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# European influence in Afghanistan
## Mohammed Zahir Shah, 1933--1973 {#mohammed_zahir_shah_19331973}
In 1933, after the assassination of Nadir Khan, Mohammed Zahir Shah became king. In 1940, the Afghan legation in Berlin asked that if Germany won the Second World War would the *Reich* give all of British India up to the Indus river to Afghanistan. Ernst von Weizsacker, the State Secretary at the *Auswärtiges Amt* wrote to the German minister in Kabul on 3 October 1940:
> \"The Afghan minister called on me on September 30 and conveyed greetings from his minister president, as well as their good wishes for a favourable outcome of the war. He inquired whether German aims in Asia coincided with Afghan hopes; he alluded to the oppression of Arab countries and referred to the 15m Afghans (Pashtuns, mainly in the North West Frontier province) who were forced to suffer on Indian territory.\
> \
> My statement that Germany\'s goal was the liberation of the peoples of the region referred to, who were under the British yoke was received with satisfaction by the Afghan minister. He stated that justice for Afghanistan would be created only when the country\'s frontier had been extended to the Indus; this would also apply if India should secede from Britain. The Afghan remarked that Afghanistan had given proof of her loyal attitude by vigorously resisting English pressure to break off relations with Germany.\"
No Afghan government ever accepted the Durand Line which divided the ethnically Pashtun population into the North-West Frontier Province of the British Indian Empire (modern north-western Pakistan) and Afghanistan, and it was the hope of Kabul that if Germany won the war, then all of the Pashtun people might be united into one realm
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# Division ring
In algebra, a **division ring**, also called a **skew field** (or, occasionally, a **sfield**), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring`{{refn|In this article, rings have a {{math|1}}.}}`{=mediawiki} in which every nonzero element `{{mvar|a}}`{=mediawiki} has a multiplicative inverse, that is, an element usually denoted `{{math|''a''{{sup|–1}}}}`{=mediawiki}, such that `{{math|1=''a{{space|thin}}a''{{sup|–1}} = ''a''{{sup|–1}}{{space|thin}}''a'' = 1}}`{=mediawiki}. So, (right) *division* may be defined as `{{math|1=''a'' / ''b'' = ''a''{{space|thin}}''b''<sup>–1</sup>}}`{=mediawiki}, but this notation is avoided, as one may have `{{math|1=''a{{space|thin}}b''{{sup|–1}} ≠ ''b''{{sup|–1}}{{space|thin}}''a''}}`{=mediawiki}.
A commutative division ring is a field. Wedderburn\'s little theorem asserts that all finite division rings are commutative and therefore finite fields.
Historically, division rings were sometimes referred to as fields, while fields were called \"commutative fields\".`{{refn|Within the English language area the terms "skew field" and "sfield" were mentioned 1948 by Neal McCoy<ref>1948, Rings and Ideals. Northampton, Mass., Mathematical Association of America</ref> as "sometimes used in the literature", and since 1965 ''skewfield'' has an entry in the [[OED]]. The German term {{ill|Schiefkörper|de|vertical-align=sup}} is documented, as a suggestion by [[van der Waerden]], in a 1927 text by [[Emil Artin]],{{refn|{{citation|last1=Artin|first1=Emil|year=1965|title=Collected Papers|editor1=Serge Lang|editor2=John T. Tate|location=New York|publisher=Springer}}}} and was used by [[Emmy Noether]] as lecture title in 1928.<ref>{{citation|last1=Brauer|first1=Richard|year=1932|title=Über die algebraische Struktur von Schiefkörpern|journal=Journal für die reine und angewandte Mathematik|volume=166 |issue=4|pages=103–252}}</ref>}}`{=mediawiki} In some languages, such as French, the word equivalent to \"field\" (\"corps\") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as \"corps commutatif\" (commutative field) or \"corps gauche\" (skew field).
All division rings are simple. That is, they have no two-sided ideal besides the zero ideal and itself.
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# Division ring
## Relation to fields and linear algebra {#relation_to_fields_and_linear_algebra}
All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring of quaternions. If one allows only rational instead of real coefficients in the constructions of the quaternions, one obtains another division ring. In general, if `{{math|''R''}}`{=mediawiki} is a ring and `{{math|''S''}}`{=mediawiki} is a simple module over `{{math|''R''}}`{=mediawiki}, then, by Schur\'s lemma, the endomorphism ring of `{{math|''S''}}`{=mediawiki} is a division ring; every division ring arises in this fashion from some simple module.
Much of linear algebra may be formulated, and remains correct, for modules over a division ring `{{math|''D''}}`{=mediawiki} instead of vector spaces over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a basis, and Gaussian elimination can be used. So, everything that can be defined with these tools works on division algebras. Matrices and their products are defined similarly. However, a matrix that is left invertible need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See *`{{slink|Generalized inverse#One-sided inverse}}`{=mediawiki}*.)
Determinants are not defined over noncommutative division algebras. Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such as quasideterminants allow some results`{{what|date=February 2025}}`{=mediawiki} to be recovered.
Working in coordinates, elements of a finite-dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite-dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring `{{math|''D''<sup>op</sup>}}`{=mediawiki} in order for the rule `{{math|(''AB'')<sup>T</sup> {{=}}`{=mediawiki} *B*^T^*A*^T^}} to remain valid.
Every module over a division ring is free; that is, it has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by matrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the *opposite* side of vectors as scalars are. The Gaussian elimination algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is the dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same and define the rank of a matrix.
Division rings are the only rings over which every module is free: a ring `{{math|''R''}}`{=mediawiki} is a division ring if and only if every `{{math|''R''}}`{=mediawiki}-module is free.
The center of a division ring is commutative and therefore a field. Every division ring is therefore a division algebra over its center. Division rings can be roughly classified according to whether or not they are finite dimensional or infinite dimensional over their centers. The former are called *centrally finite* and the latter *centrally infinite*. Every field is one dimensional over its center. The ring of Hamiltonian quaternions forms a four-dimensional algebra over its center, which is isomorphic to the real numbers.
## Examples
- As noted above, all fields are division rings.
- The quaternions form a noncommutative division ring.
- The subset of the quaternions `{{math|''a'' + ''bi'' + ''cj'' + ''dk''}}`{=mediawiki}, such that `{{mvar|a}}`{=mediawiki}, `{{mvar|b}}`{=mediawiki}, `{{mvar|c}}`{=mediawiki}, and `{{mvar|d}}`{=mediawiki} belong to a fixed subfield of the real numbers, is a noncommutative division ring. When this subfield is the field of rational numbers, this is the division ring of *rational quaternions*.
- Let $\sigma: \Complex \to \Complex$ be an automorphism of the field `{{nowrap|<math>\Complex</math>.}}`{=mediawiki} Let $\Complex((z,\sigma))$ denote the ring of formal Laurent series with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate `{{nowrap|<math>z</math>,}}`{=mediawiki} for `{{nowrap|<math>\alpha\in\Complex</math>,}}`{=mediawiki} define $z^i\alpha := \sigma^i(\alpha) z^i$ for each index `{{nowrap|<math>i\in\mathbb{Z}</math>.}}`{=mediawiki} If $\sigma$ is a non-trivial automorphism of complex numbers (such as the conjugation), then the resulting ring of Laurent series is a noncommutative division ring known as a *skew Laurent series ring*; if `{{math|1=''σ'' = [[identity function|id]]}}`{=mediawiki} then it features the standard multiplication of formal series. This concept can be generalized to the ring of Laurent series over any fixed field `{{nowrap|<math>F</math>,}}`{=mediawiki} given a nontrivial `{{nowrap|<math>F</math>-automorphism}}`{=mediawiki} `{{nowrap|<math>\sigma</math>.}}`{=mediawiki}
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# Division ring
## Main theorems {#main_theorems}
**Wedderburn\'s little theorem**: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.)
**Frobenius theorem**: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.
## Related notions {#related_notions}
Division rings *used to be* called \"fields\" in an older usage. In many languages, a word meaning \"body\" is used for division rings, in some languages designating either commutative or noncommutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article on fields.
The name \"skew field\" has an interesting semantic feature: a modifier (here \"skew\") *widens* the scope of the base term (here \"field\"). Thus a field is a particular type of skew field, and not all skew fields are fields.
While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.
A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws
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# Dia (software)
**Dia** (`{{IPAc-en|ˈ|d|iː|ə}}`{=mediawiki}) is free and open source general-purpose diagramming software, developed originally by Alexander Larsson. It uses a controlled single document interface (SDI) similar to GIMP and Inkscape.
## Features
Dia has a modular design with several shape packages available for different needs: flowchart, network diagrams, circuit diagrams, and more. It does not restrict symbols and connectors from various categories from being placed together.
Dia has special objects to help draw entity-relationship models, Unified Modeling Language (UML) diagrams, flowcharts, network diagrams, and simple electrical circuits. It is also possible to add support for new shapes by writing simple XML files, using a subset of Scalable Vector Graphics (SVG) to draw the shape.
Dia loads and saves diagrams in a custom XML format which is, by default, gzipped to save space. It can print large diagrams spanning multiple pages and can also be scripted using the Python programming language.
## Exports
Dia can export diagrams to various formats, including:
- EPS (Encapsulated PostScript)
- SVG (Scalable Vector Graphics)
- DXF (Autocad\'s Drawing Interchange format)
- CGM (Computer Graphics Metafile, defined by ISO standard 8632)
- WMF (Windows Meta File)
- PNG (Portable Network Graphics)
- JPEG (Joint Photographic Experts Group)
- VDX (Microsoft\'s XML for Visio Drawing)
## Development
Dia was originally created by Alexander Larsson but he moved on to work on GNOME and other projects. James Henstridge took over as lead developer, but he also moved on to other projects. He was followed by Cyrille Chepelov, then Lars Ræder Clausen.
Dia is currently maintained by Hans Breuer, Steffen Macke and Sameer Sahasrabuddhe.
It is written in C, and has an extension system which also supports writing extensions in Python
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# Design Science License
**Design Science License** (**DSL**) is a copyleft license for any type of free content such as text, images, music. Unlike other open source licenses, the DSL was intended to be used on any type of copyrightable work, including documentation and source code. It was the first "generalized copyleft" license. The DSL was written by Michael Stutz.
The DSL came out in the 1990s, before the formation of the Creative Commons. Once the Creative Commons arrived, Stutz considered the DSL experiment \"over\" and no longer recommended its use
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# Dispute resolution
**Dispute resolution** or **dispute settlement** is the process of resolving disputes between parties. The term *dispute resolution* is *conflict resolution* through legal means.
Prominent venues for dispute settlement in international law include the International Court of Justice (formerly the Permanent Court of International Justice); the United Nations Human Rights Committee (which operates under the ICCPR) and European Court of Human Rights; the Panels and Appellate Body of the World Trade Organization; and the International Tribunal for the Law of the Sea. Half of all international agreements include a dispute settlement mechanism.
States are also known to form their own arbitration tribunals to settle disputes. Prominent private international courts, which adjudicate disputes between commercial private entities, include the International Court of Arbitration (of the International Chamber of Commerce) and the London Court of International Arbitration.
## Methods
Methods of dispute resolution include:
- lawsuits (litigation) (legislative)
- arbitration
- collaborative law
- mediation
- conciliation
- negotiation
- facilitation
- avoidance
One could theoretically include violence or even war as part of this spectrum, but dispute resolution practitioners do not usually do so; violence rarely ends disputes effectively, and indeed, often only escalates them. Also, violence rarely causes the parties involved in the dispute to no longer disagree on the issue that caused the violence. For example, a country successfully winning a war to annex part of another country\'s territory does not cause the former waring nations to no longer seriously disagree to whom the territory rightly belongs to and tensions may still remain high between the two nations.
Dispute resolution processes fall into two major types:
1. Adjudicative processes, such as litigation or arbitration, in which a judge, jury or arbitrator determines the outcome.
2. Consensual processes, such as collaborative law, mediation, conciliation, or negotiation, in which the parties attempt to reach agreement.
Not all disputes, even those in which skilled intervention occurs, end in resolution. Such intractable disputes form a special area in dispute resolution studies.
Dispute resolution is an important requirement in international trade, including negotiation, mediation, arbitration and litigation.`{{Full citation needed|date=December 2016}}`{=mediawiki}
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# Dispute resolution
## Legal dispute resolution {#legal_dispute_resolution}
The legal system provides resolutions for many different types of disputes. Some disputants will not reach agreement through a collaborative process. Some disputes need the coercive power of the state to enforce a resolution. Perhaps more importantly, many people want a professional advocate when they become involved in a dispute, particularly if the dispute involves perceived legal rights, legal wrongdoing, or threat of legal action against them.
The most common form of judicial dispute resolution is litigation. Litigation is initiated when one party files suit against another. In the United States, litigation is facilitated by the government within federal, state, and municipal courts. While litigation is often used to resolve disputes, it is strictly speaking a form of conflict adjudication and not a form of conflict resolution per se. This is because litigation only determines the legal rights and obligations of parties involved in a dispute and does not necessarily solve the disagreement between the parties involved in the dispute. For example, supreme court cases can rule on whether US states have the constitutional right to criminalize abortion but will not cause the parties involved in the case to no longer disagree on whether states do indeed have the constitutional authority to restrict access to abortion as one of the parties may disagree with the supreme courts reasoning and still disagree with the party that the supreme court sided with. Litigation proceedings are very formal and are governed by rules, such as rules of evidence and procedure, which are established by the legislature. Outcomes are decided by an impartial judge and/or jury, based on the factual questions of the case and the application law. The verdict of the court is binding, not advisory; however, both parties have the right to appeal the judgment to a higher court. Judicial dispute resolution is typically adversarial in nature, for example, involving antagonistic parties or opposing interests seeking an outcome most favorable to their position.
Due to the antagonistic nature of litigation, collaborators frequently opt for solving disputes privately.
Retired judges or private lawyers often become arbitrators or mediators; however, trained and qualified non-legal dispute resolution specialists form a growing body within the field of alternative dispute resolution (ADR). In the United States, many states now have mediation or other ADR programs annexed to the courts, to facilitate settlement of lawsuits.
## Extrajudicial dispute resolution {#extrajudicial_dispute_resolution}
Some use the term *dispute resolution* to refer only to alternative dispute resolution (ADR), that is, extrajudicial processes such as arbitration, collaborative law, and mediation used to resolve conflict and potential conflict between and among individuals, business entities, governmental agencies, and (in the public international law context) states. ADR generally depends on agreement by the parties to use ADR processes, either before or after a dispute has arisen. ADR has experienced steadily increasing acceptance and utilization because of a perception of greater flexibility, costs below those of traditional litigation, and speedy resolution of disputes, among other perceived advantages. However, some have criticized these methods as taking away the right to seek redress of grievances in the courts, suggesting that extrajudicial dispute resolution may not offer the fairest way for parties not in an equal bargaining relationship, for example in a dispute between a consumer and a large corporation. In addition, in some circumstances, arbitration and other ADR processes may become as expensive as litigation or more so
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# Catan: Cities & Knights
***Catan: Cities & Knights*** (*Städte und Ritter*), formerly *The Cities and Knights of Catan*, is an expansion to the board game *The Settlers of Catan* for three to four players (five to six player play is also possible with the *Settlers* and *Cities & Knights* five to six player extensions; two-player play is possible with the *Traders & Barbarians* expansion). It contains features taken from *The Settlers of Catan*, with emphasis on city development and the use of knights, which are used as a method of attacking other players as well as helping opponents defend Catan against a common foe. *Cities & Knights* can also be combined with the *Catan: Seafarers* expansion or with *Catan: Traders & Barbarians* scenarios (again, five to six player play only possible with the applicable five to six player extension(s)).
## Differences from *The Settlers of Catan* {#differences_from_the_settlers_of_catan}
Because of the new rules introduced in *Cities & Knights*, the game is played to 13 victory points, as opposed to 10 as in the base game *The Settlers of Catan*.
The following cards are not used in Cities & Knights:
- the Development Cards---which have been replaced by Progress Cards.
- the Building Cost Cards---the information on these cards is provided by the City Upgrade Calendar.
- the \"Largest Army\" Card---having a large army is still an advantage, but does not earn victory points so directly as in the regular version of *The Settlers of Catan*. Instead of soldier cards, one is now able to purchase the eponymous knights.
## Commodities
One of the main additions to the game is commodities, which are a type of secondary resource produced only by cities. Like resources, commodities are associated with a type of terrain, can be stolen by the robber (with *Seafarers*, also the pirate), count against the resource hand limit, and may not be collected if the robber is on the terrain. Resources may be traded for commodities, and commodities may be traded for resources. Commodities can then be used to build city improvements (provided the player has a city), which provide additional benefits.
The commodities are paper (which comes from forest terrain), coin (from mountain terrain), and cloth (from pasture terrain).
When combining *Cities & Knights* with *Barbarian Attack*, the written rules are ambiguous with regards to whether commodities are collected along with normal resources when collecting from a Gold River tile, as well as whether or not commodities can be collected directly from Gold River tiles. However, online rules state that \"Gold can only buy you resources, not commodities.\"
A city on grain or brick gives two of each, as in the original *Settlers*. A city on wool, ore, or wood, produces one corresponding resource as well as one corresponding commodity (cloth, coin, or paper). Grain and brick, however, are used for new purchasing options: grain activates knights, and brick can be used to build city walls.
In total there are 36 commodity cards: 12 paper (from forest), 12 cloth (from pasture), and 12 coin (from mountains).
## City improvements {#city_improvements}
A player with a city may use commodities to build city improvements, which allow several advantages. There are city improvements in five levels, and in three different categories. Each category of improvements requires a different commodity and higher levels require more cards of that commodity. At the third level, players earn a special ability, depending on the type of improvement.
The first player with an improvement at the fourth level can claim any of their cities as a metropolis, worth four victory points instead of two for that city. Each type of improvement has only one associated metropolis, and no city can be a metropolis of two different types (because of this, a player without a non-metropolis city may not build improvements beyond the third level). If a player is the first to build an improvement to the final level (out-building the current holder of the metropolis), they take the metropolis from its current holder.
## Knights
The other significant concept in *Cities & Knights* is the concept of knights, which replace the concept of soldiers and the largest army. Knights are units that require continuous maintenance through their activation mechanism, but have a wide variety of functions. Knights can be promoted through three ranks, although promotion to the final rank is a special ability granted by the city improvement the Fortress.
Knights are placed on the board in a similar manner to settlements, and can be used to block opposing roads, active or not. However, knights must be activated in order to perform other functions, which immediately deactivate the knight. Knights cannot perform actions on the same turn they are activated, but can be reactivated on the same turn as performing an action. These actions include:
- Moving along a road (with *Seafarers*, a line of ships)
- Displacing opposing knights of a lower rank, forcing the lower ranked knight to retreat
- Dispelling the robber (with *Seafarers*, also the pirate) if it is stationed nearby
If a knight is promoted or forced to retreat, its active status does not change.
The standard *Cities & Knights* game comes with 24 knights, 6 of each color. The 5/6 player extension adds a further 12 knights, 6 each of two new colors.
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# Catan: Cities & Knights
## Barbarian attacks {#barbarian_attacks}
*Cities & Knights* introduces a third die, known as the event die, which serves two functions. The first applies to the concept of barbarians, a periodic foe that all players must work together to defend against. Three of the sides of the event die have a picture of a ship on them. The other three sides have a symbol of a city gate, allowing players who have sufficiently built up a city to obtain progress cards (see below).
The barbarians are represented by a ship positioned on a track representing the distance between the ship and Catan (i.e. the board). Each time the event die shows a black ship, the barbarian ship takes one step closer to Catan. When the barbarians arrive at Catan, a special phase is immediately performed before all other actions (including collecting resources). In this special phase, the barbarians\' attack strength, corresponding to the combined number of cities and metropolises held by all players, is compared to Catan\'s defense strength, corresponding to the combined levels (i.e. 1 point for each basic, 2 for each strong, and 3 for each mighty) of all activated knights in play.
If the barbarians are successful in their attack (if they have a strength greater than Catan), then the players must pay the consequence. The player(s) who had the least defense will be attacked, and will have one city reduced to a settlement. If they only have settlements, or metropolises, then they are immune to barbarians and do not count as the player contributing the least defense.
Should Catan prevail, the player who contributes the most to Catan\'s defense receives a special *Defender of Catan* card, worth a victory point. Regardless of the outcome, all knights are immediately deactivated, and the barbarian ship returns to its starting point on the track. In the event of a tie among the greatest contributors of knights, none of the tied players earn a Defender of Catan card. Instead, each of the tied players draw a progress card (explained below) of the type of their choosing. There are 6 Defender of Catan cards.
As the likelihood of having the barbarian move closer to Catan is very high, a variant in common usage is that the robber (and with *Seafarers*, the pirate) does not move until the first barbarian attack, nor can a knight move the robber before that point.
**Examples where cities are lost:**
1. Player A has 3 cities and 1 active strong knight. Player B has 1 city and 2 active basic knights. Player C has 2 cities and 1 active basic knight. When the barbarians attack, player C will lose one of their cities, because the attack strength (6 cities) is greater than all knights combined (5 knights).
2. Player A has 3 cities and 2 active basic knights. Player B has 1 city, which is a metropolis, and no active knights. Player C has 2 cities and 1 active mighty knight. Player B\'s city is a metropolis, and metropolises cannot be destroyed by the barbarians, so Player A loses a city because they have the next fewest active knights.
3. Player A has 2 cities and 2 active basic knights. Player B has 3 cities and 1 active strong knight. Player C has 2 cities and 2 active basic knights. All players will lose a city, because they all tie in the number of knights activated, and the barbarian attack strength (7 cities) is greater than number of active knights (6 knights).
4. Player A has 3 cities and 1 active mighty knight and 1 active basic knight. Player B has 2 cities, which both are metropolises, and 1 active basic knight. Player C has 1 city, which is a metropolis, and no active knights. First in line to lose a city is player C, but because their city is a metropolis we need to look at the person next in line. This would be player B, but the same applies for them: they have activated only 1 knight, but both of their cities are metropolises. This leaves player A to lose a city.
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# Catan: Cities & Knights
## Progress cards {#progress_cards}
The other significant outcome of the event die is Progress cards, which replace development cards. Because of the mechanics of progress cards explained below, one of the two white dice used in *Settlers* is replaced by a red die.
Progress cards are organized into three categories, corresponding to the three types of improvements. Yellow progress cards aid in commercial development, green progress cards aid in technological advancements, and blue progress cards allow for political moves. When a castle appears on the event die, progress cards of the corresponding type may be drawn depending on the value of the red die. Higher levels of city improvements increase the chance that progress cards will be drawn, with the highest level of city improvement allowing progress cards to be drawn regardless of the value on the red die.
Progress cards, unlike the development cards they replace, can be played on the turn that they are drawn, and more than one progress card can be played per turn. However, they can generally only be played after the dice are rolled. Progress cards granting victory points are an exception, being played immediately (without regards to whose turn it is), while the Alchemist progress card, which allows a player to select the roll of the white and red dice, necessitates the card being played instead of rolling the numerical dice. (The event die is still rolled as normal.)
Players are allowed to keep four progress cards (five in a five to six player game), and any additional ones must be discarded on the spot (unless the 5th card is a victory point, which is played immediately and the original progress cards remain). The only exception to this rule is when the player receives a 5th non-victory point progress card during their turn, in which case the player may choose to play any one of the five progress cards in hand, bringing the progress card count back down to four. While this clarification is not overtly stated in the Cities & Knights rule book, it is enforced in the online version of the game.
In total, there are 54 progress cards: 18 science, 18 politics, and 18 trade.
Out of all the available progress cards, progress cards containing victory points can be only earned in the science and politics categories.
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# Catan: Cities & Knights
## City walls {#city_walls}
City walls are a minor addition to *Cities & Knights* that increase the number of resource and commodity cards a player is allowed in their hand before having to discard on a roll of 7. However, they do not protect the player from the robber or barbarians. Only cities and metropolises may have walls, and each city or metropolis can only have one wall, up to three walls per player. Each wall that the player has deployed permits the player to hold two more cards before being required to discard on a roll of seven. This results in a maximum of 13 cards.
If the barbarians pillage your city, then the city wall is also destroyed and the wall is removed from the board.
The game comes with 12 city walls, 3 of each color.
## The Merchant {#the_merchant}
The merchant is another addition to *Cities & Knights*. Like the robber, the merchant is placed on a single land hex. Unlike the robber, the merchant has a beneficial effect.
The merchant can only be deployed through the use of a Merchant progress card (of which there are six), on a land hex near a city or a settlement. The player with the control of the merchant can trade the resource (not commodity) of that type at a two-to-one rate, as if the player had a control of a corresponding two-to-one harbor.
The player with the control of the merchant also earns a victory point. Both the victory point and the trade privilege are lost if another player takes control of the merchant.
## City Upgrade Calendar {#city_upgrade_calendar}
In place of *The Settlers of Catan* standard improvement cost card, *Cities & Knights* gives a calendar type flip-chart to each player, matching that player\'s color. The top of the chart has the standard costs from the *Settlers* game (for settlements, upgrade to city, and roads). It does not include the Development Card cost as those cards are not used in a *Cities & Knights* game. It does include the costs of hiring a knight, upgrading a knight\'s level or strength, and the cost to activate a knight. It also includes the cost of a ship, which are not used in a regular game of *Cities & Knights*, but presumably this is to cater for players who have combined *Cities & Knights* and *Seafarers*.
Those are only the rudimentary costs of the game however. The calendar also shows the costs of the next city improvement in each of the three categories --- as a city is improved in a category, that segment has its card flipped down calendar style to reveal the newly built improvement, any advantages gained by the improvement, and the updated cost of upgrading to the next level in that category. Each segment, as it is flipped down, also shows the updated dice pattern needed to earn the player a progress card in that category.
## Catan Legend of the Conquerors {#catan_legend_of_the_conquerors}
*Catan: Legend of the Conquerors* is a scenario released in 2017 for the expansion *Catan: Cities & Knights*. A blog post was made in connection with the release. The game adds swamp hexes to the board. The game also adds a cannon which can be combined with a knight to increase the strength of a knight by one, which makes the maximum possible strength 4, when applied to a mighty knight of strength 3 ($1+3=4$). To build a cannon, you pay 1 lumber and 1 ore for a foundry. When you combine a cannon and a knight you have a cannoneer. The game also adds a horse farm that you can build for one lumber and one grain. The horse farm gives you a horse that you can use to turn one of your knights into cavalry. A cavalry unit can move between road networks, even if there is no connection between them. The cannon and horse cannot be combined. The blog post writes, \"Some strategists may like the idea of equipping a knight with a horse and a cannon, thus making it some kind of overpowering "mounted cannoneer." However, you are not allowed to place both playing pieces adjacent to a knight. This being said, it\'s also hard to imagine a knight on a horse holding a cannon in his arms and firing it in all directions\..
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# Catan: Seafarers
***Catan: Seafarers***, or ***Seafarers of Catan*** in older editions, (*Die Seefahrer von Catan*) is an expansion of the board game *Catan* for three to four players (five-to-six-player play is also possible with both of the respective five-to-six-player extensions). The main feature of this expansion is the addition of ships, gold fields, and the pirate to the game, allowing play between multiple islands. The expansion also provides numerous scenarios, some of which have custom rules. The *Seafarers* rules and scenarios are also, for the most part, compatible with *Catan: Cities & Knights* and *Catan: Traders & Barbarians*.
The concepts introduced in *Seafarers* were part of designer Klaus Teuber\'s original design for *Settlers*.
## Ships
*Seafarers* introduces the concept of ships, which serve as roads over water or along the coast. Each ship costs one lumber and one wool to create (lumber for the hull and wool for the sails). A settlement must first be built before a player can switch from building roads to building ships, or vice versa. Thus, a chain of ships is always anchored at a settlement on the coast. A shipping line that is not anchored at both ends by different settlements can also move the last ship at the open end, although this can only be done once per turn and may not be done with any ships that were created on the same turn.
The \"Longest Road\" card is now renamed the \"Longest Trade Route\" since this is now calculated by counting the number of contiguous ships *plus* roads that a player has. A settlement or city is necessary between a road and a ship for the two to be considered continuous for the purposes of this card.
The Road Building card allows a player to build 2 roads, 2 ships, or one of each when used.
Along with the concept of ships, *Seafarers* also introduces the notion of the pirate, which acts as a waterborne robber which steals from nearby ships (similar to how the robber steals from nearby settlements). The pirate can also prevent ships from being built or moved nearby, but it does not interfere with harbors. The pirate does not prevent settlements from being built.
When a seven is rolled or a Knight card is played, the player may move either the robber OR the pirate.
## Gold Rivers {#gold_rivers}
*Seafarers* also introduces the \"Gold River\" or \"Gold Field\" terrain, which grants nearby players one resource of their choice for every settlement adjacent to a gold tile and 2 resources for every city. Since being able to choose any resource type allows more building power, gold rivers are often either marked with number token of only 2 or 3 dots and/or are far away from starting positions to offset this.
When combined with *Cities & Knights*, the rules state that you are not allowed to take commodities instead of resources if a city is nearby.
## Exploration
Some scenarios have extra rules encompassing the concept of exploration, which is done by having the hex tiles placed face down. Should a player build next to unexplored terrain, the terrain tile is turned face up, and the player is rewarded with a resource should the tile revealed be resource-producing. In other scenarios, the board is divided into islands, and if the player builds a settlement on an island other than the ones they begin on, the settlement is worth extra victory points.
The *Cities and Knights* manual recommends that players not use the *Cities & Knights* rules in scenarios where exploration is a factor.
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# Catan: Seafarers
## Scenarios
Unlike *The Settlers of Catan* and *Catan: Cities & Knights*, in which the only random element of setup is the placement of land tiles, number tokens, and harbors in an identically shaped playing area, *Catan: Seafarers* has a number of different scenarios or maps from which to choose. Each map uses a different selection of tiles laid out in a specific pattern, which may not use all of the tiles. Other attributes also set each map apart, for example, restrictions on the placement of initial settlements, whether tiles are distributed randomly, the number of victory points needed to win, and special victory point awards, usually for building on islands across the sea.
*Seafarers* provides scenarios for three or four players (the older fourth edition used the same maps for three- and four-player versions of the scenarios), while the extension provides scenarios for six players (the older third edition also included separate maps for five- and six-player scenarios). The scenarios between the older editions of *Seafarers* and the newest are generally incompatible, knowing the different frames included with the game. (In particular, older editions of *Settlers* did not come with a frame for their board; a separate add-on was made available for players of the older-edition *Settlers* games, containing the newer edition frames, so as to make them compatible with the newer edition of *Seafarers*; the older edition of *Seafarers* included a square frame, and while both older and newer editions of the frames have the same width across, the newer editions are not square-shaped, and are longer down the middle of the board compared to the sides.)
### Heading to New Shores {#heading_to_new_shores}
*Heading to New Shores* (*New Shores* in older editions) is the scenario resembling Teuber\'s original design for the game. The game board consists of the main *Settlers* island as well as a few smaller islands, which award a special victory point to each player for their first settlements on them. This scenario is meant for players new to *Seafarers*, with elements of *Seafarers* incorporated into the more familiar main board.
### The Four Islands {#the_four_islands}
*The Four Islands* is the first scenario introduced where new mechanics introduced to *Seafarers* is brought into the forefront. In this scenario, the map is split up into four islands of roughly equal size and resource distribution. (The six-player version found in the extension has the map split into six islands; the scenario is titled *The Six Islands*, but is played identically. Older editions of the extension had a five-player version with five islands, called *The Five Islands*.) Players may claim up to two of the islands as their home islands, and settling on any of the other islands awards a special victory point.
### The Fog Island {#the_fog_island}
*The Fog Island* (*Oceans* in older editions) is the first scenario where exploration is used. The board starts off with a portion of the map left blank: when players expand into the blank region, terrain hexes are drawn at random from a supply and placed in the empty space, and, if a land hex is \"discovered\", a number token may be assigned. As a reward for discovering land, the player making the discovery is rewarded with a bonus resource card corresponding to the type of land hex discovered.
### Through the Desert {#through_the_desert}
*Through the Desert* (*Into the Desert* in older edition) is similar to *The Four Islands*, but consists of a large continent and smaller outlying islands. On the large island, there exists a \"wall of deserts\" that separates the island into a large main area and separate smaller strips of land. As the name of the scenario implies, expanding through the desert into these smaller strips of land, or by sea to the outlying islands, award bonus victory points.
### The Forgotten Tribe {#the_forgotten_tribe}
*The Forgotten Tribe*, originally titled *Friendly Neighbors*, was a downloadable scenario (but only in the German language) which was incorporated into newer editions of *Seafarers*.
The map consists of a main island and smaller outlying islands, where the namesake forgotten tribe resides. Players may not expand into the outlying islands, but by building ships so that they border the outlying islands, players may be awarded with victory points, development cards, or harbors that players may place on the coast of the main island at a later time.
### Cloth for Catan {#cloth_for_catan}
Introduced in the newer editions, *Cloth for Catan* continues the adventures with the Forgotten Tribe. The scenario was previously available for older editions as a downloadable scenario (but only in German), titled *Coffee for Catan*.
Players begin with settlements on the outside of the map, but may build ships to reach the Forgotten Tribe\'s islands, which are in the center. By connecting to the Forgotten Tribe\'s settlements (represented by number tokens), players may earn cloth tokens when the number token for the Forgotten Tribe\'s villages are rolled. Cloth tokens, in turn, are worth one victory point for each pair obtained.
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# Catan: Seafarers
## Scenarios
### The Pirate Island {#the_pirate_island}
*The Pirate Island*, introduced in newer editions, is the first scenario which changes the mechanics of new gameplay elements introduced in *Seafarers*. *The Pirate Island* had previously been available as a downloadable scenario (but only in German) suitable for the older editions.
In this scenario, players begin with a pre-placed settlement on a main island. Ships may only be built in one single line, which must pass through a fixed waypoint (different for each player) en route to a pirate fortress (each player has their own pirate fortress). Once ships connect to the pirate fortress, they may attempt to attack the pirate fortress once per turn. Ships may be lost if the attack is unsuccessful, but after three successful attacks, the pirate fortress is converted into a settlement. Players must convert their pirate fortresses and have 10 victory points before being able to claim victory.
Furthermore, the pirate mechanics have also changed: the pirate moves through the middle of the map in a fixed path every turn, and attacks the owner of any nearby settlements. Players win resources if they are able to fend off the pirate attack (which depends on the number rolled by the dice, as well as the number of warships in the defending player\'s possession; warships are created from using Knight cards on existing ships), but lose resources if they are unsuccessful. Maritime expansion is only permitted by building a settlement at the waypoint, however, this increases the chances of a pirate attack.
### The Wonders of Catan {#the_wonders_of_catan}
*The Wonders of Catan* was a downloadable scenario for older editions of *Seafarers* in both German and English, and was incorporated into *Seafarers* in newer editions.
