing or distributing? Do we have any of the processes in place to actually make this happen, or is this just something you noticed and decided to act upon? -- LV (Dark Mark) 19:21, 6 December 2005 (UTC)

 * That was discussed at Talk:Wikijunior Solar System/Archive 2. I think it'd be a way better idea to publish Solar System first, but unforunately our grant proposal for some reason explicitly says that our first one will be on animals.  Not sure if they'd really hold us to that or not.  Anyone know who is actually in communication with the Beck Foundation?  — Laura Scudder | Talk 20:36, 8 December 2005 (UTC)

 * Nevermind, I answered my own question. I've left a message with the Grant Coordinator, m:User:Danny.  — Laura Scudder | Talk 20:52, 8 December 2005 (UTC)

I don't end up doing as much with Wikibooks as I'd like, just proofreading and finding images. Did you ever get a response from Danny since August? I can't seem to find any. — Laura Scudder | Talk 01:34, 9 December 2005 (UTC)

Thanks
Yeah, that is essentially what i wanted, although i still have some fine-tuning to do. I havent used CSS in years, so i'm a little rusty at it. -- 18:20, 7 December 2005 (UTC)

Big cat maps
I'm glad you like them. I'm planning on doing one for each cat, but it takes longer with really big ranges. Hopefully I'll be done today or else later this weekend. — Laura Scudder | Talk 18:58, 9 December 2005 (UTC)

Marbled Cat
It is correctly considered a big cat, though it's the size of your average house cat. More closely related to the leopard than the cougar, it's really quite an oddity. I added it in a hurry. You can still smell the tire skid marks where I rushed to get this one in under the wire. By all means take a lookie. John Burkitt 02:07, 11 December 2005 (UTC)

Wikibooks Debate
Hello. I have seen the Staff Lounge discussions and the matter seems to have ended. Wales came in and did some rather unusual things, there was a reaction, and now everything has passed. Perhaps you had some justification mentioning the debate three weeks ago, but I think you were pushing your opinion when you announced that the Muggles' Guide was inappropriate for Wikibooks. The community decided the appropriateness of the Guide months ago and that was settled. One user will not come in and totally change all policies. Those discussing with you in Staff Lounge made it quite clear that you had a misrepresentation of Wikibooks policies and I think many more would agree with them. I believe your actions on Wikipedia were inappropriate since they effectively did scare others about the project. In the end, I just want to make sure there isn't trouble caused that doesn't need to be. Hit me at my Talk page if you need. -Matt 18:38, 12 December 2005 (UTC)
 * I believe that since you wrote the statements yourself (obviously), you don't see how they can be seen as inappropriate. Your point of view should not affect what you say to others at Wikipedia. As a reader, I see you saying "this book is going to be deleted so we need to figure out what's going to happen with some of the content." You were not very suggestive and were instead telling the audience what would happen. You should have exercised more grace and given a "heads up" instead of a plan of action based on your opinion of what would happen. Your statements of how the Guide "is" this and "isn't" that shouldn't be placed in the same location as your so-called "informative" message. Bolding the word "might" doesn't help explain the point of view you laced in with your message. Wikibooks changes will come at a gradual pace (and quite gradual at that since I think the Wales edit issues were rather random and current policy overwhelmed them) and suggesting the immediate backup of content contradicts your previous "informative only" remarks. Even after things calmed down, you didn't go back and modify your statement to be less rash. If you came back to make one change you could have made another. Your opinions here at Wikibooks have come under a great deal of criticism (evident through the Staff Lounge discussions) and I see this as a continuation. I disagree with your actions and I don't think our conversation will move any further. You have already begun throwing out personal insults towards me and so I think I'll end it here. My explanations have agreement with Wikibooks admins and several other big-time editors here so I'm fine with my stance. Hopefully you'll understand policy better the more time you spend here. -Matt 22:15, 12 December 2005 (UTC)
 * Just one last thing, since I don't want a name-calling fight (a "no personal attacks" policy doesn't need to be here either, I just think it's appropriate). From what I've seen, there's been clashes with you here on Wikibooks. I'd rather end this as "we agree to disagree." I think the Guide fits policy, you think it might not. That's the end. Don't get so personal. I'll take the time to make sure I've read all your comments. I'd love to see many of the policies you're proposing go into official effect and maybe I'll start getting into policy discussion like you do. "What Wikibooks Is" is a constant circular discussion here and I'd rather have friends at Wikibooks than enemies. Oh, and I just like cleared talk pages. If someone wants archives, they can view the history. -Matt 22:50, 12 December 2005 (UTC)

Christmas Break
Massive edit today to add some of the more popular itty-bitty-kitty species. Am taking a Christmas Break from Big Cats until after Season's Greetings. John Burkitt 03:56, 21 December 2005 (UTC)

One Last Thing
I edited the talk page on BIG CATS not for CENSORSHIP but for massive LENGTH. It has started triggering alerts when I add to it that the page is so long that some browsers can't edit it. That is a serious concern for the democracy of the group. There are several dead horses that no longer need kicking that date back months. I removed all superfluous comments I made that added nothing to the discussion, turned a few bullet point lists into concise paragraphs, and removed a couple of completely settled discussions like whether SOMEBODY should add Margays (I did). That page is becoming Gone With The Wind. Please remain aware of this situation so nobody gets locked out. John Burkitt 15:13, 21 December 2005 (UTC)

Return
Yeah, I've been here before. It's just that I work in Wikipedia most actively. Thanks for the welcome message anyways! -- WB 23:45, 29 December 2005 (UTC)

Re: English?

