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Accelerando Technical Companion

This is a technical companion to Charlie Stross's novel Accelerando. Stross's book can be quite dense in unusual technical terms and concepts, which can sometimes be quite confusing to readers unfamiliar with them. The purpose of this companion is to help alleviate any confusions the reader may have, as well as to introduce new confusions by giving the reader an idea of the current state and expected future of the technologies described in the novel. Wherever possible, brief information on relevant research papers is provided.

Accelerando is not a "post-Singularity" novel but rather a "through-Singularity" novel as it takes the reader from our days (the first chapter "Lobsters" can be situated slightly in the future, maybe around 2020) through a Singularity to a sketched post-Singularity world. In the novel Stross focuses more on the social impact of technologies pushing to a Singularity and only hints at the technologies themselves. The aim of this technical companion is filling the gaps and providing Accelerando readers with a technical background.

From a recent online interview with Charlie Stross: "Accelerando starts out with a post-dot-com hustler, Manfred Macx. Manfred is already borderline posthuman, and he's struggling to keep up; within a generation he's an obsolete burnout case, his daughter Amber has transmigrated into a simulation space hosted aboard a one kilogram starship, and they're dismantling the solar system to build more brains. A generation later ..."

The novel is available as a free download from the official site, and will also be available for purchase in bookstores on July 1, 2005.

The first part of this technical companion is a glossary, intended to explain and elaborate on concepts without giving away plot details. Keep in mind, however, that simply knowing that the novel involves a particular term may be a spoiler in some sense, so some may wish to defer consultation of the glossary. Indeed, part of the appeal of Accelerando is the sense of confusion one gets by being exposed to the technical flurry.

The second part is a chapter guide, where chapter-specific commentary is given. This is bound to be chock-full of spoilers.

A
ackle: Transliteration of the pronunciation of "ACL", an initialism for "Access Control List", a computer science term for lists defining permission levels. For more information, see Access control list.

agalmics: A form of economics concerning the "study and practice of the production and allocation of non-scarce goods," primarily via free-market trading, open-source initiatives, and flexible standards for intellectual property. Consult Robert Levin's "The Marginalization of Scarcity" for more information.

anarcho-capitalism: A political philosophy which proposes the replacement of governments with the free market. For more information see anarcho-capitalism.

Arianespace: A European commercial launch service. For more information see Arianespace.

avabits: One avabit is Avogadro's Number of bits; about 6 * 10^23 bits (over 68 billion terabytes).

B
BDU: A Battle Dress Uniform is a US armed forces military attire.

Boötes Void: A roughly spherical region of relatively empty space 250 million light-years in diameter. Discovered in 1981 by Robert Kirshner, Augustus Oemler, Jr., Paul Schechter and Stephen Shectman in a survey of galactic redshifts. See: Bootes void

Borganism: A hive mind. Can also refer to the government. Formally, an organization of autonomous organisms that exhibit collectivism: individual "units" that have merged to yield a unified construct.

Bose-Einstein condensate: A Bose–Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero (0 kelvins or -273.15 degrees Celsius). Under such supercooled conditions, a large fraction of the atoms collapse into the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale. See: Bose-Einstein condensate

Brown Dwarf: Brown dwarfs are sub-stellar objects with a mass below that necessary to maintain hydrogen-burning nuclear fusion reactions in their cores, as do stars on the main sequence, but which have fully convective surfaces and interiors, with no chemical differentiation by depth. Brown dwarfs occupy the mass range between that of large gas-giant planets and the lowest mass stars (anywhere between 75 and 80 Jupiter masses). See: Brown Dwarf

C
Cartesian theatre: Descartes suggested that the brain served to pre-process sensory information and deliver it to the "seat of the soul," or homunculus (Latin, literally, "little man"). He proposed that this function might be somewhat like the viewing of a play, wherein the sole audience member, the soul, views the perceptions delivered by the brain as if in a theatre. Thus the "Cartesian Theatre."

C: The speed of light.

CCAA:

cellular automata: A discrete model studied in computability theory, mathematics, and theoretical biology. It consists of an infinite, regular grid of cells, each in one of a finite number of states. The grid can be in any finite number of dimensions. Time is also discrete, and the state of a cell at time t is a function of the states of a finite number of cells (called its neighborhood) at time t-1. See: cellular automata

cladistic tree: A tree-like relationship-diagram, called a "cladogram", that is drawn up to show different hypotheses of biological relationships. See: cladistics

computronium: Computronium is a class of materials optimized to perform computation. In Accelerando, post-Singularity posthumans convert much of the Solar System's mass to Computronium to sustain the huge volume of computation, including "running" uploaded humans, characteristic of a post-Singularity civilization. See also computronium.

D
Deep Space Network (DSN): From the official site: "The NASA Deep Space Network - or DSN - is an international network of antennas that supports interplanetary spacecraft missions and radio and radar astronomy observations for the exploration of the solar system and the universe." Deep Space Network.

Dyson sphere: A sphere which surrounds a star, capturing nearly all of its energy. It's thought that such spheres would tend to leak a large amount of infrared radiation, which serves as a cue for detecting them with telescopes. For more information see Dyson sphere.

E
eigenface: In layperson's terms, eigenfaces are a set of "standardized face ingredients", derived from statistical analysis of many pictures of faces. Any human face can be considered to be a combination of these standard faces. For more information see eigenface.

entanglement: see quantum entanglement below.

exocortex: See exocortex. See also Chapter Guide, below.

extropian: Extropians share an optimistic view of the future, expecting considerable advances in computational power, life extension, nanotechnology and the like. Many extropians foresee the eventual realization of unlimited maximum life spans, and the recovery, thanks to future advances in biomedical technology, of those whose bodies/brains have been preserved by means of cryonics. See Extropianism.

F
field circus: In hacker jargon, field circus is a derogatory pun on the field service organization of a hardware manufacturer. See

FLOPS: Floating Point Operations per second, a measure of processor speed. See FLOPS

J
Jerusalem, Spider: Protagonist in the cyberpunk comic book series Transmetropolitan. He used glasses similar to Manfred's and explored issues such as uploading that are also present in Accelerando. See: Transmetropolitan.

K
Kardashev Type One civilization: According to the Kardashev scale, a civilization that is able to harness all of the power available from a single planet (world).

Kardashev Type Two civilization: According to the Kardashev scale, a civilization that is able to harness all of the power available from a single star (sun).

Kardashev Type Three civilization: According to the Kardashev scale, a civilization that is able to harness all of the power available from a single galaxy.

Kardashev Type Four civilization: According to the Kardashev scale, a civilization that is able to harness all of the power available from a single brane / universe / dimension.

L
L5: One of the five positions with respect to the Earth and Moon where a small object affected only by gravity can theoretically be relatively stationary. See Lagrangian point.

light cone: In general relativity, the future light cone is the boundary of the causal future of a point and the past light cone is the boundary of its causal past. See light cone.

Lobster stomatogastric ganglion (STG): This is a well-studied network of neurons found in the lobster which controls the rhythmic contractions of stomach muscles and intra-stomach teeth. Neurobiologists treat this ganglion as a model system for understanding central pattern generators, a generic term for networks of neurons which control repetitive behavior. The neurons of the ganglion have been simulated on computers by many researchers at varying degrees of abstraction. (neuroethology link) see links to papers

M
Marder E. and Bucher D. (2001). Central pattern generators and the control of rhythmic movements. Current Biology (link)

Matrioshka Brain: A set of nested Dyson shells composed of computronium. The innermost shell is powered by the radiation from a star, while outer shells are powered by residual radiation and waste heat generated by inner shells. For more information see Matrioshka Brain.

Metacortex: See Chapter Guide, below.

MIPS Million instructions per second, a measure of processor speed.

N
Nanotechnology fabrication: See: Molecular nanotechnology

Neotenous/neoteny: childlike, maintaining some juvenile characteristics into adulthood. A page on XD38 personality types lists a few criteria that fit the protagonist.

Neural network: A very general term used to refer to any interconnected network of artificial neurons which perform some sort of computation. For more information see artificial neural network.

NHSNet: A wide area network run by the UK's National Health Service.

P
Panulirus interruptus: California spiny lobster; see Lobster stomatogastric ganglion for more details.

Parasite network: A global decentralized network of wireless antennas, proposed by Jim Griffin.

Psephology: The study of elections. See psephology.

Q
Quantum entanglement: A phenomenon in quantum mechanics where the states of two particles are locked together, even across a distance. See Quantum entanglement

Quantum key exchange: A cryptographic protocol that allows two parties to share a secret key, based on quantum mechanics. Since the act of measuring a quantum system disrupts the system itself, it's impossible for a third party to eavesdrop the exchange without being detected. See Quantum key distribution

R
reputation market: See Cory Doctorow's Whuffie.

rubberized concrete: Roger Jones and others, working at LANL in 1997, reported on the successful use of supercritical carbon dioxide to impregnate concrete with polymers, dyes, and other interesting molecules that modify its physical properties. The news report from Los Alamos (Los Alamos paves the way for better cement) was reported in New Scientist and elsewhere. (see also High Performance Concrete)

S
singularity: See technological singularity. Accelerando's author Charles Stross edits a personal wiki dedicated to the Singularity: " Singularity! A Tough Guide to the Rapture of the Nerds. "

Slashdotting: A large influx of visitors to a web server, due to a link from the front page of slashdot.org, a popular web site for technology-related news. The influx can be quite taxing on server processing power and bandwidth, and can often incapacitate a site. For more information see Slashdot effect.

small-world network: A class of random graphs where most nodes are also neighbors of one another, but every node can be reached from every other by a small number of hops or steps. See small-world network, Six Degrees of Kevin Bacon, and.

strange attractor: In dynamical systems, an attractor is a set to which the system evolves after a long enough time. For the set to be an attractor, trajectories that get close enough to the attractor must remain close even if slightly disturbed. Geometrically, a strange attractor is a complicated set with fractal structures. For more information see strange attractor.

syncitium: A concept introduced by Camillo Golgi (the same Italian biologist whose name is given to a component of the living cell). Golgi believed (incorrectly) that all neurons are connected by fibers which intermingle and mesh, and syncitium is the name he gave to the whole. The hyphae (strands) of many fungi are syncitial: long open cellular tubes with multiple nuclei but no internal cell walls, as opposed to one cell-controlling nucleus per partitioned-off cell. This seems to be the sense in which Stross uses the term.