In this scenario, there are a number of \"wonders\", each with a large cost of building as well as a prerequisite. If a player meets the prerequisite for a wonder, they may claim the wonder for themselves. A player may only claim one wonder, and each wonder may only be claimed by one player. Wonders must be built in four parts, and each wonder has a different build cost. The winner is the first player to complete their wonder, or the first player to have 10 victory points and have more parts of their wonder complete than any other player.
### The Great Crossing {#the_great_crossing}
*The Great Crossing* was a scenario in the older editions of *Seafarers*, which has been dropped in newer editions. The map is divided into two islands, Catan and Transcatania. Players begin with both settlements on one of the islands, and must build ships connecting settlements between the two islands. Players earn victory points for connecting their settlements with settlements (not necessarily theirs) from the opposite island using ships, or to another player\'s shipping lines which connect two settlements together.
### Greater Catan {#greater_catan}
*Greater Catan* was a scenario included in the older editions of *Seafarers* but is not included in newer editions. Due to the sheer amounts of equipment needed, two copies of *Settlers* and *Seafarers* are required to set up this scenario. The map consists of a standard *Settlers* island, along with a smaller chain of outlying islands. Only the main island initially has number tokens: number tokens are assigned to the outlying islands as they are expanded. However, the supply of number tokens is smaller than the number of hexes in the scenario: when the number tokens run out and players expand into a new part of the outlying islands, number tokens are moved from the main island to the outlying islands.
Hexes on the main island for which there are no number tokens do not produce resources, but number tokens are moved in such a way so as to avoid rendering a city unproductive; furthermore, whenever possible number tokens must be reassigned from hexes bordering a player\'s own settlements and cities, so as to prevent harming another player\'s economy without harming a player\'s own economy at the same time.
### New World {#new_world}
*New World* is a scenario that blankets all other scenarios that may be created from the parts of *Settlers* and *Seafarers*. This scenario uses an entirely random map, and players are encouraged to try and create a tile layout that plays well. The only difference between versions in *Seafarers*, the extension, and the older editions therein is the size of the frames.
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# Catan: Seafarers
## Reception
*The Seafarers of Catan* was reviewed in the online second volume of *Pyramid*
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# Dynamical system
In mathematics, a **dynamical system** is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The *evolution rule* of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
The study of dynamical systems is the focus of *dynamical systems theory*, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.
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# Dynamical system
## Overview
The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times---each advancing time a small step. The iteration procedure is referred to as *solving the system* or *integrating the system*. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a *trajectory* or *orbit*.
Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.
For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:
- The systems studied may only be known approximately---the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
- The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
- The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
- The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.
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# Dynamical system
## History
Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, \"New Methods of Celestial Mechanics\" (1892--1899) and \"Lectures on Celestial Mechanics\" (1905--1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.
In 1913, George David Birkhoff proved Poincaré\'s \"Last Geometric Theorem\", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his *[Dynamical Systems](https://archive.org/details/dynamicalsystems00birk/)*. Birkhoff\'s most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.
Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.
Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky\'s theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.
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# Dynamical system
## Formal definition {#formal_definition}
In the most general sense, a **dynamical system** is a tuple (*T*, *X*, Φ) where *T* is a monoid, written additively, *X* is a non-empty set and Φ is a function
$$\Phi: U \subseteq (T \times X) \to X$$ with
$$\mathrm{proj}_{2}(U) = X$$ (where $\mathrm{proj}_{2}$ is the 2nd projection map) and for any *x* in *X*:
$$\Phi(0,x) = x$$
$$\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),$$ for $\, t_1,\, t_2 + t_1 \in I(x)$ and $\ t_2 \in I(\Phi(t_1, x))$, where we have defined the set $I(x) := \{ t \in T : (t,x) \in U \}$ for any *x* in *X*.
In particular, in the case that $U = T \times X$ we have for every *x* in *X* that $I(x) = T$ and thus that Φ defines a monoid action of *T* on *X*.
The function Φ(*t*,*x*) is called the **evolution function** of the dynamical system: it associates to every point *x* in the set *X* a unique image, depending on the variable *t*, called the **evolution parameter**. *X* is called **phase space** or **state space**, while the variable *x* represents an **initial state** of the system.
We often write
$$\Phi_x(t) \equiv \Phi(t,x)$$
$$\Phi^t(x) \equiv \Phi(t,x)$$ if we take one of the variables as constant. The function
$$\Phi_x:I(x) \to X$$ is called the **flow** through *x* and its graph is called the **trajectory** through *x*. The set
$$\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}$$ is called the **orbit** through *x*. The orbit through *x* is the image of the flow through *x*. A subset *S* of the state space *X* is called Φ-**invariant** if for all *x* in *S* and all *t* in *T*
$$\Phi(t,x) \in S.$$ Thus, in particular, if *S* is Φ-**invariant**, $I(x) = T$ for all *x* in *S*. That is, the flow through *x* must be defined for all time for every element of *S*.
More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor.
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# Dynamical system
## Formal definition {#formal_definition}
### Geometrical definition {#geometrical_definition}
In the geometrical definition, a dynamical system is the tuple $\langle \mathcal{T}, \mathcal{M}, f\rangle$. $\mathcal{T}$ is the domain for time -- there are many choices, usually the reals or the integers, possibly restricted to be non-negative. $\mathcal{M}$ is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. *f* is an evolution rule *t* → *f*^ *t*^ (with $t\in\mathcal{T}$) such that *f^ t^* is a diffeomorphism of the manifold to itself. So, f is a \"smooth\" mapping of the time-domain $\mathcal{T}$ into the space of diffeomorphisms of the manifold to itself. In other terms, *f*(*t*) is a diffeomorphism, for every time *t* in the domain $\mathcal{T}$ .
#### Real dynamical system {#real_dynamical_system}
A *real dynamical system*, *real-time dynamical system*, *continuous time dynamical system*, or *flow* is a tuple (*T*, *M*, Φ) with *T* an open interval in the real numbers **R**, *M* a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a *differentiable dynamical system*. If the manifold *M* is locally diffeomorphic to **R**^*n*^, the dynamical system is *finite-dimensional*; if not, the dynamical system is *infinite-dimensional*. This does not assume a symplectic structure. When *T* is taken to be the reals, the dynamical system is called *global* or a *flow*; and if *T* is restricted to the non-negative reals, then the dynamical system is a *semi-flow*.
#### Discrete dynamical system {#discrete_dynamical_system}
A *discrete dynamical system*, *discrete-time dynamical system* is a tuple (*T*, *M*, Φ), where *M* is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When *T* is taken to be the integers, it is a *cascade* or a *map*. If *T* is restricted to the non-negative integers we call the system a *semi-cascade*.
#### Cellular automaton {#cellular_automaton}
A *cellular automaton* is a tuple (*T*, *M*, Φ), with *T* a lattice such as the integers or a higher-dimensional integer grid, *M* is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in *M* represents the \"space\" lattice, while the one in *T* represents the \"time\" lattice.
#### Multidimensional generalization {#multidimensional_generalization}
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.
#### Compactification of a dynamical system {#compactification_of_a_dynamical_system}
Given a global dynamical system (**R**, *X*, Φ) on a locally compact and Hausdorff topological space *X*, it is often useful to study the continuous extension Φ\* of Φ to the one-point compactification *X\** of *X*. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (**R**, *X\**, Φ\*).
In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.
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# Dynamical system
## Formal definition {#formal_definition}
### Measure theoretical definition {#measure_theoretical_definition}
A dynamical system may be defined formally as a measure-preserving transformation of a measure space, the triplet (*T*, (*X*, Σ, *μ*), Φ). Here, *T* is a monoid (usually the non-negative integers), *X* is a set, and (*X*, Σ, *μ*) is a probability space, meaning that Σ is a sigma-algebra on *X* and μ is a finite measure on (*X*, Σ). A map Φ: *X* → *X* is said to be Σ-measurable if and only if, for every σ in Σ, one has $\Phi^{-1}\sigma \in \Sigma$. A map Φ is said to **preserve the measure** if and only if, for every *σ* in Σ, one has $\mu(\Phi^{-1}\sigma ) = \mu(\sigma)$. Combining the above, a map Φ is said to be a **measure-preserving transformation of *X***, if it is a map from *X* to itself, it is Σ-measurable, and is measure-preserving. The triplet (*T*, (*X*, Σ, *μ*), Φ), for such a Φ, is then defined to be a **dynamical system**.
The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi$ for every integer *n* are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.
#### Relation to geometric definition {#relation_to_geometric_definition}
The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov--Bogolyubov theorem) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution.
For hyperbolic dynamical systems, the Sinai--Ruelle--Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
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# Dynamical system
## Construction of dynamical systems {#construction_of_dynamical_systems}
The concept of *evolution in time* is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as the following:
$$\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})$$
$$\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0$$
where
- $\dot{\boldsymbol{x}}$ represents the velocity of the material point **x**
- *M* is a finite dimensional manifold
- **v**: *T* × *M* → *TM* is a vector field in **R**^*n*^ or **C**^*n*^ and represents the change of velocity induced by the known forces acting on the given material point in the phase space *M*. The change is not a vector in the phase space *M*, but is instead in the tangent space *TM*.
There is no need for higher order derivatives in the equation, nor for the parameter *t* in *v*(*t*,*x*), because these can be eliminated by considering systems of higher dimensions.
Depending on the properties of this vector field, the mechanical system is called
- **autonomous**, when **v**(*t*, **x**) = **v**(**x**)
- **homogeneous** when **v**(*t*, **0**) = 0 for all *t*
The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above
$$\boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0)$$
The dynamical system is then (*T*, *M*, Φ).
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
$$\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0$$
where $\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}$ is a functional from the set of evolution functions to the field of the complex numbers.
This equation is useful when modeling mechanical systems with complicated constraints.
Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds---those that are locally Banach spaces---in which case the differential equations are partial differential equations.
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# Dynamical system
## Examples
- Arnold\'s cat map
- Baker\'s map is an example of a chaotic piecewise linear map
- Billiards and outer billiards
- Bouncing ball dynamics
- Circle map
- Complex quadratic polynomial
- Double pendulum
- Dyadic transformation
- Dynamical system simulation
- Hénon map
- Irrational rotation
- Kaplan--Yorke map
- List of chaotic maps
- Lorenz system
- Quadratic map simulation system
- Rössler map
- Swinging Atwood\'s machine
- Tent map
## Linear dynamical systems {#linear_dynamical_systems}
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the *N*-dimensional Euclidean space, so any point in phase space can be represented by a vector with *N* numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if *u*(*t*) and *w*(*t*) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will *u*(*t*) + *w*(*t*).
### Flows
For a flow, the vector field v(*x*) is an affine function of the position in the phase space, that is,
$$\dot{x} = v(x) = A x + b,$$ with *A* a matrix, *b* a vector of numbers and *x* the position vector. The solution to this system can be found by using the superposition principle (linearity). The case *b* ≠ 0 with *A* = 0 is just a straight line in the direction of *b*:
: $\Phi^t(x_1) = x_1 + b t.$
When *b* is zero and *A* ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if *x*~0~ = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point *x*~0~,
: $\Phi^t(x_0) = e^{t A} x_0.$
When *b* = 0, the eigenvalues of *A* determine the structure of the phase space. From the eigenvalues and the eigenvectors of *A* it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
The distance between two different initial conditions in the case *A* ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.
### Maps
A discrete-time, affine dynamical system has the form of a matrix difference equation:
: $x_{n+1} = A x_n + b,$
with *A* a matrix and *b* a vector. As in the continuous case, the change of coordinates *x* → *x* + (1 − *A*)^ --1^*b* removes the term *b* from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system *A*^ *n*^*x*~0~. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.
As in the continuous case, the eigenvalues and eigenvectors of *A* determine the structure of phase space. For example, if *u*~1~ is an eigenvector of *A*, with a real eigenvalue smaller than one, then the straight lines given by the points along *α* *u*~1~, with *α* ∈ **R**, is an invariant curve of the map. Points in this straight line run into the fixed point.
There are also many other discrete dynamical systems.
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# Dynamical system
## Local dynamics {#local_dynamics}
The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a *singular point* of the vector field (a point where *v*(*x*) = 0) will remain a singular point under smooth transformations; a *periodic orbit* is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.
### Rectification
A flow in most small patches of the phase space can be made very simple. If *y* is a point where the vector field *v*(*y*) ≠ 0, then there is a change of coordinates for a region around *y* where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.
The *rectification theorem* says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space *M* the dynamical system is *integrable*. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where *v*(*x*) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.
### Near periodic orbits {#near_periodic_orbits}
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point *x*~0~ in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to *v*(*x*~0~). These points are a Poincaré section *S*(*γ*, *x*~0~), of the orbit. The flow now defines a map, the Poincaré map *F* : *S* → *S*, for points starting in *S* and returning to *S*. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes *x*~0~.
The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map *F*. By a translation, the point can be assumed to be at *x* = 0. The Taylor series of the map is *F*(*x*) = *J* · *x* + O(*x*^2^), so a change of coordinates *h* can only be expected to simplify *F* to its linear part
: $h^{-1} \circ F \circ h(x) = J \cdot x.$
This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If *λ*~1~, \..., *λ*~*ν*~ are the eigenvalues of *J* they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form *λ*~*i*~ -- Σ (multiples of other eigenvalues) occurs in the denominator of the terms for the function *h*, the non-resonant condition is also known as the small divisor problem.
### Conjugation results {#conjugation_results}
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of *J* and the degree of smoothness required from *h*. As *J* does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of *J* are not in the unit circle, the dynamics near the fixed point *x*~0~ of *F* is called *hyperbolic* and when the eigenvalues are on the unit circle and complex, the dynamics is called *elliptic*.
In the hyperbolic case, the Hartman--Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map *J* · *x*. The hyperbolic case is also *structurally stable*. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of *J* in the complex plane, implying that the map is still hyperbolic.
The Kolmogorov--Arnold--Moser (KAM) theorem gives the behavior near an elliptic point.
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# Dynamical system
## Bifurcation theory {#bifurcation_theory}
When the evolution map Φ^*t*^ (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value *μ*~0~ is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.
Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter *μ*. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.
The bifurcations of a hyperbolic fixed point *x*~0~ of a system family *F~μ~* can be characterized by the eigenvalues of the first derivative of the system *DF*~*μ*~(*x*~0~) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of *DF~μ~* on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.
Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle--Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.
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# Dynamical system
## Ergodic systems {#ergodic_systems}
In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton\'s laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset *A* into the points Φ^ *t*^(*A*) and invariance of the phase space means that
: $\mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ).$
In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.
In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.
For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let *F* be a phase space volume-preserving map and *A* a subset of the phase space. Then almost every point of *A* returns to *A* infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann\'s derivation of the increase in entropy in a dynamical system of colliding atoms.
One of the questions raised by Boltzmann\'s work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region *A* is vol(*A*)/vol(Ω).
The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable *a* is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ^ t^. This introduces an operator *U*^ *t*^, the transfer operator,
: $(U^t a)(x) = a(\Phi^{-t}(x)).$
By studying the spectral properties of the linear operator *U* it becomes possible to classify the ergodic properties of Φ^ *t*^. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ^ *t*^ gets mapped into an infinite-dimensional linear problem involving *U*.
The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−β*H*). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.
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# Dynamical system
## Nonlinear dynamical systems and chaos {#nonlinear_dynamical_systems_and_chaos}
Simple nonlinear dynamical systems, including piecewise linear systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called *chaos*. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent spaces perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the *stable manifold*) and another of the points that diverge from the orbit (the *unstable manifold*).
This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like \"Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?\" or \"Does the long-term behavior of the system depend on its initial condition?\"
The chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex---even chaotic---behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The Pomeau--Manneville scenario of the logistic map and the Fermi--Pasta--Ulam--Tsingou problem arose with just second-degree polynomials; the horseshoe map is piecewise linear.
### Solutions of finite duration {#solutions_of_finite_duration}
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of \"backwards-in-time unpredictability\" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for Lipschitz continuous differential equations according to the proof of the Picard-Lindelof theorem. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.
As example, the equation:
$$y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1$$ Admits the finite duration solution:
$$y(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$$ that is zero for $t \geq 2$ and is not Lipschitz continuous at its ending time $t = 2
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# Dhimmi
***`{{Transliteration|ar|ALA|Dhimmī}}`{=mediawiki}*** (*ذمي* *`{{Transliteration|ar|DIN|ḏimmī}}`{=mediawiki}*, `{{IPA|ar|ˈðimmiː|IPA}}`{=mediawiki}, collectively *أهل الذمة* *`{{Transliteration|ar|DIN|ʾahl aḏ-ḏimmah}}`{=mediawiki}/`{{Transliteration|ar|ALA|dhimmah}}`{=mediawiki}* \"the people of the covenant\") or **`{{Transliteration|ar|DIN|muʿāhid}}`{=mediawiki}** (*معاهد*) is a historical term for non-Muslims living in an Islamic state with legal protection. The word literally means \"protected person\", referring to the state\'s obligation under *sharia* to protect the individual\'s life, property, as well as freedom of religion, in exchange for loyalty to the state and payment of the *jizya* tax, in contrast to the *zakat*, or obligatory alms, paid by the Muslim subjects. *Dhimmi* were exempt from military service and other duties assigned specifically to Muslims if they paid the poll tax (*jizya*) but were otherwise equal under the laws of property, contract, and obligation. Dhimmis were subject to specific restrictions as well, which were codified in agreements like the *Pact of ʿUmar*. These included prohibitions on building new places of worship, repairing existing ones in areas where Muslims lived, teaching children the Qurʾān, and preventing relatives from converting to Islam. They were also required to wear distinctive clothing, refrain from carrying weapons, and avoid riding on saddles.
Historically, dhimmi status was originally applied to Jews, Christians, and Sabians, who are considered \"People of the Book\" in Islamic theology. Later, this status was also applied to Zoroastrians, Sikhs, Hindus, Jains, and Buddhists.
Jews, Christians and others were required to pay the *jizyah*, and forced conversions were forbidden.
During the rule of al-Mutawakkil, the tenth Abbasid Caliph, numerous restrictions reinforced the second-class citizen status of dhimmīs and forced their communities into ghettos. For instance, they were required to distinguish themselves from their Muslim neighbors by their dress. They were not permitted to build new churches or synagogues or repair old churches without Muslim consent according to the Pact of Umar.
Under *Sharia*, the *dhimmi* communities were usually governed by their own laws in place of some of the laws applicable to the Muslim community. For example, the Jewish community of Medina was allowed to have its own Halakhic courts, and the Ottoman millet system allowed its various dhimmi communities to rule themselves under separate legal courts. These courts did not cover cases that involved religious groups outside of their own communities, or capital offences. *Dhimmi* communities were also allowed to engage in certain practices that were usually forbidden for the Muslim community, such as the consumption of alcohol and pork.
Some Muslims reject the *dhimma* system by arguing that it is a system which is inappropriate in the age of nation-states and democracies. There is a range of opinions among 20th-century and contemporary Islamic theologians about whether the notion of *dhimma* is appropriate for modern times, and, if so, what form it should take in an Islamic state.
There are differences among the Islamic Madhhabs regarding which non-Muslims can pay jizya and have dhimmi status. The Hanafi and Maliki Madhabs generally allow non-Muslims to have dhimmi status. In contrast, the Shafi\'i and Hanbali Madhabs only allow Christians, Unitarians, Jews, Sabeans and Zoroastrians to have dhimmi status, and they maintain that all other non-Muslims must either convert to Islam or be fought.`{{dubious|date=November 2024}}`{=mediawiki}
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# Dhimmi
## The \"Dhimma contract\" {#the_dhimma_contract}
Based on Quranic verses and Islamic traditions, *sharia* law distinguishes between Muslims, followers of other Abrahamic religions, and Pagans or people belonging to other polytheistic religions. As monotheists, Jews and Christians have traditionally been considered \"People of the Book\", and afforded a special legal status known as *dhimmi* derived from a theoretical contract---\"dhimma\" or \"residence in return for taxes\". Islamic legal systems based on *sharia* law incorporated the religious laws and courts of Christians, Jews, and Hindus, as seen in the early caliphate, al-Andalus, Indian subcontinent, and the Ottoman Millet system.`{{page needed|date=January 2016}}`{=mediawiki}
In Yemenite Jewish sources, a treaty was drafted between Muhammad and his Jewish subjects, known as *kitāb ḏimmat al-nabi*, written in the 17th year of the Hijra (638 CE), which gave express liberty to the Jews living in Arabia to observe the Sabbath and to grow-out their side-locks, but required them to pay the *jizya* (poll-tax) annually for their protection. Muslim governments in the Indus basin readily extended the *dhimmi* status to the Hindus and Buddhists of India. Eventually, the largest school of Islamic jurisprudence applied this term to all Non-Muslims living in Muslim lands outside the sacred area surrounding Mecca, Arabia.
In medieval Islamic societies, the *qadi* (Islamic judge) usually could not interfere in the matters of non-Muslims unless the parties voluntarily chose to be judged according to Islamic law, thus the *dhimmi* communities living in Islamic states usually had their own laws independent from the *sharia* law, as with the Jews who would have their own rabbinical courts. These courts did not cover cases that involved other religious groups, or capital offences or threats to public order. By the 18th century, however, *dhimmi* frequently attended the Ottoman Muslim courts, where cases were taken against them by Muslims, or they took cases against Muslims or other *dhimmi*. Oaths sworn by *dhimmi* in these courts were tailored to their beliefs. Non-Muslims were allowed to engage in certain practices (such as the consumption of alcohol and pork) that were usually forbidden by Islamic law, in point of fact, any Muslim who pours away their wine or forcibly appropriates it is liable to pay compensation. Some Islamic theologians held that Zoroastrian \"self-marriages\", considered incestuous under *sharia*, should also be tolerated. Ibn Qayyim Al-Jawziyya (1292--1350) opined that most scholars of the Hanbali school held that non-Muslims were entitled to such practices, as long as they were not presented to sharia courts and the religious minorities in question held them to be permissible. This ruling was based on the precedent that there were no records of the Islamic prophet Muhammad forbidding such self-marriages among Zoroastrians, despite coming into contact with Zoroastrians and knowing about this practice. Religious minorities were also free to do as they wished in their own homes, provided they did not publicly engage in illicit sexual activity in ways that could threaten public morals.
There are parallels for this in Roman and Jewish law. According to law professor H. Patrick Glenn of McGill University, \"\[t\]oday it is said that the dhimmi are \'excluded from the specifically Muslim privileges, but on the other hand they are excluded from the specifically Muslim duties\' while (and here there are clear parallels with western public and private law treatment of aliens---Fremdenrecht, la condition de estrangers), \'\[f\]or the rest, the Muslim and the dhimmi are equal in practically the whole of the law of property and of contracts and obligations\'.\" Quoting the Qur\'anic statement, \"Let Christians judge according to what We have revealed in the Gospel\", Muhammad Hamidullah writes that Islam decentralized and \"communalized\" law and justice. However, the classical *dhimma* contract is no longer enforced. Western influence over the Muslim world has been instrumental in eliminating the restrictions and protections of the *dhimma* contract.
### The Dhimma contract and Sharia law {#the_dhimma_contract_and_sharia_law}
The *dhimma* contract is an integral part of traditional Islamic law. From the 9th century AD, the power to interpret and refine law in traditional Islamic societies was in the hands of the scholars (*ulama*). This separation of powers served to limit the range of actions available to the ruler, who could not easily decree or reinterpret law independently and expect the continued support of the community. Through succeeding centuries and empires, the balance between the ulema and the rulers shifted and reformed, but the balance of power was never decisively changed. At the beginning of the 19th century, the Industrial Revolution and the French Revolution introduced an era of European world hegemony that included the domination of most of the Muslim lands. At the end of the Second World War, the European powers found themselves too weakened to maintain their empires. The wide variety in forms of government, systems of law, attitudes toward modernity and interpretations of sharia are a result of the ensuing drives for independence and modernity in the Muslim world.
Muslim states, sects, schools of thought and individuals differ as to exactly what sharia law entails. In addition, Muslim states today utilize a spectrum of legal systems. Most states have a mixed system that implements certain aspects of sharia while acknowledging the supremacy of a constitution. A few, such as Turkey, have declared themselves secular. Local and customary laws may take precedence in certain matters, as well. Islamic law is therefore polynormative, and despite several cases of regression in recent years, the trend is towards liberalization. Questions of human rights and the status of minorities cannot be generalized with regards to the Muslim world. They must instead be examined on a case-by-case basis, within specific political and cultural contexts, using perspectives drawn from the historical framework.
### The end of the Dhimma contract {#the_end_of_the_dhimma_contract}
The status of the *dhimmi* \"was for long accepted with resignation by the Christians and with gratitude by the Jews\" but the rising power of Christendom and the radical ideas of the French Revolution caused a wave of discontent among Christian dhimmis. The continuing and growing pressure from the European powers combined with pressure from Muslim reformers gradually relaxed the inequalities between Muslims and non-Muslims.
On 18 February 1856, the Ottoman Reform Edict of 1856 (*Hatt-i Humayan*) was issued, building upon the 1839 edict. It came about partly as a result of pressure from and the efforts of the ambassadors of France, Austria and the United Kingdom, whose respective countries were needed as allies in the Crimean War. It again proclaimed the principle of equality between Muslims and non-Muslims, and produced many specific reforms to this end. For example, the *jizya* tax was abolished and non-Muslims were allowed to join the army.
According to some scholars, discrimination against *dhimmis* did not end with the Edict of 1856, and they remained second-class citizens at least until the end of World War I. H.E.W. Young, the British Council in Mosul, wrote in 1909, \"The attitude of the Muslims toward the Christians and the Jews is that of a master towards slaves, whom he treats with a certain lordly tolerance so long as they keep their place. Any sign of pretension to equality is promptly repressed.\"
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# Dhimmi
## The \"Dhimma contract\" {#the_dhimma_contract}
### Views of modern Islamic scholars on the status of non-Muslims in an Islamic society {#views_of_modern_islamic_scholars_on_the_status_of_non_muslims_in_an_islamic_society}
- The Iranian Shi\'a Muslim Ayatollah Ruhollah Khomeini indicates in his book *Islamic Government: Governance of the Jurist* that non-Muslims should be required to pay the poll tax, in return for which they would profit from the protection and services of the state; they would, however, be excluded from all participation in the political process.`{{failed verification|date=December 2020}}`{=mediawiki} Bernard Lewis remarks about Khomeini that one of his main grievances against the Shah, Mohammad Reza Pahlavi, was that his legislation allowed the theoretical possibility of non-Muslims exercising political or judicial authority over Muslims.
- The Egyptian theologian Yusuf al-Qaradawi, chairman of the International Union of Muslim Scholars, has stated in his Al Jazeera program *Sharia and Life*, which has an estimated audience of 35 to 60 million viewers: \"When we say *dhimmis* (*ahl al-dhimma*) it means that \[\...\] they are under the covenant of God and His Messenger and the Muslim community and their responsibility (*ḍamān*), and it is everyone\'s duty to protect them, and this is what is intended by the word. At present many of our brethren are offended by the word *dhimmis*, and I have stated in what I wrote in my books that I don\'t see anything to prevent contemporary Islamic ijtihad from discarding this word *dhimmis* and calling them non-Muslim citizens.\"
- Another Egyptian Islamist, Mohammad Salim al-Awa argued the concept of dhimmi must be re-interpreted in the context of Egyptian nationalism. Al-Awa and other Muslim scholars based this on the idea that while the previous *dhimma* condition result from the Islamic conquest, the modern Egyptian state results from a joint Muslim-Christian campaign to end the British occupation of Egypt. In modern-day Egypt, he argues, the constitution replaces the *dhimma* contract.
- Muhammad Husayn Tabataba\'i, a 20th-century Shia scholar writes that dhimmis should be treated \"in a good and decent manner\". He addresses the argument that good treatment of dhimmis was abrogated by Quranic verse 9:29 by stating that, in the literal sense, this verse is not in conflict with good treatment of dhimmis.
- Javed Ahmad Ghamidi, a Pakistani theologian, writes in *Mizan* that certain directives of the Quran were specific only to Muhammad against peoples of his times, besides other directives, the campaign involved asking the polytheists of Arabia for submission to Islam as a condition for exoneration and the others for jizya and submission to the political authority of the Muslims for exemption from capital punishment and for military protection as the dhimmis of the Muslims. Therefore, after Muhammad and his companions, there is no concept in Islam obliging Muslims to wage war for propagation or implementation of Islam.
- The Iranian Shia jurist Grand Ayatollah Naser Makarem Shirazi states in *Selection of the Tafsir Nemooneh* that the main philosophy of jizya is that it is only a financial aid to those Muslims who are in the charge of safeguarding the security of the state and dhimmis\' lives and properties on their behalf.
- Prominent Islamic thinkers like Fahmi Huwaidi and Tarek El-Bishry have based their justification for full citizenship of non-Muslims in an Islamic states on the precedent set by Muhammad in the Constitution of Medina. They argue that in this charter the People of Book, have the status of citizens (*muwatinun*) rather than dhimmis, sharing equal rights and duties with Muslims.
- Legal scholar L. Ali Khan also points to the Constitution of Medina as a way forward for Islamic states in his 2006 paper titled *The Medina Constitution*. He suggests this ancient document, which governed the status of religions and races in the first Islamic state, in which Jewish tribes are \"placed on an equal footing with \[\...\] Muslims\" and granted \"the freedom of religion,\" can serve as a basis for the protection of minority rights, equality, and religious freedom in the modern Islamic state.
- Tariq Ramadan, Professor of Islamic Studies at the University of Oxford, advocates the inclusion of academic disciplines and Islamic society, along with traditional Islamic scholars, in an effort to reform Islamic law and address modern conditions. He speaks of remaining faithful to the higher objectives of sharia law. He posits universal rights of dignity, welfare, freedom, equality and justice in a religiously and culturally pluralistic Islamic (or other) society, and proposes a dialogue regarding the modern term \"citizenship,\" although it has no clear precedent in classical fiqh. He further includes the terms \"non-citizen\", \"foreigner\", \"resident\" and \"immigrant\" in this dialogue, and challenges not only Islam, but modern civilization as a whole, to come to terms with these concepts in a meaningful way with regards to problems of racism, discrimination and oppression.
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# Dhimmi
## Dhimmi communities {#dhimmi_communities}
Jews and Christians living under early Muslim rule were considered dhimmis, a status that was later also extended to other non-Muslims like Hindus and Buddhists. They were allowed to \"freely practice their religion, and to enjoy a large measure of communal autonomy\" and guaranteed their personal safety and security of property, in return for paying tribute and acknowledging Muslim rule. Islamic law and custom prohibited the enslavement of free dhimmis within lands under Islamic rule. Taxation from the perspective of dhimmis who came under the Muslim rule, was \"a concrete continuation of the taxes paid to earlier regimes\" (but much lower under the Muslim rule). They were also exempted from the zakat tax paid by Muslims. The dhimmi communities living in Islamic states had their own laws independent from the Sharia law, such as the Jews who had their own Halakhic courts. The dhimmi communities had their own leaders, courts, personal and religious laws, and \"generally speaking, Muslim tolerance of unbelievers was far better than anything available in Christendom, until the rise of secularism in the 17th century\". \"Muslims guaranteed freedom of worship and livelihood, provided that they remained loyal to the Muslim state and paid a poll tax\". \"Muslim governments appointed Christian and Jewish professionals to their bureaucracies\", and thus, Christians and Jews \"contributed to the making of the Islamic civilization\".
However, dhimmis faced social and symbolic restrictions, and a pattern of stricter, then more lax, enforcement developed over time. Marshall Hodgson, a historian of Islam, writes that during the era of the High Caliphate (7th--13th Centuries), zealous Shariah-minded Muslims gladly elaborated their code of symbolic restrictions on the dhimmis.
From an Islamic legal perspective, the pledge of protection granted dhimmis the freedom to practice their religion and spared them forced conversions. The dhimmis also served a variety of useful purposes, mostly economic, which was another point of concern to jurists.`{{page needed|date=December 2015}}`{=mediawiki} Religious minorities were free to do whatever they wished in their own homes, but could not \"publicly engage in illicit sex in ways that threaten public morals\". In some cases, religious practices that Muslims found repugnant were allowed. One example was the Zoroastrian practice of incestuous \"self-marriage\" where a man could marry his mother, sister or daughter. According to the medieval Islamic legal scholar Ibn Qayyim al-Jawziyya, non-Muslims had the right to engage in such religious practices even if it offended Muslims, under the conditions that such cases not be presented to Islamic Sharia courts and that these religious minorities believed that the practice in question is permissible according to their religion. This ruling was based on the precedent that Muhammad did not forbid such self-marriages among Zoroastrians despite coming in contact with them and having knowledge of their practices.
The Arabs generally established garrisons outside towns in the conquered territories, and had little interaction with the local dhimmi populations for purposes other than the collection of taxes. The conquered Christian, Jewish, Mazdean and Buddhist communities were otherwise left to lead their lives as before.
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# Dhimmi
## Dhimmi communities {#dhimmi_communities}
### Christians
According to historians Lewis and Stillman, local Christians in Syria, Iraq, and Egypt were non-Chalcedonians and many may have felt better off under early Muslim rule than under that of the Byzantine Orthodox of Constantinople. In 1095, Pope Urban II urged western European Christians to come to the aid of the Christians of Palestine. The subsequent Crusades brought Roman Catholic Christians into contact with Orthodox Christians whose beliefs they discovered to differ from their own perhaps more than they had realized, and whose position under the rule of the Muslim Fatimid Caliphate was less uncomfortable than had been supposed. Consequently, the Eastern Christians provided perhaps less support to the Crusaders than had been expected. When the Arab East came under Ottoman rule in the 16th century, Christian populations and fortunes rebounded significantly. The Ottomans had long experience dealing with Christian and Jewish minorities, and were more tolerant towards religious minorities than the former Muslim rulers, the Mamluks of Egypt.
However, Christians living under Islamic rule have suffered certain legal disadvantages and at times persecution. In the Ottoman Empire, in accordance with the *dhimmi* system implemented in Muslim countries, they, like all other Christians and also Jews, were accorded certain freedoms. The dhimmi system in the Ottoman Empire was largely based upon the Pact of Umar. The client status established the rights of the non-Muslims to property, livelihood and freedom of worship but they were in essence treated as second-class citizens in the empire and referred to in Turkish as *gavours*, a pejorative word meaning \"infidel\" or \"unbeliever\". The clause of the Pact of Umar which prohibited non-Muslims from building new places of worship was historically imposed on some communities of the Ottoman Empire and ignored in other cases, at discretion of the local authorities. Although there were no laws mandating religious ghettos, this led to non-Muslim communities being clustered around existing houses of worship.
In addition to other legal limitations, dhimmis, including the Christians among them, were not considered equals to Muslims and several prohibitions were placed on them. Their testimony against Muslims was inadmissible in courts of law wherein a Muslim could be punished; this meant that their testimony could only be considered in commercial cases. They were forbidden to carry weapons or ride atop horses and camels. Their houses could not overlook those of Muslims; and their religious practices were severely circumscribed (e.g., the ringing of church bells was strictly forbidden).