 *

Thank you for your message. Yes, I speak a bit english. I've left a buerocrat request on that page you recomended. Gubbubu 18:58, 11 January 2006 (UTC)

Blocking vandals
Hey, LV, if you block such obvious vandals as WoW, check Block log. Actually, I blocked him an hour ago ;-)

BTW: I was able to block and revert this vandal shortly after his appearance, IRC anti-vandal bot warned me about it. Without this bot this guy would make severe mess. Please take this in consideration and if you are online, join #wikibooks-en-vandalism, set highlighter on for 'wikibooks' and leave IRC client hidden in your taskbar. I reccomend you X-Chat for IRC connections, it's cross-platform, easy yet powerful. I'm also surprised that you're so active but have never (as far as I can remember) visited #wikibooks. --Derbeth talk 19:30, 13 January 2006 (UTC)

Thanks for the greeting...
Hi LV - Thanks for the greeting on my talk page. I have actually been registered here for about a year (UserID 3362, out of about 17,500!) but thanks for the thought... :) --HappyDog 18:40, 16 January 2006 (UTC)

 * :-) Appreciated! --HappyDog 19:41, 17 January 2006 (UTC)

Personal Greeting
Thanks for the policy and saying hi and stuff :) If only the 'pedia were as personal! So, Wikiversity is a good start to choosing which Wikibooks to read and in what order and so forth I guess.--Tyciol 10:01, 30 January 2006 (UTC)

Fair Use Policy
You seem to be interested in new policies, and this is one that I've come up with based on an eariler discussion which took place on the Foundation-l mailing list. Apparently Wikipedia is going through fits and starts with fair use, as is Wikisource. This policy is actually based off of a suggestion by one of the admins on the Italian Wikipedia, and I think it is a very good idea. Add comments and hack away at this policy as you see fit. --Rob Horning 16:59, 17 February 2006 (UTC)

Good to see you here, LV - it multiplies my faith in this project. BD2412 T 00:37, 6 May 2006 (UTC)

No personal attacks Change of text
You voted for enforcement of No personal attacks. The text of this policy has been changed by Jguk during the vote, I have further modified these changes to restore some of the original "bite". This is highly unusual but without these changes Jguk is unwilling to allow the policy to move to enforced. I am happy that the changes are not sufficiently substantial to undermine the policy, if you disagree then please make your views known. Changes to text during a vote could be construed as a delaying tactic so I have proposed a move to "enforced" status by 1st June. RobinH 10:03, 20 May 2006 (UTC)

Hmmm... Spaaaaaam.
I fail to see how a few links on a single page = spamming wikibooks, but if it really bothers you that much I'm sure I can leave the links off. :) Gerard Foley 22:11, 23 May 2006 (UTC)

Email...
As an active admin, you should have a specified email address so people can email their concerns to you. If you wouldn't mind, please activate it. Thanks. -- LV (Dark Mark) 21:06, 1 June 2006 (UTC)

 * Sorry, I thought (erroneously) that I had activated it. Have now corrected matters. Kind regards, Jguk 22:09, 1 June 2006 (UTC)

Editing disputes policy
This policy needs a few more people to take a look at it and vote. Any help would be most useful. RobinH 14:36, 5 June 2006 (UTC)

General voting rules/Proposal
I haven't seen you around in a while! I hope you haven't gotten sick of us. anyway, I created a new proposed version of the voting policy. It's gone through some heavy editing and discussion, but I think we are finally close to a proposal that we can put to a vote. Once we get the voting policy enforced, it will be easier to get other policies voted on in a timely, organized manner. If you are still around, and still want to help out, I hope you can come give us your comments/suggestions on this new proposal. --Whiteknight (talk) (projects) 19:23, 4 August 2006 (UTC)

General voting rules/Proposal
I want to call the attention of all wikibookians, especially active admins to the discussion about the new voting policy. Wikibooks currently does not have a concrete voting policy in effect, so the adaptation of this policy will potentially create immediate differences in the way business is conducted here. Due to large amounts of discussion, changes, and dissention, there are several versions of this policy being considered:

 * Version 1 Written by User:Whiteknight
 * Version 2 Written by User:Jguk
 * Version 3 User:RobinH's revision of Jguk's version
 * Version 4 Written by User:Kellen
 * Version 5 Whiteknight's Revision of Kellen's Version.

Also, there are additional issues being discussed, such as the inclusion of a "minimum contributions" clause, that will limit users with less then 20 votes from editing. Also, there is a proposed clause that would allow this 20 vote minimum to be raised in response to abuse.