T
Thompson hack: a type of back door (security hole) in a computer program which is undetectable even by examining the source code because it is introduced by the compiler, which itself contains this type of back door. See Backdoor (computing), reference 9, "Reflections on Trusting Trust".

timing channel attack: An attack against an encryption scheme based on measuring the time taken to execute different operations.

Tipler, Frank J.: Currently a professor at Tulane University, a proponent of Omega Point Theory. For more information see Frank J. Tipler and Omega point.

Turing Oracle: A method for determining whether a given program ever halts. There is no algorithm to do this. Access to a Turing Oracle is near-godlike computing power. For instance, one could prove Fermat's Last Theorem (or almost any other theorem) by writing a program that tries all possible x, y, z, and n until it finds a solution of x^n+y^n=z^n, then asking the Turing Oracle whether the program will ever find a solution. It would also factor large integers, fold proteins, etc.

Turing Test: Proposal for a way to judge whether a computer is intelligent, proposed by Alan Turing in 1950. See Turing Test.

U
uploading: The transfer of a mind to a computer. For more information see mind transfer.

V
van Eck radiation: The electromagnetic emissions of a computer display or processor, the act of detecting and reconstructing which is known as van Eck phreaking, in a form of electronic eavesdropping. See van Eck radiation, Cryptonomicon.

van Eck, W. (1985). Electromagnetic Radiation from Video Display Units: An Eavesdropping Risk. Computers & Security (link)

Variable specific impulse magnetoplasma rocket: The variable specific impulse magnetoplasma rocket (VASIMR) is a hypothetical form of spacecraft propulsion that uses radio waves and magnetic fields to accelerate a propellant. See Variable specific impulse magnetoplasma rocket.

The scientist and astronaut Franklin Chang-Diaz created VASIMR.

W
Wunch one of the minor villains of the book, are described as the elements of a run-amok financial institution, roughly termed "economics 2.0". It is likely that the Wunch are so named as a spoonerism play on "wunch of bankers."

Units of time
Since measuring time in units like days, months, and years are of limited usefulness in regions not on the planet Earth, one often sees the usage of time measures based on the second, such as megaseconds or gigaseconds. As most people aren't used to thinking in these units, conversions are listed here to and from the more conventional units.

See also Orders of magnitude (time).

Second-style -> conventional

 * 1 kilosecond: 16.7 minutes
 * 1 megasecond: 11.6 days
 * 1 gigasecond: 32 years
 * 1 terasecond: 32,000 years
 * 1 petasecond: 32,000,000 years
 * 1 light second: 299,792,458 metres (not actually a unit of time)

Conventional -> second-style

 * 1 hour: 3.6 kiloseconds
 * 1 Earth day: 86.4 kiloseconds
 * 1 week: 604.8 kiloseconds
 * 1 Earth month (30 days): 2.6 megaseconds
 * 1 Earth year: 31.6 megaseconds
 * 1 century: 3.16 gigaseconds

Chapter 1: Lobsters
Chapter 1 introduces two similar, but clearly separate, concepts that apply to various characters throughout the remainder of the book: metacortices and exocortices. Though etymologically similar, the two are quite different—the former, in effect, being composed in various ways of the latter. An EXOcortex can best be described as the portion of a trans- or posthuman entity's brain (or cortex) which exists outside of that entity's primary computing structure, usually the brain inhabiting a person's "meatbody." For example, a person's exocortex could very well be composed of all the external memory modules, processor, and devices that the person's biological brain interacts with on a realtime basis, thereby in effect making those external devices a functional part of the individual's "mind." A METAcortex, on the other hand, is a processing construction built entirely out of the connections between other processing constructions—a sort of "higher brain" composed of lesser brains, all of which contribute to its functionality.

A simple metaphor taken from contemporary IT describes this distinction nicely. Any technology you, as an individual, use to magnify or supplement your brain's natural functionality on a daily basis can be considered components of your EXOcortex; for example, a desktop and/or laptop computer...or even a pocket calculator! These devices, however, are almost certainly networked with millions of similar devices (other folks' computers/servers/etc.) via the Internet—which, as a collective processing and data-storage organ, can be considered a METAcortex encompassing but also transcending (in functionality) the sum of all its parts.

It's the agalmic future

Agalmics is a form of economics concerning the "study and practice of the production and allocation of non-scarce goods," primarily via free-market trading, open-source initiatives, and flexible standards for intellectual property. Consult Robert Levin's "The Marginalization of Scarcity" for more information.

Manny refers to his robot pet cat as "his aineko." This is a bastardized Japanese word meaning "pet cat" (the correct Japanese would be "aibyō" (愛猫); the second character can be read as either "neko" or "byō", but in this context, the latter is correct). The word is a nice pun on top of "AI neko" or "AI cat", as Manny does use the cat as a storage of PDA-like information, and "ai" can also mean "love". The name is also an apparent reference to the Sony Aibo (a name that can be read several ways, but means "pet robot").

Annette, when she is first introduced, works for Arainespace, a European launch service provider.

Chapter 2: Troubadour
As he arrives at the Luton airport at the beginning of the chapter, Manny makes an offhand reference to President Santorum's America, most likely referring to a future in which a decadent U.S. is presided by current (as of the book's writing) chairman of the Senate Republican Conference, Rick Santorum (of homosexual controversy fame).

Chapter 3: Tourist
widespread public paranoia about limbic spam

The limbic system is a group of brain structures involved in various emotions such as aggression, fear, pleasure and also in the formation of memory. Presumably limbic spam would be unwanted signals which delivers a stimulus directly to this area for advertising purposes.

pointy-haired management

A clear reference to the Pointy-Haired Boss of Dilbert comics.

Should have known you'd read Vinge

Vernor Vinge is credited with popularizing the idea of a transhumanist technological Singularity in fiction, and explores it in his novels Marooned in Realtime and (indirectly) A Fire Upon the Deep.

Chapter 5: Router
Hyundai +4904/-56 is a brown dwarf discovered around 2010 by an infrared deep scan, moving quickly with respect to the stellar neighborhood of the Sun. Amber Macx and her friends journey to Hyundai +4904/-56 following the discovery of a "router" orbiting the star. The router is a wormhole in a galactic network.

pocket universe (usage: "arguments proliferate indefinitely in the pocket universe of the ship") in the context, the ship is merely a container, with the crew/passengers being pure software; they and their chosen environment(s) are simulations run in real-time upon the computing matrix this is the entirety of the cargo/hull ("simulation space populated by upload minds").

cultural delta can be loosely described as the rate of change imposed upon culture/society by the speed and depth of new technology; for a small population semi-isolated from the main body of society, it is a reflection that no matter how hip and intelligent any given group could be, it is impossible for a 'small town' to be as creative as a 'large city';

fork repeatedly is what malware such as a virus would do upon penetration; ceaselessly duplicate itself until running out resources (primarily RAM); if expressed as a cliché, it would be: "when you reach a fork in the road, take both". See Fork (operating system), and the attack itself is called a Fork bomb

starwhisp (Originally spelled starwisp by the coiner of the term, Robert L. Forward) being a class of human-built space exploration probes; the name is intended to suggest the relative size (small) and mass (low—generally of the order of grams, or at most, a few kilograms) when compared to such fictional behemoths as the USS Enterprise.

immortagens (context: "cheap immortagens")

adjuvants (context: "out-of-control personality adjuvants") See Adjuvant.

watchdog thread It is common in embedded computing to have a watchdog timer. This timer must be serviced periodically by the main program, or else it will force a reset of the system. This is to prevent bugs in the main program which would normally hang the system.

web of trust Web of trust

Basilisk attacks are based on a concept popularized by British SF author David Langford. A basilisk is a visual representation which causes harm (usually death) by triggering a thought which the human brain is physically or logically incapable of thinking.

network raids

nanosats

intelligence amplification

Chapter 7: Curator
Utility fog Nanobots with extending arms reaching in several different directions, and can perform lattice reconfiguration. Utility fog is sometimes thought of as a nanotechnological version of the Swiss Army Knife.

Chapter 8: Elector
God bothering (applied theological engineering) probably refers to the use of technology in the act of prayer. A god in this case would be a post-human, and this bothering would be an attempt to get the god's attention. An example might be a DDOS attack.

Nono, god bothering means bothering people in an attempt to infect them with your particular religious meme! No wonder it should be the greatest crime of the future when even murder is allowed, because once infected it's almost impossible to rescue people from this!

eigensister? Alternate states later collapsing. So two shades - duplicate mindstates - of the same person sent to experience different things and later converging on each other would be eigensisters. When they merged they would presumably collapse into a single state. Which seems non-trivial.

title: Accelerando Technical Companion


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Portuguese/Contents/L1

Level One - Learn To Speak Portuguese
Go Back To Contents Page Bem vindo! Welcome to this Portuguese Language Course. This is a collaborative effort by many contributors and so please feel free to edit these pages to improve as you see fit. This chapter is intended for the absolute beginner in Portuguese. No prior experience nor grammatical knowledge is required. Boa sorte! (Good luck!)

 * Lesson One - Saying Hello Learn how to say hello, plus one of the ways to say "I am", "You are" etc.
 * Lesson Two - I Like... How to say you like something, how to describe things and how to say "the"
 * Lesson Three - What Are You Going To Do? Tell people what you're going to do, and on what day of the week.
 * Lesson Four - Where Do You Come From? A bit more on describing things, plus nationalities.
 * Lesson Five - Going Shopping Clothes, how to sound polite, plus how to say you want something.
 * Lesson Six - The Train Learn how to ask a question in Portuguese
 * Lesson Seven - What Time Is It? Tell the time and count to ten!
 * Lesson Eight - Meeting People Knowledge - how to say "I know",
 * Lesson Nine - Getting The Groceries 'This' and 'that' and 'those', more numbers, plus comparing things.
 * Lesson Ten - My House How to say "I have" or "There is"

Ir Ao Segundo Nível Go To Level Two

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

title: Portuguese/Contents/L1


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Portuguese/Contents/L2

Level Two - Learn To Speak Portuguese
Bem vindo to the second part of our Portuguese language course. If you are still just a beginner, go back to Level One. If you are after grammar references, go back to the Contents page.

