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# Apollo 12 ## Mission highlights {#mission_highlights} ### Lunar orbit and Moon landing {#lunar_orbit_and_moon_landing} Apollo 12 entered a lunar orbit of 170.2 by with an SPS burn of 352.25 seconds at mission time 83:25:26.36. On the first lunar orbit, there was a television transmission that resulted in good-quali...
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# Apollo 12 ## Mission highlights {#mission_highlights} ### Lunar surface activities {#lunar_surface_activities} When Conrad, the shortest man of the initial groups of astronauts, stepped onto the lunar surface his first words were \"Whoopie! Man, that may have been a small one for Neil, but that\'s a long one for me...
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# Apollo 12 ## Mission highlights {#mission_highlights} ### Splashdown *Yankee Clipper* returned to Earth on November 24, 1969, splashing down in the South Pacific Ocean southeast of Samoa at 244:36:25 (20:58:24 UTC, 10:58:24 am HST, local time at the landing site). The landing was hard, resulting in a camera becomin...
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# Apollo 12 ## Aftermath and spacecraft location {#aftermath_and_spacecraft_location} thumb\|upright=1.3\|right\|Apollo 12 CM *Yankee Clipper* on display at the Virginia Air and Space Center in Hampton, Virginia After the mission, Conrad urged his crewmates to join him in the Skylab program, seeing in it the best ch...
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# Apollo 15 **Apollo 15** (July 26`{{snd}}`{=mediawiki}August 7, 1971) was the ninth crewed mission in the Apollo program and the fourth Moon landing. It was the first J mission, with a longer stay on the Moon and a greater focus on science than earlier landings. Apollo 15 saw the first use of the Lunar Roving Vehicle...
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# Apollo 15 ## Crew and key Mission Control personnel {#crew_and_key_mission_control_personnel} ### Crew Scott was born in 1932 in San Antonio, Texas, and, after spending his freshman year at the University of Michigan on a swimming scholarship, transferred to the United States Military Academy, from which he gradua...
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# Apollo 15 ## Planning and training {#planning_and_training} Schmitt and other scientist-astronauts advocated for a greater place for science on the early Apollo missions. They were often met with disinterest from other astronauts, or found science displaced by higher priorities. Schmitt realized that what was neede...
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# Apollo 15 ## Hardware ### Spacecraft Apollo 15 used command and service module CSM-112, which was given the call sign *Endeavour*, named after HMS *Endeavour*, and Lunar Module LM-10, call sign *Falcon*, named after the United States Air Force Academy mascot. Scott explained the choice of the name *Endeavour* on t...
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# Apollo 15 ## Hardware ### Lunar Roving Vehicle {#lunar_roving_vehicle} thumb\|upright=1.49\|right\|alt=Astronaut works on the Moon at the lunar rover\|Irwin with the Lunar Roving Vehicle on the Moon. Mons Hadley is in the background. *Main article: Lunar Roving Vehicle* A vehicle that could operate on the surface ...
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# Apollo 15 ## Hardware ### Particles and Fields Subsatellite {#particles_and_fields_subsatellite} The Apollo 15 Particles and Fields Subsatellite (PFS-1) was a small satellite released into lunar orbit from the SIM bay just before the mission left orbit to return to Earth. Its main objectives were to study the plasm...
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# Apollo 15 ## Mission highlights {#mission_highlights} ### Launch and outbound trip {#launch_and_outbound_trip} *Main article: Journey of Apollo 15 to the Moon* Apollo 15 was launched on July 26, 1971, at 9:34 am EDT from the Kennedy Space Center at Merritt Island, Florida. The time of launch was at the very start...
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# Apollo 15 ## Mission highlights {#mission_highlights} ### Lunar orbit and landing {#lunar_orbit_and_landing} thumb\|upright=1.2\|alt=Control panel of lunar lander\|The interior of *Falcon* thumb\|upright=1.2\|The Apollo 15 command and service module in lunar orbit, photographed from *Falcon* On Apollo 11 and 12, th...
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# Apollo 15 ## Mission highlights {#mission_highlights} ### Lunar surface {#lunar_surface} #### Stand-up EVA and first EVA {#stand_up_eva_and_first_eva} With *Falcon* due to remain on the lunar surface for almost three days, Scott deemed it important to maintain the circadian rhythm they were used to, and as they ha...