### Jews
Because the early Islamic conquests initially preserved much of the existing administrative machinery and culture, in many territories they amounted to little more than a change of rulers for the subject populations, which \"brought peace to peoples demoralized and disaffected by the casualties and heavy taxation that resulted from the years of Byzantine-Persian warfare\".
María Rosa Menocal, argues that the Jewish dhimmis living under the caliphate, while allowed fewer rights than Muslims, were still better off than in the Christian parts of Europe. Jews from other parts of Europe made their way to al-Andalus, where in parallel to Christian sects regarded as heretical by Catholic Europe, they were not just tolerated, but where opportunities to practice faith and trade were open without restriction save for the prohibitions on proselytization.
Bernard Lewis states:
Professor of Jewish medieval history at Hebrew University of Jerusalem, Hayim Hillel Ben-Sasson, notes:
According to the French historian Claude Cahen, Islam has \"shown more toleration than Europe towards the Jews who remained in Muslim lands.\"
Comparing the treatment of Jews in the medieval Islamic world and medieval Christian Europe, Mark R. Cohen notes that, in contrast to Jews in Christian Europe, the \"Jews in Islam were well integrated into the economic life of the larger society\", and that they were allowed to practice their religion more freely than they could do in Christian Europe.
According to the scholar Mordechai Zaken, tribal chieftains (also known as aghas) in tribal Muslim societies such as the Kurdish society in Kurdistan would tax their Jewish subjects. The Jews were in fact civilians protected by their chieftains in and around their communities; in return they paid part of their harvest as dues, and contributed their skills and services to their patron chieftain.
### Hindus and Buddhists {#hindus_and_buddhists}
By the 10th century, the Turks of Central Asia had invaded the Indic plains, and spread Islam in Northwestern parts of India. At the end of the 12th century, the Muslims advanced quickly into the Ganges Plain. In one decade, a Muslim army led by Turkic slaves consolidated resistance around Lahore and brought northern India, as far as Bengal, under Muslim rule. From these Turkic slaves would come sultans, including the founder of the sultanate of Delhi. By the 15th century, major parts of Northern India was ruled by Muslim rulers, mostly descended from invaders. In the 16th century, India came under the influence of the Mughals. Babur, the first ruler of the Mughal empire, established a foothold in the north which paved the way for further expansion by his successors. Although the Mughal emperor Akbar has been described as a universalist, most Mughal emperors were oppressive of native Hindu, Buddhist and later Sikh populations. Aurangzeb specifically was inclined towards a highly fundamentalist approach.
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# Dhimmi
## Restrictions
There were a number of restrictions on dhimmis. In a modern sense the dhimmis would be described as second-class citizens. According to historian Marshall Hodgson, from very early times Muslim rulers would very often humiliate and punish dhimmis (usually Christians or Jews that refused to convert to Islam). It was official policy that dhimmis should "feel inferior and to know 'their place\".
Although *dhimmis* were allowed to perform their religious rituals, they were obliged to do so in a manner not conspicuous to Muslims. Loud prayers were forbidden, as were the ringing of church bells and the blowing of the shofar. They were also not allowed to build or repair churches and synagogues without Muslim consent. Moreover, dhimmis were not allowed to seek converts among Muslims.`{{page needed|date=January 2016}}`{=mediawiki} In the Mamluk Egypt, where non-Mamluk Muslims were not allowed to ride horses and camels, dhimmis were prohibited even from riding donkeys inside cities. Sometimes, Muslim rulers issued regulations requiring dhimmis to attach distinctive signs to their houses.
Most of the restrictions were social and symbolic in nature, and a pattern of stricter, then more lax, enforcement developed over time. The major financial disabilities of the dhimmi were the jizya poll tax and the fact dhimmis and Muslims could not inherit from each other. That would create an incentive to convert if someone from the family had already converted. Ira M. Lapidus states that the \"payment of the poll tax seems to have been regular, but other obligations were inconsistently enforced and did not prevent many non-Muslims from being important political, business, and scholarly figures. In the late ninth and early tenth centuries, Jewish bankers and financiers were important at the \'Abbasid court.\" The jurists and scholars of Islamic sharia law called for humane treatment of the dhimmis.
A Muslim man may marry a Jewish or Christian dhimmī woman, who may keep her own religion (though her children were automatically considered Muslims and had to be raised as such), but a Muslim woman cannot marry a dhimmī man unless he converts to Islam. Dhimmīs are prohibited from converting Muslims under severe penalties, while Muslims are encouraged to convert dhimmīs. `{{unreliable source?|date=September 2021}}`{=mediawiki}
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# Dhimmi
## Restrictions
### Jizya tax {#jizya_tax}
Payment of the *jizya* obligated Muslim authorities to protect dhimmis in civil and military matters. Sura 9 (At-Tawba), verse 29 stipulates that *jizya* be exacted from non-Muslims as a condition required for jihad to cease. Islamic jurists required adult, free, healthy males among the dhimma community to pay the jizya, while exempting women, children, the elderly, slaves, those affected by mental or physical handicaps, and travelers who did not settle in Muslim lands. According to Abu Yusuf dhimmi should be imprisoned until they pay the jizya in full. Other jurists specified that dhimmis who don\'t pay jizya should have their heads shaved and made to wear a dress distinctive from those dhimmis who paid the jizya and Muslims.
Lewis states there are varying opinions among scholars as to how much of a burden jizya was. According to Norman Stillman: \"*jizya* and *kharaj* were a \"crushing burden for the non-Muslim peasantry who eked out a bare living in a subsistence economy.\" Both agree that ultimately, the additional taxation on non-Muslims was a critical factor that drove many dhimmis to leave their religion and accept Islam. However, in some regions the jizya on populations was significantly lower than the zakat, meaning dhimmi populations maintained an economic advantage. According to Cohen, taxation, from the perspective of dhimmis who came under Muslim rule, was \"a concrete continuation of the taxes paid to earlier regimes\".`{{page needed|date=January 2016}}`{=mediawiki} Lewis observes that the change from Byzantine to Arab rule was welcomed by many among the dhimmis who found the new yoke far lighter than the old, both in taxation and in other matters, and that some, even among the Christians of Syria and Egypt, preferred the rule of Islam to that of Byzantines. Montgomery Watt states, \"the Christians were probably better off as dhimmis under Muslim-Arab rulers than they had been under the Byzantine Greeks.\" In some places, for example Egypt, the jizya was a tax incentive for Christians to convert to Islam.
Some scholars have tried compute the relative taxation on Muslims vs non-Muslims in the early Abbasid period. According to one estimate, Muslims had an average tax rate of 17--20 dirhams per person, which rose to 30 dirhams per person when in kind levies are included. Non-Muslims paid either 12, 24 or 48 dirhams per person, depending on their taxation category, though most probably paid 12.
The importance of dhimmis as a source of revenue for the Rashidun Caliphate is illustrated in a letter ascribed to Umar I and cited by Abu Yusuf: \"if we take dhimmis and share them out, what will be left for the Muslims who come after us? By God, Muslims would not find a man to talk to and profit from his labors.\"
The early Islamic scholars took a relatively humane and practical attitude towards the collection of *jizya*, compared to the 11th century commentators writing when Islam was under threat both at home and abroad.
The jurist Abu Yusuf, the chief judge of the caliph Harun al-Rashid, rules as follows regarding the manner of collecting the jizya
In the border provinces, dhimmis were sometimes recruited for military operations. In such cases, they were exempted from jizya for the year of service.
### Administration of law {#administration_of_law}
Religious pluralism existed in medieval Islamic law and ethics. The religious laws and courts of other religions, including Christianity, Judaism and Hinduism, were usually accommodated within the Islamic legal framework, as exemplified in the Caliphate, Al-Andalus, Ottoman Empire and Indian subcontinent. In medieval Islamic societies, the qadi (Islamic judge) usually could not interfere in the matters of non-Muslims unless the parties voluntarily chose to be judged according to Islamic law. The dhimmi communities living in Islamic states usually had their own laws independent from the Sharia law, such as the Jews who had their own Halakha courts.
Dhimmis were allowed to operate their own courts following their own legal systems. However, dhimmis frequently attended the Muslim courts in order to record property and business transactions within their own communities. Cases were taken out against Muslims, against other dhimmis and even against members of the dhimmi\'s own family. Dhimmis often took cases relating to marriage, divorce or inheritance to the Muslim courts so these cases would be decided under sharia law. Oaths sworn by dhimmis in the Muslim courts were sometimes the same as the oaths taken by Muslims, sometimes tailored to the dhimmis\' beliefs.
Muslim men could generally marry dhimmi women who are considered People of the Book, however Islamic jurists rejected the possibility any non-Muslim man might marry a Muslim woman. Bernard Lewis notes that \"similar position existed under the laws of Byzantine Empire, according to which a Christian could marry a Jewish woman, but a Jew could not marry a Christian woman under pain of death\".
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# Dhimmi
## Relevant texts {#relevant_texts}
### Quranic verses as a basis for Islamic policies toward dhimmis {#quranic_verses_as_a_basis_for_islamic_policies_toward_dhimmis}
Lewis states
- Al-Baqara 256 \"Let there be no compulsion in religion: \...\",`{{qref|2|256|s=y}}`{=mediawiki} means non-Muslims should not be forced to adopt Islam
- The phrase \"Unto you your religion, and unto me my religion.\", from `{{qref|109|6|c=y}}`{=mediawiki} has been used as a \"proof-text for pluralism and coexistence\".
- has served to justify the tolerated position accorded to the followers of Christianity, Judaism, and Sabianism under Muslim rule.
### Hadith
A hadith by Muhammad, \"Whoever killed a `{{Transliteration|ar|DIN|muʿāhid}}`{=mediawiki} (a person who is granted the pledge of protection by the Muslims) shall not smell the fragrance of Paradise though its fragrance can be smelt at a distance of forty years (of traveling).\", is cited as a foundation for the right of non-Muslim citizens to live peacefully and undisturbed in an Islamic state. Anwar Shah Kashmiri writes in his commentary on Sahih al-Bukhari *Fayd al-Bari* on this hadith: \"You know the gravity of sin for killing a Muslim, for its odiousness has reached the point of disbelief, and it necessitates that \[the killer abides in Hell\] forever. As for killing a non-Muslim citizen \[`{{Transliteration|ar|DIN|muʿāhid}}`{=mediawiki}\], it is similarly no small matter, for the one who does it will not smell the fragrance of Paradise.\"
A similar hadith in regard to the status of the dhimmis: \"Whoever wrongs one with whom a compact (treaty) has been made *\[i.e., a dhimmi\]* and lays on him a burden beyond his strength, I will be his accuser.\"
### Constitution of Medina {#constitution_of_medina}
The Constitution of Medina, a formal agreement between Muhammad and all the significant tribes and families of Medina (including Muslims, Jews and pagans), declared that non-Muslims in the Ummah had the following rights:
1. The security (*dhimma*) of God is equal for all groups,
2. Non-Muslim members have equal political and cultural rights as Muslims. They will have autonomy and freedom of religion.
3. Non-Muslims will take up arms against the enemy of the Ummah and share the cost of war. There is to be no treachery between the two.
4. Non-Muslims will not be obliged to take part in religious wars of the Muslims.
### Khaybar agreement {#khaybar_agreement}
A precedent for the dhimma contract was established with the agreement between Muhammad and the Jews after the Battle of Khaybar, an oasis near Medina. Khaybar was the first territory attacked and conquered by Muslims. When the Jews of Khaybar surrendered to Muhammad after a siege, Muhammad allowed them to remain in Khaybar in return for handing over to the Muslims one half their annual produce.
### Pact of Umar {#pact_of_umar}
The Pact of Umar, traditionally believed to be between caliph Umar and the conquered Jerusalem Christians in the seventh century, was another source of regulations pertaining to dhimmis. However, Western orientalists doubt the authenticity of the pact, arguing it is usually the victors and not the vanquished who impose rather than propose, the terms of peace, and that it is highly unlikely that the people who spoke no Arabic and knew nothing of Islam could draft such a document. Academic historians believe the Pact of Umar in the form it is known today was a product of later jurists who attributed it to Umar in order to lend greater authority to their own opinions. The similarities between the Pact of Umar and the Theodosian and Justinian Codes of the Eastern Roman Empire suggest that perhaps much of the Pact of Umar was borrowed from these earlier codes by later Islamic jurists. At least some of the clauses of the pact mirror the measures first introduced by the Umayyad caliph Umar II or by the early Abbasid caliphs.
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# Dhimmi
## Cultural interactions and cultural differences {#cultural_interactions_and_cultural_differences}
During the Middle Ages, local associations known as *futuwwa* clubs developed across the Islamic lands. There were usually several futuwwah in each town. These clubs catered to varying interests, primarily sports, and might involve distinctive manners of dress and custom. They were known for their hospitality, idealism and loyalty to the group. They often had a militaristic aspect, purportedly for the mutual protection of the membership. These clubs commonly crossed social strata, including among their membership local notables, dhimmi and slaves -- to the exclusion of those associated with the local ruler, or amir.
Muslims and Jews were sometimes partners in trade, with the Muslim taking days off on Fridays and Jews taking off on Saturdays.
Andrew Wheatcroft describes how some social customs such as different conceptions of dirt and cleanliness made it difficult for the religious communities to live close to each other, either under Muslim or under Christian rule.
## In modern times {#in_modern_times}
The dhimma and the jizya poll tax are no longer imposed in Muslim majority countries. In the 21st century, jizya is widely regarded as being at odds with contemporary secular conceptions of citizens\' civil rights and equality before the law, although there have been occasional reports of religious minorities in conflict zones and areas subject to political instability being forced to pay jizya.
In 2009 it was claimed that the Taliban imposed the *jizya* on Pakistan\'s minority Sikh community after occupying some of their homes and kidnapping a Sikh leader.
In 2013, the Muslim Brotherhood in Egypt occupied the town of Dalga immediately following the overthrow of Mohammed Morsi on 3 July, and reportedly imposed *jizya* on the 15,000 Christian Copts living there. However, in autumn of that same year Egyptian authorities were able to retake control of the town following two prior failed attempts.
In February 2014, the Islamic State of Iraq and the Levant (ISIL) announced that it intended to extract jizya from Christians in the city of Raqqa, Syria, which it controlled at the time. Christians who refused to accept the dhimma contract and pay the tax were to have to either convert to Islam, leave or be executed. Wealthy Christians would have to pay half an ounce of gold, the equivalent of \$664 twice a year; middle-class Christians were to have to pay half that amount and poorer ones were to be charged one-fourth that amount. In June 2014 the Institute for the Study of War reported that ISIL claims to have collected jizya and fay.`{{clarification needed|reason=Only mention of fay in this article. Is this the same as the topic mentioned at Fay (disambiguation)? Does it need to be mentioned?|date=May 2025}}`{=mediawiki} On 18 July 2014 ISIL ordered the Christians in Mosul to accept the dhimma contract and pay the jizya or convert to Islam. If they refused to accept either of the options they would be killed
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# Doctor V64
The **Doctor V64** (also referred to simply as the **V64**) is a development and backup device made by Bung Enterprises Ltd that is used in conjunction with the Nintendo 64. The Doctor V64 also had the ability to play video CDs and audio CDs. Additionally, it could apply stereo 3D effects to the audio.
## History
The V64 was released in 1996 and was priced around US\$450. It was one of the first commercially available backup devices for the Nintendo 64, appearing not long after the console\'s international release. The Partner N64 development kit, which was manufactured by Silicon Graphics and sold officially by Nintendo, was a comparatively expensive development machine. The V64 served as a lower-cost development machine, though its unofficial status would later lead to conflict with Nintendo. Some third-party developers used a number of V64s in their development process, with games such as Turok: Dinosaur Hunter utilizing the device during development.
### Specifications
The CPU of the V64 is a 6502, and the operating system is contained in a BIOS.
The V64 unit contains a CD-ROM drive which sits underneath the Nintendo 64 and plugs into the expansion slot on the underside of the Nintendo 64. The expansion slot is essentially a mirror image of the cartridge slot on the top of the unit, with the same electrical connections; thus, the Nintendo 64 reads data from the Doctor V64 in the same manner as it would from a cartridge plugged into the normal slot.
## Usage
### Game booting {#game_booting}
In order to get around Nintendo\'s lockout chip, when using the V64, a game cartridge is plugged into the Nintendo 64 through an adapter which connects only the lockout chip. The game cart used for the operation had to contain the same lockout chip used by the game back up.
### Saving game progress {#saving_game_progress}
The second problem concerned saving progress. Most N64 games are saved to the cart itself instead of external memory cards. If the player wanted to keep their progress, then the cartridge used had to have the same type of non-volatile memory hardware. Alternatively, Bung produced the \"DX256\" and \"DS1\" add-ons to allow (EEPROM and SRAM respectively) saves to be made without using the inserted cartridge. These devices were inserted into the top-slot of the N64 with the game cartridge being then inserted into the top of them to just provide the security bypass. Save slots on the DX256 were selected using an alpha and numeric encoder knobs on the front of the device.
### Uploading game images {#uploading_game_images}
The Doctor V64 could be used to read the data from a game cartridge and transfer the data to a PC via the parallel port. This allowed developers and homebrew programmers to upload their game images to the Doctor V64 without having to create a CD backup each time. It also allowed users to upload game images taken from the Internet.
## Doctor V64 Jr. {#doctor_v64_jr.}
Following the Doctor V64\'s success, Bung released the Doctor V64 Jr. in December 1998. This was a condensed, cost-efficient version of the original V64. The Doctor V64 Jr. has no CD drive and plugs into the normal cartridge slot on the top of the Nintendo 64. Data is loaded into the Doctor V64 Jr.\'s battery-backed RAM from a PC via a parallel port connection. The Doctor V64 Jr. has up to 512 megabits (64 MB) of memory storage. This was done to provide for future Nintendo 64 carts that employed larger memory storage, but the high costs associated with ordering large storage carts kept this occurrence at a minimum. Only a handful of 512-megabit games were released for the Nintendo 64 system.
## Promotions
In 1998 and 1999, there was a homebrew competition known as \"Presence of Mind\" (POM), an N64 demo competition led by dextrose.com. The contest consisted of submitting a user-developed N64 program, game, or utility. Bung Enterprises promoted the event and supplied prizes (usually Doctor V64 related accessories). Though a contest was planned for 2000, the interest in the N64 was already fading, and so did the event. POM contest demo entries can still be found on the Internet.
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# Doctor V64
## Legal issues {#legal_issues}
### Role in piracy {#role_in_piracy}
The Doctor V64 unit was the first commercially available backup device for the Nintendo 64 unit. Though the unit was sold as a development machine, it could be modified to enable the creation and use of commercial game backups. Unlike official development units, the purchase of V64s was not restricted to software companies only. For this reason, the unit became a popular choice among those looking to proliferate unlicensed copies of games.
Original Doctor V64 units sold by Bung did not allow the playing of backups. A person would have to modify the unit by themselves in order to make it backup friendly. This usually required a user to download and install a modified Doctor V64 BIOS. Additionally, the cartridge adapter had to be opened and soldered in order to allow for the operational procedure. Though Bung never sold backup enabled V64s, many re-sellers would modify the units themselves.
### Conflicts with Nintendo {#conflicts_with_nintendo}
During the N64\'s lifetime, Nintendo revised the N64\'s model, making the serial port area smaller. This slight change in the N64\'s plastic casing made the connection to the Doctor V64 difficult to achieve without user modification. This revision may have been a direct reaction from Nintendo to discourage the use of V64 devices, and may also explain why Bung decided to discontinue the use of this port in the later Doctor V64 Jr. models.
Nintendo made many legal efforts worldwide in order to stop the sale of Doctor V64 units. They sued Bung directly as well as specific store retailers in Europe and North America for copyright infringement. Eventually, Nintendo managed to have the courts prohibit the sale of Doctor V64 units in the United States.
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# Doctor V64
## Main menu {#main_menu}
The Doctor V64 implemented text-based menu-driven screens. The menus consisted of white text superimposed over a black background. Utilizing the buttons on the V64 unit, a user would navigate the menus and issue commands. Though the menu was mainly designed for game developers, it is possible to back up cartridges with it (through the use of an unofficial V64 BIOS). Some of the menu items related to game backups were removed from the V64\'s BIOS near the end of its life due to pressure from Nintendo. These items are only available by obtaining a patched V64 BIOS.
Menu option Effect
------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Alternate and BootCrack This option would load a workaround for booting games. It only worked on certain types of game images.
Load Boot Crack Routine An advanced option that allowed uploading of program code for the use of boot related problems.
Backup Card Auto → DRAM This option would read a game cartridge and store it in the V64\'s RAM.
Backup Card Auto → PC Same as the previous option, but would transfer the data to a PC through the V64\'s parallel port.
Manual Slide Show Switch manual between Screenshots the user made in VCD Movie.
Auto Slide Show Switch automatic between Screenshots the user made in VCD Movie.
V64 Self Test Diagnostics routine; would check all of V64\'s subsystems.
Fully Test 128M DRAM Diagnostics routine; would check only the first 128MB of memory.
Fully Test 256M DRAM Diagnostics routine; specifically for units with 256MB of memory.
Upload DRAM Data → PC Used in conjunction with the option \"Backup Card Auto → DRAM\", this option would transfer the contents of Doctor V64\'s RAM to a PC through the use of V64\'s parallel port.
Fix CRC Code → run game Another boot-related command to enable the playing of game images. It would only work on certain types of backups.
Show Game Name in DRAM An advanced option that would read the backup image and extract the game\'s name, displaying it on the screen.
Upload V64 BIOS to PC Another advanced option for DV64 developers. It would transfer the Doctor V64\'s own program code to the PC through a parallel connection.
DX256 Upload to PC This command allowed the operation of specific features of the DX256 cartridge adapter (an alternate cartridge adapter sold by Bung).
PC Download to DX256 This command allowed the operation of specific features to be downloaded onto the DX256 cartridge adaptor.
Swap Byte Order in DRAM This command would convert the game image\'s endianness of a game image already loaded in RAM. Later bios revisions would do this automatically, deprecating this option.
: Complete V64 menu listing
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# Doctor V64
## Detailed specifications {#detailed_specifications}
### CD-ROM access speed {#cd_rom_access_speed}
Most early V64 models shipped with a standard IDE 8X CD-ROM . During the manufacturing lifetime of the device, latter V64 models shipped with 16X and eventually 20X drives. V64 units could be purchased without a CD-ROM drive. It is possible to replace the unit with a faster IDE CD-ROM unit (such as the 52X model in the image on this page).
Many Doctor V64s shipped internationally were ordered without an installed CD-ROM drive, to save on shipping costs associated with weight, to avoid import duty on the drive, and to allow users to customize the units in response to the ever-increasing speeds of drives available. The variance in the power draw of different manufacturers drives at different speeds caused issues with disc spin-ups exceeding the wattage rating of the included Bung PSU. This led to users swapping out the Bung PSU for a more powerful model, or selecting low draw drives (mainly Panasonic drives sometimes badged as Creative).
### CD-Media {#cd_media}
V64s can read CD-Rs and CD-RWs (provided the installed CD-ROM unit supports rewritable media). Supported media has to be recorded in Mode 1, ISO 9660 format. Doctor V64s only support the 8.3 DOS naming convention. As such, Joliet file system is not supported.
### RAM
Depending on the model, V64s came with either 128 megabits (16 MB) or 256 megabits (32 MB) of RAM. Original V64 units shipped with 128 megabits of RAM. V64 units started shipping with 256 megabits when developers started using bigger sized memory carts for their games. Users had the option of buying a memory upgrade from Bung and other re-sellers.
### Power supply {#power_supply}
The Doctor V64 uses a 4 Pin MiniDIN jack (as used for S-Video) for connecting the power supply cord. Power supplies included with Doctor V64s were very unreliable. Bung replaced the power supply with a sturdier version in later V64 units. Replacing broken power supplies became one of the most common maintenance problems with the V64. It is possible to modify an AT PC power supply for V64 use. Only 4 cables have to be connected to the V64 for it to function.
## Additional information {#additional_information}
- The ROM extensions \".v64\" and \".z64\" started out as the preferred naming conventions by Doctor V64 and Z64 users, respectively. It would also imply the file\'s \"endianness\" as those units employed little endian (V64) and big endian (Z64) byte alignment. \".n64\" was used as well but not as much (it became more popular as N64 emulators began to appear). The terms \".v64\" and \".z64\" are still widely used today by the emulation community.
- Acclaim Entertainment subsidiary Iguana Entertainment used Doctor V64 units as their development hardware of choice during the N64 era. They were best known for developing the *Turok*, *NBA Jam*, *NFL Quarterback Club*, and *South Park* video games
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# Diophantus
**Diophantus of Alexandria** (*Diophantos*) (`{{IPAc-en|d|aɪ|oʊ|ˈ|f|æ|n|t|ə|s}}`{=mediawiki}; `{{fl|250 CE}}`{=mediawiki}) was a Greek mathematician who was the author of the *Arithmetica* in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Although Joseph-Louis Lagrange called Diophantus \"the inventor of algebra\" he did not invent it; however, his exposition became the standard within the Neoplatonic schools of Late antiquity, and its translation into Arabic in the 9th century AD and had influence in the development of later algebra: Diophantus\' method of solution matches medieval Arabic algebra in its concepts and overall procedure. The 1621 edition of *Arithmetica* by Bachet gained fame after Pierre de Fermat wrote his famous \"Last Theorem\" in the margins of his copy.
In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him. Some problems from the *Arithmetica* have inspired modern work in both abstract algebra and number theory.
## Biography
The exact details of Diophantus\' life are obscure. Although he probably flourished in the third century CE, he may have lived anywhere between 170 BCE, roughly contemporaneous with Hypsicles, the latest author he quotes from, and 350 CE, when Theon of Alexandria quotes from him. Paul Tannery suggested that a reference to an \"Anatolius\" as a student of Diophantus in the works of Michael Psellos may refer to the early Christian bishop Anatolius of Alexandria, who may possibly the same Anatolius mentioned by Eunapius as a teacher of the pagan Neopythagorean philosopher Iamblichus, either of which would place him in the 3rd century CE.
The only definitive piece of information about his life is derived from a set of mathematical puzzles attributed to the 5th or 6th century CE grammarian Metrodorus preserved in book 14 of the Greek Anthology. One of the problems (sometimes called Diophantus\' epitaph) states:
> Here lies Diophantus, the wonder behold. Through art algebraic, the stone tells how old: \'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father\'s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.\'
This puzzle implies that Diophantus\' age `{{math|''x''}}`{=mediawiki} can be expressed as
: `{{sfrac|''x''|6}}`{=mediawiki} + `{{sfrac|''x''|12}}`{=mediawiki} + `{{sfrac|''x''|7}}`{=mediawiki} + 5 + `{{sfrac|''x''|2}}`{=mediawiki} + 4}}
which gives `{{math|''x''}}`{=mediawiki} a value of 84 years. However, the accuracy of the information cannot be confirmed.
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# Diophantus
## *Arithmetica*
*Arithmetica* is the major work of Diophantus and the most prominent work on premodern algebra in Greek mathematics. It is a collection of 290 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. *Arithmetica* was originally written in thirteen books, but only six of them survive in Greek, while another four books survive in Arabic, which were discovered in 1968. The books in Arabic correspond to books 4 to 7 of the original treatise, while the Greek books correspond to books 1 to 3 and 8 to 10.
*Arithmetica* is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him. Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.
Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the *Arithmetica* problems lead to quadratic equations.
### Notation
Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown.
Similar to medieval Arabic algebra, Diophantus uses three stages to solution of a problem by algebra:
1. An unknown is named and an equation is set up
2. An equation is simplified to a standard form (*al-jabr* and *al-muqābala* in Arabic)
3. Simplified equation is solved
Diophantus does not give classification of equations in six types like Al-Khwarizmi in extant parts of *Arithmetica*. He does says that he would give solution to three terms equations later, so this part of work is possibly just lost.
The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So for example, what would be written in modern notation as $x^3 - 2x^2 + 10x -1 = 5,$ which can be rewritten as $\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5,$ would be written in Diophantus\'s notation as
$$\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;$$*ἴ*$\sigma\;\, \Mu \overline{\varepsilon}$
Symbol What it represents
--------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\overline{\alpha}$ 1 (Alpha is the 1st letter of the Greek alphabet)
$\overline{\beta}$ 2 (Beta is the 2nd letter of the Greek alphabet)
$\overline{\varepsilon}$ 5 (Epsilon is the 5th letter of the Greek alphabet)
$\overline{\iota}$ 10 (Iota is the 9th letter of the `{{em|modern}}`{=mediawiki} Greek alphabet but it was the 10th letter of an ancient archaic Greek alphabet that had the letter digamma (uppercase: Ϝ, lowercase: ϝ) in the 6th position between epsilon ε and zeta ζ.)
\"equals\" (short for *ἴσος\]\]*)
$\pitchfork$ represents the subtraction of everything that follows $\pitchfork$ up to *ἴσ*
$\Mu$ the zeroth power (that is, a constant term)
$\zeta$ the unknown quantity (because a number $x$ raised to the first power is just $x,$ this may be thought of as \"the first power\")
$\Delta^{\upsilon}$ the second power, from Greek *δύναμις*, meaning strength or power
$\Kappa^{\upsilon}$ the third power, from Greek *κύβος*, meaning a cube
$\Delta^{\upsilon}\Delta$ the fourth power
$\Delta\Kappa^{\upsilon}$ the fifth power
$\Kappa^{\upsilon}\Kappa$ the sixth power
Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus\'s equation into a modern equation would be the following: ${x^3}1 {x}10 - {x^2}2 {x^0}1 = {x^0}5$ where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as: $\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5$
### Contents
In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form $4n + 3$ cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph-Louis Lagrange proved it using results due to Leonhard Euler.
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# Diophantus
## Other works {#other_works}
Another work by Diophantus, *On Polygonal Numbers* is transmitted in an incomplete form in four Byzantine manuscripts along with the *Arithmetica*. Two other lost works by Diophantus are known: *Porisms* and *On Parts*.
Recently, Wilbur Knorr has suggested that another book, *Preliminaries to the Geometric Elements*, traditionally attributed to Hero of Alexandria, may actually be by Diophantus.
### On polygonal numbers {#on_polygonal_numbers}
This work on polygonal numbers, a topic that was of great interest to the Pythagoreans consists of a preface and five propositions in its extant form. The treatise breaks off in the middle of a proposition about how many ways a number can be a polygonal number.
### The *Porisms* {#the_porisms}
The *Porisms* was a collection of lemmas along with accompanying proofs. Although *The Porisms* is lost, we know three lemmas contained there, since Diophantus quotes them in the *Arithmetica* and refers the reader to the *Porisms* for the proof.
One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any `{{math|''a''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki}, with `{{math|''a'' > ''b''}}`{=mediawiki}, there exist `{{math|''c'' and ''d''}}`{=mediawiki}, all positive and rational, such that
: *c*`{{sup|3}}`{=mediawiki} + *d*`{{sup|3}}`{=mediawiki}}}.
### *On Parts* {#on_parts}
This work, on fractions, is known by a single reference, a Neoplatonic scholium to Iamblichus\' treatise on Nicomachus\' *Introduction to Arithmetic*. Next to a line where Iamblichus writes \"Some of the Pythagoreans said that the unit is the borderline between number and parts\" the scholiast writes \"So Diophantus writes in *On Parts*, for parts involve progress in diminution carried to infinity.\"
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# Diophantus
## Influence
Diophantus\' work has had a large influence in history. Although Joseph-Louis Lagrange called Diophantus \"the inventor of algebra\", he did not invent it, however his work *Arithmetica* created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. Diophantus and his works influenced mathematics in the medieval Islamic world, and editions of *Arithmetica* exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries.
### Later antiquity {#later_antiquity}
After its publication, Diophantus\' work continued to be read in the Greek-speaking Mediterranean from the 4th through the 7th centuries. The earliest known reference to Diophantus, in the 4th century, is the *Commentary on the Almagest* Theon of Alexandria, which quotes from the introduction to the *Arithmetica*. According to the Suda, Hypatia, who was Theon\'s daughter and frequent collaborator, wrote a now lost commentary on Diophantus\' *Arithmetica*, which suggests that this work may have been closely studied by Neoplatonic mathematicians in Alexandria during Late antiquity. References to Diophantus also survive in a number of Neoplatonic scholia to the works of Iamblichus. A 6th century Neoplatonic commentary on Porphyry\'s *Isagoge* by Pseudo-Elias also mentions Diophantus; after outlining the quadrivium of arithmetic, geometry, music, and astronomy and four other disciplines adjacent to them (\"logistic\", \"geodesy\", \"music in matter\" and \"spherics\"), it mentions that Nicomachus (author of the *Introduction to Arithmetic*) occupies the first place in arithmetic but Diophantus occupies the first place in \"logistic\", showing that, despite the title of *Arithmetica*, the more algebraic work of Diophantus was already seen as distinct from arithmetic prior to the medieval era.
### Medieval era {#medieval_era}
Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the Dark Ages, since the study of ancient Greek, and literacy in general, had greatly declined. The portion of the Greek *Arithmetica* that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370--1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 -- 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.
*Arithmetica* became known to mathematicians in the Islamic world in the ninth century, when Qusta ibn Luqa translated it into Arabic.
In 1463 German mathematician Regiomontanus wrote:\"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden.\" *Arithmetica* was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book *Algebra*. The *editio princeps* of *Arithmetica* was published in 1575 by Xylander.
### Fermat
The Latin translation of *Arithmetica* by Bachet in 1621 became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it and made notes in the margins. The 1621 edition of *Arithmetica* by Bachet gained fame after Pierre de Fermat wrote his famous \"Last Theorem\" in the margins of his copy:
> If an integer `{{math|''n''}}`{=mediawiki} is greater than 2, then `{{math|''a''{{sup|''n''}} + ''b''{{sup|''n''}} {{=}}`{=mediawiki} *c*`{{sup|''n''}}`{=mediawiki}}} has no solutions in non-zero integers `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki}, and `{{math|''c''}}`{=mediawiki}. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.
Fermat\'s proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat\'s son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat\'s annotations---including the \"Last Theorem\"---were printed in this version.
Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (1370--1437) had written \"Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem\" next to the same problem.
Diophantus was among the first to recognise positive rational numbers as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης (*parisotēs*) to refer to an approximate equality. This term was rendered as *adaequalitas* in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves.
### Diophantine analysis {#diophantine_analysis}
Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in *Arithmetica* lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: `{{math|''ax''{{sup|2}} + ''bx'' {{=}}`{=mediawiki} *c*}}, `{{math|''ax''{{sup|2}} {{=}}`{=mediawiki} *bx* + *c*}}, and `{{math|''ax''{{sup|2}} + ''c'' {{=}}`{=mediawiki} *bx*}}. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki}, `{{math|''c''}}`{=mediawiki} to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions \"useless\", \"meaningless\", and even \"absurd\". To give one specific example, he calls the equation `{{math|4 {{=}}`{=mediawiki} 4*x* + 20}} \'absurd\' because it would lead to a negative value for `{{math|''x''}}`{=mediawiki}. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.