Lição Onze - De Quem é Isso?]] How to say something is yours. Lição Doze - O Fim de Semana]] How to say what you are doing at the moment - in Brazilian and European Portuguese Lição Treze - No Médico]] Different parts of the body. Lição Catorze - Comer no Restaurante ]] Food, plus conjugating verbs ending in "ar" Lição Quinze - Comer no Restaurante, Segunda Lição]] Continuing last topic, with verbs in "ir" and "er".
 * [[Portuguese/Contents/L2/Lesson Eleven - Whose Is This?|Lesson Eleven - Whose Is This?
 * [[Portuguese/Contents/L2/Lesson Twelve - The Weekend|Lesson Twelve - The Weekend
 * [[Portuguese/Contents/L2/Lesson Thirteen - At The Doctor|Lesson Thirteen - At The Doctor
 * [[Portuguese/Contents/L2/Lesson Fourteen - Eating Out|Lesson Fourteen - Eating Out
 * [[Portuguese/Contents/L2/Lesson Fifteen - Eating Out Part II|Lesson Fifteen - Eating Out Part II

title: Portuguese/Contents/L2


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Portuguese/Contents/L2/Lesson Fourteen - Eating Out

By now, we have seen many verbs, like gostar, ser, estar and ter. Up until now, we have looked at the various conjugations of these verbs individually. However, for the vast majority of Portuguese verbs, there are simple rules to follow in order to get the various conjugations for a verb. In Portuguese, there are three main types of verbs: those that end with -ar, those that end in er and ir. Of these, some are irregular - that means the rules don't apply. However, most of them are regular. Now we'll have a look at the most common types of verbs, those that end in -ar.


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Wikijunior:Solar System/Mars/Deimos

Deimos Facts:
 * Deimos is one of the smallest moons in the Solar System.
 * Deimos has a satellite named after it, Deimos-1.
 * Most people could sprint out of Deimos' gravitational zone.

How big is Deimos?
Deimos is one of the smallest moons in the solar system, at only 12 km in diameter, or about 1/2 the diameter of Phobos. Surprisingly, however, Deimos was spotted and identified as a moon before Phobos, partly because of its greater distance from Mars, which meant that it was not so hidden in the glare of Mars' light. The surface area of Deimos is roughly the size of a medium-sized city on Earth. It is also very irregular in shape due to its small size.

From the viewpoint of somebody standing on Mars, Deimos would appear almost as just a very bright star, and you would not be able to identify any surface features without a telescope.

What is its surface like?

The surface of Deimos has no atmosphere of any kind, and it is full of craters from meteors hitting the surface, just like the Earth's Moon. The surface of Deimos is made up of very dark rocks called carbonaceous chondrite, which includes a lot of the element carbon. There is also water ice on the surface of Deimos, as well as most of the interior.

Deimos has two named craters: Swift and Voltaire. Swift is named after Irishman Johnathan Swift (1667–1745), the author of Gulliver's Travels. Swift is the smaller out of the two named craters, having a 1000 m diameter, while Voltaire has a 1900 m diameter. The person Voltaire (1694–1778) was a French supporter of freedom of religion and freedom of expression.

Because Deimos is so small, the gravity on Deimos is very, very small. It is about 1/2500th of the force of gravity here on Earth. This is so little gravity that it would be very dangerous for people to walk on the surface without a tether or some other restraint to keep them from pushing themselves completely away from Deimos. If there were a structure on Deimos built for humans, it would resemble an orbital space station inside.

Because Deimos is tidally locked to Mars, it always keeps one side facing Mars during its entire orbit. This means that if you were on the side facing Mars, you would always see Mars in the same part of the sky all of the time. The other side would never see Mars in the sky. On the side facing Mars, the view of Mars itself takes up almost one eleventh of the sky, so Mars itself would be a very significant object to look at.

One of the reasons why scientists are interested in exploring Deimos is because it is believed to be an asteroid that was captured by Mars many millions of years ago. By studying Deimos and its brother moon Phobos, scientists hope to get a very close view of what other asteroids of a very similar size also look like elsewhere in the Solar System.

How long is a day on Deimos?
A "day" on Deimos lasts about 30.5 hours.

How long is its orbit around Mars?
Deimos is tidally locked to Mars, so a day on Deimos is precisely the same as the time it takes to orbit Mars. From the surface of Mars, Deimos still appears to rise from the east and set in the west like the Sun, planets, and everything else in the sky (except Phobos), but it lingers in the sky for a very long time, taking almost 3 sols (Martian days) before it sets in the western sky.

Does Deimos cause a solar eclipse on Mars?
Just as the Earth's Moon can come between the Earth and the Sun, Deimos occasionally comes between Mars and the Sun. This is also called a transit, and in this case perhaps transit is a more appropriate term than eclipse.

Because Deimos is so tiny and relatively far away from Mars as well, the portion of the Sun that is covered by Deimos during an eclipse is very small, and somebody standing on the ground on Mars would hardly even notice it. It is similar to taking a flashlight and putting a small pebble on it. You can't see much difference in the light that comes from the flashlight.

Who is it named after?
Deimos (Ancient Greek Δεῖμος) is named after the Greek son of Ares (the Greek name for Mars), who is also called "panic" or even "dread". In Greek mythology, Deimos was the personification of terror. Deimos, together with Phobos and others, would often accompany Ares into battle with the gods.

How was it discovered?
Asaph Hall was an astronomer with the United States Naval Observatory, where he studied many of the planets and objects in the Solar System. In 1877, he discovered both Deimos and, less than two hours later, Phobos and classified them as moons of Mars. The name for Deimos was suggested by Henry Madan based on the book Iliad, a classical Greek book about mythology.

Wikijunior Sonnensystem/ Mars/ Deimos

title: Wikijunior:Solar System/Mars/Deimos


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User:Nateji77~enwikibooks

ESL/EFL Lesson Plans

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text:
Business English/Activities and Dynamics

Games
Tip: A good goal for games is that the game in itself is fun enough to play even if you are not trying to practice a language.

Guess the person
Have each student write the names of about eight famous people on slips of paper, fold up the slips and collect them in one pile. Form teams of at least two people. One person takes a slip of paper and tries to get his team to guess the name written there without saying the name or its spelling. Each team should have about one minute per round. For every name that the team guesses correctly, it receives one point (they can keep the slip of paper to count). At the end of the time, the team that guessed the most names correctly, wins.


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Introduction to Philosophy/Origins of Philosophy

Etymology
The term philosophy is derived from the Greek words phylos meaning "to love" and sophie meaning "wisdom".

Introduction to Philosophy
In the Phaedo, Socrates says that philosophy is a preparation for the death that awaits us all. When the mind is engaged in philosophy it is free of concerns and dwells in the domain of ideas. Our minds enter a spiritual region transcending the death of our corporeal experience. Another, better known, view of Socrates is that of philosophy as ‘the love of wisdom’. This love discovers truth, and we become wise by practical application in our daily lives of what has been discovered.

These definitions highlight the nature of philosophical inquiry. Philosophers ask questions. These questions try to understand the metaphysical and physical universe including humans and their world

Pre-Socratics
The history of philosophy in the west begins with the Greeks, and particularly with a group of philosophers commonly called the pre-Socratics. This is not to deny the occurrence of other pre-philosophical rumblings in Egyptian and Babylonian cultures. Certainly great thinkers and writers existed in each of these cultures, and we have evidence that some of the earliest Greek philosophers may have had contact with at least some of the products of Egyptian and Babylonian thought. However, the early Greek thinkers added at least one element which differentiates their thoughts from all those who came before them. For the first time in history, we discover in their writings something more than dogmatic assertions about the ordering of the world - we find reasoned arguments for various beliefs about the world.

Thales
Thales (in Greek: Θαλης) of Miletus (circa 624 BC - 546 BC), also known as Thales the Milesian, was a pre-Socratic Greek philosopher. Many regard him as the first philosopher in the Greek tradition. He has also been traditionally considered the father of science, although it is also contended that the beginnings of science may be traced to Ancient Egypt.

Life
Thales lived in the city of Miletus, in Ionia, now western Turkey. According to Herodotus, he was of Phoenician descent.

The well-traveled Ionians had many dealings with Egypt and Babylon; Thales may have studied in Egypt as a young man. In any event, Thales almost certainly had exposure to Egyptian mythology, astronomy, and mathematics, as well as to other traditions alien to the Homeric traditions of Greece. Perhaps because of this, his inquiries into the nature of things extends beyond traditional mythology.

Several anecdotes suggest that Thales was not solely a thinker; he was involved in business and politics. One story recounts that he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. Another version of this same story states that he bought the presses to demonstrate to his fellow Milesians that he could use his intelligence to enrich himself. Herodotus recorded that Thales advised the city-states of Ionia to form a federation.

Thales is said to have died in his seat, while watching an athletic contest.

Theories and influence
Before Thales, the Greeks explained the origin and nature of the world through myths of anthropomorphic gods and heroes. Phenomena like lightning or earthquakes were attributed to the actions of the gods.

By contrast, Thales attempted to find naturalistic explanations of the world, without reference to the supernatural. He explained earthquakes by imagining that the Earth floats on water and that earthquakes occur when the Earth is rocked by waves. Herodotus cites him as having predicted the solar eclipse of 585 BC that put an end to the fighting between the Lydians and the Medes.

Thales's most famous belief was his cosmological doctrine, which held that the world originated from water. It is sometimes assumed that Thales considered everything to be made from water. According to others, however, it's likely that while Thales saw water as an origin, he never pondered whether water continued to be the substance of the world.

Many philosophers followed Thales's lead in searching for explanations in nature rather than in the supernatural; others returned to supernatural explanations, but couched them in the language of philosophy rather than myth or religion.

Thales is credited for having first popularized Geometry to ancient Greek culture, mainly that of spatial relationships.