I am sending this same message to all currently active admins. I hope you are able to come in and give your opinions on this matter. --Whiteknight (talk) (projects) 20:48, 11 August 2006 (UTC)

copyvio status confusion
was tagged as copyvio, but also has a copyleft tag on it. The image is used in WJ Big Cats, so I hate to delete it, but the user who posted it (User:John Burkitt) hasn't been around since 2005. Any thoughts? -- SB_Johnny | talk 12:51, 24 August 2006 (UTC)

Your De-Adminship
Hello. I would like to inform you that you will have your sysop rights removed here on 8 Oct 2007 due to inactivity. If you would like to discuss the matter, please see WB:RFA. You can re-apply for adminship at a later date if you wish. Thanks. -within focus 01:06, 8 September 2007 (UTC)
 * Hi, Lord Voldemort,
 * After a request on meta I have removed your adminship on this wiki, according to local removal policy. Effeietsanders 14:50, 11 October 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Lord Voldemort. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Lord Voldemort~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:23, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:12, 19 April 2015 (UTC)

title: User talk:Lord Voldemort~enwikibooks


text: User talk:206.163.0.82

title: User talk:206.163.0.82


text:
Guide to X11/Introduction

Introduction
This book, the Wikibooks Guide to X11 is about the X Window System. The target audience of this book is X11 users.

X11 is the graphical environment for most Unix or Unix-like systems, including *BSD and GNU/Linux; it provides access to screen, keyboard, and mouse. X11 is the Unix and Linux graphics drivers. Are you using Firefox, OpenOffice.org, GNOME, GIMP, or KDE with those systems? Then you are an X11 user.

This book is about the X Window System. Some users describe it as the Unix and Linux graphics drivers.

This chapter will explain the basics of X11, including:
 * Why are there so many window managers and desktop environments?
 * What is a console, and how does X provide access to it?
 * What is the purpose of X servers and X clients?
 * Is it possible for different X clients to interoperate?

X11 has diverse desktop environments
Most of the Unix and Linux distributions provide both a command line and X Window System. In Mac OS X, "X" is the version number "ten"; however, Apple and other parties provide X11 for Mac OS X, which allows you to use the X11 and native Macintosh window systems simultaneously.

However, just because so many systems provide X Window System, does not mean that it always looks the same. Some users of X have the GNOME desktop environment, some have the KDE desktop environment. Some use Xfce desktop environment while working with both GNOME and KDE programs. It is not as if all *BSD users have GNOME and all Linux users have KDE; instead, both *BSD and Linux users are split between those who use GNOME, those who use KDE, those who use other environments, and those who do not run X.

There are other Wikibooks, Using GNOME and Using KDE, that document those two environments.

X11 gives choice of window manager to user
There are many different window managers for X. Most windows have decorations around them which provide features like moving, closing, resizing, and otherwise organizing the window. However, these decorations are not the same on every computer that runs X.

Window managers are those programs which make those decorations. Most window managers draw title bars, but some do not. Some window managers make no decorations, but provide menus or keystrokes for managing windows. Some window managers allow you to overlap windows, switch between windows, and hide windows temporarily.

Some window managers are older than both GNOME and KDE, and some users continue to prefer lightweight window managers to the heavy desktop environments that occupy more memory and computing resources. Many X11 users are running Metacity, the GNOME window manager, or KWin, the KDE window manager.

What is extraordinary about X is that there are many window managers to pick from. (This book lists at least 20 window managers.) As the user, you pick which window managers to install, configure, and use. You can have your own custom environment, unless you are not able to install or configure the window manager that you want.

However, though X allows only one window manager, it allows more than one window. Unlike Mac or Windows, X11 does not provide a standard interface or color scheme, so each window might be drawing its menus, scrollbars, and buttons in a different way. At least the window decorations will be the same on every window. For example, one can run Firefox and OpenOffice.org, and also programs from GNOME and KDE, all at the same time on one X server, because each program has its own window or windows. But if you are running Metacity, then you cannot also be running KWin at the same time, as these two are Windows Managers, and not desktop environments.

X provides access to the console
At the interior, the X window system has a special protocol - version 11 of the X protocol. The role of this protocol is give console access to programs like Firefox, OpenOffice.org, Nautilus, and Konqueror.

Each home computer or workstation has a console, which consists of a screen, mouse, and keyboard. In fact, some consoles have multiple screens and multiple input devices, like two mice or a mouse and touchpad. Typically, the Unix kernel only provides a console terminal. The kernel loads a text-only video driver and a keyboard driver and makes a text-only terminal, good for running a command line such as the Bourne shell and certain text-only programs like vi and NetHack.

If we want to draw graphics or use the mouse, we must run a program that knows how to do this. We start an X11 program called the X server. It loads a graphics video driver, a keyboard driver, and a mouse driver. Then, the X server allows programs to make a window. A window is a rectangular area where the program appears; one can type and click windows. X11 allows programs to share the console because it allows more than one window on the screen. (For example, one might have two Firefox windows and one OpenOffice.org window open.) This is the same function that Mac and Windows has.

X clients connect to X servers
When you start the X server, it changes the screen to grey and makes a mouse cursor. Then it waits. Like a web server, which waits for web browsers to connect, or an ssh server, which waits for remote logins, the X server waits for X clients to connect. An X client is a program like Firefox and OpenOffice.org which wants to make one or more windows on the screen.