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# Diophantus
## Influence
### Rediscovery of books IV-VII {#rediscovery_of_books_iv_vii}
In 1968, Fuat Sezgin found four previously unknown books of *Arithmetica* at the shrine of Imam Rezā in the holy Islamic city of Mashhad in northeastern Iran. The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa (820--912). Norbert Schappacher has written:
> \[The four missing books\] resurfaced around 1971 in the Astan Quds Library in Meshed (Iran) in a copy from 1198. It was not catalogued under the name of Diophantus (but under that of Qusta ibn Luqa) because the librarian was apparently not able to read the main line of the cover page where Diophantus's name appears in geometric Kufi calligraphy.
## Editions and translations {#editions_and_translations}
- Bachet de Méziriac, C.G. *Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus*. Paris: Lutetiae, 1621.
- Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac, *Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus*. Cum comm. C(laude) G(aspar) Bacheti et observationibus P(ierre) de Fermat. Acc. doctrinae analyticae inventum novum, coll. ex variis eiu. Tolosae 1670, `{{doi|10.3931/e-rara-9423}}`{=mediawiki}.
- Tannery, P. L. *Diophanti Alexandrini Opera omnia: cum Graecis commentariis*, Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: [vol. 1](https://archive.org/details/diophantialexan03plangoog), [vol. 2](https://archive.org/details/diophantialexan00plangoog))
- Sesiano, Jacques. *The Arabic text of Books IV to VII of Diophantus' translation and commentary*. Thesis. Providence: Brown University, 1975
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# Duke Kahanamoku
thumb\|upright=1.2\|Signature
**Duke Paoa Kahinu Mokoe Hulikohola Kahanamoku** (August 24, 1890 -- January 22, 1968) was a Hawaiian competition swimmer, lifeguard, and popularizer of the sport of surfing. A Native Hawaiian, he was born three years before the overthrow of the Hawaiian Kingdom. He lived to see the territory\'s admission as a state and became a United States citizen. He was a five-time Olympic medalist in swimming, winning medals in 1912, 1920 and 1924.
Kahanamoku joined fraternal organizations: he was a Scottish Rite Freemason in the Honolulu lodge, and a Shriner. He worked as a law enforcement officer, an actor, a beach volleyball player, and a businessman.
## Family background {#family_background}
According to Kahanamoku, he was born in Honolulu at Haleʻākala, the home of Bernice Pauahi Bishop, which was later converted into the Arlington Hotel.
He was born into a family of Native Hawaiians headed by Duke Halapu Kahanamoku and Julia Paʻakonia Lonokahikina Paoa. He had five brothers, and three sisters. His brothers were Sargent, Samuel, David, William and Louis, all of whom participated in competitive aquatic sports. His sisters were Bernice, Kapiolani and Maria.
\"Duke\" was not a title or a nickname, but a given name. He was named after his father, Duke Halapu Kahanamoku, who was christened by Bernice Pauahi Bishop in honor of Prince Alfred, Duke of Edinburgh, who was visiting Hawaii at the time. His father was a policeman. His mother Julia Pa`{{okina}}`{=mediawiki}akonia Lonokahikina Paoa was a deeply religious woman with a strong sense of family ancestry.
His parents were from prominent Hawaiian *ohana* (families). The Kahanamoku and the Paoa ohana were considered to be lower-ranking nobles, who were in service to the *aliʻi nui*, or royalty. His paternal grandfather was Kahanamoku and his grandmother, Kapiolani Kaoeha (sometimes spelled *Kahoea*), a descendant of Alapainui. They were *kahu*, retainers and trusted advisors of the Kamehamehas, to whom they were related. His maternal grandparents Paoa, son of Paoa Hoolae and Hiikaalani, and Mele Uliama, were also of aliʻi descent.
In 1893, his family moved to Kālia, Waikiki (near the present site of Hilton Hawaiian Village), to be closer to his mother\'s parents and family. Kahanamoku grew up with his siblings and 31 Paoa cousins. He attended the Waikiki Grammar School, Kaahumanu School, and the Kamehameha Schools, although he never graduated because he had to quit to help support the family.
## Early years {#early_years}
Growing up on the outskirts of Waikiki, Kahanamoku spent much of his youth at the beach, where he developed his surfing and swimming skills. In his youth, Kahanamoku preferred a traditional surf board, which he called his *\"papa nui\"*, constructed after the fashion of ancient Hawaiian olo boards. Made from the wood of a koa tree, it was 16 ft long and weighed 114 lb. The board was without a skeg, which had yet to be invented. In his later surfing career, he would often use smaller boards but always preferred those made of wood.
Kahanamoku was a powerful swimmer. On August 11, 1911, he was timed at 55.4 seconds in the 100 yd freestyle, beating the existing world record by 4.6 seconds, in the salt water of Honolulu Harbor. He broke the record in the 220 yd and equaled it in the 50 yd. But the Amateur Athletic Union (AAU), in disbelief, would not recognize these feats until many years later. The AAU initially claimed that the judges must have been using alarm clocks rather than stopwatches and later claimed that ocean currents aided Kahanamoku.
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# Duke Kahanamoku
## Career
Kahanamoku easily qualified for the U.S. Olympic swimming team in 1912. At the 1912 Summer Olympics in Stockholm, he won a gold medal in the 100-meter freestyle, and a silver medal with the second-place U.S. team in the men\'s 4×200-meter freestyle relay.
During the 1920 Olympics in Antwerp, Kahanamoku won gold medals in both the 100 meters (bettering fellow Hawaiian Pua Kealoha) and in the relay. He finished the 100 meters with a silver medal during the 1924 Olympics in Paris, with the gold going to Johnny Weissmuller and the bronze to Kahanamoku\'s brother, Samuel. By then age 34, Kahanamoku won no more Olympic medals. But he served as an alternate for the U.S. water polo team at the 1932 Summer Olympics.
### Post-Olympic career {#post_olympic_career}
Between Olympic competitions, and after retiring from the Olympics, Kahanamoku traveled internationally to give swimming exhibitions. It was during this period that he popularized the sport of surfing, previously known only in Hawaii, by incorporating surfing exhibitions into his touring exhibitions as well. He attracted people to surfing in mainland America first in 1912 while in Southern California. He trained and loaned equipment to new surfers, such as Dorothy Becker.
His surfing exhibition at Sydney, Australia\'s Freshwater Beach on December 24, 1914, is widely regarded as a seminal event in the development of surfing in Australia. The board that Kahanamoku built from a piece of pine from a local hardware store is retained by the Freshwater Surf Life Saving Club. A statue of Kahanamoku was erected in his honor on the Northern headland of Freshwater Lake, New South Wales.
During his time living in Southern California, Kahanamoku performed in Hollywood as a background actor and a character actor in several films. He made connections in this way with people who could further publicize the sport of surfing. Kahanamoku was involved with the Los Angeles Athletic Club, acting as a lifeguard and competing in both swimming and water polo teams.
While living in Newport Beach, California, on June 14, 1925, Kahanamoku rescued eight men from a fishing vessel that capsized in heavy surf while it was attempting to enter the city\'s harbor. Using his surfboard, Kahanamoku made repeated trips from shore to the capsized ship, and helped rescue several people. Two other surfers saved four more fishermen, while five succumbed to the seas before they could be rescued. At the time the Newport Beach police chief called Kahanamoku\'s efforts \"The most superhuman surfboard rescue act the world has ever seen.\" The widespread publicity surrounding the rescue influenced lifeguards across the US to begin the use of surfboards as standard equipment for water rescues.
Kahanamoku was the first person to be inducted into both the Swimming Hall of Fame and the Surfing Hall of Fame. The Duke Kahanamoku Invitational Surfing Championships in Hawaii, the first major professional surfing contest event ever held in the huge surf on the North Shore of Oahu, was named in his honor. He is a member of the U.S. Olympic Hall of Fame.
Later Kahanamoku was elected to serve as the Sheriff of Honolulu, Hawaii from 1932 to 1961, completing 13 consecutive terms. During World War II, he served as a military police officer for the United States; Hawai\'i was not yet a state and was administered.
In the postwar period, Kahanamoku appeared in a number of television programs and films, including *Mister Roberts* (1955). He was well-liked throughout the Hollywood community.
Kahanamoku became a friend and surfing companion of heiress Doris Duke. She built a home (now a museum) on Oahu named Shangri-la. Kahanamoku gave private surfing lessons to Franklin D. Roosevelt Jr. and John Aspinwall Roosevelt, the children of Franklin D. Roosevelt.
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# Duke Kahanamoku
## *Duncan v. Kahanamoku* {#duncan_v._kahanamoku}
In 1946, Kahanamoku was the *pro forma* defendant in the landmark Supreme Court case *Duncan v. Kahanamoku*. While Kahanamoku was a military police officer during World War II, he arrested Duncan, a civilian shipfitter, for public intoxication.
At the time, Hawaii, not yet a state, was being administered by the United States under the Hawaiian Organic Act. This effectively instituted martial law on the island. After Duncan was tried by a military tribunal, he appealed to the Supreme Court. In a *post hoc* ruling, the court ruled that trial by military tribunal for the civilian was, in this case, unconstitutional.
## Personal life {#personal_life}
On August 2, 1940, Kahanamoku married dance instructor Nadine Alexander, who had relocated to Hawaii from Cleveland, Ohio, after she had been hired to teach at the Royal Hawaiian Hotel. Duke was 50 years old, Nadine was 35.
He was initiated, passed and raised to the degree of Master Mason in Hawaiian Lodge Masonic Lodge No 21 and was also a Noble (member) of the Shriners fraternal organization. He was a Republican.
## Death and legacy {#death_and_legacy}
Kahanamoku died of a heart attack on January 22, 1968, at age 77. For his burial at sea, a long motorcade of mourners, accompanied by a 30-man police escort, traveled in procession across town to Waikiki Beach. Reverend Abraham Akaka, the pastor of Kawaiahao Church, performed the service. A group of beach boys sang Hawaiian songs, including \"Aloha Oe\", and Kahanamoku\'s ashes were scattered into the ocean.
### Statues and monuments {#statues_and_monuments}
In 1994, a statue of Kahanamoku by Barry Donohoo was inaugurated in Freshwater, NSW, Australia. It is the showpiece of the Australian Surfers Walk of Fame.
On February 28, 2015, a monument featuring a replica of Kahanamoku\'s surfboard was unveiled at New Brighton beach, Christchurch, New Zealand in honor of the 100th anniversary of Kahanamoku\'s visit to New Brighton.
A statue of Kahanamoku was installed in Huntington Beach, California. A nearby restaurant is named for him and is close to Huntington Beach pier. The City of Huntington Beach identifies with the legacy of surfing, and a museum dedicated to that sport is located here.
In April 2022, NSW Heritage announced that Kahanamoku would be included in the first batch of Blue Plaques to be issued, to recognize his contribution to recreation and surfing.
A sculpture of Kahanamoku flanked by a male knee paddler and a female prone paddler commemorating the Catalina Classic Paddleboard Race was installed on the Manhattan Beach Pier in 2023.
### Additional tributes {#additional_tributes}
Hawaii music promoter Kimo Wilder McVay capitalized on Kahanamoku\'s popularity by naming his Waikiki showroom \"Duke Kahanamoku\'s\" at the International Market Place and giving Kahanamoku a financial interest in the showroom in exchange for the use of his name. It was a major Waikiki showroom in the 1960s and is remembered as the home of Don Ho & The Aliis from 1964 through 1969. The showroom continued to be known as Duke Kahanamoku\'s until Hawaii showman Jack Cione bought it in the mid-1970s and renamed it Le Boom Boom.
The Duke Kahanamoku Aquatic Complex (DKAC) serves as the home for the University of Hawai'i\'s swimming and diving and women\'s water polo teams. The facility, located on the university\'s lower campus, includes a 50-meter training pool and a separate 25-yard competition and diving pool. The long course pool is four feet at both ends, seven feet in the middle, and an average depth of six feet.
Kahanamoku\'s name is also used by Duke\'s Canoe Club & Barefoot Bar, `{{as of|2016|lc=y}}`{=mediawiki} known as Duke\'s Waikiki, a beachfront bar and restaurant in the Outrigger Waikiki on the Beach Hotel. There is a chain of restaurants named after him in California, Florida and Hawaii called Duke\'s.
On August 24, 2002, the 112th anniversary of Kahanamoku\'s birth, the U.S. Postal Service issued a first-class commemorative stamp with Duke\'s picture on it. The First Day Ceremony was held at the Hilton Hawaiian Village in Waikiki and was attended by thousands. At this ceremony, attendees could attach the Duke stamp to an envelope and get it canceled with a First Day of Issue postmark. These first day covers are very collectible.
On August 24, 2015, a Google Doodle honored the 125th anniversary of Duke Kahanamoku\'s birthday.
In 2021, a 88-minute feature film was made about Kahanamoku\'s life. It was later broadcast by PBS as part of their American Masters series
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# Don Tennant
**Donald G. Tennant** (November 23, 1922 -- December 8, 2001) was an American advertising agency executive.
He worked at the Leo Burnett agency in Chicago, Illinois. The agency placed anthropomorphic faces of \'critters\' on packaged goods. Tennant was in charge of the Marlboro account and invented the Marlboro Man
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# Dolly (sheep)
**Dolly** (5 July 1996 -- 14 February 2003) was a female Finn-Dorset sheep and the first mammal that was cloned from an adult somatic cell. She was cloned by associates of the Roslin Institute in Scotland, using the process of nuclear transfer from a cell taken from a mammary gland. Her cloning proved that a cloned organism could be produced from a mature cell from a specific body part. Contrary to popular belief, she was not the first animal to be cloned.
The employment of adult somatic cells in lieu of embryonic stem cells for cloning emerged from the foundational work of John Gurdon, who cloned African clawed frogs in 1958 with this approach. The successful cloning of Dolly led to widespread advancements within stem cell research, including the discovery of induced pluripotent stem cells.
Dolly lived at the Roslin Institute throughout her life and produced several lambs. She was euthanized at the age of six years due to a progressive lung disease. No cause which linked the disease to her cloning was found.
Dolly\'s body was preserved and donated by the Roslin Institute in Scotland to the National Museum of Scotland, where it has been regularly exhibited since 2003.
## Genesis
Dolly was cloned by Keith Campbell, Ian Wilmut and colleagues at the Roslin Institute, part of the University of Edinburgh, Scotland, and the biotechnology company PPL Therapeutics, based near Edinburgh. The funding for Dolly\'s cloning was provided by PPL Therapeutics and the Ministry of Agriculture. She was born on 5 July 1996. She has been called \"the world\'s most famous sheep\" by sources including BBC News and *Scientific American*.
The cell used as the donor for the cloning of Dolly was taken from a mammary gland, and the production of a healthy clone, therefore, proved that a cell taken from a specific part of the body could recreate a whole individual. On Dolly\'s name, Wilmut stated \"Dolly is derived from a mammary gland cell and we couldn\'t think of a more impressive pair of glands than Dolly Parton\'s.\"
## Birth
Dolly was born on 5 July 1996 and had three mothers: one provided the egg, another the DNA, and a third carried the cloned embryo to term. She was created using the technique of somatic cell nuclear transfer, where the cell nucleus from an adult cell is transferred into an unfertilized oocyte (developing egg cell) that has had its cell nucleus removed. The hybrid cell is then stimulated to divide by an electric shock, and when it develops into a blastocyst it is implanted in a surrogate mother. Dolly was the first clone produced from a cell taken from an adult mammal. The production of Dolly showed that genes in the nucleus of such a mature differentiated somatic cell are still capable of reverting to an embryonic totipotent state, creating a cell that can then go on to develop into any part of an animal.
Dolly\'s existence was announced to the public on 22 February 1997. It gained much attention in the media. A commercial with Scottish scientists playing with sheep was aired on TV, and a special report in *Time* magazine featured Dolly. *Science* featured Dolly as the breakthrough of the year. Even though Dolly was not the first animal cloned, she received media attention because she was the first cloned from an adult cell.
## Life
Dolly lived her entire life at the Roslin Institute in Midlothian. There she was bred with a Welsh Mountain ram and produced six lambs in total. Her first lamb, named Bonnie, was born in April 1998. The next year, Dolly produced twin lambs Sally and Rosie; further, she gave birth to triplets Lucy, Darcy and Cotton in 2000. In late 2001, at the age of four, Dolly developed arthritis and started to have difficulty walking. This was treated with anti-inflammatory drugs.
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## Death
On 14 February 2003, Dolly was euthanised because she had a progressive lung disease and severe arthritis. A Finn Dorset such as Dolly has a life expectancy of around 11 to 12 years, but Dolly lived 6.5 years. A post-mortem examination showed she had a form of lung cancer called ovine pulmonary adenocarcinoma, also known as Jaagsiekte, which is a fairly common disease of sheep and is caused by the retrovirus JSRV. Roslin scientists stated that they did not think there was a connection with Dolly being a clone, and that other sheep in the same flock had died of the same disease. Such lung diseases are a particular danger for sheep kept indoors, and Dolly had to sleep inside for security reasons.
Some in the press speculated that a contributing factor to Dolly\'s death was that she could have been born with a genetic age of six years, the same age as the sheep from which she was cloned. One basis for this idea was the finding that Dolly\'s telomeres were short, which is typically a result of the aging process. The Roslin Institute stated that intensive health screening did not reveal any abnormalities in Dolly that could have come from advanced aging.
In 2016, scientists reported no defects in thirteen cloned sheep, including four from the same cell line as Dolly. The first study to review the long-term health outcomes of cloning, the authors found no evidence of late-onset, non-communicable diseases other than some minor examples of osteoarthritis and concluded \"We could find no evidence, therefore, of a detrimental long-term effect of cloning by SCNT on the health of aged offspring among our cohort.\"
After her death Dolly\'s body was preserved via taxidermy and is currently on display at the National Museum of Scotland in Edinburgh.
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# Dolly (sheep)
## Legacy
After cloning was successfully demonstrated through the production of Dolly, many other large mammals were cloned, including pigs, deer, horses and bulls. The attempt to clone argali (mountain sheep) did not produce viable embryos. The attempt to clone a banteng bull was more successful, as were the attempts to clone mouflon (a form of wild sheep), both resulting in viable offspring. The reprogramming process that cells need to go through during cloning is not perfect and embryos produced by nuclear transfer often show abnormal development. Making cloned mammals was highly inefficient`{{snd}}`{=mediawiki}in 1996, Dolly was the only lamb that survived to adulthood from 277 attempts. By 2014, Chinese scientists were reported to have 70--80% success rates cloning pigs, and in 2016, a Korean company, Sooam Biotech, was producing 500 cloned embryos a day. Wilmut, who led the team that created Dolly, announced in 2007 that the nuclear transfer technique may never be sufficiently efficient for use in humans.
Cloning may have uses in preserving endangered species, and may become a viable tool for reviving extinct species. In January 2009, scientists from the Centre of Food Technology and Research of Aragon in northern Spain announced the cloning of the Pyrenean ibex, a form of wild mountain goat, which was officially declared extinct in 2000. Although the newborn ibex died shortly after birth due to physical defects in its lungs, it is the first time an extinct animal has been cloned, and may open doors for saving endangered and newly extinct species by resurrecting them from frozen tissue.
In July 2016, four identical clones of Dolly (Daisy, Debbie, Dianna, and Denise) were alive and healthy at nine years old.
*Scientific American* concluded in 2016 that the main legacy of Dolly has not been cloning of animals but in advances into stem cell research. Gene targeting was added in 2000, when researchers cloned female lamb Diana from sheep DNA altered to contain the human gene for alpha 1-antitrypsin. The human gene was specifically activated in the ewe's mammary gland, so Diana produced milk containing human alpha 1-antitrypsin. After Dolly, researchers realised that ordinary cells could be reprogrammed to induced pluripotent stem cells, which can be grown into any tissue.
The first successful cloning of a primate species was reported in January 2018, using the same method which produced Dolly. Two identical clones of a macaque monkey, Zhong Zhong and Hua Hua, were created by researchers in China and were born in late 2017.
In January 2019, scientists in China reported the creation of five identical cloned gene-edited monkeys, again using this method, and the gene-editing CRISPR-Cas9 technique allegedly used by He Jiankui in creating the first ever gene-modified human babies Lulu and Nana. The monkey clones were made in order to study several medical diseases
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# Equivalence relation
Equivalence}} `{{stack|{{Binary relations}}}}`{=mediawiki}
In mathematics, an **equivalence relation** is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number $a$ is equal to itself (reflexive). If $a = b$, then $b = a$ (symmetric). If $a = b$ and $b = c$, then $a = c$ (transitive).
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
## Notation
Various notations are used in the literature to denote that two elements $a$ and $b$ of a set are equivalent with respect to an equivalence relation $R;$ the most common are \"$a \sim b$\" and \"`{{math|''a'' ≡ ''b''}}`{=mediawiki}\", which are used when $R$ is implicit, and variations of \"$a \sim_R b$\", \"`{{math|''a'' ≡<sub>''R''</sub> ''b''}}`{=mediawiki}\", or \"${a\mathop{R}b}$\" to specify $R$ explicitly. Non-equivalence may be written \"`{{math|''a'' ≁ ''b''}}`{=mediawiki}\" or \"$a \not\equiv b$\".
## Definitions
A binary relation $\,\sim\,$ on a set $X$ is said to be an equivalence relation, if it is reflexive, symmetric and transitive. That is, for all $a, b,$ and $c$ in $X:$
- $a \sim a$ (reflexivity).
- $a \sim b$ if and only if $b \sim a$ (symmetry).
- If $a \sim b$ and $b \sim c$ then $a \sim c$ (transitivity).
$X$ together with the relation $\,\sim\,$ is called a setoid. The equivalence class of $a$ under $\,\sim,$ denoted $[a],$ is defined as $[a] = \{x \in X : x \sim a\}.$
### Alternative definition using relational algebra {#alternative_definition_using_relational_algebra}
In relational algebra, if $R\subseteq X\times Y$ and $S\subseteq Y\times Z$ are relations, then the composite relation $SR\subseteq X\times Z$ is defined so that $x \, SR \, z$ if and only if there is a $y\in Y$ such that $x \, R \, y$ and $y \, S \, z$. This definition is a generalisation of the definition of functional composition. The defining properties of an equivalence relation $R$ on a set $X$ can then be reformulated as follows:
- $\operatorname{id} \subseteq R$. (reflexivity). (Here, $\operatorname{id}$ denotes the identity function on $X$.)
- $R=R^{-1}$ (symmetry).
- $RR\subseteq R$ (transitivity).
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# Equivalence relation
## Examples
### Simple example {#simple_example}
On the set $X = \{a, b, c\}$, the relation $R = \{(a, a), (b, b), (c, c), (b, c), (c, b)\}$ is an equivalence relation. The following sets are equivalence classes of this relation: $[a] = \{a\}, ~~~~ [b] = [c] = \{b, c\}.$
The set of all equivalence classes for $R$ is $\{\{a\}, \{b, c\}\}.$ This set is a partition of the set $X$. It is also called the quotient set of $X$ by $R$.
### Equivalence relations {#equivalence_relations}
The following relations are all equivalence relations:
- \"Is equal to\" on the set of numbers. For example, $\tfrac{1}{2}$ is equal to $\tfrac{4}{8}.$
- \"Is similar to\" on the set of all triangles.
- \"Is congruent to\" on the set of all triangles.
- Given a function $f:X \to Y$, \"has the same image under $f$ as\" on the elements of $f$\'s domain $X$. For example, $0$ and $\pi$ have the same image under $\sin$, viz. $0$. In particular:
- \"Has the same absolute value as\" on the set of real numbers
- \"Has the same cosine as\" on the set of all angles.
- Given a natural number $n$, \"is congruent to, modulo $n$\" on the integers.
- \"Have the same length and direction\" (equipollence) on the set of directed line segments.
- \"Has the same birthday as\" on the set of all people.
### Relations that are not equivalences {#relations_that_are_not_equivalences}
- The relation \"≥\" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7.
- The relation \"has a common factor greater than 1 with\" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.
- The empty relation *R* (defined so that *aRb* is never true) on a set *X* is vacuously symmetric and transitive; however, it is not reflexive (unless *X* itself is empty).
- The relation \"is approximately equal to\" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions *f* and *g* are approximately equal near some point if the limit of *f − g* is 0 at that point, then this defines an equivalence relation.
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# Equivalence relation
## Connections to other relations {#connections_to_other_relations}
- A partial order is a relation that is reflexive, `{{em|[[Antisymmetric relation|antisymmetric]]}}`{=mediawiki}, and transitive.
- Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class.
- A strict partial order is irreflexive, transitive, and asymmetric.
- A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is total, that is, if for all $a,$ there exists some $b \text{ such that } a \sim b.$ Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and total relation.
- A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation.
- A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite.
- A preorder is reflexive and transitive.
- A congruence relation is an equivalence relation whose domain $X$ is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the normal subgroups).
- Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle.
- Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
## Well-definedness under an equivalence relation {#well_definedness_under_an_equivalence_relation}
If $\,\sim\,$ is an equivalence relation on $X,$ and $P(x)$ is a property of elements of $X,$ such that whenever $x \sim y,$ $P(x)$ is true if $P(y)$ is true, then the property $P$ is said to be well-defined or a `{{em|class invariant}}`{=mediawiki} under the relation $\,\sim.$
A frequent particular case occurs when $f$ is a function from $X$ to another set $Y;$ if $x_1 \sim x_2$ implies $f\left(x_1\right) = f\left(x_2\right)$ then $f$ is said to be a `{{em|morphism}}`{=mediawiki} for $\,\sim,$ a `{{em|class invariant under}}`{=mediawiki} $\,\sim,$ or simply `{{em|invariant under}}`{=mediawiki} $\,\sim.$ This occurs, e.g. in the character theory of finite groups. The latter case with the function $f$ can be expressed by a commutative triangle. See also invariant. Some authors use \"compatible with $\,\sim$\" or just \"respects $\,\sim$\" instead of \"invariant under $\,\sim$\".
More generally, a function may map equivalent arguments (under an equivalence relation $\,\sim_A$) to equivalent values (under an equivalence relation $\,\sim_B$). Such a function is known as a morphism from $\,\sim_A$ to $\,\sim_B.$
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# Equivalence relation
## Related important definitions {#related_important_definitions}
Let $a, b \in X$, and $\sim$ be an equivalence relation. Some key definitions and terminology follow:
### Equivalence class {#equivalence_class}
A subset $Y$ of $X$ such that $a \sim b$ holds for all $a$ and $b$ in $Y$, and never for $a$ in $Y$ and $b$ outside $Y$, is called an *equivalence class* of $X$ by $\sim$. Let $[a] := \{x \in X : a \sim x\}$ denote the equivalence class to which $a$ belongs. All elements of $X$ equivalent to each other are also elements of the same equivalence class.
### Quotient set {#quotient_set}
The set of all equivalence classes of $X$ by $\sim,$ denoted $X / \mathord{\sim} := \{[x] : x \in X\},$ is the *quotient set* of $X$ by $\sim.$ If $X$ is a topological space, there is a natural way of transforming $X / \sim$ into a topological space; see *Quotient space* for the details.`{{undue weight inline|date=October 2024}}`{=mediawiki}
### Projection
The *projection* of $\,\sim\,$ is the function $\pi : X \to X/\mathord{\sim}$ defined by $\pi(x) = [x]$ which maps elements of $X$ into their respective equivalence classes by $\,\sim.$
: **Theorem** on projections: Let the function $f : X \to B$ be such that if $a \sim b$ then $f(a) = f(b).$ Then there is a unique function $g : X / \sim \to B$ such that $f = g \pi.$ If $f$ is a surjection and $a \sim b \text{ if and only if } f(a) = f(b),$ then $g$ is a bijection.
### Equivalence kernel {#equivalence_kernel}
The **equivalence kernel** of a function $f$ is the equivalence relation \~ defined by $x \sim y \text{ if and only if } f(x) = f(y).$ The equivalence kernel of an injection is the identity relation.
### Partition
A *partition* of *X* is a set *P* of nonempty subsets of *X*, such that every element of *X* is an element of a single element of *P*. Each element of *P* is a *cell* of the partition. Moreover, the elements of *P* are pairwise disjoint and their union is *X*.
#### Counting partitions {#counting_partitions}
Let *X* be a finite set with *n* elements. Since every equivalence relation over *X* corresponds to a partition of *X*, and vice versa, the number of equivalence relations on *X* equals the number of distinct partitions of *X*, which is the *n*th Bell number *B~n~*:
: $B_n = \frac{1}{e} \sum_{k=0}^\infty \frac{k^n}{k!} \quad$ (Dobinski\'s formula).
## Fundamental theorem of equivalence relations {#fundamental_theorem_of_equivalence_relations}
A key result links equivalence relations and partitions:
- An equivalence relation \~ on a set *X* partitions *X*.
- Conversely, corresponding to any partition of *X*, there exists an equivalence relation \~ on *X*.
In both cases, the cells of the partition of *X* are the equivalence classes of *X* by \~. Since each element of *X* belongs to a unique cell of any partition of *X*, and since each cell of the partition is identical to an equivalence class of *X* by \~, each element of *X* belongs to a unique equivalence class of *X* by \~. Thus there is a natural bijection between the set of all equivalence relations on *X* and the set of all partitions of *X*.
## Comparing equivalence relations {#comparing_equivalence_relations}
If $\sim$ and $\approx$ are two equivalence relations on the same set $S$, and $a \sim b$ implies $a \approx b$ for all $a, b \in S,$ then $\approx$ is said to be a **coarser** relation than $\sim$, and $\sim$ is a **finer** relation than $\approx$. Equivalently,
- $\sim$ is finer than $\approx$ if every equivalence class of $\sim$ is a subset of an equivalence class of $\approx$, and thus every equivalence class of $\approx$ is a union of equivalence classes of $\sim$.
- $\sim$ is finer than $\approx$ if the partition created by $\sim$ is a refinement of the partition created by $\approx$.
The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.
The relation \"$\sim$ is finer than $\approx$\" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.
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# Equivalence relation
## Generating equivalence relations {#generating_equivalence_relations}
- Given any set $X,$ an equivalence relation over the set $[X \to X]$ of all functions $X \to X$ can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
- An equivalence relation $\,\sim\,$ on $X$ is the equivalence kernel of its surjective projection $\pi : X \to X / \sim.$ Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over $X,$ a partition of $X,$ and a projection whose domain is $X,$ are three equivalent ways of specifying the same thing.
- The intersection of any collection of equivalence relations over *X* (binary relations viewed as a subset of $X \times X$) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation *R* on *X*, the equivalence relation `{{em|generated by R}}`{=mediawiki} is the intersection of all equivalence relations containing *R* (also known as the smallest equivalence relation containing *R*). Concretely, *R* generates the equivalence relation
$$a \sim b$$ if there exists a natural number $n$ and elements $x_0, \ldots, x_n \in X$ such that $a = x_0$, $b = x_n$, and $x_{i-1} \mathrel{R} x_i$ or $x_i \mathrel{R} x_{i-1}$, for $i = 1, \ldots, n.$
: The equivalence relation generated in this manner can be trivial. For instance, the equivalence relation generated by any total order on *X* has exactly one equivalence class, *X* itself.
- Equivalence relations can construct new spaces by \"gluing things together.\" Let *X* be the unit Cartesian square $[0, 1] \times [0, 1],$ and let \~ be the equivalence relation on *X* defined by $(a, 0) \sim (a, 1)$ for all $a \in [0, 1]$ and $(0, b) \sim (1, b)$ for all $b \in [0, 1],$ Then the quotient space $X / \sim$ can be naturally identified (homeomorphism) with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.
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# Equivalence relation
## Algebraic structure {#algebraic_structure}
Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.
### Group theory {#group_theory}
Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hence permutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathematical structure of equivalence relations.
Let \'\~\' denote an equivalence relation over some nonempty set *A*, called the universe or underlying set. Let *G* denote the set of bijective functions over *A* that preserve the partition structure of *A*, meaning that for all $x \in A$ and $g \in G, g(x) \in [x].$ Then the following three connected theorems hold:
- \~ partitions *A* into equivalence classes. (This is the `{{em|Fundamental Theorem of Equivalence Relations}}`{=mediawiki}, mentioned above);
- Given a partition of *A*, *G* is a transformation group under composition, whose orbits are the cells of the partition;{{#tag:ref\|
*Proof*. Let function composition interpret group multiplication, and function inverse interpret group inverse. Then *G* is a group under composition, meaning that $x \in A$ and $g \in G, [g(x)] = [x],$ because *G* satisfies the following four conditions:
- *G is closed under composition*. The composition of any two elements of *G* exists, because the domain and codomain of any element of *G* is *A*. Moreover, the composition of bijections is bijective;
- *Existence of identity function*. The identity function, *I*(*x*) = *x*, is an obvious element of *G*;
- *Existence of inverse function*. Every bijective function *g* has an inverse *g*^−1^, such that *gg*^−1^ = *I*;
- *Composition associates*. *f*(*gh*) = (*fg*)*h*. This holds for all functions over all domains.
Let *f* and *g* be any two elements of *G*. By virtue of the definition of *G*, \[*g*(*f*(*x*))\] = \[*f*(*x*)\] and \[*f*(*x*)\] = \[*x*\], so that \[*g*(*f*(*x*))\] = \[*x*\]. Hence *G* is also a transformation group (and an automorphism group) because function composition preserves the partitioning of $A. \blacksquare$}}
- Given a transformation group *G* over *A*, there exists an equivalence relation \~ over *A*, whose equivalence classes are the orbits of *G*.
In sum, given an equivalence relation \~ over *A*, there exists a transformation group *G* over *A* whose orbits are the equivalence classes of *A* under \~.
This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe *A*. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, *A* → *A*.
Moving to groups in general, let *H* be a subgroup of some group *G*. Let \~ be an equivalence relation on *G*, such that $a \sim b \text{ if and only if } a b^{-1} \in H.$ The equivalence classes of \~---also called the orbits of the action of *H* on *G*---are the right **cosets** of *H* in *G*. Interchanging *a* and *b* yields the left cosets.
Related thinking can be found in Rosen (2008: chpt. 10).
### Categories and groupoids {#categories_and_groupoids}
Let *G* be a set and let \"\~\" denote an equivalence relation over *G*. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of *G*, and for any two elements *x* and *y* of *G*, there exists a unique morphism from *x* to *y* if and only if $x \sim y.$
The advantages of regarding an equivalence relation as a special case of a groupoid include:
- Whereas the notion of \"free equivalence relation\" does not exist, that of a free groupoid on a directed graph does. Thus it is meaningful to speak of a \"presentation of an equivalence relation,\" i.e., a presentation of the corresponding groupoid;
- Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies;
- In many contexts \"quotienting,\" and hence the appropriate equivalence relations often called congruences, are important. This leads to the notion of an internal groupoid in a category.
### Lattices
The equivalence relations on any set *X*, when ordered by set inclusion, form a complete lattice, called **Con** *X* by convention. The canonical map **ker** : *X*\^*X* → **Con** *X*, relates the monoid *X*\^*X* of all functions on *X* and **Con** *X*. **ker** is surjective but not injective. Less formally, the equivalence relation **ker** on *X*, takes each function *f* : *X* → *X* to its kernel **ker** *f*. Likewise, **ker(ker)** is an equivalence relation on *X*\^*X*.