The best explanation of Thales's view is the following passage from Aristotle's Metaphysics (983 BC). The passage is given in translation with key phrases transliterated from the Greek for the reader's benefit. The reader will see in the transliteration words from the theory of matter and form that were adopted by science with quite different meanings. The translation is somewhat literal, for purposes of accuracy.
 * "That from which is everything that exists (ta onta) and from which it first becomes (ex hou gignetai protou) and into which it is rendered at last (eis ho phtheiretai teleutaion), its substance remaining under it (tes men ousias hypomenouses), but transforming in qualities (pathesi metaballouses), that they say is the element (stoicheion) and principle (archen) of things that are (ton onton)."

And again:

 * "For it is necessary (dei) that there be some nature (physin), either one or more than one, from which become (gignetai) the other things (t'alla) of the object being preserved (sozomenes ekeines)... Thales says that it is water (hydor)."

Aristotle's depiction of the change problem and the definition of substance could not be more clear. If an object changes, is it the same or different? In either case how can there be a change (metabollein) from one to the other? The answer is the substance (ousia or physis), which "is saved", but acquires or loses different qualities (pathe, the things you "experience").

A deeper dip into the waters of the theory of matter and form is properly reserved to other articles. The question for this article is, how far does Aristotle reflect Thales? He was probably not far off, and Thales was probably an incipient matter-and-formist.

The essentially non-philosophic DL states that Thales taught as follows:
 * "Water constituted (hypestesato, "stood under") the principle of all things."

Heraclitus Homericus (Quaes. Hom. 22, not the same as Heraclitus of Ephesus) states that Thales drew his conclusion from seeing moist substance (hygra physis) turn into air, slime and earth. It seems clear that Thales viewed the Earth as solidifying from the water on which it floated and which surrounded it as Ocean.

Thales applied his method to objects that changed to become other objects, such as water into earth (he thought). But what about the changing itself? Thales did address the topic, approaching it through magnets and amber--which, when electrified by rubbing, attracts in the same way.

How was the power to move other things without the mover’s changing to be explained? Thales saw a commonality with the powers of living things to act. The magnet and the amber must be alive, and if that were so, there could be no difference between the living and the dead. When asked why he didn’t die if there was no difference, he replied “because there is no difference.”

Aristotle defined the soul as the principle of life, that which permeates the matter and makes it live, giving it the animation, or power to act. The idea did not originate with him, as the Greeks in general believed in the distinction between mind and matter, which was ultimately to lead to a distinction not only between body and soul but also between matter and energy.

If things were alive, they must have souls. This belief was no innovation, as the ordinary ancient populations of the Mediterranean did believe that natural actions were caused by divinities. Accordingly, the sources say that Thales believed all things possessed divinities. In their zeal to make him the first in everything they said he was the first to hold the belief, which even they must have known was not true.

However, Thales was looking for something more general, a universal substance of mind. That also was true to the polytheism of the times. Zeus was the very personification of supreme mind, dominating all the subordinate manifestations. From Thales on, however, philosophers had a tendency to depersonify or objectify mind, as though it were the substance of animation per se and not actually a god like the other gods. The end result was a total removal of mind from substance, opening the door to a non-divine principle of action. This tradition persisted until Einstein, whose cosmology is quite a different one and does not distinguish between matter and energy.

Classical thought, however, had proceded only a little way along that path. Instead of referring to the person, Zeus, they talked about the great mind:

 * "Thales", says Cicero, "assures that water is the principle of all things; and that God is that Mind which shaped and created all things from water." (Cicero:"De Nat.Deorum,"i.,10.)

The universal mind appears as a Roman belief in Vergil as well:

 * "In the beginning, SPIRIT within strengthens Heaven and Earth,
 * The watery fields, and the lucid globe of Lina, and then --
 * Titan stars; and mind infused through the limbs
 * Agitates the whole mass, and mixes itself with GREAT MATTER"
 * (Virgil: "Aeneid," vi., 724 ff.)

Socrates
Socrates (c.470 – 399 BC) (Greek Σωκράτης Sōkrátēs) was a Greek (Athenian) philosopher.

Like Thales and the other pre-Socratics, Socrates too had the ambition to ask questions of life. However, where the pre-Socratics were more concerned with the cosmological questions, Socrates was concerned with questions of the following nature: What is piety? What kind of life is worthwhile for a human to live? Can virtue be taught? What is justice? Is there more than one's virtue? What is human excellence?

Socrates did not actually write any of these ideas down. The only written information about his philosophy can be found in the dialogues of Plato and Xenophon. These dialogues deal mostly with questions of the good life, human excellence, and the cultivation of knowledge. One of Plato's most important and best known works is "The Republic" in which we find the allegory of The Cave that explains the difference between perceived reality and "real" reality which, according to Plato, can only be found in the realm of ideas. More: Greek Philosophy: Socrates


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Talk:Puzzles/Action sequences/Barrel of Ale

Does anyone have a solution to this? If not, I may have to consider this problem unsolvable. Infobacker 20:20, 30 Jun 2005 (UTC)

Hint: You can empty barrel (it's a waste of ale, yes, but nothing stops you from doing this nonetheless) --Divinity 12:36, 15 July 2005 (UTC)


text:
Cookbook:Peanut Butter and Jelly Sandwich

Peanut butter and jelly sandwiches, also know as PB&Js, are a type of sandwich. They are very quickly made and only require a couple common ingredients. This makes them quite popular, especially in the United States. This recipe will detail how to make the most simple, and indeed traditional form of these sandwiches—feel free to add any ingredients that you think would go well.

Ingredients

 * 2 slices of white bread
 * Peanut butter
 * Jelly or jam

Procedure

 * 1) Using a spreading knife, spread an even coat of peanut butter on one side of each slice of bread.
 * 2) When the bread is evenly covered, spread an even coat of jelly atop the peanut butter.
 * 3) After applying the jelly, put one of the slices on a plate (jelly side up) and place the other slice on top of the first (jelly side down).

Notes, tips, and variations

 * Instead of white bread, you can use whole-grain bread, or really any type of bread. Go nuts.
 * Try adding in some sliced bananas—they go very well with PB&Js.
 * If using a crab apple jelly, a dash of cinnamon will nicely complement the flavours of the sandwich.
 * Try adding a bit of honey, or even entirely substituting it for the jelly.
 * Try adding some sweet or savory cream cheese.
 * Try toasting the bread before putting the other ingredients on it.


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User:Mesolimbo~enwikibooks

More information about Mesolimbo can be found [here].

title: User:Mesolimbo~enwikibooks


text:
Physics with Calculus/Introduction/For Mathematicians

Math is a wonderful discipline. Too often though, mathematicians focus solely on theoretical mathematics, but this text can aid us in realizing the wonderful applications of math and, in this case, calculus. —Professor M.

Although physics is often thought of as an "applied mathematics" class, the relationship between physics and mathematics is far more interesting than just applying mathematics to the physical world. Often in the development of both physics and mathematics, the innovations in one area have driven new innovations in the other. One interesting example of this is the fact that when Newton wanted to test his theory of gravity he realized a major problem. His law of gravitation calculated the force between the two interacting objects as if all of the mass of the two objects was concentrated at a point at the center of the object. Newton was faced with a serious problem. How could he prove that all of the mass of an extended object could be truly considered to be at a point at the center of the object? This required a new kind of mathematics, and the integral calculus was devised. This illustrates the fact that, historically, the motivation for developing mathematics was often based on real problems. It is often not until later that all of the theoretical aspects of the mathematics are worked out. Although Newton is often credited with inventing the calculus, it is clear that he was very often more interested in developing mathematical models that would solve his scientific problems, and then proving his mathematical assumptions in a theoretical way. It was left to others that followed to put the calculus on the same solid foundation that had been developed for geometry and other branches of mathematics.

Innovations in mathematics can often drive new discoveries in science too. An interesting example of this the study of fractals. When they were first studied there was no practical use for them, but now physics and other sciences use them in the study of systems that are known to be chaotic.

So why would a mathematician want to study physics? It seems that one very powerful answer to this question is that it helps one to see into the minds of many of the great thinkers of the past. What motivated them? What where the challenges that they faced as they developed their ideas? They would often turn to mathematics as the language to express their ideas and solve their problems. The study of mathematics strengthens the tools of the scientist, and the study of science adds fullness to our understanding of mathematics.


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Sanskrit/Basic grammar

'Basic' grammar in Sanskrit may be an oxymoron, but here are some fundamentals.

Sanskrit is an inflected language, meaning that the nouns are declined and the verbs are conjugated.

The Sanskrit language tends toward the subject-object-verb word ordering, but does allow for multiple type of sentence formulations particularly in the sutra form.

There are three principal numbers used in the inflection of the Sanskrit language. These are singular (1 thing), dual (2 things), and plural (many things).

In declining Sanskrit nouns, there are eight cases used. These are nominative, accusative, instrumental, dative, ablative, genitive, locative, and vocative.

Nouns are declined according to their endings, with the exception of pronouns which have specific declensions.