Unlike Mac or Windows, the X server can accept network connections. While you use the X server on one computer, you can login remotely to another computer and then ask it to connect to your X server. This lets you see and use programs that are on other computers, even if those computers use X on different operating systems.

When an X client connects to the X server (on the same or different computer), it speaks version 11 of the X protocol. This is how the client tells the server what to draw, and the server tells the client what the keyboard and mouse input is. The many X clients never need to know about the hardware or operating system of the X server, because the X server controls the console. Clients like Firefox and OpenOffice.org only need to know X protocol.

X sometimes lets programs interoperate
X11 programs share a clipboard and fonts, but sometimes they have not much else in common. This led to incompatibilities in not only appearance, but function: drag and drop between applications was sometimes impossible, and use of the clipboard was inconsistent.

Now, with efforts at freedesktop.org and elsewhere, some programs not of the same desktop environment can cooperate. Desktop environments, such as GNOME and KDE, and less-used environments like Xfce, provide applications that work well with each other. The results of these efforts include:
 * standard rules for using the clipboard
 * cooperation between window manager and applications
 * drag-and-drop between GNOME, KDE, and other windows
 * improvements to the X server

title: Guide to X11/Introduction


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User:20040302

Go find me at en.wikipedia.org

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title: Template:Guide to X11/stub


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Guide to X11/Configuring

Configuring
You would do $ XFree86 -configure $ Xorg -configure

or $ xf86cfg $ xorgcfg

Especially with XFree86 4.0 or Xorg, the "xf86config" and "xorgconfig" scripts are obsolete (unless you have old hardware).

This makes your XF86Config or xorg.conf file.

Debian and Ubuntu
If you use Debian, you can create a new config file if you reconfigure the xserver-xorg package. Running this will overwrite your current configuration, so create a backup copy, e.g.: cp /etc/X11/xorg.conf /etc/X11/xorg.conf.new.backup Reconfigure with: dpkg-reconfigure xserver-xorg

Easier Alternatives / Troubleshooting

 * Usually, the first thing you want to do if X is misbehaving is shut it down. Usually this can be done with ctrl-alt-backspace, but sometimes KDM/GDM will start back up (the login manager). Press ctrl-n or whatever for console login.
 * Login in as root and type "init 3" to go to runlevel 3. This will allow you to install graphic drivers.
 * If you are using opensuse, you should run sax2, or if that doesn't work, the option for non-accelerated x, sax2-vesa.
 * If you have an nvidia graphics card, they have an x configure script. the command is nvidia-xconfig, start typing "nvidia" and press tab twice. If you don't get anything, you need to install the graphics driver.

If you have Opensuse, you might receive help from Novell's Bugzilla team. They can't always immediately resolve the issue, but often you'll find that either because of your efforts or someone else, things are fixed in a later revision.

Manual Configuration
For manual configuration of almost anything, search for the gentoo or gentoo wikis. For this subject, http://www.gentoo.org/doc/en/xorg-config.xml should be useful.

title: Guide to X11/Configuring


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Guide to X11/Starting Sessions

Starting Sessions
This chapter describes how to start an X11 session.

This chapter will answer the following questions:
 * What are the different methods for starting X11?
 * How does one start X11 and a window manager?
 * How does one control which window manager and programs will start?
 * Is there a way to make fonts appear nicer?

Methods to start X11
Recall that X11 consists of an X server and several clients. If you only start the X server, you obtain a blank screen. What one needs is a way to start the X server plus some default clients such as a window manager.

First, some Unix consoles give a text-only login prompt. In this case, first login as normal to obtain your Unix shell, then use a xinit or startx command to start the X server and default clients at your shell.

Second, some Unix consoles have a graphical login prompt! In this case, the system boot scripts have started an X server already; a program called the display manager has drawn a graphical user interface that lets you type a username and password. In this case, when you login, you start using X11 immediately.

Third, some systems are using a graphical environment that is not X11. For example, Mac OS X has its native Quartz system. In Mac OS X, you start the X11.app program, which is an X server that lets X clients appear on the same desktop as your other Mac programs. Similar programs exist for Microsoft Windows (which is not a Unix platform).

xinit and startx
There is a program startx, which launches X11. It is simple to use: $ startx

This gives you some default environment. However, if you want to customise your default environment, using xinit might be easier. The startx script looks for several files (~/.Xresources, ~/.Xmodmap, ...) and loads them; the files it uses are different on each system, so you might have to read the startx script on your system to understand what it does.

In contrast, the xinit program only starts an X server and runs a ~/.xinitrc script. There is no need to remember whether your system calls it ".Xresources" or ".xresources" or ".xrdb"; you can simply specify the exact behavior that you want in your "~/.xinitrc".

Using xinit without an .xinitrc
If you broke your ".xinitrc", or do not have one yet, then you can start one. For example:

these will work $ xinit /usr/X11R6/bin/xterm $ xinit $(which xterm) this will NOT work $ xinit xterm

This will start an X server and the program called "xterm", which you can use to start more X clients. The last line fails (even if "xterm" is in your PATH) because xinit assumes that it is an argument, not a program. You must have a slash / somewhere in the name of the program.

If you have KDE, you can find your startkde script: $ which startkde /usr/local/bin/startkde

Then you can start KDE: $ xinit /usr/local/bin/startkde

This will let you use Konqueror to find your ".xinitrc" and Kate to edit it. (When browsing your home directory, check "Show Hidden Files" in the View menu.)