## Equivalence relations and mathematical logic {#equivalence_relations_and_mathematical_logic}
Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples:
- *Reflexive and transitive*: The relation ≤ on **N**. Or any preorder;
- *Symmetric and transitive*: The relation *R* on **N**, defined as *aRb* ↔ *ab* ≠ 0. Or any partial equivalence relation;
- *Reflexive and symmetric*: The relation *R* on **Z**, defined as *aRb* ↔ \"*a* − *b* is divisible by at least one of 2 or 3.\" Or any dependency relation.
Properties definable in first-order logic that an equivalence relation may or may not possess include:
- The number of equivalence classes is finite or infinite;
- The number of equivalence classes equals the (finite) natural number *n*;
- All equivalence classes have infinite cardinality;
- The number of elements in each equivalence class is the natural number *n*
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# Evolutionary linguistics
**Evolutionary linguistics** or **Darwinian linguistics** is a sociobiological approach to the study of language. Evolutionary linguists consider linguistics as a subfield of sociobiology and evolutionary psychology. The approach is also closely linked with evolutionary anthropology, cognitive linguistics and biolinguistics. Studying languages as the products of nature, it is interested in the biological origin and development of language. Evolutionary linguistics is contrasted with humanistic approaches, especially structural linguistics.
A main challenge in this research is the lack of empirical data: there are no archaeological traces of early human language. Computational biological modelling and clinical research with artificial languages have been employed to fill in gaps of knowledge. Although biology is understood to shape the brain, which processes language, there is no clear link between biology and specific human language structures or linguistic universals.
For lack of a breakthrough in the field, there have been numerous debates about what kind of natural phenomenon language might be. Some researchers focus on the innate aspects of language. It is suggested that grammar has emerged adaptationally from the human genome, bringing about a language instinct; or that it depends on a single mutation which has caused a language organ to appear in the human brain. This is hypothesized to result in a crystalline grammatical structure underlying all human languages. Others suggest language is not crystallized, but fluid and ever-changing. Others, yet, liken languages to living organisms. Languages are considered analogous to a parasite or populations of mind-viruses. There is so far little scientific evidence for any of these claims, and some of them have been labelled as pseudoscience.
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# Evolutionary linguistics
## History
### 1863--1945: social Darwinism {#social_darwinism}
Although pre-Darwinian theorists had compared languages to living organisms as a metaphor, the comparison was first taken literally in 1863 by the historical linguist August Schleicher who was inspired by Charles Darwin\'s *On the Origin of Species*. At the time there was not enough evidence to prove that Darwin\'s theory of natural selection was correct. Schleicher proposed that linguistics could be used as a testing ground for the study of the evolution of species. A review of Schleicher\'s book *Darwinism as Tested by the Science of Language* appeared in the first issue of *Nature* journal in 1870. Darwin reiterated Schleicher\'s proposition in his 1871 book *The Descent of Man*, claiming that languages are comparable to species, and that language change occurs through natural selection as words \'struggle for life\'. Darwin believed that languages had evolved from animal mating calls. Darwinists considered the concept of language creation as unscientific.
August Schleicher and his friend Ernst Haeckel were keen gardeners and regarded the study of cultures as a type of botany, with different species competing for the same living space. Similar ideas became later advocated by politicians who wanted to appeal to working class voters, not least by the national socialists who subsequently included the concept of struggle for living space in their agenda. Highly influential until the end of World War II, social Darwinism was eventually banished from human sciences, leading to a strict separation of natural and sociocultural studies.
This gave rise to the dominance of structural linguistics in Europe. There had long been a dispute between the Darwinists and the French intellectuals with the topic of language evolution famously having been banned by the Paris Linguistic Society as early as in 1866. Ferdinand de Saussure proposed structuralism to replace evolutionary linguistics in his *Course in General Linguistics*, published posthumously in 1916. The structuralists rose to academic political power in human and social sciences in the aftermath of the student revolts of Spring 1968, establishing the Sorbonne as an international centrepoint of humanistic thinking.
### From 1959 onwards: genetic determinism {#from_1959_onwards_genetic_determinism}
In the United States, structuralism was however fended off by the advocates of behavioural psychology; a linguistics framework nicknamed as \'American structuralism\'. It was eventually replaced by the approach of Noam Chomsky who published a modification of Louis Hjelmslev\'s formal structuralist theory, claiming that syntactic structures are innate. An active figure in peace demonstrations in the 1950s and 1960s, Chomsky rose to academic political power following Spring 1968 at the MIT.
Chomsky became an influential opponent of the French intellectuals during the following decades, and his supporters successfully confronted the post-structuralists in the *Science Wars* of the late 1990s. The shift of the century saw a new academic funding policy where interdisciplinary research became favoured, effectively directing research funds to biological humanities. The decline of structuralism was evident by 2015 with Sorbonne having lost its former spirit.
Chomsky eventually claimed that syntactic structures are caused by a random mutation in the human genome, proposing a similar explanation for other human faculties such as ethics. But Steven Pinker argued in 1990 that they are the outcome of evolutionary adaptations.
### From 1976 onwards: Neo-Darwinism {#from_1976_onwards_neo_darwinism}
At the same time when the Chomskyan paradigm of biological determinism defeated humanism, it was losing its own clout within sociobiology. It was reported likewise in 2015 that generative grammar was under fire in applied linguistics and in the process of being replaced with *usage-based linguistics*; a derivative of Richard Dawkins\'s memetics. It is a concept of linguistic units as replicators. Following the publication of memetics in Dawkins\'s 1976 nonfiction bestseller *The Selfish Gene*, many biologically inclined linguists, frustrated with the lack of evidence for Chomsky\'s Universal Grammar, grouped under different brands including a framework called Cognitive Linguistics (with capitalised initials), and \'functional\' (adaptational) linguistics (not to be confused with functional linguistics) to confront both Chomsky and the humanists. The replicator approach is today dominant in evolutionary linguistics, applied linguistics, cognitive linguistics and linguistic typology; while the generative approach has maintained its position in general linguistics, especially syntax; and in computational linguistics.
## View of linguistics {#view_of_linguistics}
Evolutionary linguistics is part of a wider framework of Universal Darwinism. In this view, linguistics is seen as an ecological environment for research traditions struggling for the same resources. According to David Hull, these traditions correspond to species in biology. Relationships between research traditions can be symbiotic, competitive or parasitic. An adaptation of Hull\'s theory in linguistics is proposed by William Croft. He argues that the Darwinian method is more advantageous than linguistic models based on physics, structuralist sociology, or hermeneutics.
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# Evolutionary linguistics
## Approaches
Evolutionary linguistics is often divided into functionalism and formalism, concepts which are not to be confused with functionalism and formalism in the humanistic reference. Functional evolutionary linguistics considers languages as adaptations to human mind. The formalist view regards them as crystallised or non-adaptational.
### Functionalism (adaptationism) {#functionalism_adaptationism}
The adaptational view of language is advocated by various frameworks of cognitive and evolutionary linguistics, with the terms \'functionalism\' and \'Cognitive Linguistics\' often being equated. It is hypothesised that the evolution of the animal brain provides humans with a mechanism of abstract reasoning which is a \'metaphorical\' version of image-based reasoning. Language is not considered as a separate area of cognition, but as coinciding with general cognitive capacities, such as perception, attention, motor skills, and spatial and visual processing. It is argued to function according to the same principles as these.
It is thought that the brain links action schemes to form--meaning pairs which are called constructions. Cognitive linguistic approaches to syntax are called cognitive and construction grammar. Also deriving from memetics and other cultural replicator theories, these can study the natural or social selection and adaptation of linguistic units. Adaptational models reject a formal systemic view of language and consider language as a population of linguistic units.
The bad reputation of social Darwinism and memetics has been discussed in the literature, and recommendations for new terminology have been given. What correspond to replicators or mind-viruses in memetics are called *linguemes* in Croft\'s *theory of Utterance Selection* (TUS), and likewise linguemes or constructions in construction grammar and usage-based linguistics; and metaphors, frames or schemas in cognitive and construction grammar. The reference of memetics has been largely replaced with that of a Complex Adaptive System. In current linguistics, this term covers a wide range of evolutionary notions while maintaining the Neo-Darwinian concepts of replication and replicator population.
Functional evolutionary linguistics is not to be confused with functional humanistic linguistics.
### Formalism (structuralism) {#formalism_structuralism}
Advocates of formal evolutionary explanation in linguistics argue that linguistic structures are crystallised. Inspired by 19th century advances in crystallography, Schleicher argued that different types of languages are like plants, animals and crystals. The idea of linguistic structures as frozen drops was revived in tagmemics, an approach to linguistics with the goal to uncover divine symmetries underlying all languages, as if caused by the Creation.
In modern biolinguistics, the X-bar tree is argued to be like natural systems such as ferromagnetic droplets and botanic forms. Generative grammar considers syntactic structures similar to snowflakes. It is hypothesised that such patterns are caused by a mutation in humans.
The formal--structural evolutionary aspect of linguistics is not to be confused with structural linguistics.
## Evidence
There was some hope of a breakthrough with the discovery of the *FOXP2* gene. There is little support, however, for the idea that *FOXP2* is \'the grammar gene\' or that it had much to do with the relatively recent emergence of syntactical speech. The idea that people have a language instinct is disputed. Memetics is sometimes discredited as pseudoscience and neurological claims made by evolutionary cognitive linguists have been likened to pseudoscience. All in all, there does not appear to be any evidence for the basic tenets of evolutionary linguistics beyond the fact that language is processed by the brain, and brain structures are shaped by genes.
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# Evolutionary linguistics
## Criticism
Evolutionary linguistics has been criticised by advocates of (humanistic) structural and functional linguistics. Ferdinand de Saussure commented on 19th century evolutionary linguistics: `{{Blockquote|text=
"Language was considered a specific sphere, a fourth natural kingdom; this led to methods of reasoning which would have caused astonishment in other sciences. Today one cannot read a dozen lines written at that time without being struck by absurdities of reasoning and by the terminology used to justify these absurdities”<ref>{{cite book |last=de Saussure |first=Ferdinand |title=Course in general linguistics |place=New York |publisher=Philosophy Library |date=1959 |orig-year=First published 1916 |url=https://monoskop.org/images/0/0b/Saussure_Ferdinand_de_Course_in_General_Linguistics_1959.pdf |isbn=9780231157278 |author-link=Ferdinand de Saussure |access-date=2020-03-04 |archive-date=2020-04-14 |archive-url=https://web.archive.org/web/20200414113626/https://monoskop.org/images/0/0b/Saussure_Ferdinand_de_Course_in_General_Linguistics_1959.pdf |url-status=dead }}</ref>|}}`{=mediawiki}
Mark Aronoff, however, argues that historical linguistics had its golden age during the time of Schleicher and his supporters, enjoying a place among the hard sciences, and considers the return of Darwinian linguistics as a positive development. Esa Itkonen nonetheless deems the revival of Darwinism as a hopeless enterprise: `{{Blockquote|text=
"There is ... an application of intelligence in linguistic change which is absent in biological evolution; and this suffices to make the two domains totally disanalogous ... [Grammaticalisation depends on] cognitive processes, ultimately serving the goal of problem solving, which intelligent entities like humans must perform all the time, but which biological entities like genes cannot perform. Trying to eliminate this basic difference leads to confusion.”<ref name=Itkonen_1999>{{cite journal |last=Itkonen |first=Esa |year=1999 |title=Functionalism yes, biologism no |journal=Zeitschrift für Sprachwissenschaft |volume=18 |issue=2 |pages=219–221 |doi=10.1515/zfsw.1999.18.2.219|s2cid=146998564 |doi-access=free }}</ref>|}}`{=mediawiki}
Itkonen also points out that the principles of natural selection are not applicable because language innovation and acceptance have the same source which is the speech community. In biological evolution, mutation and selection have different sources. This makes it possible for people to change their languages, but not their genotype
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# Ethical non-naturalism
**Ethical non-naturalism** (or **moral non-naturalism**) is the meta-ethical view which claims that:
1. Ethical sentences express propositions.
2. Some such propositions are true.
3. Those propositions are made true by objective features of the world, independent of human opinion.
4. These moral features of the world are *not* reducible to any set of non-moral features.
This makes ethical non-naturalism a non-definist form of moral realism, which is in turn a form of cognitivism. Ethical non-naturalism stands in opposition to ethical naturalism, which claims that moral terms and properties are reducible to non-moral terms and properties, as well as to all forms of moral anti-realism, including ethical subjectivism (which denies that moral propositions refer to objective facts), error theory (which denies that any moral propositions are true), and non-cognitivism (which denies that moral sentences express propositions at all).
## Definitions and examples {#definitions_and_examples}
According to G. E. Moore, \"Goodness is a simple, undefinable, non-natural property.\" To call goodness \"non-natural\" does not mean that it is supernatural or divine. It does mean, however, that goodness cannot be reduced to natural properties such as needs, wants or pleasures. Moore also stated that a reduction of ethical properties to a divine command would be the same as stating their naturalness. This would be an example of what he referred to as \"the naturalistic fallacy.\"
Moore claimed that goodness is \"indefinable\", i.e., it cannot be defined in any other terms. This is the central claim of non-naturalism. Thus, the meaning of sentences containing the word \"good\" cannot be explained entirely in terms of sentences not containing the word \"good.\" One cannot substitute words referring to pleasure, needs or anything else in place of \"good.\"
Some properties, such as hardness, roundness and dampness, are clearly natural properties. We encounter them in the real world and can perceive them. On the other hand, other properties, such as being good and being right, are not so obvious. A great novel is considered to be a good thing; goodness may be said to be a property of that novel. Paying one\'s debts and telling the truth are generally held to be right things to do; rightness may be said to be a property of certain human actions.
However, these two types of property are quite different. Those natural properties, such as hardness and roundness, can be perceived and encountered in the real world. On the other hand, it is not immediately clear how to physically see, touch or measure the goodness of a novel or the rightness of an action.
## A difficult question {#a_difficult_question}
Moore did not consider goodness and rightness to be natural properties, i.e., they cannot be defined in terms of any natural properties. How, then, can we know that anything is good and how can we distinguish good from bad?
Moral epistemology, the part of epistemology (and/or ethics) that studies how we know moral facts and how moral beliefs are justified, has proposed an answer. British epistemologists, following Moore, suggested that humans have a special faculty, a faculty of moral intuition, which tells us what is good and bad, right and wrong.
Ethical intuitionists assert that, if we see a good person or a right action, and our faculty of moral intuition is sufficiently developed and unimpaired, we simply intuit that the person is good or that the action is right. Moral intuition is supposed to be a mental process different from other, more familiar faculties like sense-perception, and that moral judgments are its outputs. When someone judges something to be good, or some action to be right, then the person is using the faculty of moral intuition. The faculty is attuned to those non-natural properties. Perhaps the best ordinary notion that approximates moral intuition would be the idea of a conscience.
## Another argument for non-naturalism {#another_argument_for_non_naturalism}
Moore also introduced what is called the open-question argument, a position he later rejected.
Suppose a definition of \"good\" is \"pleasure-causing.\" In other words, if something is good, it causes pleasure; if it causes pleasure, then it is, by definition, good. Moore asserted, however, that we could always ask, \"But are pleasure-causing things good?\" This would always be an open question. There is no foregone conclusion that, indeed, pleasure-causing things are good. In his initial argument, Moore concluded that any similar definition of goodness could be criticized in the same way
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# Demand-pull inflation
`{{Macroeconomics sidebar}}`{=mediawiki}
**Demand-pull inflation** occurs when aggregate demand in an economy is more than aggregate supply. It involves inflation rising as real gross domestic product rises and unemployment falls, as the economy moves along the Phillips curve. This is commonly described as \"too much money chasing too few goods\". More accurately, it should be described as involving \"too much money spent chasing too few goods\", since only money that is spent on goods and services can cause inflation. This would not be expected to happen, unless the economy is already at a full employment level. It is the opposite of cost-push inflation.
## How it occurs {#how_it_occurs}
In Keynesian theory, increased employment results in increased aggregate demand (AD), which leads to further hiring by firms to increase output. Due to capacity constraints, this increase in output will eventually become so small that the price of the good will rise. At first, unemployment will go down, shifting AD1 to AD2, which increases demand (noted as \"Y\") by (Y2 − Y1). This increase in demand means more workers are needed, and then AD will be shifted from AD2 to AD3, but this time much less is produced than in the previous shift, but the price level has risen from P2 to P3, a much higher increase in price than in the previous shift. This increase in price is what causes inflation in an overheating economy.
Demand-pull inflation is in contrast with cost-push inflation, when price and wage increases are being transmitted from one sector to another. However, these can be considered as different aspects of an overall inflationary process---demand-pull inflation explains how price inflation starts, and cost-push inflation demonstrates why inflation once begun is so difficult to stop.
## Causes of demand-pull inflation {#causes_of_demand_pull_inflation}
- There is a quick increase in consumption and investment along with extremely confident firms.
- There is a sudden increase in exports due to huge under-valuation of the currency.
- There is a lot of government spending.
- The expectation that inflation will rise often leads to a rise in inflation. Workers and firms will increase their prices to \'catch up\' to inflation.
- There is excessive monetary growth, when there is too much money in the system chasing too few goods. The \'price\' of a good will thus increase.
- There is a rise in population
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# Cost-push inflation
`{{Macroeconomics sidebar}}`{=mediawiki} **Cost-push inflation** is a purported type of inflation caused by increases in the cost of important goods or services where no suitable alternative is available.
## Cause
As businesses face higher prices for underlying inputs, they are forced to increase prices of their outputs. It is contrasted with the theory of demand-pull inflation. Both accounts of inflation have at various times been put forward, with inconclusive evidence as to which explanation is superior. Cost-push inflation can also result from a rise in expected inflation, which in turn the workers will demand higher wages, thus causing inflation.
## Examples
One example of cost-push inflation is the oil crisis of the 1970s, which some economists see as a major cause of the inflation experienced in the Western world in that decade. It is argued that this inflation resulted from increases in the cost of petroleum imposed by the member states of OPEC. Since petroleum is so important to industrialized economies, a large increase in its price can lead to the increase in the price of most products, raising the price level. Some economists argue that such a change in the price level can raise the inflation rate over longer periods, due to adaptive expectations and the price/wage spiral, so that a supply shock can have persistent effects.
## Debated existence and opposition to the concept {#debated_existence_and_opposition_to_the_concept}
The existence of cost-push inflation is disputed. Dallas S. Batten described it as a myth, writing \"Though the cost-push argument is appealing on the surface, neither economic theory nor empirical evidence indicates that businesses and labor can cause continually rising prices\", and identifying the real cause as \"increased aggregate demand resulting from increased money growth\".
Milton Friedman criticised the concept of cost-push inflation, writing \"To each businessman separately it looks as if he has to raise prices because costs have gone up. But then, we must ask, \'Why did his costs go up? \... The answer is, because \... total demand all over was increasing.\" Friedman wrote, \"the inflation arises from one and only one reason: an increase in a quantity of money
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# Extractor (mathematics)
An $(N,M,D,K,\epsilon)$ -**extractor** is a bipartite graph with $N$ nodes on the left and $M$ nodes on the right such that each node on the left has $D$ neighbors (on the right), which has the added property that for any subset $A$ of the left vertices of size at least $K$, the distribution on right vertices obtained by choosing a random node in $A$ and then following a random edge to get a node x on the right side is $\epsilon$-close to the uniform distribution in terms of total variation distance.
A disperser is a related graph.
An equivalent way to view an extractor is as a bivariate function
$$E : [N] \times [D] \rightarrow [M]$$
in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness $X$ that gives $n$ bits with min-entropy $\log K$, the distribution $E(X,U_D)$ is $\epsilon$-close to $U_M$, where $U_T$ denotes the uniform distribution on $[T]$.
Extractors are interesting when they can be constructed with small $K,D,\epsilon$ relative to $N$ and $M$ is as close to $KD$ (the total randomness in the input sources) as possible.
Extractor functions were originally researched as a way to *extract* randomness from weakly random sources. *See* randomness extractor.
Using the probabilistic method it is easy to show that extractor graphs with really good parameters exist. The challenge is to find explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor (and disperser) graphs have found many applications in computer science
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# Expander graph
In graph theory, an **expander graph** is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.
## Definitions
Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not \"too large\" has a \"large\" boundary. Different formalisations of these notions give rise to different notions of expanders: *edge expanders*, *vertex expanders*, and *spectral expanders*, as defined below.
A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected finite graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. Informally, a graph is a good expander if it has low degree and high expansion parameters.
### Edge expansion {#edge_expansion}
The *edge expansion* (also *isoperimetric number* or Cheeger constant) `{{math|''h''(''G'')}}`{=mediawiki} of a graph `{{mvar|G}}`{=mediawiki} on `{{mvar|n}}`{=mediawiki} vertices is defined as
: $h(G) = \min_{0 < |S| \le \frac{n}{2} } \frac{|\partial S|}{|S|},$
: where $\partial S := \{ \{ u, v \} \in E(G) \ : \ u \in S, v \notin S \},$
which can also be written as `{{math|1=∂''S'' = ''E''(''S'', {{overline|''S''}})}}`{=mediawiki} with `{{math|1={{overline|''S''}} := ''V''(''G'') \ ''S''}}`{=mediawiki} the complement of `{{mvar|S}}`{=mediawiki} and
$$E(A,B) = \{ \{ u, v \} \in E(G) \ : \ u \in A , v \in B \}$$ the edges between the subsets of vertices `{{math|''A'',''B'' ⊆ ''V''(''G'')}}`{=mediawiki}.
In the equation, the minimum is over all nonempty sets `{{mvar|S}}`{=mediawiki} of at most `{{math|{{frac|''n''|2}}}}`{=mediawiki} vertices and `{{math|∂''S''}}`{=mediawiki} is the *edge boundary* of `{{mvar|S}}`{=mediawiki}, i.e., the set of edges with exactly one endpoint in `{{mvar|S}}`{=mediawiki}.
Intuitively,
: $\min {|\partial S|} = \min E({S}, \overline{S})$
is the minimum number of edges that need to be cut in order to split the graph in two. The edge expansion normalizes this concept by dividing with smallest number of vertices among the two parts. To see how the normalization can drastically change the value, consider the following example. Take two complete graphs with the same number of vertices `{{mvar|n}}`{=mediawiki} and add `{{mvar|n}}`{=mediawiki} edges between the two graphs by connecting their vertices one-to-one. The minimum cut will be `{{mvar|n}}`{=mediawiki} but the edge expansion will be 1.
Notice that in `{{math|min {{abs|∂''S''}}}}`{=mediawiki}, the optimization can be equivalently done either over `{{math|0 ≤ {{abs|''S''}} ≤ {{frac|''n''|2}}}}`{=mediawiki} or over any non-empty subset, since $E(S, \overline{S}) = E(\overline{S}, S)$. The same is not true for `{{math|''h''(''G'')}}`{=mediawiki} because of the normalization by `{{math|{{abs|''S''}}}}`{=mediawiki}. If we want to write `{{math|''h''(''G'')}}`{=mediawiki} with an optimization over all non-empty subsets, we can rewrite it as
: $h(G) = \min_{\emptyset \subsetneq S\subsetneq V(G) } \frac{E({S}, \overline{S})}{\min\{|S|, |\overline{S}|\}}.$
### Vertex expansion {#vertex_expansion}
\[\[<File:Vertex> expansion.svg\|thumb\|220px\|Here, a subset `{{mvar|S}}`{=mediawiki} of the graph `{{mvar|G}}`{=mediawiki} (denoted red) has 4 vertices, and 2 vertices outside the subset that are neighbors of `{{mvar|S}}`{=mediawiki} (denoted green). The number of neighboring vertices divided by the size of the subset is denoted $|\partial_{out} S|/|S|$, which here is $2/4 = 0.5$.
The **vertex expansion** (or vertex isoperimetric number) is the minimum $|\partial_{out} S|/|S|$of all subsets of the graph `{{mvar|G}}`{=mediawiki} which are not empty and whose size is less than or equal to half the size of `{{mvar|G}}`{=mediawiki}. For this graph `{{mvar|G}}`{=mediawiki}, this subset `{{mvar|S}}`{=mediawiki} has the smallest value $|\partial_{out} S|/|S|$, and therefore 0.5 is the vertex expansion of `{{mvar|G}}`{=mediawiki}.\]\]
The *vertex isoperimetric number* `{{math|''h''{{sub|out}}(''G'')}}`{=mediawiki} (also *vertex expansion* or *magnification*) of a graph `{{mvar|G}}`{=mediawiki} is defined as
: $h_{\text{out}}(G) = \min_{0 < |S|\le \frac{n}{2}} \frac{|\partial_{\text{out}}(S)|}{|S|},$
where `{{math|∂{{sub|out}}(''S'')}}`{=mediawiki} is the *outer boundary* of `{{mvar|S}}`{=mediawiki}, i.e., the set of vertices in `{{math|''V''(''G'') \ ''S''}}`{=mediawiki} with at least one neighbor in `{{mvar|S}}`{=mediawiki}. In a variant of this definition (called *unique neighbor expansion*) `{{math|∂{{sub|out}}(''S'')}}`{=mediawiki} is replaced by the set of vertices in `{{mvar|V}}`{=mediawiki} with *exactly* one neighbor in `{{mvar|S}}`{=mediawiki}.
The *vertex isoperimetric number* `{{math|''h''{{sub|in}}(''G'')}}`{=mediawiki} of a graph `{{mvar|G}}`{=mediawiki} is defined as
: $h_{\text{in}}(G) = \min_{0 < |S|\le \frac{n}{2}} \frac{|\partial_{\text{in}}(S)|}{|S|},$
where $\partial_{\text{in}}(S)$ is the *inner boundary* of `{{mvar|S}}`{=mediawiki}, i.e., the set of vertices in `{{mvar|S}}`{=mediawiki} with at least one neighbor in `{{math|''V''(''G'') \ ''S''}}`{=mediawiki}.
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# Expander graph
## Definitions
### Spectral expansion {#spectral_expansion}
When `{{mvar|G}}`{=mediawiki} is `{{mvar|d}}`{=mediawiki}-regular, a linear algebraic definition of expansion is possible based on the eigenvalues of the adjacency matrix `{{math|1=''A'' = ''A''(''G'')}}`{=mediawiki} of `{{mvar|G}}`{=mediawiki}, where `{{mvar|A{{sub|ij}}}}`{=mediawiki} is the number of edges between vertices `{{mvar|i}}`{=mediawiki} and `{{mvar|j}}`{=mediawiki}. Because `{{mvar|A}}`{=mediawiki} is symmetric, the spectral theorem implies that `{{mvar|A}}`{=mediawiki} has `{{mvar|n}}`{=mediawiki} real-valued eigenvalues `{{math|''λ''{{sub|1}} ≥ ''λ''{{sub|2}} ≥ … ≥ ''λ''{{sub|''n''}}}}`{=mediawiki}. It is known that all these eigenvalues are in `{{math|[−''d'', ''d'']}}`{=mediawiki} and more specifically, it is known that `{{math|1=''λ''{{sub|''n''}} = −''d''}}`{=mediawiki} if and only if `{{mvar|G}}`{=mediawiki} is bipartite.
More formally, we refer to an `{{mvar|n}}`{=mediawiki}-vertex, `{{mvar|d}}`{=mediawiki}-regular graph with
$$\max_{i \neq 1}|\lambda_i| \leq \lambda$$ as an `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-*graph*. The bound given by an `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graph on `{{math|''λ''{{sub|''i''}}}}`{=mediawiki} for `{{math|''i'' ≠ 1}}`{=mediawiki} is useful in many contexts, including the expander mixing lemma.
Spectral expansion can be *two-sided*, as above, with $\max_{i \neq 1}|\lambda_i| \leq \lambda$, or it can be *one-sided*, with $\max_{i \neq 1}\lambda_i \leq \lambda$. The latter is a weaker notion that holds also for bipartite graphs and is still useful for many applications, such as the Alon--Chung lemma.
Because `{{mvar|G}}`{=mediawiki} is regular, the uniform distribution $u\in\R^n$ with `{{math|1=''u{{sub|i}}'' = {{frac|1|''n''}}}}`{=mediawiki} for all `{{math|1=''i'' = 1, …, ''n''}}`{=mediawiki} is the stationary distribution of `{{mvar|G}}`{=mediawiki}. That is, we have `{{math|1=''Au'' = ''du''}}`{=mediawiki}, and `{{mvar|u}}`{=mediawiki} is an eigenvector of `{{mvar|A}}`{=mediawiki} with eigenvalue `{{math|1=''λ''{{sub|1}} = ''d''}}`{=mediawiki}, where `{{mvar|d}}`{=mediawiki} is the degree of the vertices of `{{mvar|G}}`{=mediawiki}. The *spectral gap* of `{{mvar|G}}`{=mediawiki} is defined to be `{{math|''d'' − ''λ''{{sub|2}}}}`{=mediawiki}, and it measures the spectral expansion of the graph `{{mvar|G}}`{=mediawiki}.
If we set
$$\lambda=\max\{|\lambda_2|, |\lambda_n|\}$$ as this is the largest eigenvalue corresponding to an eigenvector orthogonal to `{{mvar|u}}`{=mediawiki}, it can be equivalently defined using the Rayleigh quotient:
$$\lambda=\max_{v \perp u , v \neq 0} \frac{\|Av\|_2}{\|v\|_2},$$ where
$$\|v\|_2=\left(\sum_{i=1}^n v_i^2\right)^{1/2}$$ is the 2-norm of the vector $v\in\R^n$.
The normalized versions of these definitions are also widely used and more convenient in stating some results. Here one considers the matrix `{{math|{{sfrac|1|''d''}}''A''}}`{=mediawiki}, which is the Markov transition matrix of the graph `{{mvar|G}}`{=mediawiki}. Its eigenvalues are between −1 and 1. For not necessarily regular graphs, the spectrum of a graph can be defined similarly using the eigenvalues of the Laplacian matrix. For directed graphs, one considers the singular values of the adjacency matrix `{{mvar|A}}`{=mediawiki}, which are equal to the roots of the eigenvalues of the symmetric matrix `{{math|''A''{{sup|T}}''A''}}`{=mediawiki}.
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# Expander graph
## Definitions
### Expander families {#expander_families}
A family $(G_i)_{i \in \mathbb N}$ of $d$-regular graphs of increasing size is an **expander family** if $h(G_i)$ is bounded away from zero.
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# Expander graph
## Relationships between different expansion properties {#relationships_between_different_expansion_properties}
The expansion parameters defined above are related to each other. In particular, for any `{{mvar|d}}`{=mediawiki}-regular graph `{{mvar|G}}`{=mediawiki},
$$h_{\text{out}}(G) \le h(G) \le d \cdot h_{\text{out}}(G).$$ Consequently, for constant degree graphs, vertex and edge expansion are qualitatively the same.
### Cheeger inequalities {#cheeger_inequalities}
When `{{mvar|G}}`{=mediawiki} is `{{mvar|d}}`{=mediawiki}-regular, meaning each vertex is of degree `{{mvar|d}}`{=mediawiki}, there is a relationship between the isoperimetric constant `{{math|''h''(''G'')}}`{=mediawiki} and the gap `{{math|''d'' − ''λ''{{sub|2}}}}`{=mediawiki} in the spectrum of the adjacency operator of `{{mvar|G}}`{=mediawiki}. By standard spectral graph theory, the trivial eigenvalue of the adjacency operator of a `{{mvar|d}}`{=mediawiki}-regular graph is `{{math|1=''λ''{{sub|1}} = ''d''}}`{=mediawiki} and the first non-trivial eigenvalue is `{{math|''λ''{{sub|2}}}}`{=mediawiki}. If `{{mvar|G}}`{=mediawiki} is connected, then `{{math|''λ''{{sub|2}} < ''d''}}`{=mediawiki}. An inequality due to Dodziuk and independently Alon and Milman states that
: $\tfrac{1}{2}(d - \lambda_2) \le h(G) \le \sqrt{2d(d - \lambda_2)}.$
In fact, the lower bound is tight. The lower bound is achieved in limit for the hypercube `{{mvar|Q{{sub|n}}}}`{=mediawiki}, where `{{math|1=''h''(''G'') = 1}}`{=mediawiki} and `{{math|1=''d'' – ''λ''{{sub|2}} = 2}}`{=mediawiki}. The upper bound is (asymptotically) achieved for a cycle, where `{{math|1=''h''(''C{{sub|n}}'') = 4/''n'' = Θ(1/''n'')}}`{=mediawiki} and `{{math|1=''d'' – ''λ''{{sub|2}} = 2 – 2cos(2<math>\pi</math>/''n'') ≈ (2<math>\pi</math>/''n''){{sup|2}} = Θ(1/''n''{{sup|2}})}}`{=mediawiki}. A better bound is given in as
: $h(G) \le \sqrt{d^2 - \lambda_2^2}.$
These inequalities are closely related to the Cheeger bound for Markov chains and can be seen as a discrete version of Cheeger\'s inequality in Riemannian geometry.
Similar connections between vertex isoperimetric numbers and the spectral gap have also been studied:
: $h_{\text{out}}(G)\le \left(\sqrt{4 (d-\lambda_2)} + 1\right)^2 -1$
: $h_{\text{in}}(G) \le \sqrt{8(d-\lambda_2)}.$
Asymptotically speaking, the quantities `{{math|{{frac|''h''{{sup|2}}|''d''}}}}`{=mediawiki}, `{{math|''h''{{sub|out}}}}`{=mediawiki}, and `{{math|''h''{{sub|in}}{{sup|2}}}}`{=mediawiki} are all bounded above by the spectral gap `{{math|''O''(''d'' – ''λ''{{sub|2}})}}`{=mediawiki}.
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# Expander graph
## Constructions
There are four general strategies for explicitly constructing families of expander graphs. The first strategy is algebraic and group-theoretic, the second strategy is analytic and uses additive combinatorics, the third strategy is combinatorial and uses the zig-zag and related graph products, and the fourth strategy is based on lifts. Noga Alon showed that certain graphs constructed from finite geometries are the sparsest examples of highly expanding graphs.
### Margulis--Gabber--Galil
Algebraic constructions based on Cayley graphs are known for various variants of expander graphs. The following construction is due to Margulis and has been analysed by Gabber and Galil. For every natural number `{{mvar|n}}`{=mediawiki}, one considers the graph `{{mvar|G{{sub|n}}}}`{=mediawiki} with the vertex set $\mathbb Z_n \times \mathbb Z_n$, where $\mathbb Z_n=\mathbb Z/n\mathbb Z$: For every vertex $(x,y)\in\mathbb Z_n \times \mathbb Z_n$, its eight adjacent vertices are
$$(x \pm 2y,y), (x \pm (2y+1),y), (x,y \pm 2x), (x,y \pm (2x+1)).$$
Then the following holds:
> **Theorem.** For all `{{mvar|n}}`{=mediawiki}, the graph `{{mvar|G{{sub|n}}}}`{=mediawiki} has second-largest eigenvalue $\lambda(G)\leq 5 \sqrt{2}$.