The short a masculine declension:

The short a neuter declension:

The long ā feminine declension:

title: Sanskrit/Basic grammar


text:
Talk:Puzzles/Lateral puzzles/6

What's to say it's not an average *wooden* cabin and one of them shot the others and then shot himself?
 * Obvoiusly the idiot who wrote this.

title: Talk:Puzzles/Lateral puzzles/6


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Calculus/Tables of Integrals

Rules

 * $$\int c\cdot f(x)\mathrm{d}x=c\cdot\int f(x)\mathrm{d}x$$
 * $$\int\big(f(x)\pm g(x)\big)\mathrm{d}x=\int f(x)\mathrm{d}x\pm\int g(x)\mathrm{d}x$$
 * $$\int f(g(x)) g'(x) \mathrm{d}x = \int f(u) \mathrm{d}u = F(u) + C = F(g(x)) + C$$ where $$ F' = f$$
 * $$\int u\,dv=uv-\int v\,du$$

Powers

 * $$\int \mathrm{d}x=x+C$$
 * $$\int a\,\mathrm{d}x=ax+C$$
 * $$\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+C\qquad(\text{for }n\ne-1)$$
 * $$\int\frac{\mathrm{d}x}{x}=\ln|x|+C$$
 * $$\int\frac{\mathrm{d}x}{ax+b}=\frac{\ln|ax+b|}{a}+C\qquad(\text{for }a\ne0)$$

Basic Trigonometric Functions

 * $$\int\sin(x)\mathrm{d}x=-\cos(x)+C$$
 * $$\int\cos(x)\mathrm{d}x=\sin(x)+C$$
 * $$\int\tan(x)\mathrm{d}x=\ln|\sec(x)|+C$$
 * $$\int\sin^2(x)\mathrm{d}x=\int\frac{1-\cos(2x)}{2}\mathrm{d}x=\frac{x}{2}-\frac{\sin(2x)}{4}+C$$
 * $$\int\cos^2(x)\mathrm{d}x=\int\frac{1+\cos(2x)}{2}\mathrm{d}x=\frac{x}{2}+\frac{\sin(2x)}{4}+C$$
 * $$\int\tan^2(x)\mathrm{d}x=\tan(x)-x+C$$

Reciprocal Trigonometric Functions

 * $$\int\sec(x)\mathrm{d}x=\ln\Big|\sec(x)+\tan(x)\Big|+C=\ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\right|+C=2\mathrm{artanh}\left(\tan\left(\frac{x}{2}\right)\right)+C$$
 * $$\int\csc(x)\mathrm{d}x=-\ln\Big|\csc(x)+\cot(x)\Big|+C=\ln\left|\tan\left(\frac{x}{2}\right)\right|+C$$
 * $$\int\cot(x)\mathrm{d}x=\ln|\sin(x)|+C$$

 * $$\int\sec^2(ax)\mathrm{d}x=\frac{\tan(ax)}{a}+C$$
 * $$\int\csc^2(ax)\mathrm{d}x=-\frac{\cot(ax)}{a}+C$$
 * $$\int\cot^2(ax)\mathrm{d}x=-x-\frac{\cot(ax)}{a}+C$$
 * $$\int\sec(x)\tan(x)\mathrm{d}x=\sec(x)+C$$
 * $$\int\sec(x)\csc(x)\mathrm{d}x=\ln|\tan(x)|+C$$

Reduction formulae

 * $$\int\sin^n(x)\mathrm{d}x=-\frac{\sin^{n-1}(x)\cos(x)}{n}+\frac{n-1}{n}\int\sin^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n>0)$$
 * $$\int\cos^n(x)\mathrm{d}x=-\frac{\cos^{n-1}(x)\sin(x)}{n}+\frac{n-1}{n}\int\cos^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n>0)$$
 * $$\int\tan^n(x)\mathrm{d}x=\frac{\tan^{n-1}(x)}{(n-1)}-\int\tan^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\sec^n(x)\mathrm{d}x=\frac{\sec^{n-1}(x)\sin(x)}{n-1}+\frac{n-2}{n-1}\int\sec^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\csc^n(x)\mathrm{d}x=-\frac{\csc^{n-1}(x)\cos(x)}{n-1}+\frac{n-2}{n-1}\int\csc^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\cot^n(x)\mathrm{d}x=-\frac{\cot^{n-1}(x)}{n-1}-\int\cot^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$a^2\int x^n\sin(ax)\mathrm{d}x = nx^{n-1}\sin(ax)-ax^n\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm{d}x$$
 * $$a^2\int x^n\cos(ax)\mathrm{d}x = ax^n\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm{d}x$$

Explicit forms

 * $$\int \sin^n(x)\mathrm{d}x = -\cos(x)_2F_1\left(\frac{1}{2}, \frac{1-n}{2}; \frac{3}{2}; \cos^2(x)\right) + C$$
 * $$\int \cos^n(x)\mathrm{d}x = -\frac{1}{n+1}\mathrm{sgn}(\sin(x))\cos^{n+1}(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{n+3}{2}; \cos^2(x)\right) + C\qquad(\text{for }n\ne-1)$$
 * $$\int \tan^n(x)\mathrm{d}x = \frac{1}{n+1}\tan^{n+1}(x)_2F_1\left(1, \frac{n+1}{2}; \frac{n+3}{2}; -\tan^2(x)\right) + C\qquad(\text{for }n\ne-1)$$
 * $$\int \csc^n(x)\mathrm{d}x = -\cos(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{3}{2}; \cos^2(x)\right) + C$$
 * $$\int \sec^n(x)\mathrm{d}x = \sin(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{3}{2}; \sin^2(x)\right) + C$$
 * $$\int \cot^n(x)\mathrm{d}x = -\frac{1}{n+1}\cot^{n+1}(x)_2F_1\left(1, \frac{n+1}{2}; \frac{n+3}{2}; -\cot^2(x)\right) + C\qquad(\text{for }n\ne-1)$$

Where $${}_2F_1$$ is the hypergeometric function and $$\mathrm{sgn}$$ is the sign function.

Inverse Trigonometric Functions

 * $$\int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin(x)+C$$
 * $$\int\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin\left(\tfrac{x}{a}\right)+C\qquad(\text{for }a\ne0)$$
 * $$\int\frac{\mathrm{d}x}{1+x^2}=\arctan(x)+C$$
 * $$\int\frac{\mathrm{d}x}{a^2+x^2}=\frac{\arctan\left(\tfrac{x}{a}\right)}{a}+C\qquad(\text{for }a\ne0)$$

Exponential and Logarithmic Functions

 * $$\int e^x\mathrm{d}x=e^x+C$$
 * $$\int e^{ax}\mathrm{d}x=\frac{e^{ax}}{a}+C\qquad(\text{for }a\ne0)$$
 * $$\int a^x\mathrm{d}x=\frac{a^x}{\ln(a)}+C\qquad(\text{for }a>0,a\ne1)$$
 * $$\int\ln(x)\mathrm{d}x=x\ln(x)-x+C$$
 * $$\int e^{x}\sin(x)\mathrm{d}x = \frac{e^{x}}{2}(\sin(x) - \cos(x)) + C$$
 * $$\int e^{x}\cos(x)\mathrm{d}x = \frac{e^{x}}{2}(\sin(x) + \cos(x)) + C$$

Reduction formulae

 * $$\int x^ne^{ax}\mathrm{d}x = \frac{1}{a}x^ne^{ax} - \frac{n}{a}\int x^{n-1}e^{ax}\mathrm{d}x$$

Inverse Trigonometric Functions

 * $$\int\arcsin(x)\mathrm{d}x=x\arcsin(x)+\sqrt{1-x^2}+C$$
 * $$\int\arccos(x)\mathrm{d}x=x\arccos(x)-\sqrt{1-x^2}+C$$
 * $$\int\arctan(x)\mathrm{d}x=x\arctan(x)-\frac{1}{2}\ln|1+x^2|+C$$
 * $$\int\arccsc(x)\mathrm{d}x=x\arccsc(x)+\ln\left|x + x\sqrt{1-\frac{1}{x^2}}\right| + C$$
 * $$\int\arcsec(x)\mathrm{d}x=x\arcsec(x)-\ln\left|x + x\sqrt{1-\frac{1}{x^2}}\right| + C$$
 * $$\int\arccot(x)\mathrm{d}x=x\arccot(x)+\frac{1}{2}\ln|1+x^2| + C$$

Hyperbolic functions

 * $$\int \sinh(x)\mathrm{d}x = -i\int \sin(ix) \mathrm{d}x = \cos(ix) + C = \cosh(x) + C$$
 * $$\int \cosh(x)\mathrm{d}x = \int \cos(ix) \mathrm{d}x = -i\sin(ix) + C = \sinh(x) + C$$
 * $$\int \tanh(x)\mathrm{d}x = -i\int \tan(ix) \mathrm{d}x = \log\left|\cos(ix)\right| + C = \log\left|\cosh(x)\right| + C$$

Reciprocals
2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) + C$$
 * $$\int \mathrm{csch}(x)\mathrm{d}x = i\int \csc(ix) \mathrm{d}x = \log\left|-i\tan\left(\frac{ix}{2}\right)\right| + C = \log\left|\tanh\left(\frac{x}{2}\right)\right| + C$$
 * $$\int \mathrm{sech}(x)\mathrm{d}x = \int \sec(ix) \mathrm{d}x = 2\mathrm{artanh}\left(-i\tan\left(\frac{x}{2}i\right)\right) + C =
 * $$\int \mathrm{coth}(x) \mathrm{d}x = i\int \cot(ix) \mathrm{d}x = \log\left|-i\sin(ix)\right| + C = \log\left|\sinh(x)\right| + C$$

Inverses

 * $$\int \mathrm{arsinh}(x)\mathrm{d}x = x\mathrm{arsinh}(x) - \sqrt{x^2 + 1} + C$$
 * $$\int \mathrm{arcosh}(x)\mathrm{d}x = x\mathrm{arcosh}(x) - \sqrt{x^2 - 1} + C$$
 * $$\int \mathrm{artanh}(x)\mathrm{d}x = x\mathrm{artanh}(x) + \frac{1}{2}\ln(1-x^2) + C$$
 * $$\int \mathrm{arcsch}(x)\mathrm{d}x = x\mathrm{arcsch}(x) + |\mathrm{arsinh}(x)| + C$$
 * $$\int \mathrm{arsech}(x)\mathrm{d}x = x\mathrm{arsech}(x) + \arcsin(x) + C$$
 * $$\int \mathrm{artanh}(x)\mathrm{d}x = x\mathrm{arcoth}(x) + \frac{1}{2}\ln(x^2 - 1) + C$$

Misc

 * $$\int |f(x)|\mathrm{d}x = \mathrm{sgn}(f(x))\int f(x)\mathrm{d}x$$, where $$\mathrm{sgn}$$ is the sign function.