If you see a message like: $ xinit /bin/ksh: xinit: not found

...then you need to add /usr/X11R6/bin to your PATH environment variable.

How to write .xinitrc
If you do not specify a program with a slash, then "xinit" uses your .xinitrc script in your home directory.

$ xinit           # starts .xinitrc $ xinit xterm     # passes "xterm" as argument to .xinitrc

Before using "xinit", one needs a file called .xinitrc in their home directory. Because the filename starts with a dot, the file is normally invisible. There are ways to handle invisible files from the shell:

$ cd              # changes current directory to home directory $ ls -a           # list all files, including dotfiles $ ls .xinitrc     # list the .xinitrc file $ vi .xinitrc     # edit .xinitrc using vi $ emacs .xinitrc  # edit it using Emacs

An ".xinitrc" is simply a Bourne shell script. You could try running it with sh .xinitrc, but it would fail unless you already started the X server. You put in commands like you would type at the Unix shell. If you fail to understand this, try starting with the examples below.

Here is a simple, one-line example of an ".xinitrc": exec startkde

A "startkde" command starts the KDE desktop environment. This command does not finish until you logout of KDE. Normally the shell would wait for KDE to finish, then run the next command. The "exec" prefix to this command tells the shell that this is the last command, so the shell does not need to wait to run a next command. The "exec" is optional in this case.

Here is a more complex example of an ".xinitrc": ulimit -Sn 512 /usr/bin/mixerctl monitor.master=5,5 /usr/X11R6/bin/xmodmap -e 'keycode 107 = BackSpace' case "$#" in 0)  exec sh /etc/X11/xinit/xinitrc   ;; *) session="$1" shift exec "$session" "$@" ;; esac exec /usr/X11R6/bin/xterm
 * 1) NOTREACHED

The first line, "ulimit -Sn 512", is a Bourne shell command to increase the soft limit "-S" on open files "-n" to 512. (Use ulimit -a, or if your shell is not a Bourne shell, use sh -c 'ulimit -a' , to see your default limit.) The next command, /usr/bin/mixerctl monitor.master=5,5, sets the sound volume on a OpenBSD or NetBSD system to 5 on both the left and right channels. On other Unix systems, the command will be different. There are many other commands that one could add.

The third line, "/usr/X11R6/bin/xmodmap -e 'keycode 107 = BackSpace'" (or just "xmodmap -e 'keycode 107 = BackSpace'"), adjusts the keymap. For users that only make a few adjustments, this might by easier than loading a .Xmodmap file. (You can load such a file with xmodmap ~/.Xmodmap.)

The remaining lines form a Bourne Shell "case" statement. These process the command-line arguments to enable the following:

$ xinit gnome-session # start GNOME $ xinit startkde,GNOME # start KDE $ xinit fvwm,KDE# start window manager fvwm $ xinit xtermfvwm      # start an xterm without a window manager

Note that because none of the programs have a slash in them, they are handled by the ".xinitrc" script, not directly by "xinit".

xdm
Example ~/.xsession: . ~/.profile managers=startkde,fvwm,wmaker prompt="Welcome, $USER@$(hostname), select window manager:" xmodmap -e 'keycode 107 = BackSpace' test -f /usr/local/GNUstep/System/Makefiles/GNUstep.sh && . /usr/local/GNUstep/System/Makefiles/GNUstep.sh manager="$(xmessage -buttons "$managers" -print -center "$prompt")" test -z "$manager" && exec fvwm exec "$manager"
 * 1) miscellaneous setup

Making screenshots
The standard utility to dump an image of an X Window is <tt>xwd(1)</tt>, xwd produces an XWD image. It can be invoked in the following way: xwd -root -out root.xwd xwd can also be used to dump a single window selected with the mouse or provided with the <tt>-id</tt> option followed by the corresponding window id; for further info see <tt>man 1 xwd</tt>. When run remotely, xwd is useful for taking screen shots of modal menus in action. The GIMP can be used to convert an XWD file to other more common formats such as PNG; equivalently, the command-line utility xwdtopnm, part of the Netpbm utility suite, can be used to convert the XWD file to PNM format, from which it can be converted to any common format, for example with the following pipeline: xwd | xwdtopnm | pnmtopng > screendump.png It is also possible to convert an XWD image into many other formats using the <tt>convert</tt> utility (available here):, where XYZ is a typical filename-extension for a desired format.

title: Guide to X11/Starting Sessions


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Guide to X11/Xt Clients

Xt Clients
Some X clients are Xt clients, that is, they use the X Toolkit Intrinsics. This is a mechanism to divide a window into smaller windows called widgets such as buttons and menus. You can configure these widgets using X resources.

For example, a typical "xterm" window contains a scroll bar and the emulated screen of the terminal. The "xman" program, for viewing manual pages, contains three widgets of buttons on its tiny main screen. By using the "Manual Page" button, you obtain a larger window with several widgets: menu buttons, a scroll bar, and a text box containing the manual page.

However, developers of libraries like GTK+ and Qt have decided not to use these intrinsics. Thus, X resources are not available to GTK+ and Qt programs.