### Ramanujan graphs {#ramanujan_graphs}
By a theorem of Alon and Boppana, all sufficiently large `{{mvar|d}}`{=mediawiki}-regular graphs satisfy $\lambda_2 \ge 2\sqrt{d-1} - o(1)$, where `{{math|''λ''{{sub|2}}}}`{=mediawiki} is the second largest eigenvalue in absolute value. As a direct consequence, we know that for every fixed `{{mvar|d}}`{=mediawiki} and $\lambda< 2 \sqrt{d-1}$ , there are only finitely many `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graphs. Ramanujan graphs are `{{mvar|d}}`{=mediawiki}-regular graphs for which this bound is tight, satisfying
$$\lambda = \max_{|\lambda_i| < d} |\lambda_i| \le 2\sqrt{d-1}.$$ Hence Ramanujan graphs have an asymptotically smallest possible value of `{{math|''λ''{{sub|2}}}}`{=mediawiki}. This makes them excellent spectral expanders.
Lubotzky, Phillips, and Sarnak (1988), Margulis (1988), and Morgenstern (1994) show how Ramanujan graphs can be constructed explicitly.
In 1985, Alon, conjectured that most `{{mvar|d}}`{=mediawiki}-regular graphs on `{{mvar|n}}`{=mediawiki} vertices, for sufficiently large `{{mvar|n}}`{=mediawiki}, are almost Ramanujan. That is, for `{{math|''ε'' > 0}}`{=mediawiki}, they satisfy
$$\lambda \le 2\sqrt{d-1}+\varepsilon$$.
In 2003, Joel Friedman both proved the conjecture and specified what is meant by \"most `{{mvar|d}}`{=mediawiki}-regular graphs\" by showing that random `{{mvar|d}}`{=mediawiki}-regular graphs have $\lambda \le 2\sqrt{d-1}+\varepsilon$ for every `{{math|''ε'' > 0}}`{=mediawiki} with probability `{{math|1 – ''O''(''n''{{sup|-τ}})}}`{=mediawiki}, where
$$\tau = \left\lceil\frac{\sqrt{d-1} +1}{2} \right\rceil.$$ A simpler proof of a slightly weaker result was given by Puder.
Marcus, Spielman and Srivastava, gave a construction of bipartite Ramanujan graphs based on lifts.
In 2024 a preprint by Jiaoyang Huang, Theo McKenzieand and Horng-Tzer Yau proved that
$$\lambda \le 2\sqrt{d-1}$$. with the fraction of eigenvalues that hit the Alon-Boppana bound approximately 69% from proving that edge universality holds, that is they follow a Tracy-Widom distribution associated with the Gaussian Orthogonal Ensemble
### Zig-zag product {#zig_zag_product}
Reingold, Vadhan, and Wigderson introduced the zig-zag product in 2003. Roughly speaking, the zig-zag product of two expander graphs produces a graph with only slightly worse expansion. Therefore, a zig-zag product can also be used to construct families of expander graphs. If `{{mvar|G}}`{=mediawiki} is a `{{math|(''n'', ''d'', ''λ''{{sub|1}})}}`{=mediawiki}-graph and `{{mvar|H}}`{=mediawiki} is an `{{math|(''m'', ''d'', ''λ''{{sub|2}})}}`{=mediawiki}-graph, then the zig-zag product `{{math|''G'' ◦ ''H''}}`{=mediawiki} is a `{{math|(''nm'', ''d''{{sup|2}}, ''φ''(''λ''{{sub|1}}, ''λ''{{sub|2}}))}}`{=mediawiki}-graph where `{{mvar|φ}}`{=mediawiki} has the following properties.
1. If `{{math|''λ''{{sub|1}} < 1}}`{=mediawiki} and `{{math|''λ''{{sub|2}} < 1}}`{=mediawiki}, then `{{math|''φ''(''λ''{{sub|1}}, ''λ''{{sub|2}}) < 1}}`{=mediawiki};
2. .
Specifically,
$$\phi(\lambda_1, \lambda_2)=\frac{1}{2}(1-\lambda^2_2)\lambda_2 +\frac{1}{2}\sqrt{(1-\lambda^2_2)^2\lambda_1^2 +4\lambda^2_2}.$$
Note that property (1) implies that the zig-zag product of two expander graphs is also an expander graph, thus zig-zag products can be used inductively to create a family of expander graphs.
Intuitively, the construction of the zig-zag product can be thought of in the following way. Each vertex of `{{mvar|G}}`{=mediawiki} is blown up to a \"cloud\" of `{{mvar|m}}`{=mediawiki} vertices, each associated to a different edge connected to the vertex. Each vertex is now labeled as `{{math|(''v'', ''k'')}}`{=mediawiki} where `{{mvar|v}}`{=mediawiki} refers to an original vertex of `{{mvar|G}}`{=mediawiki} and `{{mvar|k}}`{=mediawiki} refers to the `{{mvar|k}}`{=mediawiki}th edge of `{{mvar|v}}`{=mediawiki}. Two vertices, `{{math|(''v'', ''k'')}}`{=mediawiki} and `{{math|(''w'',''ℓ'')}}`{=mediawiki} are connected if it is possible to get from `{{math|(''v'', ''k'')}}`{=mediawiki} to `{{math|(''w'', ''ℓ'')}}`{=mediawiki} through the following sequence of moves.
1. *Zig* -- Move from `{{math|(''v'', ''k'')}}`{=mediawiki} to `{{math|(''v'', ''k' '')}}`{=mediawiki}, using an edge of `{{mvar|H}}`{=mediawiki}.
2. Jump across clouds using edge `{{mvar|k'}}`{=mediawiki} in `{{mvar|G}}`{=mediawiki} to get to `{{math|(''w'', ''ℓ′'')}}`{=mediawiki}.
3. *Zag* -- Move from `{{math|(''w'', ''ℓ′'')}}`{=mediawiki} to `{{math|(''w'', ''ℓ'')}}`{=mediawiki} using an edge of `{{mvar|H}}`{=mediawiki}.
### Lifts
An `{{mvar|r}}`{=mediawiki}-lift of a graph is formed by replacing each vertex by `{{mvar|r}}`{=mediawiki} vertices, and each edge by a matching between the corresponding sets of $r$ vertices. The lifted graph inherits the eigenvalues of the original graph, and has some additional eigenvalues. Bilu and Linial showed that every `{{mvar|d}}`{=mediawiki}-regular graph has a 2-lift in which the additional eigenvalues are at most $O(\sqrt{d\log^3 d})$ in magnitude. They also showed that if the starting graph is a good enough expander, then a good 2-lift can be found in polynomial time, thus giving an efficient construction of `{{mvar|d}}`{=mediawiki}-regular expanders for every `{{mvar|d}}`{=mediawiki}.
Bilu and Linial conjectured that the bound $O(\sqrt{d\log^3 d})$ can be improved to $2\sqrt{d-1}$, which would be optimal due to the Alon--Boppana bound. This conjecture was proved in the bipartite setting by Marcus, Spielman and Srivastava, who used the method of interlacing polynomials. As a result, they obtained an alternative construction of bipartite Ramanujan graphs. The original non-constructive proof was turned into an algorithm by Michael B. Cohen. Later the method was generalized to `{{mvar|r}}`{=mediawiki}-lifts by Hall, Puder and Sawin.
### Randomized constructions {#randomized_constructions}
There are many results that show the existence of graphs with good expansion properties through probabilistic arguments. In fact, the existence of expanders was first proved by Pinsker who showed that for a randomly chosen `{{mvar|n}}`{=mediawiki} vertex left `{{mvar|d}}`{=mediawiki} regular bipartite graph, `{{math|{{abs|''N''(''S'')}} ≥ (''d'' – 2){{abs|''S''}}}}`{=mediawiki} for all subsets of vertices `{{math|{{abs|''S''}} ≤ ''c{{sub|d}}n''}}`{=mediawiki} with high probability, where `{{mvar|c{{sub|d}}}}`{=mediawiki} is a constant depending on `{{mvar|d}}`{=mediawiki} that is `{{math|''O''(''d''{{sup|-4}})}}`{=mediawiki}. Alon and Roichman showed that for every `{{math|1 > ''ε'' > 0}}`{=mediawiki}, there is some `{{math|''c''(''ε'') > 0}}`{=mediawiki} such that the following holds: For a group `{{mvar|G}}`{=mediawiki} of order `{{mvar|n}}`{=mediawiki}, consider the Cayley graph on `{{mvar|G}}`{=mediawiki} with `{{math|''c''(''ε'') log{{sub|2}} ''n''}}`{=mediawiki} randomly chosen elements from `{{mvar|G}}`{=mediawiki}. Then, in the limit of `{{mvar|n}}`{=mediawiki} getting to infinity, the resulting graph is almost surely an `{{math|''ε''}}`{=mediawiki}-expander.
In 2021, Alexander modified an MCMC algorithm to look for randomized constructions to produce Ramanujan graphs with a fixed vertex size and degree of regularity. The results show the Ramanujan graphs exist for every vertex size and degree pair up to 2000 vertices.
In 2024 Alon produced an explicit construction for near Ramanujan graphs of every vertex size and degree pair.
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# Expander graph
## Applications and useful properties {#applications_and_useful_properties}
The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded degree is precisely an asymptotic robust graph with the number of edges growing linearly with size (number of vertices), for all subsets.
Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks (`{{harvtxt|Ajtai|Komlós|Szemerédi|1983}}`{=mediawiki}) and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL = L (`{{harvtxt|Reingold|2008}}`{=mediawiki}) and the PCP theorem (`{{harvtxt|Dinur|2007}}`{=mediawiki}). In cryptography, expander graphs are used to construct hash functions.
In a [2006 survey of expander graphs](https://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/), Hoory, Linial, and Wigderson split the study of expander graphs into four categories: extremal problems, typical behavior, explicit constructions, and algorithms. Extremal problems focus on the bounding of expansion parameters, while typical behavior problems characterize how the expansion parameters are distributed over random graphs. Explicit constructions focus on constructing graphs that optimize certain parameters, and algorithmic questions study the evaluation and estimation of parameters.
### Expander mixing lemma {#expander_mixing_lemma}
The expander mixing lemma states that for an `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graph, for any two subsets of the vertices `{{math|''S'', ''T'' ⊆ ''V''}}`{=mediawiki}, the number of edges between `{{mvar|S}}`{=mediawiki} and `{{mvar|T}}`{=mediawiki} is approximately what you would expect in a random `{{mvar|d}}`{=mediawiki}-regular graph. The approximation is better the smaller `{{math|''λ''}}`{=mediawiki} is. In a random `{{mvar|d}}`{=mediawiki}-regular graph, as well as in an Erdős--Rényi random graph with edge probability `{{math|{{frac|''d''|''n''}}}}`{=mediawiki}, we expect `{{math|{{frac|''d''|''n''}} • {{abs|''S''}} • {{abs|''T''}}}}`{=mediawiki} edges between `{{mvar|S}}`{=mediawiki} and `{{mvar|T}}`{=mediawiki}.
More formally, let `{{math|''E''(''S'', ''T'')}}`{=mediawiki} denote the number of edges between `{{mvar|S}}`{=mediawiki} and `{{mvar|T}}`{=mediawiki}. If the two sets are not disjoint, edges in their intersection are counted twice, that is,
: $E(S,T)=2|E(G[S\cap T])| + E(S\setminus T,T) + E(S,T\setminus S).$
Then the expander mixing lemma says that the following inequality holds:
$$\left|E(S, T) - \frac{d \cdot |S| \cdot |T|}{n}\right| \leq \lambda \sqrt{|S| \cdot |T|}.$$ Many properties of `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graphs are corollaries of the expander mixing lemmas, including the following.
- An independent set of a graph is a subset of vertices with no two vertices adjacent. In an `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graph, an independent set has size at most `{{math|{{frac|''λn''|''d''}}}}`{=mediawiki}.
- The chromatic number of a graph `{{mvar|G}}`{=mediawiki}, `{{math|''χ''(''G'')}}`{=mediawiki}, is the minimum number of colors needed such that adjacent vertices have different colors. Hoffman showed that `{{math|{{frac|''d''|''λ''}} ≤ ''χ''(''G'')}}`{=mediawiki}, while Alon, Krivelevich, and Sudakov showed that if `{{math|''d'' < {{frac|2''n''|3}}}}`{=mediawiki}, then
$\chi(G) \leq O \left( \frac{d}{\log(1+d/\lambda)} \right).$
- The diameter of a graph is the maximum distance between two vertices, where the distance between two vertices is defined to be the shortest path between them. Chung showed that the diameter of an `{{math|(''n'', ''d'', ''λ'')}}`{=mediawiki}-graph is at most
$\left\lceil \log \frac{n}{ \log(d/\lambda)} \right\rceil.$
### Expander walk sampling {#expander_walk_sampling}
The Chernoff bound states that, when sampling many independent samples from a random variable in the range `{{math|[−1, 1]}}`{=mediawiki}, with high probability the average of our samples is close to the expectation of the random variable. The expander walk sampling lemma, due to `{{harvtxt|Ajtai|Komlós|Szemerédi|1987}}`{=mediawiki} and `{{harvtxt|Gillman|1998}}`{=mediawiki}, states that this also holds true when sampling from a walk on an expander graph. This is particularly useful in the theory of derandomization, since sampling according to an expander walk uses many fewer random bits than sampling independently.
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# Expander graph
## Applications and useful properties {#applications_and_useful_properties}
### AKS sorting network and approximate halvers {#aks_sorting_network_and_approximate_halvers}
Sorting networks take a set of inputs and perform a series of parallel steps to sort the inputs. A parallel step consists of performing any number of disjoint comparisons and potentially swapping pairs of compared inputs. The depth of a network is given by the number of parallel steps it takes. Expander graphs play an important role in the AKS sorting network, which achieves depth `{{math|''O''(log ''n'')}}`{=mediawiki}. While this is asymptotically the best known depth for a sorting network, the reliance on expanders makes the constant bound too large for practical use.
Within the AKS sorting network, expander graphs are used to construct bounded depth `{{mvar|ε}}`{=mediawiki}-halvers. An `{{mvar|ε}}`{=mediawiki}-halver takes as input a length `{{mvar|n}}`{=mediawiki} permutation of `{{math|(1, …, ''n'')}}`{=mediawiki} and halves the inputs into two disjoint sets `{{mvar|A}}`{=mediawiki} and `{{mvar|B}}`{=mediawiki} such that for each integer `{{math|''k'' ≤ {{frac|''n''|2}}}}`{=mediawiki} at most `{{mvar|εk}}`{=mediawiki} of the `{{mvar|k}}`{=mediawiki} smallest inputs are in `{{mvar|B}}`{=mediawiki} and at most `{{mvar|εk}}`{=mediawiki} of the `{{mvar|k}}`{=mediawiki} largest inputs are in `{{mvar|A}}`{=mediawiki}. The sets `{{mvar|A}}`{=mediawiki} and `{{mvar|B}}`{=mediawiki} are an `{{mvar|ε}}`{=mediawiki}-halving.
Following `{{harvtxt|Ajtai|Komlós|Szemerédi|1983}}`{=mediawiki}, a depth `{{mvar|d}}`{=mediawiki} `{{mvar|ε}}`{=mediawiki}-halver can be constructed as follows. Take an `{{mvar|n}}`{=mediawiki} vertex, degree `{{mvar|d}}`{=mediawiki} bipartite expander with parts `{{mvar|X}}`{=mediawiki} and `{{mvar|Y}}`{=mediawiki} of equal size such that every subset of vertices of size at most `{{mvar|εn}}`{=mediawiki} has at least `{{math|{{sfrac|1 – ''ε''|''ε''}}}}`{=mediawiki} neighbors.
The vertices of the graph can be thought of as registers that contain inputs and the edges can be thought of as wires that compare the inputs of two registers. At the start, arbitrarily place half of the inputs in `{{mvar|X}}`{=mediawiki} and half of the inputs in `{{mvar|Y}}`{=mediawiki} and decompose the edges into `{{mvar|d}}`{=mediawiki} perfect matchings. The goal is to end with `{{mvar|X}}`{=mediawiki} roughly containing the smaller half of the inputs and `{{mvar|Y}}`{=mediawiki} containing roughly the larger half of the inputs. To achieve this, sequentially process each matching by comparing the registers paired up by the edges of this matching and correct any inputs that are out of order. Specifically, for each edge of the matching, if the larger input is in the register in `{{mvar|X}}`{=mediawiki} and the smaller input is in the register in `{{mvar|Y}}`{=mediawiki}, then swap the two inputs so that the smaller one is in `{{mvar|X}}`{=mediawiki} and the larger one is in `{{mvar|Y}}`{=mediawiki}. It is clear that this process consists of `{{mvar|d}}`{=mediawiki} parallel steps.
After all `{{mvar|d}}`{=mediawiki} rounds, take `{{mvar|A}}`{=mediawiki} to be the set of inputs in registers in `{{mvar|X}}`{=mediawiki} and `{{mvar|B}}`{=mediawiki} to be the set of inputs in registers in `{{mvar|Y}}`{=mediawiki} to obtain an `{{mvar|ε}}`{=mediawiki}-halving. To see this, notice that if a register `{{mvar|u}}`{=mediawiki} in `{{mvar|X}}`{=mediawiki} and `{{mvar|v}}`{=mediawiki} in `{{mvar|Y}}`{=mediawiki} are connected by an edge `{{mvar|uv}}`{=mediawiki} then after the matching with this edge is processed, the input in `{{mvar|u}}`{=mediawiki} is less than that of `{{mvar|v}}`{=mediawiki}. Furthermore, this property remains true throughout the rest of the process. Now, suppose for some `{{math|''k'' ≤ {{frac|''n''|2}}}}`{=mediawiki} that more than `{{mvar|εk}}`{=mediawiki} of the inputs `{{math|(1, …, ''k'')}}`{=mediawiki} are in `{{mvar|B}}`{=mediawiki}. Then by expansion properties of the graph, the registers of these inputs in `{{mvar|Y}}`{=mediawiki} are connected with at least `{{math|{{sfrac|1 – ''ε''|''ε''}}''k''}}`{=mediawiki} registers in `{{mvar|X}}`{=mediawiki}. Altogether, this constitutes more than `{{mvar|k}}`{=mediawiki} registers so there must be some register `{{mvar|A}}`{=mediawiki} in `{{mvar|X}}`{=mediawiki} connected to some register `{{mvar|B}}`{=mediawiki} in `{{mvar|Y}}`{=mediawiki} such that the final input of `{{mvar|A}}`{=mediawiki} is not in `{{math|(1, …, ''k'')}}`{=mediawiki}, while the final input of `{{mvar|B}}`{=mediawiki} is. This violates the previous property however, and thus the output sets `{{mvar|A}}`{=mediawiki} and `{{mvar|B}}`{=mediawiki} must be an `{{mvar|ε}}`{=mediawiki}-halving
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# Ericales
The **Ericales** are a large and diverse order of flowering plants in the asterid group of the eudicots. Well-known and economically important members of this order include tea and ornamental camellias, persimmon, ebony, blueberry, cranberry, lingonberry, huckleberry, kiwifruit, Brazil nut, argan, sapote, azaleas and rhododendrons, heather, heath, impatiens, phlox, Jacob\'s ladder, primroses, cyclamens, shea, sapodilla, pouterias, and trumpet pitchers.
The order includes 22 families, according to the APG IV system of classification.
The Ericales include trees, bushes, lianas, and herbaceous plants. Together with ordinary autophytic plants, they include chlorophyll-deficient mycoheterotrophic plants (e.g., *Sarcodes sanguinea*) and carnivorous plants (e.g., genus *Sarracenia*). Mycorrhizal associations are quite common among the order representatives, and three kinds of mycorrhiza are found exclusively among Ericales (namely, ericoid, arbutoid and monotropoid mycorrhiza). In addition, some families among the order are notable for their exceptional ability to accumulate aluminum.
Many species have five petals, often grown together. Fusion of the petals as a trait was traditionally used to place the order in the subclass Sympetalae.
Ericales are a cosmopolitan order. Areas of distribution of families vary largely -- while some are restricted to tropics, others exist mainly in Arctic or temperate regions. The entire order contains over 8,000 species, of which the Ericaceae account for 2,000--4,000 species (by various estimates).
According to molecular studies, the lineage that led to Ericales diverged from other plants about 127 million years or diversified 110 million years ago.
## Economic importance {#economic_importance}
The most commercially used plant in the order is tea (*Camellia sinensis*) from the family Theaceae. The order also includes some edible fruits, including kiwifruit (esp. *Actinidia chinensis* var. *deliciosa*), persimmon (genus *Diospyros*), blueberry, huckleberry, cranberry, Brazil nut, and Mamey sapote. The order also includes shea (*Vitellaria paradoxa*), which is the major dietary lipid source for millions of sub-Saharan Africans. Many Ericales species are cultivated for their showy flowers: well-known examples are azalea, rhododendron, camellia, heather, polyanthus, cyclamen, phlox, and busy Lizzie.
## Gallery of photos {#gallery_of_photos}
Balsam I IMG 9566.jpg\|*Impatiens balsamina* of Balsaminaceae family Primula rosea I IMG 7210.jpg\|*Primula rosea* of Primulaceae family Fuyu persimmon fruits, one cut open.jpg\|*Diospyros kaki* or oriental persimmon of *Diospyros* genus and Ebenaceae family
## Classification
22 families are recognized as members of the Ericales in the APG IV system of classification:
- Family Actinidiaceae (kiwifruit family)
- Family Balsaminaceae (balsam family)
- Family Clethraceae (clethra family)
- Family Cyrillaceae (cyrilla family)
- Family Diapensiaceae
- Family Ebenaceae (ebony and persimmon family)
- Family Ericaceae (heath, rhododendron, and blueberry family)
- Family Fouquieriaceae (ocotillo family)
- Family Lecythidaceae (Brazil nut family)
- Family Marcgraviaceae
- Family Mitrastemonaceae
- Family Pentaphylacaceae
- Family Polemoniaceae (phlox family)
- Family Primulaceae (primrose and snowbell family)
- Family Roridulaceae
- Family Sapotaceae (sapodilla family)
- Family Sarraceniaceae (American pitcher plant family)
- Family Sladeniaceae
- Family Styracaceae (silverbell family)
- Family Symplocaceae (sapphireberry family)
- Family Tetrameristaceae
- Family Theaceae (tea and camellia family)
Likely phylogenetic relationships between the families of the Ericales: `{{clade
|label1=Ericales
|1={{clade
|1={{clade
|1={{clade
|1={{clade
|1={{clade
|1={{clade
|1={{clade
|1={{clade
|label1=ericoids
|1={{clade
|1={{clade
|1=[[Cyrillaceae]]
|2=[[Ericaceae]]
}}
|2=[[Clethraceae]]
}}
|label2=sarracenioids
|2={{clade
|1={{clade
|1=[[Roridulaceae]]
|2=[[Actinidiaceae]]
}}
|2=[[Sarraceniaceae]]
}}
}}
|2={{clade
|label1= styracoids
|1={{clade
|1={{clade
|1=[[Styracaceae]]
|2=[[Diapensiaceae]]
}}
|2=[[Symplocaceae]]
}}
}}
}}
|2=[[Theaceae]]
}}
|2=[[Pentaphylacaceae]]
}}
|2={{clade
|label1=primuloids
|1={{clade
|1={{clade
|1=[[Primulaceae]]
|2=[[Ebenaceae]]
}}
|2=[[Sapotaceae]]
}}
}}
}}
|2={{clade
|label1=polemonioids
|1={{clade
|1=[[Polemoniaceae]]
|2=[[Fouquieriaceae]]
}}
}}
}}
|2={{clade
|1=[[Lecythidaceae]]
|2=[[Mitrastemonaceae]]
}}
}}
|2={{clade
|label1=balsaminoids
|1={{clade
|1={{clade
|1=[[Marcgraviaceae]]
|2=[[Tetrameristaceae]]
}}
|2=[[Balsaminaceae]]
}}
}}
}}
}}`{=mediawiki}
## Previously included families {#previously_included_families}
These families are not recognized in the APG III system but have been in common use in the recent past:
- Family Myrsinaceae (cyclamen and scarlet pimpernel family) → Primulaceae
- Family Pellicieraceae → Tetrameristaceae
- Family Maesaceae → Primulaceae
- Family Ternstroemiaceae → Pentaphylacaceae
- Family Theophrastaceae → Primulaceae
These make up an early diverging group of asterids
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# Easter egg
**Easter eggs**, also called **Paschal eggs**, are eggs that are decorated for the Christian holiday of Easter, which celebrates the resurrection of Jesus. As such, Easter eggs are commonly used during the season of Eastertide (Easter season). The oldest tradition, which continues to be used in Central and Eastern Europe, is to dye and paint chicken eggs.
Although eggs, in general, were a traditional symbol of fertility and rebirth, in Christianity, for the celebration of Eastertide, Easter eggs symbolize the empty tomb of Jesus, from which Jesus was resurrected. In addition, one ancient tradition was the staining of Easter eggs with the colour red \"in memory of the blood of Christ, shed as at that time of his crucifixion.\"
This custom of the Easter egg, according to many sources, can be traced to early Christians of Mesopotamia, and from there it spread into Eastern Europe and Siberia through the Orthodox Churches, and later into Europe through the Catholic and Protestant Churches. Additionally, the widespread usage of Easter eggs, according to mediaevalist scholars, is due to the prohibition of eggs during Lent after which, on Easter, they are blessed for the occasion.
A modern custom in some places is to substitute chocolate eggs wrapped in coloured foil, hand-carved wooden eggs, or plastic eggs filled with confectionery such as chocolate.
## History
The practice of decorating eggshells is quite ancient, with decorated, engraved ostrich eggs found in Africa which are 60,000 years old. In the pre-dynastic period of Egypt and the early cultures of Mesopotamia and Crete, eggs were associated with death and rebirth, as well as with kingship, with decorated ostrich eggs, and representations of ostrich eggs in gold and silver, were commonly placed in graves of the ancient Sumerians and Egyptians as early as 5,000 years ago. These cultural relationships may have influenced early Christian and Islamic cultures in those areas, as well as through mercantile, religious, and political links from those areas around the Mediterranean.
Eggs in Christianity carry a Trinitarian symbolism as shell, yolk, and albumen are three parts of one egg. According to many sources, the Christian custom of Easter eggs started among the early Christians of Mesopotamia, who stained them with red colouring \"in memory of the blood of Christ, shed at His crucifixion\". The Christian Church officially adopted the custom, regarding the eggs as a symbol of the resurrection of Jesus, with the Roman Ritual, the first edition of which was published in 1610 but which has texts of much older date, containing among the Easter Blessings of Food, one for eggs, along with those for lamb, bread, and new produce. `{{blockquote|Lord, let the grace of your blessing + come upon these eggs, that they be healthful food for your faithful who eat them in thanksgiving for the resurrection of our Lord Jesus Christ, who lives and reigns with you forever and ever.|author=|title=|source=}}`{=mediawiki}
Sociology professor Kenneth Thompson discusses the spread of the Easter egg throughout Christendom, writing that \"use of eggs at Easter seems to have come from Persia into the Greek Christian Churches of Mesopotamia, thence to Russia and Siberia through the medium of Orthodox Christianity. From the Greek Church the custom was adopted by either the Roman Catholics or the Protestants and then spread through Europe.\" Both Thompson, as well as British orientalist Thomas Hyde state that in addition to dyeing the eggs red, the early Christians of Mesopotamia also stained Easter eggs green and yellow.
Peter Gainsford maintains that the association between eggs and Easter most likely arose in western Europe during the Middle Ages as a result of the fact that Catholic Christians were prohibited from eating eggs during Lent, but were allowed to eat them when Easter arrived.
Influential 19th century folklorist and philologist Jacob Grimm speculates, in the second volume of his *Deutsche Mythologie*, that the folk custom of Easter eggs among the continental Germanic peoples may have stemmed from springtime festivities of a Germanic goddess known in Old English as Ēostre (namesake of modern English *Easter*) and possibly known in Old High German as \**Ostara* (and thus namesake of Modern German *Ostern* \'Easter\'). However, despite Grimm\'s speculation, there is no evidence to connect eggs with a speculative deity named Ostara. The use of eggs as favors or treats at Easter originated when they were prohibited during Lent. A common practice in England in the medieval period was for children to go door-to-door begging for eggs on the Saturday before Lent began. People handed out eggs as special treats for children prior to their fast.
Although one of the Christian traditions are to use dyed or painted chicken eggs, a modern custom is to substitute chocolate eggs, or plastic eggs filled with candy such as jelly beans; as many people give up sweets as their Lenten sacrifice, individuals enjoy them at Easter after having abstained from them during the preceding forty days of Lent. These eggs can be hidden for children to find on Easter morning, which may be left by the Easter Bunny. They may also be put in a basket filled with real or artificial straw to resemble a bird\'s nest.
While the practice of giving away easter eggs is, to this day, popular, it was briefly banned in 1916s Hungary in the Easter Egg Act, due to the scarcity caused by the ongoing war, and the ban was only lifted when the war ended. The contemporary news reports emphasised however, that the locsolás was still legal to practice.
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# Easter egg
## Traditions and customs {#traditions_and_customs}
### Lenten tradition {#lenten_tradition}
The Easter egg tradition may also have merged into the celebration of the end of the privations of Lent. Traditionally, eggs are among the foods forbidden on fast days, including all of Lent, an observance which continues among the Eastern Christian Churches but has fallen into disuse in Western Christianity.
Historically, it has been traditional to use up all of the household\'s eggs before Lent began.
This established the tradition of Pancake Day being celebrated on Shrove Tuesday. This day, the Tuesday before Ash Wednesday when Lent begins, is also known as Mardi Gras, a French phrase which translates as \"Fat Tuesday\" to mark the last consumption of eggs and dairy before Lent begins.
In the Orthodox Church, Great Lent begins on Clean Monday, rather than Wednesday, so the household\'s dairy products would be used up in the preceding week, called Cheesefare Week.
During Lent, since chickens would not stop producing eggs during this time, a larger than usual store might be available at the end of the fast. This surplus, if any, had to be eaten quickly to prevent spoiling. Then, with the coming of Easter, the eating of eggs resumes. Some families cook a special meatloaf with eggs in it to be eaten with the Easter dinner.
To avoid waste, it was common for families to hard boil or pickle eggs that their chickens produced during lent, and for this reason the Spanish dish hornazo (traditionally eaten on and around Easter) contains hard-boiled eggs as a primary ingredient. In Spain it is common for godparents to give a Easter mona to their godchildren during Easter period. In Hungary, eggs are used sliced in potato casseroles around the Easter period.
### Symbolism and related customs {#symbolism_and_related_customs}
Some Christians symbolically link the cracking open of Easter eggs with the empty tomb of Jesus.
In the Orthodox churches, Easter eggs are blessed by the priest at the end of the Paschal Vigil (which is equivalent to Holy Saturday), and distributed to the faithful. The egg is seen by followers of Christianity as a symbol of resurrection: while being dormant it contains a new life sealed within it.
Similarly, in the Roman Catholic Church in Poland, the so-called święconka, i.e. blessing of decorative baskets with a sampling of Easter eggs and other symbolic foods, is one of the most enduring and beloved Polish traditions on Holy Saturday.
During Paschaltide, in some traditions the Pascal greeting with the Easter egg is even extended to the deceased. On either the second Monday or Tuesday of Pascha, after a memorial service people bring blessed eggs to the cemetery and bring the joyous paschal greeting, \"Christ has risen\", to their beloved departed (see Radonitza).
In Greece, women traditionally dye the eggs with onion skins and vinegar on Thursday (also the day of Communion). These ceremonial eggs are known as kokkina avga. They also bake tsoureki for the Easter Sunday feast. Red Easter eggs are sometimes served along the centerline of tsoureki (braided loaf of bread).
In Egypt, it is a tradition to decorate boiled eggs during Sham el-Nessim holiday, which falls every year after the Eastern Christian Easter.
Coincidentally, every Passover, Jews place a hard-boiled egg on the Passover ceremonial plate, and the celebrants also eat hard-boiled eggs dipped in salt water as part of the ceremony.
### Colouring
The dyeing of Easter eggs in different colours is commonplace, with colour being achieved through boiling the egg in natural substances (such as, onion peel (brown colour), oak or alder bark or walnut nutshell (black), beet juice (pink) etc.), or using artificial colourings.
A greater variety of colour was often provided by tying on the onion skin with different coloured woollen yarn. In the North of England these are called pace-eggs or paste-eggs, from a dialectal form of Middle English *pasche*. King Edward I\'s household accounts in 1290 list an item of \'one shilling and sixpence for the decoration and distribution of 450 Pace-eggs!\', which were to be coloured or gilded and given to members of the royal household. Traditionally in England, eggs were wrapped in onion skins and boiled to make their shells look like mottled gold, or wrapped in flowers and leaves first in order to leave a pattern, which parallels a custom practised in traditional Scandinavian culture. Eggs could also be drawn on with a wax candle before staining, often with a person\'s name and date on the egg. Pace Eggs were generally eaten for breakfast on Easter Sunday breakfast. Alternatively, they could be kept as decorations, used in egg-jarping (egg tapping) games, or given to Pace Eggers. In more recent centuries in England, eggs have been stained with coffee grains or simply boiled and painted in their shells.
In the Orthodox and Eastern Catholic Churches, Easter eggs are dyed red to represent the blood of Christ, with further symbolism being found in the hard shell of the egg symbolizing the sealed Tomb of Christ---the cracking of which symbolized his resurrection from the dead. The tradition of red easter eggs was used by the Russian Orthodox Church. The tradition to dyeing the easter eggs in an Onion tone exists in the cultures of Armenia, Bulgaria, Georgia, Lithuania, Ukraine, Belarus, Russia, Czechia, Romania, Serbia, Slovakia, Slovenia, and Israel. The colour is made by boiling onion peel in water.
### Patterning
When boiling them with onion skins, leaves can be attached prior to dyeing to create leaf patterns. The leaves are attached to the eggs before they are dyed with a transparent cloth to wrap the eggs with like inexpensive muslin or nylon stockings, leaving patterns once the leaves are removed after the dyeing process. These eggs are part of Easter custom in many areas and often accompany other traditional Easter foods. Passover haminados are prepared with similar methods.
Pysanky are Ukrainian Easter eggs, decorated using a wax-resist (batik) method. The word comes from the verb *pysaty*, \"to write\", as the designs are not painted on, but written with beeswax. Lithuanians create intricately detailed margučiai using a hot wax application and dipping method, and also by dipping the eggs first and then etching designs into the shells.
Decorating eggs for Easter using wax resistant batik is a popular method in some other eastern European countries.