Definite integrals

 * $$\int_{[0,1]^n} \frac{\prod_{i=1}^n\mathrm{d}x_i}{1-\prod_{i=1}^n x_i} = \zeta(n) \text{ for all integers } n > 1$$, where $$\zeta$$ is the Riemann zeta function.
 * $$\int_{-\infty}^{\infty} e^{-x^2} \mathrm{d}x = \sqrt{\pi}$$
 * $$\int_0^1 t^{u-1}(1-t)^{v-1}\mathrm{d}t = \beta(u,v) = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u+v)}$$, where $$\Gamma$$ is the gamma function.
 * $$\int_0^\infty t^{s-1}e^{-t}\mathrm{d}t = \Gamma(s)$$
 * $$\int_0^{2\pi} e^{u\cos\theta}\mathrm{d}\theta = 2\pi I_0(u)$$, where $$I_0$$ is the modified Bessel function of the first kind.
 * $$\int_0^\infty \frac{\sin(x)}{x} \mathrm{d}x = \frac{\pi}{2}$$

title: Calculus/Tables of Integrals


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Portuguese/Contents/L1/Lesson Five - Going Shopping

Return to Portuguese – Return to Contents – Go to Lesson Six

Here, Ricardo is in a clothes shop. He asks the attendant, Carla, if he can have a white shirt - uma camisa branca. He also asks for something else - um chapéu. This means "a hat". We've come across the words for "a". These are called, properly, "indefinite articles" (as opposed to the words for "the", which are called "definite articles"). This is because the word "a" or "an" is indefinite - it doesn't specify which shirt or hat. Let's have a look at this table.

Some examples? Sure.

Quickly, what does that e mean by itself? Simple - e means "and". Let's have some practice.

Another major word we need to get to know is Quero. This word means "I want". Of course, there are other conjugations...

Você quer alguma coisa? - Do you want something? Sim, quero uma camisa branca. - Yes, I want a white shirt.

O senhor quer também uma camisa preta?

Também means quite simply "also". So we have "Do you want also a black shirt?". But what on earth is this phrase, o senhor? Well, we've already learnt that você is more polite than tu. However, there is another level of politeness used in Portuguese. This form of "you" is used with people you don't know well, or people you have to show respect to, like bosses or teachers. Basically, this form allows you to refer to people indirectly. So rather than address someone directly by name or with a pronoun such as "você", you address them as if you were addressing a third person, "o senhor" ("the gentleman") or "a senhora" ("the lady"). This added distance makes the speech more polite. Take note, however, that this form of "you" changes according to the gender of the person you're talking to, plus how many people you are talking to as well!

Have a look at the table below...

de:Portugiesisch: Lektionen: Einkaufen gehen


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Ada Programming/Delimiters//=

Operator
The operator /= compares two values on inequality. It is predefined for all non limited types. The operator will also be defined if a suitable operator = is available.

Note that in Ada the representation for this operator was chosen for resembling the mathematical symbol $$\ne$$, in the same way than <= resembles $$\le$$, or >= resembles $$\ge$$.

Wikibook

 * Ada Programming
 * Ada Programming/Delimiters

Ada 2005 Reference Manual

|/= /=

title: Ada Programming/Delimiters//=


text:
Ada Programming/Delimiters/less than or equal to

Operator
The operator <= compares two values on less than or equal to. It is predefined for all discrete types.

Wikibook

 * Ada Programming
 * Ada Programming/Delimiters

Ada 2005 Reference Manual

|<= <=


text:
Ada Programming/Delimiters/greater than

Operator
The operator > compares two values on being greater. It is predefined for all discrete types.

Wikibook

 * Ada Programming
 * Ada Programming/Delimiters
 * Ada Programming/Delimiters/<
 * Ada Programming/Delimiters/>>

Ada 2005 Reference Manual

|> >

title: Ada Programming/Delimiters/greater than


text:
Ada Programming/Delimiters/greater than or equal to

Operator
The operator >= compares two values on greater than or equal to. It is predefined for all discrete types.

Wikibook

 * Ada Programming
 * Ada Programming/Delimiters

Ada 2005 Reference Manual

|>= >=


text:
User:Shreshth91~enwikibooks

Hi. I'm Shreshth. I study in 9th grade in Delhi Public School, RK Puram, in New Delhi. That's in India for those of you who don't know. It's rated among the best schools in India.

I joined Wikibooks around July 1, 2005. So I'm relatively new around here. I basically use Wikibooks for help in my projects and homework. I think it's the coolest collection of books ever. I haven't got around to much editing yet. You can check out my contributions here or you can go to my talk page

I am an administrator on English Wikipedia

External Link
Delhi Public School official website

title: User:Shreshth91~enwikibooks


text:
User talk:Shreshth91~enwikibooks

 * Hi. I'm Shreshth. I study in 9th grade in Delhi Public School, RK Puram, in New Delhi. That's in India for those of you who don't know. It's rated among the best schools in India.
 * I joined Wikibooks around July 1, 2005. So I'm relatively new around here. I basically use Wikibooks for help in my projects and homework. I think it's the coolest collection of books ever. I haven't got around to much editing yet. You can check out my contributions here

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Delhi Public School official website

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title: User talk:Shreshth91~enwikibooks


text:
Software Engineering

Overlaps the other book: Introduction to Software Engineering

''The idea of this book is to couple together the different projects on the different subjects of software engineering. Currently the only book linked is Computer Programming. Other subjects should be added over time.''

As written in the Computer Programming book, coding is only a small part of software engineering. This book is intended as an introduction to the realm of software engineering.

=The Basics=

What is software engineering?
A systematic approach to the analysis, design, implementation and maintenance of software.

Software engineering is the engineering discipline through which software is developed. Commonly the process involves finding out what the client wants, composing this in a list of requirements, designing an architecture capable of supporting all of the requirements, designing, coding, testing and integrating the separate parts, testing the whole, deploying and maintaining the software. Programming is only a small part of software engineering.

The discipline is still in its infancy (early stage of growth/development) as an engineering discipline. We haven't had enough experience, nor gathered enough empirical data to systematically understand and predict the life-cycle of a software project.

The Software Engineering Body of Knowledge (SWEBOK) divides software engineering into 10 knowledge areas:

 * 1) Software requirements
 * 2) Software design
 * 3) Software construction
 * 4) Software testing
 * 5) Software maintenance
 * 6) Software configuration management
 * 7) Software engineering management
 * 8) Software engineering process
 * 9) Software engineering tools and methods
 * 10) Software quality

=The content of the field= ''Note: the following summaries are originally from Wikipedia. These are here only temporarily. The titles aren't final, nothing on this page is final yet (or even near final).''

Vision and Scope
The practice of software engineering in the corporate or industrial sector, starts with the business and ends with the business. Although computers, programming languages, and creative problem solving are what fill the engineer with interest in the field, without servicing and enabling the user, the exercise would be pointless. Thus the first stage in any software engineering process, is the /Vision and Scope/ document, or some equivalent meeting.

A vision of the system to be built is described by the user or /Product Champion/. The business context that it will service is described. The key stakeholders are enumerated including their interests, risks, etc. The success conditions are listed, so that it is understood what will be completed and how to measure success.

Business opportunities are discussed, to justify why the project should be invested with the blood, sweat, and tears of programmers, testers, project managers, and program managers. The Scope of the project is made clear and if it is to be in phases, the scope of each phase at a high level is laid out. The priorities of the project are recorded.

With an understanding of the business' vision, the engineers begin to ask questions.

Requirements Analysis
Extracting requirements of a desired software product is the first task in creating it. This process is called requirements elicitation. While customers probably believe they know what the software should do, it may require skill and experience in software engineering to recognize incomplete, ambiguous or contradictory requirements.

Compounding this problem is the absence of formal abstractions. A civil, mechanical engineer has a plan, elevation, sections to refer to and these make good sense to both the author and reader of these abstractions. However, while (E-R) or object diagrams, process flow diagrams, data flow diagrams are abstractions that have been used and built on, many of these are not universally used or practiced.

Besides, considerable effort goes into repairing and enhancing existing systems. Engineering required here is a mix of reverse and forward engineering in the sense that requirements definitions refer to a set of circumstances that already exist and these circumstances have to be stated in unambiguous terms as well.

Perhaps, lack of universally accepted abstractions make life difficult for software engineers and some people have even argued that software production process is closer to art rather than engineering!

Of late, people have started using prototypes in various degrees of working. Screen shots and report layouts are non working prototypes and form the basic level of prototyping and increasing sophistication is built in wherever possible. Making prototypes is always more time consuming than not making one. Thus, this practice is not always utilised.

Analyst Role
The role of analyst can be fulfilled typically by three types of people. A technical project manager, a software engineer, or a dedicated analyst. In some organizations it is unthinkable to engineers that they would serve the analyst role, as they are "technical" and would consider dealing directly with the customer unthinkable. This is purely a cultural stance that varies between companies and organizations. It is also quite common for software engineers to work closely with the business people and customers associated with a project, and this programmer/analyst role can be quite beneficial to the projects success rate.

As you are gathering requirements, you can subject them to analysis. Does the requirement in question conflict with others? What is its priority? Where did it come from? Does it need further clarification?

Analysis is the ability to draw inferences from requirements and represent them in a structured way. Structures themselves may vary based on what is being analysed. It is very tough to draw boundaries between analysis and requirements gathering as they both use the same abstractions many a time.

One could argue that analysis yields data storage structures and workflow structures etc. However, one would think that these are not the only outputs from analysis.

Communication Techniques
The common technique that means the software requirement analysis begins with communication between two or more parties. Review of a software requirement specification is conducted by both software developer and customer. Because the specification forms the foundation for design and subsequent software engineering activities, extreme care should be taken in conducting the review. The review is first conducted at a macroscopic level. At this level, the reviewers attempt to ensure that the specification is complete, consistent and accurate. Once the review is complete a software requirement specification is signed off, “by both customers and developer”.

Case Study
While requirement analysis phase some problems are come. The problems are identified and defined and listed here. • The problem occurs that did exist in smaller systems, which leads to a redefining of priority of the activities that go into developing software. • In the beginning of the requirements phase, the needs of clients are in the minds of various people in the client organization. • The major role of the problem statement is that the customer has a problem that may be responsible to a computer-based solution. A developer responds to the customer’s request for help. Here the one more problem is the path from communication to understanding is often full of difficulties.

Suggestions
The above case study/problem determined while requirement analyzing phase in "system engineering". This page can be improved soon. --Ananthakumar Selvaraj (discuss • contribs) 13:02, 18 March 2013 (UTC)

Use Cases
Once the vision and scope of a engineering project has been established the requirements gather process begins. One tool for communicating between the customer, analyst, developers, and testers is a set of Use Cases. A Use Case typically has a name, a short description, and then a number set of steps which a user would go through to accomplish a task. Use cases are sometimes grouped together in a UML or other format of diagram depicting users, the system, external entity and specific use cases.