The X resources are settings that you write in a text file in a certain syntax and then load with xrdb.

xrdb
To set X resources, you need to put them in some text file like ~/.Xresources and then load it with:

$ xrdb -merge ~/.Xresources

There is a difference between "XTerm.resource" and "xterm.resource".

If you want one xterm to use the resources "GameTerm.resource" instead of "XTerm.resource", you would start it with:

$ xterm -class GameTerm

xrdb with standard input
You can also input resources using standard input.

xterm
xterm has menus. You must hold Control and click with mouse button 1, 2, or 3.

title: Guide to X11/Xt Clients


text:
Blender 3D: Noob to Pro/Advanced Tutorials/Advanced Animation/Guided tour/Armature/pose

Posemode is a very versatile place where you Animate your character, create and manage constraints and apply your rig to your character.

Contrary to Editmode, Pose mode isn't an obligatory mode where you can't do anything else. It's now part of the UI like any other object. A good example of it is you can be in posemode and still select another object.

So What Can You Do?
When you are done building your armature, you can go into Posemode to add constraints and start creating actions. There are also some new tools accessible in Posemode that you may want to look at. You can easily get into "pose" mode by selecting the mode from IPO type list box in the left portion of the lower screen.

The panel has changed a bit too:

 * What's new in the panels?:
 * You can use the Automatic IK feature in the Editbutton(F9) to pose a chain of bones like it was an ik chain. It's usefulness is very limited though. It works well only if there is no other ik solver in the chain, and if your chain is isolated from the rest of the rig.
 * Ghost: in the armature panel the ghost option lets you see the action linked to the armature over time. Also called onion skinning.
 * There are two number fields to better tweak the effect of B-Bones. The in/out is used to tell the scale of the virtual handle of the bezier curve. In is the Root of the bone, and Out is the Tip. The bigger the value, the bigger the effect of rotation.
 * There is now a Constraint panel where you can add a constraint to a bone, like any other object in the scene. This will be shown later.
 * You can pose your rig using GKEY, SKEY and RKEY. Note that if the bone is part of a chain it can't be moved (except if it's the first of the chain, moving all the chain as they are all children), so you rotate the bone instead.
 * You can do ALT-SKEY on one or more bones while in Envelope display mode to tweak the envelope size in real time while animating. Useful when for example you move the hand and some part of the character isn't in the influence zone; the result will be that some vertices will stay behind.
 * You can do CTRL-CKEY to copy stuff from a bone to bones. The options are location, rotation, scale and constraint. Constraint is very handy when you wan to copy a constraint to other bone. The way it works is easy.
 * The W-KEY menu gets some neat options too:
 * Select constraint target: Will select the target of the bone's constraint currently selected.
 * Flip name: Yep, you can flip name in Posemode too.
 * Calculate/Clear path: This is a visual way to see the action linked to your armature. You can select just some bones and ask Blender to show you the paths of the bones.
 * You can pose your character and select all bones you want to see included in the action and press I-KEY. You can insert a key just for loc, rot or size. Avail will add a key to all available channels in IPO window (all channels you previously added something).
 * When you insert key for your armature, a new action is created and linked to the armature if there was no action before. You can also see the curves of each selected bone of the armature in the IPO window. We will see action window and IPO window later.
 * You can parent a bone to an external object by selecting this object then selecting the bone in question so it's active (The armature is in Posemode so you can select a bone). Do CTRL-PKEY. Then when you move the bone the object will follow. This kind of Hard relationship doesn't include any bending at all. It's useful when doing robot rigs as you are just moving objects around.


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Engineering Acoustics/How an Acoustic Guitar works

Introduction
There are three main parts of the guitar that contribute to sound production.

First of all, there are strings. Any string that is under tension will vibrate at a certain frequency. The tension and gauge in the string determine the frequency at which it vibrates. The guitar controls the length and tension of six differently weighted strings to cover a very wide range of frequencies.

Second of all, there is the body of the guitar. The guitar body is connected directly to one end of each of the strings. The body receives the vibrations of the strings and transmits them to the air around the body. It is the body’s large surface area that allows it to “push” a lot more air than a string.

Finally, there is the air inside the body. This is very important for the lower frequencies of the guitar. The air mass just inside the soundhole oscillates, compressing and decompressing the compliant air inside the body. In practice, this concept is called a Helmholtz resonator. Without this, it would difficult to produce the wonderful timbre of the guitar.

The Strings
The strings of the guitar vary in linear density, length, and tension. This gives the guitar a wide range of attainable frequencies. The larger the linear density is, the slower the string vibrates. The same goes for the length; the longer the string is the slower it vibrates. This causes a low frequency. Inversely, if the strings are less dense and/or shorter they create a higher frequency. The lowest resonance frequencies of each string can be calculated by

$$f_1 = \frac{1}{2 L} \sqrt{\frac{T}{\rho_1}} $$ where $$T$$= string tension, $$\rho_1$$=linear density, $$L$$ = string length

The string length, L, in the equation is what changes when a player presses on a string at a certain fret. This will shorten the string which in turn increases the frequency it produces when plucked. The spacing of these frets is important. The length from the nut to bridge determines how much space goes between each fret. If the length is 25 inches, then the position of the first fret should be located (25/17.817) inches from the nut. Then the second fret should be located (25-(25/17.817))/17.817 inches from the first fret. This results in the equation

When a string is plucked, a disturbance is formed and travels in both directions away from point where the string was plucked. These "waves" travel at a speed that is related to the tension and linear density and can be calculated by

The waves travel until they reach the boundaries on each end where they are reflected back. The link below displays how the waves propagate in a string.