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# Easter egg
## Traditions and customs {#traditions_and_customs}
### Use of Easter eggs in decorations {#use_of_easter_eggs_in_decorations}
In some Mediterranean countries, especially in Lebanon, chicken eggs are boiled and decorated by dye and/or painting and used as decoration around the house. Then, on Easter Day, young kids would duel with them saying \"Christ is resurrected, Indeed, He is\", breaking and eating them. This also happens in Georgia, Bulgaria, Cyprus, Greece, North Macedonia, Romania, Russia, Serbia and Ukraine. In Easter Sunday friends and family hit each other\'s egg with their own. The one whose egg does not break is believed to be in for good luck in the future.
In Germany, eggs decorate trees and bushes as Easter egg trees, and in several areas public wells as Osterbrunnen.
There used to be a custom in Ukraine, during Easter celebrations to have *krashanky* on a table in a bowl with wheatgrass. The number of the *krashanky* equalled the number of departed family members.
<File:Pysanky2011.JPG%7CUkrainian> Easter eggs <File:Sorbische> Ostereier.jpg\|Easter eggs from Sorbs <File:Marguciai2>. 2007-04-21.jpg\|Easter eggs from Lithuania <File:Sleepingbeauty.jpg%7CPerforated> egg from Germany, Sleeping Beauty <File:Egg> dekorerte.jpg\|Norwegian Easter eggs <File:Ostereier-Griechenland.JPG%7CEaster> eggs from Greece <File:Pisanki> ażurowe.jpg\|Perforated eggs <File:Oeuf> de paque.JPG\|Easter eggs from France <File:White> House Easter Egg Roll.jpg\|American Easter egg from the White House Washington, D.C. <File:Oeufs.jpg%7CPace> eggs boiled with onion skins and leaf patterns. <File:Red> and blue Easter eggs.jpg\|Easter eggs decorated with straw <File:04> Easter eggs at a Cultural Miner\'s House in Sanok.JPG\|Easter egg from Poland <File:Washi> Egg Japan US 2.png\|Washi egg from Japan
\|PLEASE DO NOT ADD MORE PICTURES TO THIS GALLERY BECAUSE THIS GALLERY HAS REACHED ITS MAXIMUM AMOUNT OF PICTURES
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# Easter egg
## Easter egg games {#easter_egg_games}
### Egg hunts {#egg_hunts}
An egg hunt is a game in which decorated eggs, which may be hard-boiled chicken eggs, chocolate eggs, or artificial eggs containing candies, are hidden for children to find. The eggs often vary in size, and may be hidden both indoors and outdoors. When the hunt is over, prizes may be given for the largest number of eggs collected, or for the largest or the smallest egg.
Some Central European nations (Czechs, Slovaks, etc.) have a tradition of men gathering eggs from women in return for whipping them with an easter whip and splashing them with water. The ritual is traditionally believed to preserve the women\'s health and beauty.
Cascarones, a Latin American tradition now shared by many US States with high Hispanic demographics, are emptied and dried chicken eggs stuffed with confetti and sealed with a piece of tissue paper. The eggs are hidden in a similar tradition to the American Easter egg hunt and when found the children (and adults) break them over each other\'s heads.
In order to enable children to take part in egg hunts despite visual impairment, eggs have been created that emit various clicks, beeps, noises, or music so that visually impaired children can easily hunt for Easter eggs.
### Egg rolling {#egg_rolling}
Egg rolling is also a traditional Easter Egg game played with eggs at Easter. In the United Kingdom, Germany, and other countries children traditionally rolled eggs down hillsides at Easter. This tradition was taken to the New World by European settlers, and continues to this day each Easter with an Easter egg roll on the White House lawn. Rutherford B. Hayes started the tradition of the Easter Egg Roll at the White House. The Easter Monday Egg Roll was normally held at the United States Capitol, however, by the mid-1870s, Congress passed a law forbidding the Capitol\'s grounds to be used for the activity due to the toll it was taking on the landscape. The law was enforced in 1877, but the rain that year canceled all outdoor activities. In 1878, Hayes was approached by many young Easter Egg rollers who asked for the event to be held at the White House. He invited any children who wanted to roll eggs to come to the White House in order to do so. The tradition still occurs every year on the South Lawn of the White House. Now, there are many other games and activities that take place such as \"Egg Picking\" and \"Egg Ball\". Different nations have different versions of the Easter Egg roll game.
### Egg tapping {#egg_tapping}
In the North of England, during Eastertide, a traditional game is played where hard boiled *pace eggs* are distributed and each player hits the other player\'s egg with their own. This is known as \"egg tapping\", \"egg dumping\", or \"egg jarping\". The winner is the holder of the last intact egg. The annual egg jarping world championship is held every year over Easter in Peterlee, Durham.
It is also practiced in Italy (where it is called *scuccetta*), Poland, Belarus, Bulgaria, Hungary, Croatia, Latvia, Lithuania, Lebanon, North Macedonia, Romania, Serbia, Slovenia (where it is called *turčanje* or *trkanje*), Ukraine, Russia, and other countries. In parts of Austria, Bavaria and German-speaking Switzerland it is called *Ostereiertitschen* or *Eierpecken*. In parts of Europe it is also called *epper*, presumably from the German name *Opfer*, meaning \"offering\" and in Greece it is known as *tsougrisma*. In South Louisiana, this practice is called pocking eggs and is slightly different. The Louisiana Creoles hold that the winner eats the eggs of the losers in each round.
In the Greek Orthodox tradition, red eggs are also cracked together when people exchange Easter greetings.
### Egg dance {#egg_dance}
Egg dance is a traditional Easter game in which eggs are laid on the ground or floor and the goal is to dance among them without damaging any eggs which originated in Germany.
### Pace egg plays {#pace_egg_plays}
The Pace Egg plays are traditional village plays, with a rebirth theme. The drama takes the form of a combat between the hero and villain, in which the hero is killed and brought back to life. The plays take place in England during Easter.
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# Easter egg
## Variants
### Chocolate
Chocolate eggs first appeared at the court of Louis XIV in Versailles and in 1725 the widow Giambone in Turin started producing chocolate eggs by filling empty chicken egg shells with molten chocolate. In 1873, J.S. Fry & Sons produced the first hollow chocolate egg making a smooth paste that could be poured into egg moulds. Manufacturing their first Easter egg in 1875, Cadbury created the modern chocolate Easter egg after developing a pure cocoa butter that could be moulded into smooth shapes.
In Western cultures, the giving of chocolate eggs is now commonplace, with 80 million Easter eggs sold in the UK alone. Formerly, the containers Easter eggs were sold in contained large amounts of plastic, although in the United Kingdom this has gradually been replaced with recyclable paper and cardboard.
In Brazil, Argentina, Chile, Uruguay and Paraguay, hollow chocolate eggs known as *Ovos de Páscoa* or *Huevos de Páscua* (Easter eggs) are popular and are commonly sold around Easter in supermarkets. Variations of this dessert containing fillings such as pistachio cream, hazelnut cream, *\[\[furrundu\]\]* or doce de leite, are known as *Ovos de Páscoa de colher* (Spoon Easter eggs) or *Ovos de colher* (Spoon eggs).
<File:Chocolate> egg.jpg\|Hollow chocolate Easter egg <File:Easter-Chocolate-egg-bunny.jpg%7CChocolate> Easter egg bunny <File:Easter> egg with candy.jpg\|Easter egg with candy <File:Gladys> as a Chocolate Easter Bunny.jpg\|Gladys as a Chocolate Easter Bunny with Easter eggs <File:Ovos> de Páscoa 2.jpg\|Brazilian *Ovos de Páscoa* for sale in a supermarket
### Marzipan eggs {#marzipan_eggs}
In the Indian state of Goa, the Goan Catholic version of marzipan is used to make easter eggs. In the Philippines, mazapán de pili (Spanish for \"pili marzipan\") is made from pili nuts.
<File:Marzipan> easter eggs.jpg\|Marzipan easter eggs
### Artificial eggs {#artificial_eggs}
The jewelled Easter eggs made by the Fabergé firm for the two last Russian Tsars are regarded as masterpieces of decorative arts. Most of these creations themselves contained hidden surprises such as clock-work birds, or miniature ships.
In Bulgaria, Poland, Romania, Russia, Ukraine, and other Central European countries\' folk traditions, Easter eggs are carved from wood and hand-painted, and making artificial eggs out of porcelain for ladies is common.
Easter eggs are frequently depicted in sculpture, including a 27 ft sculpture of a pysanka standing in Vegreville, Alberta.
<File:Fabergé> egg Rome 05.JPG\|Fabergé egg <File:Huevo> de chocolate en Bariloche (Argentina).jpg\|Giant easter egg, Bariloche, Argentina <File:Vegreville> Pysanka.jpg\|Giant pysanka from Vegreville, Alberta, Canada <File:Zagrebacko> uskrsnje jaje 4 050409.jpg\|Giant easter egg or *pisanica* in Zagreb, Croatia <File:Easter> egg sculpture in Gogolin 2014 P01.JPG\|Easter egg sculpture in Gogolin, Poland <File:Oul> uriaş din Suceava2.jpg\|Giant easter egg in Suceava, Romania
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# Easter egg
## Legends
### Christian traditions {#christian_traditions}
While the origin of Easter eggs can be explained in the symbolic terms described above, among followers of Eastern Christianity the legend says that Mary Magdalene was bringing cooked eggs to share with the other women at the tomb of Jesus, and the eggs in her basket miraculously turned bright red when she saw the risen Christ.
A different, but not necessarily conflicting legend concerns Mary Magdalene\'s efforts to spread the Gospel. According to this tradition, after the Ascension of Jesus, Mary went to the Emperor of Rome and greeted him with \"Christ has risen,\" whereupon he pointed to an egg on his table and stated, \"Christ has no more risen than that egg is red.\" After making this statement it is said the egg immediately turned blood red.
Red Easter eggs, known as `{{Transliteration|el|kokkina avga}}`{=mediawiki} (*κόκκινα αυγά*) in Greece and *krashanki* in Ukraine, are an Easter tradition and a distinct type of Easter egg prepared by various Orthodox Christian peoples. The red eggs are part of Easter custom in many areas and often accompany other traditional Easter foods. Passover haminados are prepared with similar methods. Dark red eggs are a tradition in Greece and represent the blood of Christ shed on the cross. The practice dates to the early Christian church in Mesopotamia. In Greece, superstitions of the past included the custom of placing the first-dyed red egg at the home\'s iconostasis (place where icons are displayed) to ward off evil. The heads and backs of small lambs were also marked with the red dye to protect them.
### Parallels in other faiths {#parallels_in_other_faiths}
The egg is widely used as a symbol of the start of new life, just as new life emerges from an egg when the chick hatches out.
Painted eggs are used at the Iranian spring holidays, the Nowruz that marks the first day of spring or Equinox, and the beginning of the year in the Persian calendar. It is celebrated on the day of the astronomical Northward equinox, which usually occurs on March 21 or the previous/following day depending on where it is observed. The painted eggs symbolize fertility and are displayed on the Nowruz table, called Haft-Seen together with various other symbolic objects. There are sometimes one egg for each member of the family. The ancient Zoroastrians painted eggs for Nowruz, their New Year celebration, which falls on the Spring equinox. The tradition continues among Persians of Islamic, Zoroastrian, and other faiths today. The Nowruz tradition has existed for at least 2,500 years. The sculptures on the walls of Persepolis show people carrying eggs for Nowruz to the king.
The Neopagan holiday of Ostara occurs at roughly the same time as Easter. While it is often claimed that the use of painted eggs is an ancient, pre-Christian component of the celebration of Ostara, there are no historical accounts that ancient celebrations included this practice, apart from the Old High German lullaby which is believed by most to be a modern fabrication. Rather, the use of painted eggs has been adopted under the assumption that it might be a pre-Christian survival. In fact, modern scholarship has been unable to trace any association between eggs and a supposed goddess named Ostara before the 19th century, when early folklorists began to speculate about the possibility.
There are good grounds for the association between hares (later termed Easter bunnies) and bird eggs, through folklore confusion between hares\' forms (where they raise their young) and plovers\' nests.
In Judaism, a hard-boiled egg is an element of the Passover Seder, representing festival sacrifice. The children\'s game of hunting for the afikomen (a half-piece of matzo) has similarities to the Easter egg hunt tradition, by which the child who finds the hidden matzah will be awarded a prize. In other homes, the children hide the afikoman and a parent must look for it; when the parents give up, the children demand a prize for revealing its location
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# Errol Morris
**Errol Mark Morris** (born February 5, 1948) is an American film director known for documentaries that interrogate the epistemology of their subjects, and the invention of the Interrotron. In 2003, his *The Fog of War: Eleven Lessons from the Life of Robert S. McNamara* won the Academy Award for Best Documentary Feature. His film *The Thin Blue Line* placed fifth on a *Sight & Sound* poll of the greatest documentaries ever made. Morris is known for making films about unusual subjects; *Fast, Cheap & Out of Control* interweaves the stories of an animal trainer, a topiary gardener, a robot scientist, and a naked mole-rat specialist.
## Early life and education {#early_life_and_education}
Morris was born on February 5, 1948, into a Jewish family in Hewlett, New York. His father died when he was two and he was raised by his mother, a piano teacher. He had one older brother, Noel, who was a computer programmer. After being treated for strabismus in childhood, Morris refused to wear an eye patch. As a consequence, he has limited sight in one eye and lacks normal stereoscopic vision.
In the 10th grade, Morris attended The Putney School, a boarding school in Vermont. He began playing the cello, spending a summer in France studying music under the acclaimed Nadia Boulanger, who also taught Morris\'s future collaborator Philip Glass. Describing Morris as a teenager, Mark Singer wrote that he \"read with a passion the 14-odd *Oz* books, watched a lot of television, and on a regular basis went with a doting but not quite right maiden aunt (\'I guess you\'d have to say that Aunt Roz was somewhat demented\') to Saturday matinées, where he saw such films as *This Island Earth* and *Creature from the Black Lagoon*---horror movies that, viewed again 30 years later, still seem scary to him.\"
### College
Morris attended the University of Wisconsin--Madison, graduating in 1969 with a Bachelor of Arts in history. For a brief time, Morris held small jobs, first as a cable-television salesman, and then as a term-paper writer. His unorthodox approach to applying for graduate school included \"trying to get accepted at different graduate schools just by showing up on their doorstep.\" Having unsuccessfully approached both the University of Oxford and Harvard University, Morris was able to talk his way into Princeton University, where he began studying the history of science, a topic in which he had \"absolutely no background.\" His concentration was in the history of physics, and he was bored and unsuccessful in the prerequisite physics classes he had to take. This, together with his antagonistic relationship with his advisor Thomas Kuhn (\'You won\'t even look through my telescope.\' And his response was \'Errol, it\'s not a telescope, it\'s a kaleidoscope.\') ensured that his stay at Princeton would be short.
Morris left Princeton in 1972, enrolling at Berkeley as a doctoral student in philosophy. At Berkeley, he once again found that he was not well-suited to his subject. \"Berkeley was just a world of pedants. It was truly shocking. I spent two or three years in the philosophy program. I have very bad feelings about it\", he later said.
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# Errol Morris
## Career
After leaving UC Berkeley, he became a regular at the Pacific Film Archive. As Tom Luddy, the director of the archive at the time, later remembered: \"He was a film noir nut. He claimed we weren\'t showing the real film noir. So I challenged him to write the program notes. Then, there was his habit of sneaking into the films and denying that he was sneaking in. I told him if he was sneaking in he should at least admit he was doing it.\"
### Unfinished project on Ed Gein {#unfinished_project_on_ed_gein}
Inspired by Hitchcock\'s *Psycho*, Morris visited Plainfield, Wisconsin in 1975, where he conducted multiple interviews with Ed Gein, the infamous body snatcher who resided at Mendota State Hospital in Madison. He later made plans with German film director Werner Herzog, whom Tom Luddy had introduced to Morris, to return in the summer of 1975 to secretly open the grave of Gein\'s mother to test their theory that Gein himself had already dug her up. Herzog arrived on schedule, but Morris had second thoughts and was not there. Herzog did not open the grave. Morris later returned to Plainfield, this time staying for almost a year, conducting hundreds of hours of interviews. Despite this, his plans to either write a book or make a film (which he would call *Digging up the Past*) were left unfinished at the time. In an October 2023 interview with Letterboxd, Morris mentioned that he has since returned to the project, saying \"I started rewatching *Psycho*, because I\'m making a movie about Ed Gein.\"
In the fall of 1976, Herzog visited Plainfield again, this time to shoot part of his film *Stroszek*.
### First films {#first_films}
Morris accepted \$2,000 from Herzog and used it to take a trip to Vernon, Florida. Vernon was nicknamed \"Nub City\" because its residents supposedly participated in a particularly gruesome form of insurance fraud in which they deliberately amputated a limb to collect the insurance money. Morris\'s second documentary was about the town and bore its name, although it made no mention of Vernon as \"Nub City\", but instead explored other idiosyncrasies of the town\'s residents. Morris made this omission because he received death threats while doing research; the town\'s residents were afraid that Morris would reveal their secret.
After spending two weeks in Vernon, Morris returned to Berkeley and began working on a script for a work of fiction that he called *Nub City.* After a few unproductive months, he happened upon a headline in the *San Francisco Chronicle* that read, \"450 Dead Pets Going to Napa Valley.\" Morris left for Napa Valley and began working on the film that would become his first feature, *Gates of Heaven*, which premiered in 1978. Herzog had said he would eat his shoe if Morris completed the documentary. After the film premiered, Herzog publicly followed through on the bet by cooking and eating his shoe, which was documented in the short film *Werner Herzog Eats His Shoe* by Les Blank.
*Gates of Heaven* was given a limited release in the spring of 1981. Roger Ebert was a champion of the film, including it on his ballot in the 1992 *Sight & Sound* critics\' poll. Morris returned to Vernon in 1979 and again in 1980, renting a house in town and conducting interviews with the town\'s citizens. *Vernon, Florida* premiered at the 1981 New York Film Festival. *Newsweek* called it, \"a film as odd and mysterious as its subjects, and quite unforgettable.\" The film, like *Gates of Heaven*, suffered from poor distribution. It was released on video in 1987, and DVD in 2005.
After finishing *Vernon, Florida*, Morris tried to get funding for a variety of projects. The *Road* story was about an interstate highway in Minnesota; one project was about Robert Golka, the creator of laser-induced fireballs in Utah; and another story was about Centralia, Pennsylvania, the coal town in which an inextinguishable subterranean fire ignited in 1962. He eventually got funding in 1983 to write a script about John and Jim Pardue, Missouri bank robbers who had killed their father and grandmother and robbed five banks. Morris\'s pitch went, \"The great bank-robbery sprees always take place at a time when something is going wrong in the country. Bonnie and Clyde were apolitical, but it\'s impossible to imagine them without the Depression as a backdrop. The Pardue brothers were apolitical, but it\'s impossible to imagine them without Vietnam.\" Morris wanted Tom Waits and Mickey Rourke to play the brothers, and he wrote the script, but the project eventually failed. Morris worked on writing scripts for various other projects, including a pair of ill-fated Stephen King adaptations.
In 1984, Morris married Julia Sheehan, whom he had met in Wisconsin while researching Ed Gein and other serial killers. He would later recall an early conversation with Julia: \"I was talking to a mass murderer but I was thinking of you,\" he said, and instantly regretted it, afraid that it might not have sounded as affectionate as he had wished. But Julia was actually flattered: \"I thought, really, that was one of the nicest things anyone ever said to me. It was hard to go out with other guys after that.\"
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# Errol Morris
## Career
### *The Thin Blue Line* {#the_thin_blue_line}
In 1985, Morris became interested in Dr. James Grigson, a psychiatrist in Dallas. Under Texas law, the death penalty can only be issued if the jury is convinced that the defendant is not only guilty, but will commit further violent crimes in the future if he is not put to death. Grigson had spent 15 years testifying for such cases, and he almost invariably gave the same damning testimony, often saying that it is \"one hundred per cent certain\" that the defendant would kill again. This led to Grigson being nicknamed \"Dr. Death.\" Through Grigson, Morris met the subject of his next film, 36-year-old Randall Dale Adams.
Adams was serving a life sentence that had been commuted from a death sentence on a legal technicality for the 1976 murder of Robert Wood, a Dallas police officer. Adams told Morris that he had been framed, and that David Harris, who was present at the time of the murder and was the principal witness for the prosecution, had in fact killed Wood. Morris began researching the case because it related to Dr. Grigson. He was at first unconvinced of Adams\'s innocence. After reading the transcripts of the trial and meeting David Harris at a bar, however, Morris was no longer so sure.
At the time, Morris had been making a living as a private investigator for a well-known private detective agency that specialized in Wall Street cases. Bringing together his talents as an investigator and his obsessions with murder, narration, and epistemology, Morris went to work on the case in earnest. Unedited interviews in which the prosecution\'s witnesses systematically contradicted themselves were used as testimony in Adams\'s 1986 *habeas corpus* hearing to determine if he would receive a new trial. David Harris famously confessed, in a roundabout manner, to killing Wood.
Although Adams was finally found innocent after years of being processed by the legal system, the judge in the *habeas corpus* hearing officially stated that, \"much could be said about those videotape interviews, but nothing that would have any bearing on the matter before this court.\" Regardless, *The Thin Blue Line*, as Morris\'s film would be called, was popularly accepted as the main force behind getting its subject, Randall Adams, out of prison. As Morris said of the film, \"*The Thin Blue Line* is two movies grafted together. On one simple level is the question, Did he do it, or didn\'t he? And on another level, *The Thin Blue Line*, properly considered, is an essay on false history. A whole group of people, literally everyone, believed a version of the world that was entirely wrong, and my accidental investigation of the story provided a different version of what happened.\"
*The Thin Blue Line* ranks among the most critically acclaimed documentaries ever made. According to a survey by *The Washington Post*, the film made dozens of critics\' top ten lists for 1988, more than any other film that year. It won the documentary of the year award from both the New York Film Critics Circle and the National Society of Film Critics. Despite its widespread acclaim, it was not nominated for an Oscar, which created a small scandal regarding the nomination practices of the academy. The academy cited the film\'s genre of \"non-fiction\", arguing that it was not actually a documentary. It was the first of Morris\'s films to be scored by Philip Glass.
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# Errol Morris
## Career
### *A Brief History of Time* and *Fast, Cheap & Out of Control* {#a_brief_history_of_time_and_fast_cheap_out_of_control}
Morris wanted to make a film about Albert Einstein\'s brain and approached Amblin Entertainment about it. Gordon Freeman had acquired the rights to Stephen Hawking\'s bestseller *A Brief History of Time* and Steven Spielberg suggested Morris direct it. After reading Hawking\'s book, Morris agreed to direct a documentary adaptation of it, having studied the philosophy of science at Princeton. Morris\'s film *A Brief History of Time* is less an adaptation of Hawking\'s book than a portrait of the scientist. It combines interviews with Hawking, his colleagues and his family with computer animations and clips from movies like Disney\'s *The Black Hole*. Morris said he was \"very moved by Hawking as a man\", calling him \"immensely likable, perverse, funny\...and yes, he\'s a genius.\"
Morris\'s *Fast, Cheap & Out of Control* interweaves interviews with a wild animal trainer, a topiary gardener, a robot scientist and a naked mole rat specialist with stock footage, cartoons and clips from film serials. Roger Ebert said of it, \"If I had to describe it, I\'d say it\'s about people who are trying to control things - to take upon themselves the mantle of God.\" Morris agreed there was a \"Frankenstein element\", adding \"They\'re all involved in some very odd inquiry about life. It sounds horribly pretentious laid out that way, but there\'s something mysterious in each of the stories, something melancholy as well as funny. And there\'s an edge of mortality. For the end of the movie I showed the gardener clipping the top of his camel, clipping in a heavenly light, and then walking away in the rain. You know that this garden is not going to last much longer than the gardener\'s lifetime.\" The film was scored by Caleb Sampson of the Alloy Orchestra and photographed by Robert Richardson. Morris dedicated the film to his mother and stepfather, who had recently died. It was named by several critics as one of the best films of 1997.
In 2002, Morris was commissioned to make a short film for the 75th Academy Awards. He was hired based on his advertising resume, not his career as a director of feature-length documentaries. Those interviewed ranged from Laura Bush to Iggy Pop to Kenneth Arrow to Morris\'s 15-year-old son Hamilton. Morris was nominated for an Emmy for this short film. He considered editing this footage into a feature-length film, focusing on Donald Trump discussing *Citizen Kane* (this segment was later released on the second issue of *Wholphin*). Morris went on to make a second short for the 79th Academy Awards in 2007, this time interviewing the various nominees and asking them about their Oscar experiences.
### *The Fog of War* and later films {#the_fog_of_war_and_later_films}
In 2003, Morris won the Oscar for Best Documentary for *The Fog of War*, a film about the career of Robert S. McNamara, the Secretary of Defense during the Vietnam War under Presidents John F. Kennedy and Lyndon B. Johnson. In the haunting opening about McNamara\'s relationship with U.S. General Curtis LeMay during World War II, Morris brings out complexities in the character of McNamara, which shaped McNamara\'s positions in the Cuban Missile Crisis and the Vietnam War. Like his earlier documentary, *The Thin Blue Line,* *The Fog of War* included extensive use of re-enactments, a technique which many had believed was inappropriate for documentaries prior to his Oscar win.
In early 2010, a new Morris documentary was submitted to several film festivals, including Toronto International Film Festival, Cannes Film Festival, and Telluride Film Festival. The film, *Tabloid*, features interviews with Joyce McKinney, a former Miss Wyoming, who was convicted *in absentia* for the kidnap and indecent assault of a Mormon missionary in England during 1977.
Subsequently, Morris has made documentaries such as *The Unknown Known* (2013), *American Dharma* (2018), and *The Pigeon Tunnel* (2023), revolving around interviews conducted with Donald Rumsfeld, Steve Bannon, and John le Carré, respectively.
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# Errol Morris
## Career
### Commercials
Although Morris has achieved fame as a documentary filmmaker, he is also an accomplished director of television commercials. In 2002, Morris directed a series of television ads for Apple Computer as part of a popular \"Switch\" campaign. The commercials featured ex-Windows users discussing their various bad experiences that motivated their own personal switches to Macintosh. One commercial in the series, starring Ellen Feiss, a high-schooler friend of his son Hamilton Morris, became an Internet meme. Morris has directed hundreds of commercials for various companies and products, including Adidas, AIG, Cisco Systems, Citibank, Kimberly-Clark\'s Depend brand, Levi\'s, Miller High Life, Nike, PBS, The Quaker Oats Company, Southern Comfort, EA Sports, Toyota and Volkswagen. Many of these commercials are available on his website.
In July 2004, Morris directed another series of commercials in the style of the \"Switch\" ads. This campaign featured Republicans who voted for Bush in the 2000 election giving their personal reasons for voting for Kerry in 2004. Upon completing more than 50 commercials, Morris had difficulty getting them on the air. Eventually, the liberal advocacy group MoveOn PAC paid to air a few of the commercials. Morris also wrote an editorial for *The New York Times* discussing the commercials and Kerry\'s losing campaign.
In late 2004, Morris directed a series of noteworthy commercials for Sharp Electronics. The commercials enigmatically depicted various scenes from what appeared to be a short narrative that climaxed with a car crashing into a swimming pool. Each commercial showed a slightly different perspective on the events, and each ended with a cryptic weblink. The weblink was to a fake webpage advertising a prize offered to anyone who could discover the secret location of some valuable urns. It was in fact an alternate reality game. The original commercials can be found on Morris\'s website.
Morris directed a series of commercials for Reebok that featured six prominent National Football League (NFL) players. The 30-second promotional videos were aired during the 2006 NFL season.
In 2013, Morris stated that he has made around 1,000 commercials during his career. Since then he has continued in the field, including a 2019 campaign for Chipotle.
In 2015, Morris made commercials for medical technology firm Theranos, and interviewed its founder, Elizabeth Holmes. After the company fell in disgrace, Morris was criticized by *The Telegraph* for seeming \"captivated\" by Holmes, and for contributing to Holmes\' mythical persona as a visionary. In a 2019 *New Yorker* interview, Morris reflected, \"To me, what really is interesting about Elizabeth \[Holmes\] \... did she really see herself as a fraud? Was it calculation? I have a hard time squaring that with my own experience. Could I have been self-deceived, delusional? You betcha. I\'m no different than the next guy. I\'d like to think I\'m a little different. But I\'m still fascinated by her.\"
### Writings and documentary shorts {#writings_and_documentary_shorts}
Morris has also written long-form journalism, exploring different areas of interest and published on *The New York Times* website. A collection of these essays, titled *Believing is Seeing: Observations on the Mysteries of Photography,* was published by Penguin Press on September 1, 2011. In November 2011, Morris premiered a documentary short titled \"The Umbrella Man\"---featuring Josiah \"Tink\" Thompson---about the Kennedy assassination on *The New York Times* website.
In 2012, Morris published his second book, *A Wilderness of Error: The Trials of Jeffrey MacDonald*, about Jeffrey MacDonald, the Green Beret physician convicted of killing his wife and two daughters on February 17, 1970. Morris first became interested in the case in the early 1990s and believes that MacDonald is not guilty after undertaking extensive research. Morris explained in a July 2013 interview, prior to the reopening of the case: \"What happened here is wrong. It\'s wrong to convict a man under these circumstances. And if I can help correct that, I will be a happy camper.\" He now states that he does not believe that Macdonald is guilty, but thinks it possible that Macdonald is guilty.
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# Errol Morris
## Style and legacy {#style_and_legacy}
To conduct interviews, Morris invented a machine, called the Interrotron, which allows the interviewer and his subject to make eye-contact with each other while both staring through the camera lens itself. He explains the device as follows:
Author Marsha McCreadie, in her book *Documentary Superstars: How Today\'s Filmmakers Are Reinventing the Form*, had paired Morris with Werner Herzog as practitioners and visionaries in their approach in documentary filmmaking.
Morris uses narrative elements within his films. These include but are not limited to: stylized lighting, musical score, and re-enactment. The use of these elements is rejected by many documentary filmmakers who followed the cinema vérité style of the previous generations. Cinema vérité is characterized by its rejection of artistic additions to documentary film. While Morris faced backlash from many of the older-era filmmakers, his style has been embraced by the younger generations of filmmakers, as the use of re-enactment is present in many contemporary documentary films.
Morris advocates the reflexive style of documentary filmmaking. In Bill Nichols\'s book *Introduction to Documentary* he states that reflexive documentary \"\[speaks\] not only about the historical world but about the problems and issues of representing it as well.\" Morris uses his films not only to portray social issues and non-fiction events but also to comment on the reliability of documentary making itself.
His style has been spoofed in the mockumentary series *Documentary Now*.
Even when interviewing controversial figures, Morris does not generally believe in adversarial interviews:`{{blockquote|I don't really believe in adversarial interviews. I don't think you learn very much. You create a theater, a gladiatorial theater, which may be satisfying to an audience, but if the goal is to learn something that you don't know, that's not the way to go about doing it. In fact, it's the way to destroy the possibility of ever hearing anything interesting or new. .... the most interesting and most revealing comments have come not as a result of a question at all, but having set up a situation where people actually want to talk to you, and want to reveal something to you.<ref>{{Cite web|url=https://cafe.com/stay-tuned-transcript-bannon-the-f-you-presidency-with-errol-morris/|title=Stay Tuned Transcript: Bannon & The F You Presidency (with Errol Morris)|date=2019-11-07|website=CAFE|language=en-US|access-date=2019-11-13|archive-date=November 13, 2019|archive-url=https://web.archive.org/web/20191113033751/https://cafe.com/stay-tuned-transcript-bannon-the-f-you-presidency-with-errol-morris/|url-status=dead}}</ref>}}`{=mediawiki}
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# Errol Morris
## Filmography
### Feature films {#feature_films}
- *Gates of Heaven* (1978)
- *Vernon, Florida* (1981)
- *The Thin Blue Line* (1988)
- *The Dark Wind* (1991), fiction movie
- *A Brief History of Time* (1991)
- *Fast, Cheap & Out of Control* (1997)
- *Mr. Death: The Rise and Fall of Fred A. Leuchter, Jr.* (1999)
- *The Fog of War: Eleven Lessons from the Life of Robert S. McNamara* (2003)
- *Standard Operating Procedure* (2008)
- *Tabloid* (2010)
- *The Act of Killing* (executive producer) (2012)
- *The Unknown Known* (2013)
- *The Look of Silence* (executive producer) (2014)
- *Happy Father\'s Day* (video) (2015)
- *Uncle Nick* (executive producer) (2015)
- *The B-Side: Elsa Dorfman\'s Portrait Photography* (2016)
- *National Bird* (executive producer) (2016)
- *American Dharma* (2018)
- *Enemies of the State* (executive producer) (2020)
- *My Psychedelic Love Story* (2020)
- *The Pigeon Tunnel* (2023)
- *Tune Out the Noise* (2024)
- *Separated* (2024)
- *CHAOS: The Manson Murders* (2025)
### Short films {#short_films}
- *Survivors* (2008)
- *They Were There* (Documentary short) (2011)
- *El Wingador* (Documentary short) (2012)
- *Three Short Films About Peace* (2014)
- *Leymah Gbowee: The Dream* (Documentary short) (2014)
### Television
- *Errol Morris Interrotron Stories: Digging Up the Past* (TV miniseries documentary) (1995)
- *First Person* (TV series documentary) (17 episodes) (2000)
- *Op-Docs* (TV series documentary trilogy)
- *The Umbrella Man* about Umbrella man (JFK assassination) (2011)
- *November 22, 1963* (2013)
- *A Demon in the Freezer* (2016)
- *P.O.V.* (executive producer) (2014--2016)
- *It\'s Not Crazy, It\'s Sports* (TV documentary series) (2015)
- *The Subterranean Stadium* (TV movie) (2015)
- *The Streaker* (TV movie) (2015)
- *The Heist* (TV movie) (2015)
- *Most Valuable Whatever* (TV movie) (2015)
- *Chrome* (TV movie) (2015)
- *Being Mr. Met* (TV movie) (2015)
- *Zillow Hiram\'s Home* (TV movie) (2016)
- *Wormwood* (miniseries) (2017)
- *A Wilderness of Error* (docuseries on FX) (2020)
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# Errol Morris
## Accolades
- *Gates of Heaven* (1978) was long featured on Roger Ebert\'s list of the ten greatest films ever made.
- Golden Horse for Best Foreign Film at the Taiwan International Film Festival for *The Thin Blue Line* (1988)
- New York Film Critics Circle and the National Society of Film Critics Best Documentary for *The Thin Blue Line* (1988)
- Washington Post Best Film of the Year for *The Thin Blue Line* (1988)
- Edgar Award for Best Motion Picture, from the Mystery Writers of America, for *The Thin Blue Line* (1989)
- Guggenheim Fellowship (1989)
- MacArthur Fellowship (1989)
- Emmy for Best Commercial for PBS commercial \"[Photobooth](http://www.errolmorris.com/commercials/pbs/pbs_photobooth.html)\" (2001)
- In December 2001, the United States\' National Film Preservation Foundation announced that Morris\'s *The Thin Blue Line* would be one of the 25 films selected that year for preservation in the National Film Registry at the Library of Congress, bringing the total at the time to 325.