Granularity
Use Cases are typically coarse-grained and don't dive into every detail and functional requirement of the system. A business person or software engineer should be able to read through all of the use cases and get an idea of the entire scope of what the project will deliver.

Formality of Use Cases
Agile driven projects may skip these cases, or at least only record the attributes discussed above. Plan driven methodologies may give additional attributes to Use Cases such as alternative paths after covering the main steps, trace requirements covered by the use case back to their source, and have an error handling section for each use case.

THE STAGES OF ANALYSIS

Analysis is typically accomplished in two stages. The first stage, Business Analysis includes an in-depth examination of the customer's present set of business or manufacturing processes and procedures with an eye toward how they might be improved by automation. The second stage, Systems Analysis consists of an examination of the proposed improvements and a recommendation on what computer environment and software technology is best suited to accomplish those improvements.

BUSINESS ANALYSIS STAGE

From the Business Analyst's in-depth understanding of the customer's present environment, he or she creates a set of "business use-cases", which are short descriptions of the role that each of the business's "actors" perform within each process or procedure. For instance, the waiter in a restaurant might have a use case as follows: " A. Actor approaches customer table, introduces self and recites a description of the day's specials.  B. Actor presents a beverage menu and inquires if customer would like to order a beverage.  C. If yes, actor writes each customer's order on a drink tag and proceeds to bar area.  D. Actor presents the tag to bar-keep and makes a mental note to check back within a customary time to see if the drink order is complete.  E. If no, Actor presents a dinner menu and helps customers to make selections.  As each new party of customers is seated, actor repeats procedure, beginning from step A." With a set of raw use-cases like the proceeding, the business analyst can, with the customer's collaboration, examine how each use-case might be automated and can thereby define an official system requirement which becomes one of maybe hundreds of requirements that define the future software application. As might be surmised, the effective Business Analyst must not only have extensive experience with and knowledge of the business world, but must also have expertise in the potentialities offered by modern software systems and how those systems map to the reality of the business world.

SYSTEMS ANALYSIS STAGE

The second stage of the analysis process, Systems Analysis begins from the foundation laid by the business analyst. From the set of business use cases and their resultant software requirements, the Systems Analyst makes recommendations as to the most suitable software technology to accomplish those requirements. In some cases, the hardware environment limits choices of software technologies. For instance, in the restaurant scenario, above. The customer might be adamant about providing the waiter with a handheld device to replace the drink tag and to make the trip to the bar unnecessary before the drink order is ready to serve. In such a case, the systems analyst might be constrained to using only those operating systems and APIs that are supported on the preferred handheld device.

Once the programming and run-time environment is decided, then the systems analyst can begin turning the business use-cases and requirements into a set of "system use-cases," which are descriptions of how the actors will interact with the proposed system. Those system use-cases, taken together now comprise the actual blueprint that will be used by software architects to begin laying out the classes or other software modules that will define a road map for the software coding and development process which follows.

Specification
Specification is the task of precisely describing the software to be written, in a mathematically rigorous way. In reality, most successful specifications are written to understand and fine-tune applications that were already well-developed. Specifications are the most important for external interfaces that must remain stable.

Requirements Management
Once you have gathered requirements, captured them into a specification, and the customer agrees with it, it is possible to take a snapshot or baseline of the project requirements. This allows one to manage existing requirements and process new incoming ones. The specification is placed under configuration management, such as a tool like CVS so that the requirements can be versioned and tracked.

Rejecting Requirements
It is important to reject requirements that
 * Don't fit within the project scope
 * Cannot show a positive return on investment
 * Won't make the system competitive
 * Aren't technically feasible

A requirement rejected doesn't have to be further analyzed, implemented, documented, or tested. Every requirement costs the company and must be thought through.

Change Control
Regardless of whether your methodology is agile or plan driven, you should create some form of change control to process new requirements. Failure to do so may lead to project failure due to scope creep, project overruns, poor quality, or high costs. Change control can be as simple as a development manager reviewing specification, to an elaborate tool supported process with forms and a board of people with change control authority.

Lack of communication about changes to a specification can lead to costly wasted work by developers, testers, or analysts. The time it takes to change one requirement in a specification, may cost 100 times the person hours to realize the decision.

Design and Architecture
Design and architecture are the activities involved in specifying how the software will actually work. This phase frequently described as being divided into two main phases, which might be described as "business design" and "technical design". Business design generally specifies the "why" of the system, indicating how data will be used and how it will flow. Technical design generally specifies the "how" of the system, or how its components will be arranged, what their features will be, and what kind of hardware the system will require. These two phases can often be conducted roughly simultaneously, with business design usually starting first and technical design ending last.

Of all of the phases, the design phase can often consume the most time, and many engineers consider it the most important. A poor design can cause the failure of a project, even after considerable resources are spent during the construction phase of the project building it. It is important for the design and architecture of the system to be complete enough that it can be validated and then used to develop the system.

Coding
Reducing a design to code may be the most obvious part of the software engineering process, but it is not necessarily the largest portion. In fact, many software engineers only work on the analysis and design process, then the coding work is handed off to computer programmers to implement. This trend is increasing in many companies, where software engineers are hired domestically and the implementation is out-sourced to a country where programming labor is less expensive. In spite of this trend, many software engineers still go through the entire process and do coding as well as design.

Coding Wikibook

Testing
The purpose of testing is not only to prove that the code performs in accordance with the design specifications, but to prove that it does not fail when subjected to undefined inputs.

There are several types of testing. Unit testing tests one module of code for correct inputs, outputs, and functionality. System testing tests all the modules of a software system together. User acceptance testing is a form of systems testing to see if not only the system works, but it meets the requirements of the business user. Regression testing is testing performed after a system change to make sure that all system features are still present after the change.

To perform systems and verifications testing, Use cases are developed. These cases describe a user action or a specific feature of the system. These cases are very useful during the construction phase of the system, and are usually specified as a deliverable of the design phase. The use cases are generally gathered into test scripts which describe testing activities in a reproducible way. These scripts may be executed manually by testers, or they may be executed by automated testing software.

Generally speaking, it is an accepted industry practice for system testing and validation to be performed by a separate person, team, or department, or in some cases, by an outside entity. Testing is considered a specialized skill, with a skill set that is very different from system development. It also is generally considered a bad practice for software developers to test their own systems, since they can form habits that do not exercise the system thoroughly, and because they can frequently test around system failures.

There are different kinds of testing such as white box testing or black box testing.

Documentation
An important (and often overlooked) task is documenting the internal design and external functionality of software for the purpose of future maintenance and enhancement.

Software documentation is important. Without documentation, software can be unusable, because the end user does not know how to operate a program. Without documentation, an application can be incorrectly installed on a server, causing components to fail. Documentation is most important for external interfaces.

There are several means by which software can be documented, including flowcharts, deployment guides, user's manuals, and maintenance manuals.

Development team members may write their own documentation, or a professional technical writer may write or edit the documents for readability and style.

It's important to note that testing is or should be done during or at the end of every stage. If this is not done, inputs into the next stage can not be very complete. Documentation may also be sketchy or none existent.

Maintenance
Maintaining and enhancing software to cope with newly discovered problems or new requirements can take far more time than the initial development of the software. Not only may it be necessary to add code that does not fit the original design but just determining how software works at some point after it is completed may require significant effort by a software engineer. About 2/3 of all software engineering work is maintenance, but this statistic can be misleading. A small part of that is fixing bugs. Most maintenance is extending systems to do new things, which in many ways can be considered new work. Similarly, about 2/3 of all civil engineering, architecture, and construction work is maintenance in a similar way.

=Applied Software Engineering= The process of software engineering differs between different software types. Building a word processor has different requirements and design methods than building a 3D first person shooter. These sections cover the differences and unique properties of the different software types.

SE on Server Software
Software engineering for server software is more complex then development for the desktop computers. A server is more complex, has more requirements and features than an ordinary, consumer system. Server software can have client-server as well as central architecture. Development and testing of server software is also different from the development and testing of the desktop software. Not every programming language is suitable for implementation of server software. In conclusion, SE on server software requires special methods and processes to successfully complete.

The main reason server software is more complex is that communication between 2 computers -- the server and the client -- is inherently more complex and prone to failure than communication between parts of a program running on a single computer.

Games
Designing software for gaming simulations can be quite challenging, since it requires a diverse background in software. For example, a game designer might have to write software that simulates physics as well as software that renders 3-dimensional images. However, many software engineers in the gaming industry are able to use 3rd party engines that make their job much easier. Nonetheless, they must follow the software engineering process closely, since small issues in the software can make the product fail in the consumer market. Writing software for games is a high-risk industry and only a small percentage of games are financially successful.

SE on Embedded software
Software engineers that work with embedded computer systems need more knowledge of hardware than most software engineers, since embedded systems typically introduce more constraints on memory availability and processing power than, say, gaming PCs. Designing software for embedded systems can indeed be challenging, since most embedded processors cannot handle the overhead produced by compiling high-level languages. This often forces software engineers to implement embedded software, at least partially, in assembly.

title: Software Engineering


text:
Talk:Software Engineering

This wikibook is intended as a connector book between the differend wikibooks on the differend fields of software engineering. Idealy we would get a complete work on software engineering. The idea is that seperate wikibooks will be started on the differend subjects, starting from this page or independed and then linked. Possible we could use a page URL standard (e.g. software_engineering_subfield_pagename_... ). Any ideas are welcome. --Patrik 08:36, 1 Jul 2005 (UTC)

Waterfall model?
The sections here are set up in a very traditional way, kind of implicitly supporting the waterfall model: spec, design, implement, test, etc.