Plucked String @ www.phys.unsw.edu The strings themselves do not produce very much sound because they are so thin. This is why they are connected to the top plate of the guitar body. They need to transfer the frequencies they are producing to a large surface area which can create more intense pressure disturbances.

The Body
The body of the guitar transfers the vibrations of the bridge to the air that surrounds it. The top plate contributes to most of the pressure disturbances, because the player dampens the back plate and the sides are relatively stiff. This is why it is important to make the top plate out of a light springy wood, like spruce. The more the top plate can vibrate, the louder the sound it produces will be. It is also important to keep the top plate flat, so a series of braces are located on the inside to strengthen it. Without these braces the top plate would bend and crack under the large stress created by the tension in the strings. This would also affect the magnitude of the sound being transmitted. The warped plate would not be able to "push" air very efficiently. A good experiment to try, in order to see how important this part of the guitar is in the amplification process, is as follows:

1. Start with an ordinary rubber band, a large bowl, adhesive tape, and plastic wrap.

2. Stretch the rubber band and pluck it a few times to get a good sense for how loud it is.

3. Stretch the plastic wrap over the bowl to form a sort of drum.

4. Tape down one end of the rubber band to the plastic wrap.

5. Stretch the rubber band and pluck it a few times.

6. The sound should be much louder than before.

The Air
The final part of the guitar is the air inside the body. This is very important for the lower range of the instrument. The air just inside the sound hole oscillates, compressing and expanding the air inside the body. This is just like blowing across the top of a bottle and listening to the tone it produces. This forms what is called a Helmholtz resonator. For more information on Helmholtz resonators go to Helmholtz Resonance. This link also shows the correlation to acoustic guitars in great detail. The acoustic guitar makers often tune these resonators to have a resonance frequency between F#2 and A2 (92.5 to 110.0 Hz)(Hz stands for Hertz). Having such a low resonance frequency is what aids the amplification of the lower frequency strings. To demonstrate the importance of the air in the cavity, simply play an open A on the guitar (the second string). Now, as the string is vibrating, place a piece of cardboard over the sound hole. The sound level is reduced dramatically. This is because you've stopped the vibration of the air mass just inside the sound hole, causing only the top plate to vibrate. Although the top plate still vibrates and transmits sound, it isn't as effective at transmitting lower frequency waves, thus the need for the Helmholtz resonator.

Back to Main page


text:
Optics/Lenses

See also wikipedia: List of many optics topics
 * Holding a magnifying glass against a wall at the proper distance usually will provide a picture of the lit-up parts of the room. Note that that distance varies, depending on the distance of the object concerned from the lens. Objects that are far away will require a distance called the focal length, while objects that are closer to the lens require a longer distance for a sharp picture of that object on the wall.
 * Eye glasses restore proper sight to very many people, also eyepieces of optical instruments include lenses.
 * Photography has many uses

Refraction at a spherical interface
See also wikipedia:Refraction

Locating the image
See also
 * Photography
 * Basics of the Camera
 * wikipedia:Camera

The lensmaker’s equation
See also
 * Optical Instruments
 * wikipedia:Lens
 * wikipedia:Photographic lens

Converging and diverging lenses
See also
 * wikipedia:Convex lens
 * wikipedia:Concave lens

title: Optics/Lenses


text:
Engineering Acoustics/Bessel Functions and the Kettledrum

=Abstract=

In class, we have begun to discuss the solutions of multidimensional wave equations. A particularly interesting aspect of these multidimensional solutions are those of Bessel functions for circular boundary conditions. The practical application of these solutions is the kettledrum. This page will explore in qualitative and quantitative terms how the of the kettledrum works. More specifically, the kettledrum will be introduced as a circular membrane and its solution will be discussed in visual (e.g. visualization of Bessel functions, video of kettledrums and audio forms (wav files of kettledrums playing. In addition, links to more information about this material, including references, will be included.

=What is a kettledrum=

A kettledrum is a percussion instrument with a circular drumhead mounted on a "kettle-like" enclosure. When one strikes the drumhead with a mallet, it vibrates which produces its sound. The pitch of this sound is determined by the tension of the drumhead, which is precisely tuned before playing. The sound of the kettldrum (called the Timpani in classical music) is present in many forms of music from many difference places of the world.

=The math behind the kettledrum:the brief version=

When one looks at how a kettledrum produces sound, one should look no farther than the drumhead. The vibration of this circular membrane (and the air in the drum enclosure) is what produces the sound in this instrument. The mathematics behind this vibrating drum are relatively simple. If one looks at a small element of the drum head, it looks exactly like the situation for the vibrating string (see:). The only difference is that there are two dimensions where there are forces on the element, the two dimensions that are planar to the drum. As this is the same situation, we have the same equation, except with another spatial term in the other planar dimension. This allows us to model the drumhead using a helmholtz equation. The next step (solved in detail below) is to assume that the displacement of the drumhead (in polar coordinates) is a product of two separate functions for theta and r. This allows us to turn the PDE into two ODES which are readily solved and applied to the situation of the kettledrum head. For more info, see below.