- 2002 International Documentary Association list of the 20 all-time best documentaries: *The Thin Blue Line* (#2), *Fast, Cheap & Out of Control* (#14)
- Best Documentary of the Year awards for *The Fog of War* (2003): the National Board of Review, the Los Angeles Film Critics Association, the Chicago Film Critics, and the Washington D.C. Area Film Critics.
- In 2003, *The Guardian* put him seventh in its list of the world\'s 40 best active directors.
- Academy Award for Documentary Feature *The Fog of War* (2004)
- Fellow of the American Academy of Arts and Sciences (2007)
- Jury Grand Prix Silver Bear at the 2008 Berlin International Film Festival for *Standard Operating Procedure*
- Columbia Journalism Award (2013)
- In 2019, *The Fog of War* was selected by the Library of Congress for preservation in the National Film Registry for being \"culturally, historically, or aesthetically significant\".
### Honorary degrees {#honorary_degrees}
- Middlebury College, Hon. D.F.A. (2010)
- Brandeis University, Hon. D.H.L. (2011)
- University of Wisconsin--Madison, Hon. D.H.L
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# Ethan Allen
**Ethan Allen** (`{{OldStyleDate|January 21, 1738||January 10, 1737}}`{=mediawiki} -- February 12, 1789) was an American farmer, writer, military officer and politician. He is best known as one of the founders of Vermont and for the capture of Fort Ticonderoga during the American Revolutionary War, and was also the brother of Ira Allen and the father of Fanny Allen.
Allen was born in rural Connecticut and had a frontier upbringing, but he also received an education that included some philosophical teachings. In the late 1760s, he became interested in the New Hampshire Grants, buying land there and becoming embroiled in the legal disputes surrounding the territory. Legal setbacks led to the formation of the Green Mountain Boys, whom Allen led in a campaign of intimidation and property destruction to drive New York settlers from the Grants. He and the Patriot-aligned Green Mountain Boys seized the initiative early in the Revolutionary War and captured Fort Ticonderoga in May 1775. In September 1775, Allen led a failed attempt on Montreal which resulted in his capture by the British. He was imprisoned aboard ships of the Royal Navy, then paroled in New York City, and finally released in a prisoner exchange in 1778.
Upon his release, Allen returned to the New Hampshire Grants which had declared independence in 1777, and he resumed political activity in the territory, continuing resistance to New York\'s attempts to assert control over the territory. Allen lobbied Congress for Vermont\'s official state recognition, and he participated in controversial negotiations with the British over the possibility of Vermont becoming a separate British province.
Allen wrote accounts of his exploits in the war that were widely read in the 19th century, as well as philosophical treatises and documents relating to the politics of Vermont\'s formation. His business dealings included successful farming operations, one of Connecticut\'s early iron works, and land speculation in the Vermont territory. Allen and his brothers purchased tracts of land that became Burlington, Vermont. He was married twice, fathering eight children.
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# Ethan Allen
## Early life {#early_life}
Allen was born in Litchfield, Connecticut Colony, the first child of Joseph and Mary Baker Allen, both descended from English Puritans. The family moved to the town of Cornwall shortly after his birth due to his father\'s quest for freedom of religion during the Great Awakening. As a boy, Allen already excelled at quoting the Bible and was known for disputing the meaning of passages. He had five brothers (Heman, Heber, Levi, Zimri, and Ira) and two sisters (Lydia and Lucy). His brothers Ira and Heman were also prominent figures in the early history of Vermont.
The town of Cornwall was frontier territory in the 1740s, but it began to resemble a town by the time that Allen was a teenager, with wood-frame houses beginning to replace the rough cabins of the early settlers. Joseph Allen was one of the wealthier landowners in the area by the time of his death in 1755. He ran a successful farm and had served as town selectman. Allen began studies under a minister in the nearby town of Salisbury with the goal of gaining admission to Yale College.
### First marriage and early adulthood {#first_marriage_and_early_adulthood}
Allen was forced to end his studies upon his father\'s death. He volunteered for militia service in 1757 in response to the French siege of Fort William Henry, but his unit received word that the fort had fallen while they were en route, and they turned back. The French and Indian War continued over the next several years, but Allen did not participate in any further military activities and is presumed to have tended his farm. In 1762, he became part owner of an iron furnace in Salisbury. He also married Mary Brownson from Roxbury in July 1762, who was five years his senior. They first settled in Cornwall, but moved the following year to Salisbury with their infant daughter Loraine. They bought a small farm and proceeded to develop the iron works. The expansion of the iron works was apparently costly to Allen; he was forced to sell off portions of the Cornwall property to raise funds, and eventually sold half of his interest in the works to his brother Heman. The Allen brothers sold their interest in the iron works in October 1765.
By most accounts, Allen\'s first marriage was unhappy. His wife was rigidly religious, prone to criticizing him, and barely able to read and write. In contrast, his behavior was sometimes quite flamboyant, and he maintained an interest in learning. Nevertheless, they remained together until Mary\'s death in 1783. They had five children together, only two of whom reached adulthood.
Allen and his brother Heman went to the farm of a neighbor whose pigs had escaped onto their land, and they seized the pigs. The neighbor sued to have the animals returned to him; Allen pleaded his own case and lost. Allen and Heman were fined ten shillings, and the neighbor was awarded another five shillings in damages. He was also called to court in Salisbury for inoculating himself against smallpox, a procedure that required the sanction of the town selectmen.
Allen met Thomas Young when he moved to Salisbury, a doctor living and practicing just across the provincial boundary in New York. Young taught him a great deal about philosophy and political theory, while Allen shared his appreciation of nature and life on the frontier with Young. They eventually decided to collaborate on a book intended as an attack on organized religion, as Young had convinced Allen to become a Deist. They worked on the manuscript until 1764, when Young moved away from the area taking the manuscript with him. Allen recovered the manuscript many years later, after Young\'s death. He expanded and reworked the material, and eventually published it as *Reason: the Only Oracle of Man*.
Heman remained in Salisbury where he ran a general store until his death in 1778, but Allen\'s movements are poorly documented over the next few years. He lived in Northampton, Massachusetts, in the spring of 1766, where his son Joseph was born and where he invested in a lead mine. The authorities asked him to leave Northampton in July 1767, though no official reason is known. Biographer Michael Bellesiles suggests that religious differences and Allen\'s tendency to be disruptive may have played a role in his departure. Allen briefly returned to Salisbury before settling in Sheffield, Massachusetts, with his younger brother Zimri. It is likely that his first visits to the New Hampshire Grants occurred during these years. Sheffield was the family home for ten years, although Allen was often absent for extended periods.
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# Ethan Allen
## New Hampshire Grants {#new_hampshire_grants}
New Hampshire Governor Benning Wentworth was selling land grants west of the Connecticut River as early as 1749, an area to which New Hampshire had always laid claim. Many of these grants were sold at relatively low prices to land speculators, with Wentworth also reserving for himself a share of each grant. In 1764, King George III issued an order resolving the competing claims of New York and New Hampshire in favor of New York. New York had issued land grants that overlapped some of those sold by Wentworth, and authorities there insisted that holders of the Wentworth grants pay a fee to New York to have their grants validated. This fee approached the original purchase price, and many of the holders were land-rich and cash-poor, so there was a great deal of resistance to the demand. By 1769, the situation had deteriorated to the point that surveyors and other figures of New York authority were being physically threatened and driven from the area.
A few of the holders of Wentworth grants were from northwestern Connecticut, and some of them were related to Allen, including Remember Baker and Seth Warner. In 1770, a group of them asked him to defend their case before New York\'s Supreme Court. Allen hired Jared Ingersoll to represent the grant-holder interest in the trial, which began in July 1770 and pitted Allen against politically powerful New York grant-holders, including New York\'s Lieutenant Governor Colden, James Duane who was prosecuting the case, and Robert Livingston, the Chief Justice of the Supreme Court who was presiding over the case. The trial was brief and the outcome unsurprising, as the court refused to allow the introduction of Wentworth\'s grants as evidence, citing their fraudulently issued nature. Duane visited Allen and offered him payments \"for going among the people to quiet them\". Allen denied taking any money and claimed that Duane was outraged and left with veiled threats, indicating that attempts to enforce the judgment would be met with resistance.
Many historians believe that Allen took these actions`{{clarify|WHAT actions??|date=September 2019}}`{=mediawiki} because he already held Wentworth grants of his own, although there is no evidence that he was issued any such grants until after he had been asked to take up the defense at the trial. He acquired grants from Wentworth to about 1000 acre in Poultney and Castleton prior to the trial.
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# Ethan Allen
## New Hampshire Grants {#new_hampshire_grants}
### Green Mountain Boys {#green_mountain_boys}
On Allen\'s return to Bennington, the settlers met at the Catamount Tavern to discuss their options. These discussions resulted in the formation of the Green Mountain Boys in 1770, with local militia companies in each of the surrounding towns. Allen was named their Colonel Commandant, and cousins Seth Warner and Remember Baker were captains of two of the companies. Further meetings resulted in creating committees of safety; they also laid down rules to resist New York\'s attempts to establish its authority. These included not allowing New York\'s surveyors to survey *any* land in the Grants, not just land owned through the Wentworth grants. Allen participated in some of the actions to drive away surveyors, and he also spent much time exploring the territory. He sold some of his Connecticut properties and began buying land farther north in the territory, which he sold at a profit as the southern settlements grew and people began to move farther north.
Friction increased with the provincial government in October 1771, when Allen and a company of Green Mountain Boys drove off a group of Scottish settlers near Rupert. Allen detained two of the settlers and forced them to watch them burn their newly constructed cabins. Allen then ordered them to \"go your way now, and complain to that damned scoundrel your Governor, God damn your Governor, Laws, King, Council, and Assembly\". The settlers protested his language but Allen continued the tirade, threatening to send any troops from New York to Hell. In response, New York Governor William Tryon issued warrants for the arrests of those responsible, and eventually put a price of £20 (around £3.3k today, or \$4.4k) on the heads of six participants, including Allen. Allen and his comrades countered by issuing offers of their own. `{{Quote box|align=right|width=30%|£25 REWARD—Whereas James Duane and John Kempe, of New York, have by their menaces and threats greatly disturbed the public peace and repose of the honest peasants of Bennington and the settlements to the northward, which are now and ever have been in the peace of God and the King, and are patriotic and liege subjects of Geo. the 3rd. Any person that will apprehend these common disturbers, viz: James Duane and John Kempe, and bring them to Landlord Fay's at Bennington shall have £15 reward James Duane and £10 reward for John Kempe, paid by|Ethan Allen, Remember Baker, Robert Cochran<ref name="Jellison62">Jellison, p. 62</ref>}}`{=mediawiki}
The situation deteriorated further over the next few years. Governor Tryon and the Green Mountain Boys exchanged threats, truce offers, and other writings, frequently written by Allen in florid and didactic language while the Green Mountain Boys continued to drive away surveyors and incoming tenants. Most of these incidents did not involve bloodshed, although individuals were at times manhandled, and the Green Mountain Boys sometimes did extensive property damage when driving tenants out. By March 1774, the harsh treatment of settlers and their property prompted Tryon to increase some of the rewards to £100.
### Onion River Company {#onion_river_company}
Allen joined his cousin Remember Baker and his brothers Ira, Heman, and Zimri to form the Onion River Company in 1772, a land-speculation organization devoted to purchasing land around the Winooski River, which was known then as the Onion River. The success of this business depended on the defense of the Wentworth grants. Early purchases included about 40000 acre from Edward Burling and his partners; they sold land at a profit to Thomas Chittenden, among others, and their land became the city of Burlington.
The outrage of the Wentworth proprietors was renewed in 1774 when Governor Tryon passed a law containing harsh provisions clearly targeted at the actions of the \"Bennington Mob\". Vermont historian Samuel Williams called it \"an act which for its savage barbarity is probably without parallel in the legislation of any civilized country\". Its provisions included the death penalty for interfering with a magistrate, and outlawing meetings of more than three people \"for unlawful purposes\" in the Grants. The Green Mountain Boys countered with rules of their own, forbidding anyone in the Grants from holding \"any office of honor or profit under the colony of N. York\".
Allen spent much of the summer of 1774 writing *A Brief Narrative of the Proceedings of the Government of New York Relative to Their Obtaining the Jurisdiction of that Large District of Land to the Westward of the Connecticut River*, a 200-page polemic arguing the position of the Wentworth proprietors. He had it printed in Connecticut and began selling and giving away copies in early 1775. Historian Charles Jellison describes it as \"rebellion in print\".
### Westminster massacre {#westminster_massacre}
Allen traveled into the northern parts of the Grants early in 1775 for solitude and to hunt for game and land opportunities. A few days after his return, news came that blood had finally been shed over the land disputes. Most of the resistance activity had taken place on the west side of the Green Mountains until then, but a small riot broke out in Westminster on March 13 and led to the deaths of two men. Allen and a troop of Green Mountain Boys traveled to Westminster where the town\'s convention adopted a resolution to draft a plea to the King to remove them \"out of so oppressive a jurisdiction\". It was assigned to a committee which included Allen. The American Revolutionary War began less than a week after the Westminster convention ended, while Allen and the committee worked on their petition.
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# Ethan Allen
## Revolutionary War {#revolutionary_war}
### Capture of Fort Ticonderoga {#capture_of_fort_ticonderoga}
Allen received a message from members of an irregular Connecticut militia in late April, following the battles of Lexington and Concord, that they were planning to capture Fort Ticonderoga and requesting his assistance in the effort. Allen agreed to help and began rounding up the Green Mountain Boys, and 60 men from Massachusetts and Connecticut met with Allen in Bennington on May 2, 1775, where they discussed the logistics of the expedition. By May 7, these men joined Allen and 130 Green Mountain Boys at Castleton. They elected Allen to lead the expedition, and they planned a dawn raid for May 10. Two small companies were detached to procure boats, and Allen took the main contingent north to Hand\'s Cove in Shoreham to prepare for the crossing.
On the afternoon of May 9, Benedict Arnold unexpectedly arrived, flourishing a commission from the Massachusetts Committee of Safety. He asserted his right to command the expedition, but the men refused to acknowledge his authority and insisted that they would follow only Allen\'s lead. Allen and Arnold reached an accommodation privately, the essence of which was that Arnold and Allen would both be at the front of the troops when they attacked the fort.
The troops procured a few boats around 2 a.m. for the crossing, but only 83 men made it to the other side of the lake before Allen and Arnold decided to attack, concerned that dawn was approaching. The small force marched on the fort in the early dawn, surprising the lone sentry, and Allen went directly to the fort commander\'s quarters, seeking to force his surrender. Lieutenant Jocelyn Feltham was awakened by the noise, and called to wake the fort\'s commander Captain William Delaplace. He demanded to know by what authority the fort was being entered, and Allen said, \"In the name of the Great Jehovah and the Continental Congress!\" Delaplace finally emerged from his chambers and surrendered his sword, and the rest of the fort\'s garrison surrendered without firing a shot. The only casualty had been a British soldier who became concussed when Allen hit him with a cutlass, hitting the man\'s hair comb and saving his life.
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# Ethan Allen
## Revolutionary War {#revolutionary_war}
### Raids on St. John {#raids_on_st._john}
On the following day, a detachment of the Boys under Seth Warner\'s command went to nearby Fort Crown Point and captured the small garrison there. On May 14, following the arrival of 100 men recruited by Arnold\'s captains, and the arrival of a schooner and some bateaux that had been taken at Skenesboro, Arnold and 50 of his men sailed north to raid Fort St. John, on the Richelieu River downstream from the lake, where a small British warship was reported by the prisoners to be anchored. Arnold\'s raid was a success; he seized the sloop `{{HMS|Royal George|1776|6}}`{=mediawiki}, supplies, and a number of bateaux.
Allen, shortly after Arnold\'s departure on the raid, decided, after his successes at the southern end of the lake, to take and hold Fort St. John himself. To that end, he and about 100 Boys climbed into four bateaux, and began rowing north. After two days without significant food (which they had forgotten to provision in the boats), Allen\'s small fleet met Arnold\'s on its way back to Ticonderoga near the foot of the lake. Arnold generously opened his stores to Allen\'s hungry men, and tried to dissuade Allen from his objective, noting that it was likely the alarm had been raised and troops were on their way to St. John. Allen, likely both stubborn in his determination, and envious of Arnold, persisted.
When Allen and his men landed above St. John and scouted the situation, they learned that a column of 200 or more regulars was approaching. Rather than attempt an ambush on those troops, which significantly outnumbered his tired company, Allen withdrew to the other side of the river, where the men collapsed with exhaustion and slept without sentries through the night. They were awakened when British sentries discovered them and began firing grapeshot at them from across the river. The Boys, in a panic, piled into their bateaux and rowed with all speed upriver. When the expedition returned to Ticonderoga two days later, some of the men were greatly disappointed that they felt they had nothing to show for the effort and risks they took, but the capture of Fort Ticonderoga and Crown Point proved to be important in the Revolutionary War because it secured protection from the British to the North and provided vital cannon for the colonial army.
### Promoting an invasion {#promoting_an_invasion}
Following Allen\'s failed attempt on St. John, many of his men drifted away, presumably drawn by the needs of home and farm. Arnold then began asserting his authority over Allen for control of Ticonderoga and Crown Point. Allen publicly announced that he was stepping down as commander, but remained hopeful that the Second Continental Congress was going to name \"a commander for this department \... Undoubtedly, we shall be rewarded according to our merit\". Congress, for its part, at first not really wanting any part of the affair, effectively voted to strip and then abandon the forts. Both Allen and Arnold protested these measures, pointing out that doing so would leave the northern border wide open. They both also made proposals to Congress and other provincial bodies for carrying out an invasion of Quebec. Allen, in one instance, wrote that \"I will lay my life on it, that with fifteen hundred men, and a proper artillery, I will take Montreal\". Allen also attempted correspondence with the people of Quebec and with the Indians living there in an attempt to sway their opinion toward the revolutionary cause.
On June 22, Allen and Seth Warner appeared before Congress in Philadelphia, where they argued for the inclusion of the Green Mountain Boys in the Continental Army. After deliberation, Congress directed General Philip Schuyler, who had been appointed to lead the Army\'s Northern Department, to work with New York\'s provincial government to establish (and pay for) a regiment consisting of the Boys, and that they be paid Army rates for their service at Ticonderoga. On July 4, Allen and Warner made their case to New York\'s Provincial Congress, which, despite the fact that the Royal Governor had placed a price on their heads, agreed to the formation of a regiment. Following a brief visit to their families, they returned to Bennington to spread the news. Allen went to Ticonderoga to join Schuyler, while Warner and others raised the regiment.
### Allen loses command of the Boys {#allen_loses_command_of_the_boys}
When the regimental companies in the Grants had been raised, they held a vote in Dorset to determine who would command the regiment. By a wide margin, Seth Warner was elected to lead the regiment. Brothers Ira and Heman were also given command positions, but Allen was not given any position at all in the regiment. The thorough rejection stung; Allen wrote to Connecticut Governor Jonathan Trumbull, \"How the old men came to reject me I cannot conceive inasmuch as I saved them from the incroachments of New York.\"
The rejection likely had several causes. The people of the Grants were tired of the disputes with New York, and they were tired of Allen\'s posturing and egotistic behavior, which the success at Ticonderoga had enhanced. Finally, the failure of the attempt on St. John\'s was widely seen as reckless and ill-advised, attributes they did not appreciate in a regimental leader. Warner was viewed as a more stable and quieter choice, and was someone who also commanded respect. The history of Warner\'s later actions in the revolution (notably at Hubbardton and Bennington) may be seen as a confirmation of the choice made by the Dorset meeting. In the end, Allen took the rejection in stride, and managed to convince Schuyler and Warner to permit him to accompany the regiment as a civilian scout.
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# Ethan Allen
## Revolutionary War {#revolutionary_war}
### Capture
The American invasion of Quebec departed from Ticonderoga on August 28. On September 4, the army had occupied the Île aux Noix in the Richelieu River, a few miles above Fort St. John, which they then prepared to besiege. On September 8, Schuyler sent Allen and Massachusetts Major John Brown, who had also been involved in the capture of Ticonderoga, into the countryside between St. John and Montreal to spread the word of their arrival to the habitants and the Indians. They were successful enough in gaining support from the inhabitants that Quebec\'s governor, General Guy Carleton, reported that \"they have injured us very much\".
When he returned from that expedition eight days later, Brigadier General Richard Montgomery had assumed command of the invasion due to Schuyler\'s illness. Montgomery, likely not wanting the troublemaker in his camp, again sent Allen out, this time to raise a regiment of French-speaking Canadiens. Accompanied by a small number of Americans, he again set out, traveling through the countryside to Sorel, before turning to follow the Saint Lawrence River up toward Montreal, recruiting upwards of 200 men.
On September 24, he and Brown, whose company was guarding the road between St. John\'s and Montreal, met at Longueuil, and, according to Allen\'s account of the events, came up with a plan in which both he and Brown would lead their forces to attack Montreal. Allen and about 100 men crossed the Saint Lawrence that night, but Brown and his men, who were to cross the river at La Prairie, did not. General Carleton, alerted to Allen\'s presence, mustered every man he could, and, in the Battle of Longue-Pointe, scattered most of Allen\'s force, and captured him and about 30 men. His capture ended his participation in the revolution until 1778, as he was imprisoned by the British. General Schuyler, upon learning of Allen\'s capture, wrote, \"I am very apprehensive of disagreeable consequences arising from Mr. Allen\'s imprudence. I always dreaded his impatience and imprudence.\"
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# Ethan Allen
## Imprisonment
Much of what is known of Allen\'s captivity is known only from his own account of the time; where contemporary records are available, they tend to confirm those aspects of his story.
Allen was first placed aboard `{{HMS|Gaspée|1773|6}}`{=mediawiki}, a brig anchored at Montreal. He was kept in solitary confinement and chains, and General Richard Prescott had, according to Allen, ordered him to be treated \"with much severity\". In October 1775, *Gaspée* went downriver, and her prisoners were transferred to the merchant vessel *Adamant*, which then sailed for England. Allen wrote of the voyage that he \"was put under the power of an English Merchant from London, whose name was Brook Watson: a man of malicious and cruel disposition\".
On arrival at Falmouth, England, after a crossing under filthy conditions, Allen and the other prisoners were imprisoned in Pendennis Castle, Cornwall. At first his treatment was poor, but Allen wrote a letter, ostensibly to the Continental Congress, describing his conditions and suggesting that Congress treat the prisoners it held the same way. Unknown to Allen, British prisoners now included General Prescott, captured trying to escape from Montreal, and the letter came into the hands of the British cabinet. Also faced with opposition within the British establishment to the treatment of captives taken in North America, King George decreed that the men should be sent back to America and treated as prisoners of war.
In January 1776, Allen and his men were put on board HMS *Soledad*, which sailed for Cork, Ireland. The people of Cork, when they learned that the famous Ethan Allen was in port, took up a collection to provide him and his men with clothing and other supplies. Much of the following year was spent on prison ships off the American coast. At one point, while aboard HMS *Mercury*, she anchored off New York, where, among other visitors, the captain entertained William Tryon; Allen reports that Tryon glanced at him without any sign of recognition, although it is likely the New York governor knew who he was. In August 1776, Allen and other prisoners were temporarily put ashore in Halifax, owing to extremely poor conditions aboard ship; due to food scarcity, both crew and prisoners were on short rations, and scurvy was rampant. By the end of October, Allen was again off New York, where the British, having secured the city, moved the prisoners on-shore, and, as he was considered an officer, gave Allen limited parole. With the financial assistance of his brother Ira, he lived comfortably, if out of action, until August 1777. Allen then learned of the death of his young son Joseph due to smallpox.
According to another prisoner\'s account, Allen wandered off after learning of his son\'s death. He was arrested for violating his parole, and placed in solitary confinement. There Allen remained while Vermont declared independence, and John Burgoyne\'s campaign for the Hudson River met a stumbling block near Bennington in August 1777. On May 3, 1778, he was transferred to Staten Island. Allen was admitted to General John Campbell\'s quarters, where he was invited to eat and drink with the general and several other British field officers. He stayed there for two days and was treated politely. On the third day Allen was exchanged for Colonel Archibald Campbell, who was conducted to the exchange by Colonel Elias Boudinot, the American commissary general of prisoners appointed by General George Washington. Following the exchange, Allen reported to Washington at Valley Forge. On May 14, he was breveted a colonel in the Continental Army in \"reward of his fortitude, firmness and zeal in the cause of his country, manifested during his long and cruel captivity, as well as on former occasions,\" and given military pay of \$75 per month. The brevet rank, however, meant that there was no active role, until called, for Allen. Allen\'s services were never requested, and eventually the payments stopped.
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# Ethan Allen
## Vermont Republic {#vermont_republic}
### Return home {#return_home}
Following his visit to Valley Forge, Allen traveled to Salisbury, arriving on May 25, 1778. There he learned that his brother Heman had died just the previous week and that his brother Zimri, who had been caring for Allen\'s family and farm, had died in the spring following his capture. The death of Heman, with whom Allen had been quite close, hit him quite hard.
Allen then set out for Bennington, where news of his impending return preceded him, and he was met with all of the honor due to a military war hero. There he learned that the Vermont Republic had declared independence in 1777, that a constitution had been drawn up, and that elections had been held. Allen wrote of this homecoming that \"we passed the flowing bowl, and rural felicity, sweetened with friendship, glowed in every countenance\". The next day he went to Arlington to see his family and his brother Ira, whose prominence in Vermont politics had risen considerably during Allen\'s captivity.
### Politics
Allen spent the next several years involved in Vermont\'s political and military matters. While his family remained in Arlington, he spent most of his time either in Bennington or on the road, where he could avoid his wife\'s nagging. Shortly after his arrival, Vermont\'s Assembly passed the Banishment Act, a sweeping measure allowing for the confiscation and auction by the republic of property owned by known Tories. Allen was appointed to be one of the judges responsible for deciding whose property was subject to seizure under the law. (This law was so successful at collecting revenue that Vermont did not impose any taxes until 1781.) Allen personally escorted some of those convicted under the law to Albany, where he turned them over to General John Stark for transportation to the British lines. Some of these supposed Tories protested to New York Governor George Clinton that they were actually dispossessed Yorkers. Clinton, who considered Vermont to still be a part of New York, did not want to honor the actions of the Vermont tribunals; Stark, who had custody of the men, disagreed with Clinton. Eventually the dispute made its way to George Washington, who essentially agreed with Stark since he desperately needed the general\'s services. The prisoners were eventually transported to West Point, where they remained in \"easy imprisonment\".
While Allen\'s service as a judge in Vermont was brief, he continued to ferret out Tories and report them to local Boards of Confiscation for action. He was so zealous in these efforts that they also included naming his own brother Levi, who was apparently trying to swindle Allen and Ira out of land at the time. This action was somewhat surprising, as Levi had not only attempted to purchase Allen\'s release while he was in Halifax, but he had also traveled to New York while Allen was on parole there and furnished him with goods and money. Allen and Levi engaged in a war of words, many of which were printed in the *Connecticut Courant*, even after Levi crossed British lines. They would eventually reconcile in 1783.
Early in 1779, Governor Clinton issued a proclamation stating that the state of New York would honor the Wentworth grants, if the settlers would recognize New York\'s political jurisdiction over the Vermont territory. Allen wrote another pamphlet in response, entitled *An `{{sic|Animad|versory}}`{=mediawiki} Address to the Inhabitants of the State of Vermont; with Remarks on a Proclamation under the Hand of his Excellency George Clinton, Esq; Governor of the State of New York*. In typical style, Allen castigated the governor for issuing \"romantic proclamations \... calculated to deceive woods people\", and for his \"folly and stupidity\". Clinton\'s response, once he recovered his temper, was to issue another proclamation little different from the first. Allen\'s pamphlet circulated widely, including among members of Congress, and was successful in casting the Vermonters\' case in a positive light.
In a dispatch to Clinton from Westminster, two prisoners from New York sentenced after Allen\'s intervention pleaded with the governor to free them from being at \"the disposal of Ethan `{{sic|Allin}}`{=mediawiki} which is more to be dreaded than Death with all its Terrors.\"
In 1779, Allen published the account of his time in captivity, *A Narrative of Colonel Ethan Allen\'s Captivity \... Containing His Voyages and Travels, With the most remarkable Occurrences respecting him and many other Continental Prisoners of Observations. Written by Himself and now published for the Information of the Curious in all Nations*. First published as a serial by the *Pennsylvania Packet*, the book was an instant best-seller; it is still available today. While largely accurate, it notably omits Benedict Arnold from the capture of Ticonderoga, and Seth Warner as the leader of the Green Mountain Boys.
### Negotiations with the British {#negotiations_with_the_british}
Allen appeared before the Continental Congress as early as September 1778 on behalf of Vermont, seeking recognition as an independent state. He reported that due to Vermont\'s expansion to include border towns from New Hampshire, Congress was reluctant to grant independent statehood to Vermont. Between 1780 and 1783, Allen participated, along with his brother Ira, Vermont Governor Thomas Chittenden, and others, in negotiations with Frederick Haldimand, the governor of Quebec, that were ostensibly about prisoner exchanges, but were really about establishing Vermont as a new British province and gaining military protection for its residents. The negotiations, once details of them were published, were often described by opponents of Vermont statehood as treasonous, but no such formal charges were ever laid against anyone involved.
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# Ethan Allen
## Later years {#later_years}
As the war had ended with the 1783 Treaty of Paris, and the United States, operating under the Articles of Confederation, resisted any significant action with respect to Vermont, Allen\'s historic role as an agitator became less important, and his public role in Vermont\'s affairs declined. Vermont\'s government had also become more than a clique dominated by the Allen and Chittenden families due to the territory\'s rapid population growth.
In 1782, Allen\'s brother Heber died at the relatively young age of 38. Allen\'s wife Mary died in June 1783 of consumption, to be followed several months later by their first-born daughter Loraine. While they had not always been close, and Allen\'s marriage had often been strained, Allen felt these losses deeply. A poem he wrote memorializing Mary was published in the *Bennington Gazette*.
### Publication of *Reason* {#publication_of_reason}
In these years, Allen recovered from Thomas Young\'s widow, who was living in Albany, the manuscript that he and Young had worked on in his youth and began to develop it into the work that was published in 1785 as *Reason: the Only Oracle of Man*. The work was a typical Allen polemic, but its target was religious, not political. Specifically targeted against Christianity, it was an unbridled attack against the Bible, established churches, and the powers of the priesthood. As a replacement for organized religion, he espoused a mixture of deism, Spinoza\'s naturalist views, and precursors of Transcendentalism, with man acting as a free agent within the natural world. While historians disagree over the exact authorship of the work, the writing contains clear indications of Allen\'s style.
The book was a complete financial and critical failure. Allen\'s publisher had forced him to pay the publication costs up front, and only 200 of the 1,500 volumes printed were sold. (The rest were eventually destroyed by a fire at the publisher\'s house.) The theologically conservative future president of Yale, Timothy Dwight, opined that \"the style was crude and vulgar, and the sentiments were coarser than the style. The arguments were flimsy and unmeaning, and the conclusions were fastened upon the premises by mere force.\" Allen took the financial loss and the criticism in stride, observing that most of the critics were clergymen, whose livelihood he was attacking.
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# Ethan Allen
## Later years {#later_years}
### Second marriage {#second_marriage}
Allen met his second wife, a young widow named Frances \"Fanny\" Montresor Brush Buchanan, early in 1784; and after a brief courtship, they wed on February 16, 1784. Fanny came from a notably Loyalist background (including Crean Brush, notorious for acts during the Siege of Boston, from whom she inherited land in Vermont), but they were both smitten, and the marriage was a happy one. They had three children: Fanny (1784--1819), Hannibal Montresor (1786--1813), and Ethan Alphonso (1789--1855). Fanny had a settling effect on Allen; for the remainder of his years he did not embark on many great adventures.
The notable exception to this was when land was claimed by the Connecticut-based owners of the Susquehanna Company, who had been granted titles to land claimed by Connecticut in the Wyoming Valley, in an area that is now Wilkes-Barre, Pennsylvania. The area was also claimed by Pennsylvania, which refused to recognize the Connecticut titles. Allen, after being promised land, traveled to the area and began stirring up not just Pennsylvania authorities but also his long-time nemesis, Governor Clinton of New York, by proposing that a new state be carved out of the disputed area and several counties of New York. The entire affair was more bluster than anything else, and was resolved amicably when Pennsylvania agreed to honor the Connecticut titles.
Allen was also approached by Daniel Shays in 1786 for support in what became the Shays\'s Rebellion in western Massachusetts. He was unsupportive of the cause, in spite of Shays\'s offer to crown him \"king of Massachusetts\"; he felt that Shays was just trying to erase unpayable debts.
In his later years, independent Vermont continued to experience rapid population growth, and Allen sold a great deal of his land, but also reinvested much the proceeds in more land. A lack of cash, complicated by Vermont\'s currency problems, placed a strain on Fanny\'s relatively free hand on spending, which was further exacerbated by the cost of publishing *Reason*, and of the construction of a new home near the mouth of the Onion River. He was threatened with debtors\' prison on at least one occasion, and was at times reduced to borrowing money and calling in old debts to make ends meet.
Allen and his family moved to Burlington in 1787, which was no longer a small frontier settlement but a small town, and much more to Allen\'s liking than the larger community that Bennington had become. He frequented the tavern there, and began work on *An Essay on the Universal Plenitude of Being*, which he characterized as an appendix to *Reason*. This essay was less polemic than many of his earlier writings. Allen affirmed the perfection of God and His creation, and credited intuition as well as reason as a way to bring Man closer to the universe. The work was not published until long after his death, and is primarily of interest to students of Transcendentalism, a movement the work foreshadows.
### Death
On February 11, 1789, Allen traveled to South Hero, Vermont with one of his workers to visit his cousin, Ebenezer Allen, and to collect a load of hay. After an evening spent with friends and acquaintances, he spent the night there and set out the next morning for home. While accounts of the return journey are not entirely consistent, Allen apparently suffered an apoplectic fit en route and was unconscious by the time they returned home. Allen died at home several hours later, without ever regaining consciousness. He was buried four days later in the Green Mount Cemetery in Burlington. The funeral was attended by dignitaries from the Vermont government and by large numbers of common folk who turned out to pay respects to a man many considered their champion.
Allen\'s death made nationwide headlines. The *Bennington Gazette* wrote of the local hero, \"the patriotism and strong attachment which ever appeared uniform in the breast of this *Great Man*, was worth of his exalted character; the public have to lament the loss of a man who has rendered them great service\". Although most obituaries were positive, a number of clergymen expressed different sentiments. \"Allen was an ignorant and profane Deist, who died with a mind replete with horror and despair\" was the opinion of Newark, New Jersey\'s Reverend Uzal Ogden. Yale\'s Timothy Dwight expressed satisfaction that the world no longer had to deal with a man of \"peremptoriness and effrontery, rudeness and ribaldry\". It is not recorded what New York Governor Clinton\'s reaction was to the news.
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