It would be good I think to balance this out with some info on agile programming techniques. It isn't just a fad, real companies are developing real software this way...--Kenlars99 21:33, 3 December 2005 (UTC)

--Kenlars99 21:33, 3 December 2005 (UTC)

Actually the layout supports multiple models. For example the spiral model uses spec, design, impl. and test in small repeating iterations. Ofcourse the layout may be changed to better represent what these steps (specification, design, ...) are and how they are employed in differend development models. You're more than welcome to propose a better layout as I'm not an expert in the material. The many models (functional, OO, agile, extreme programming) should get a clear treatment and compared to each other, not so much as to say which is the best, but more to explain where they are best used. --Patrik 20:14, 4 December 2005 (UTC)

Test cases
I have had a list of common test cases transwikied from Wikipedia because I thought it could be put to good use in this or a similar book. The list can be found at Transwiki:Common Test Cases. It needs a little cleanup, of course, but is in pretty good shape. Hopefully, someone for familiar with this book than me can put it to good use. Thanks --ThaddeusB (talk) 00:33, 6 June 2009 (UTC)

title: Talk:Software Engineering


text:
Ordinary Differential Equations

 * /Definition, reduction of explicit equations to first order/
 * /Existence and uniqueness of solutions/
 * /Differential inequalities/
 * Solutions to specific equations
 * /One-dimensional time-independent equations of first or second order/

OLD TOC
Ordinary Differential Equations

covering uses of and solutions to ordinary differential equations

This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. This book provides information about the whole area of differential equations, concentrating first on the simpler equations.

Table of contents

 * /Introduction/
 * /Preliminaries from calculus/
 * Form and Solutions of Differential Equations
 * First-Order Differential Equations
 * Separation of Variables
 * Linear Differential Equations
 * Exact Differential Equations
 * Substitution Methods
 * Bernoulli Equations
 * Ricatti Equations
 * Orthogonal and Oblique Trajectories
 * Equations of higher degrees
 * Equations without x or y
 * Equations that are homogeneous in x and y
 * d'Alembert's Equation
 * Clairaut Equations
 * Legendre Transformations
 * Graphing Differential Equations
 * Second-Order Differential Equations
 * Constant Coefficients
 * Series Solutions
 * Hypergeometric Equation
 * Frobenius Solution to the Hypergeometric Equation
 * Legendre Equation
 * Bessel Equation
 * Mathieu Equation
 * Continued Fraction Solutions
 * Applications of Second-Order Differential Equations
 * Higher Order Differential Equations
 * Linear equations
 * General Linear Equations
 * Infinite Series Solutions to Linear Equations
 * Integration methods
 * Laplace Transform
 * Bessel Function
 * Euler Transform
 * Mellin Transform
 * Hypergeometric Series
 * Double Integration
 * /Sturm-Liouville theory/
 * Systems of linear differential equations
 * /Nonlinear Systems/
 * /Green's Functions/
 * Existence and Uniqueness of Solutions
 * /The Picard–Lindelöf theorem/
 * /Peano's theorem/
 * /Blow-ups and moving to boundary/
 * /Global uniqueness of solution over interval/
 * /Maximum domain of solution/
 * The Successive Approximations Method of Proof
 * Applications to Linear Equations
 * The Cauchy-Lipschitz Method of Proof
 * Existence Theorems for Complex Numbers
 * Continuous Transformation Groups
 * Infinitesimal Transformations
 * Invariant Functions
 * /Glossary/
 * /List of Some Equations/
 * Help Needed
 * /Roadmap/

title: Ordinary Differential Equations


text:
Ordinary Differential Equations/Introduction

Directly integrable differential equations
An example of an ordinary differential equation is the equation
 * $$x'(t) = 0$$.

To solve this equation shall mean that we find a function $$x: I \to \mathbb R$$ defined on some interval $$I = (a, b)$$ such that the above equation holds for all $$t \in I$$. That is, as opposed to "normal" algebraic equations such as
 * $$x - 3 = \frac{1}{2}$$,

to which the solution is a number (in this case $$x = \frac{7}{2} = 3.5$$), the solution to an ordinary differential equation is a function.

There is a theorem in calculus that says that a differentiable function $$x: I \to \mathbb R$$ on a connected interval $$I = (a, b)$$ has vanishing derivative if and only if it is a constant function. Thus, the solutions to the equation
 * $$x'(t) = 0$$

are precisely the constant functions $$x \equiv c$$ ($$c \in \mathbb R$$). Note here that we have lost the uniqueness of solutions (which we had in our example for an algebraic equation); every constant function is a solution. We will later be able to restore the uniqueness of solutions (at least in some special cases) by imposing initial conditions; for example, if we additionally require that
 * $$x(t_0) = x_0 \in \mathbb R$$

(where we also require that $$t_0$$ is contained within the solution interval $$I$$), then we have only one choice for a solution to our equation: $$c = x_0$$.

Let's turn to a slightly more difficult differential equation:
 * $$x'(t) = f(t)$$

for some function $$f: \mathbb R \to \mathbb R$$. In this case, by the fundamental theorem of calculus we have
 * $$x(t) = \int_{t_0}^t f(s) ds + C$$ for some $$t_0$$ to be contained in the solution interval $$I$$,

where again $$C \in \mathbb R$$ is an arbitrary integration constant. The arbitraryness of $$C$$ of course implies that we have infinitely many solutions, but by imposing an initial condition $$x(t_0) = x_0$$, we get that $$C$$ must equal $$x_0$$, and hence we have once again obtained a unique solution if we additionally require the initial condition to be satisfied.

Exercises

 * Solve the differential equation $$x'(t) = t^2$$ under the initial condition $$x(0) = 0$$. More generally, solve the differential equation $$x'(t) = t^k$$ under the initial condition $$x(2-k) = k-2$$.

Order of an ordinary differential equation
Instead of looking at the equation
 * $$x'(t) = 0$$,

we may instead look at the equation
 * $$x(t) = 0$$, or, more generally, $$x(t) = f(t)$$

for some function $$f: \mathbb R \to \mathbb R$$. Note the difference: Previously, we had only had a first-order derivative, and now we have a second order derivative. The new equation is still an ordinary differential equation (we will give the precise definition of an ordinary differential equation at the end, when we are finished with the simple examples), but this time involving a second-order derivative. This gives rise to the following definition.

In fact, for the above examples, we can easily compute the solutions using just integration. If $$x''(t) = 0$$, then $$x'(t) = c_1$$ for some constant $$c_1 \in \mathbb R$$. Therefore, integrating that again, we get
 * $$x(t) = \int_{t_0}^t x'(s) ds = (t - t_0) c_1 + c_2$$

for some constant $$c_2 \in \mathbb R$$.

Exercises

 * What is the order of the ordinary differential equation $$x'(t) + 2x(t) + 3x'(t) = 2t$$? What is the order of the ODE $$x^{(k)}(t) = t^2$$? Based on the last question, write down an ordinary differential equation of order 23.
 * Solve the differential equation $$x''(t) = t^3$$ under the conditions $$x(0) = 1$$ and $$x(1) = 2$$.

Linear differential equations
Consider the differential equation
 * $$x'(t) = c x(t)$$,

where $$c \in \mathbb R$$ is an arbitrary constant. This differential equation has a remarkable property: Whenever $$x_1, x_2$$ are solutions to it, so is $$a x_1 + b x_2$$ for arbitrary $$a, b \in \mathbb R$$ (in fact, complex numbers are permissible as well). More generally, every linear combination of solutions is again a solution. This is a direct consequence of linearity of the derivatives as follows:
 * $$(a_1 x_1(t) + \cdots + a_n x_n(t))' = a_1 x_1'(t) + \cdots + a_n x_n'(t) = a_1 c x_1(t) + \cdots a_n c x_n(t) = c (a_1 x_1(t) + \cdots + a_n x_n(t))$$.

Differential equations with this property are called linear.

Note that since zero can be written as a linear combination of any existing solution (as $$0 \cdot x$$), a linear ODE has either no solution or zero is a solution.

There is also such a thing as an inhomogenous linear equation, which is closely related, but not equal, to linear equations.

Consider the ordinary differential equation
 * $$x'(t) = c x(t) + g(t)$$

for some function $$g(t)$$. This is an example of what's called an inhomogenous linear equation, for the following reason: Suppose it has a solution $$x$$, and $$y$$ is a solution of the above equation $$y'(t) = c y(t)$$. Then any superposition $$z(t) := x(t) + a y(t)$$ ($$a \in \mathbb R$$) is again a solution:
 * $$z'(t) = (x(t) + a y(t))' = x'(t) + a y'(t) = c x(t) + g(t) + a c y(t) = c z(t) + g(t)$$.

We capture this property in the following definition.

To contrast linear ODEs from inhomogeneous linear ODEs, we often call the former homogenous linear ODEs. Some mathematicians also use the term "linear" to refer to homogenous or inhomogenous ODEs, which is why it is advisable to use the term "homogenous" anyway.

We will see only later how to solve these things, although it's actually very easy.

Exercises
Let $$\lambda \in \mathbb C$$. Prove by direct computation that the function $$x(t) := e^{\lambda t}$$ is a solution to the ODE $$x'(t) = \lambda x(t)$$.


text:
Introduction to Philosophy/Resources For Editing

Part I - The Nature of Philosophy

 * 1) Origins of Philosophy
 * 2) What is Philosophy
 * 3) The Branches of Philosophy
 * 4) What is meant by "Doing Philosophy?"

Part II - Metaphysics

 * 1) What is Metaphysics?
 * 2) What is a Person?
 * 3) What is Buddhist Philosophy?

Part III - Epistemology

 * 1) What is Epistemology?
 * 2) Rationalism
 * 3) Empiricism
 * 4) Pragmatism

Part IV - Logic

 * 1) Logic

Part V - Ethics

 * 1) What are Ethics?
 * 2) What is a Person?
 * 3) Virtue and value-Ethics
 * 4) Deontology
 * 5) Utilitarianism
 * 6) Existentialism
 * 7) Zen Buddhism
 * 8) Transcendentalist Theology Transcendentalism

Part VI - Aesthetics

 * 1) Aesthetics

Part VII - Political Philosophy

 * 1) What is Political Philosophy?
 * 2) The Political Spectrum
 * 3) Ancient Political Philosophy
 * 4) Middles Ages Political Philosophy
 * 5) Social Contract
 * 6) Marx and Marxists
 * 7) Liberalism
 * 8) Communitarianism
 * 9) Conservativism
 * 10) Anarchism
 * 11) Post-Politics
 * 12) Theocracies