=The math behind the kettledrum:the derivation=

So starting with the trusty general Helmholtz equation:

$$\nabla^2\Psi+k^2\Psi=0$$

Where k is the wave number, the frequency of the forced oscillations divided by the speed of sound in the membrane.

Since we are dealing with a circular object, it make sense to work in polar coordinates (in terms of radius and angle) instead of rectangular coordinates. For polar coordinates the Laplacian term of the Helmholtz relation ($$\nabla^2$$) becomes $$\partial^2 \Psi/ \partial r^2 + 1/r \partial^2\Psi/ \partial r +1/r^2 \partial^2 \Psi /\partial \theta^2$$

Now lets assume that:$$\Psi (r,\theta) = R(r) \Theta(\theta)$$

This assumption follows the method of separation of variables. (see Reference 3 for more info) Substituting this result back into our trusty Helmholtz equation gives the following:

$$r^2 / R (d^2 R/dr^2 + 1/r dR/dr) + k^2 r^2 = -1/\Theta d^2 \Theta /d\theta^2$$

Since we separated the variables of the solution into two one-dimensional functions, the partial derivatives become ordinary derivatives. Both sides of this result must equal the same constant. For simplicity, I will use $$\lambda$$ as this constant. This results in the following two equations:

$$ d^2 \Theta / d\theta^2 = -\lambda^2 \Theta $$

$$ d^2 R / dr^2 + 1/r dR/dr + (k^2 - \lambda^2 / r^2) R = 0 $$

The first of these equations readily seen as the standard second order ordinary differential equation which has a harmonic solution of sines and cosines with the frequency based on $$ \lambda $$. The second equation is what is known as Bessel's Equation. The solution to this equation is cryptically called Bessel functions of order $$ \lambda $$ of the first and second kind. These functions, while sounding very intimidating, are simply oscillatory functions of the radius times the wave number that are unbounded at when kr (for the function of the second kind) approaches zero and diminish as kr get larger. (For more information on what these functions look like see References 1,2, and 3)

Now that we have the general solution to this equation, we can now model a infinite radius kettledrum head. However, since i have yet to see an infinite kettle drum, we need to constrain this solution of a vibrating membrane to a finite radius. We can do this by applying what we know about our circular membrane: along the edges of the kettledrum, the drum head is attached to the drum. This means that there can be no displacement of the membrane at the termination at the radius of the kettle drum. This boundary condition can be mathematically described as the following:

$$ R(a) = 0 $$

Where a is the arbitrary radius of the kettledrum. In addition to this boundary condition, the displacement of the drum head at the center must be finite. This second boundary condition removes the Bessel function of the second kind from the solution. This reduces the R part of our solution to:

$$ R(r) = AJ_{\lambda}(kr) $$

Where $$ J_{\lambda} $$ is a Bessel function of the first kind of order $$ \lambda $$. Apply our other boundary condition at the radius of the drum requires that the wave number k must have discrete values, ($$j_{mn}/a$$) which can be looked up. Combining all of these gives us our solution to how a drumhead behaves (which is the real part of the following):

$$ y_{\lambda n}(r,\theta,t) = A_{\lambda n} J_{\lambda n}(k_{\lambda n} r)e^{j \lambda \theta+j w_{\lambda n} t} $$

=The math behind the kettledrum:the entire drum=

The above derivation is just for the drum head. An actual kettledrum has one side of this circular membrane surrounded by an enclosed cavity. This means that air is compressed in the cavity when the membrane is vibrating, adding more complications to the solution. In mathematical terms, this makes the partial differencial equation non-homogeneous or in simpler terms, the right side of the Helmholtz equation does not equal zero. This result requires significantly more derivation, and will not be done here. If the reader cares to know more, these results are discussed in the two books under references 6 and 7.

=Sites of interest=

As one can see from the derivation above, the kettledrum is very interesting mathematically. However, it also has a rich historical music tradition in various places of the world. As this page's emphasis is on math, there are few links provided below that reference this rich history.

A discussion of Persian kettledrums: Kettle drums of Iran and other countries

A discussion of kettledrums in classical music: Kettle drum Lit.

A massive resource for kettledrum history, construction and technique" Vienna Symphonic Library

Wikibooks sister cite, references under Timpani: Wikipedia reference

=References=

1.Eric W. Weisstein. "Bessel Function of the First Kind." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

2.Eric W. Weisstein. "Bessel Function of the Second Kind." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

3.Eric W. Weisstein. "Bessel Function." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunction.html

4.Eric W. Weisstein et al. "Separation of Variables." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/SeparationofVariables.html

5.Eric W. Weisstein. "Bessel Differential Equation." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BesselDifferentialEquation.html

6. Kinsler and Frey, "Fundamentals of Acoustics", fourth edition, Wiley & Sons

7. Haberman, "Applied Partial Differential Equations", fourth edition, Prentice Hall Press

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