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# Apollo 15 ## Mission highlights {#mission_highlights} ### Command module activities {#command_module_activities} After the departure of *Falcon*, Worden in *Endeavour* executed a burn to take the CSM to a higher orbit. While *Falcon* was on the Moon, the mission effectively split, Worden and the CSM being assigned ...
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# Apollo 15 ## Mission highlights {#mission_highlights} ### Return to Earth {#return_to_earth} thumb \|alt=Video showing the lunar lander taking off\|The liftoff from the Moon as seen by the TV camera on the lunar rover *Main article: Return of Apollo 15 to Earth* *Falcon* lifted off the Moon at 17:11:22 GMT on Augu...
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# Apollo 15 ## Assessment The mission objectives for Apollo 15 were to \"perform selenological inspection, survey, and sampling of materials and surface features in a pre-selected area of the Hadley--Apennine region. Emplace and activate surface experiments. Evaluate the capability of the Apollo equipment to provide ...
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# Apollo 15 ## Controversies Further information: Apollo 15 postal covers incident Despite the successful mission, the careers of the crew were tarnished by a deal they had made before the flight to carry postal covers to the Moon in exchange for about \$7,000 each, which they planned to set aside for their children...
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# Apollo 15 ## Visibility from space {#visibility_from_space} The halo area of the Apollo 15 landing site, created by the LM\'s exhaust plume, was observed by a camera aboard the Japanese lunar orbiter SELENE and confirmed by comparative analysis of photographs in May 2008. This corresponds well to photographs taken ...
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# April 17
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# Argument from morality The **argument from morality** is an argument for the existence of God. Arguments from morality tend to be based on moral normativity or moral order. Arguments from moral normativity observe some aspect of morality and argue that God is the best or only explanation for this, concluding that Go...
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# Argument from morality ## General form {#general_form} All variations of the argument from morality begin with an observation about moral thought or experiences and conclude with the existence of God. Some of these arguments propose moral facts which they claim evident through human experience, arguing that God is ...
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# Argument from morality ## Variations ### Practical reason {#practical_reason} In his *Critique of Pure Reason*, German philosopher Immanuel Kant stated that no successful argument for God\'s existence arises from reason alone. In his *Critique of Practical Reason* he went on to argue that, despite the failure of t...
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# Atomic semantics **Atomic semantics** is a type of guarantee provided by a data register shared by several processors in a parallel machine or in a network of computers working together. Atomic semantics are very strong. An atomic register provides strong guarantees even when there is concurrency and failures. A re...
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# Antarctic Circumpolar Current **Antarctic Circumpolar Current** (**ACC**) is an ocean current that flows clockwise (as seen from the South Pole) from west to east around Antarctica. An alternative name for the ACC is the **West Wind Drift**. The ACC is the dominant circulation feature of the Southern Ocean and has a...
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# Antarctic Circumpolar Current ## Structure The ACC connects the Atlantic, Pacific, and Indian Oceans, and serves as a principal pathway of exchange among them. The current is strongly constrained by landform and bathymetric features. To trace it starting arbitrarily at South America, it flows through the Drake Pass...
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# Antarctic Circumpolar Current ## Dynamics The circumpolar current is driven by the strong westerly winds in the latitudes of the Southern Ocean. In latitudes where there are continents, winds blowing on light surface water can simply pile up light water against these continents. But in the Southern Ocean, the mome...
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# Antarctic Circumpolar Current ## Phytoplankton Antarctic sea ice cycles seasonally, in February--March the amount of sea ice is lowest, and in August--September the sea ice is at its greatest extent. Ice levels have been monitored by satellite since 1973. Upwelling of deep water under the sea ice brings substantial...
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# Antarctic Circumpolar Current ## Studies An expedition in May 2008 by 19 scientists studied the geology and biology of eight Macquarie Ridge sea mounts, as well as the Antarctic Circumpolar Current to investigate the effects of climate change of the Southern Ocean. The circumpolar current merges the waters of the A...
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# Delian League Delyan}} The **Delian League** was a confederacy of Greek city-states, numbering between 150 and 330, founded in 478 BC under the leadership (hegemony) of Athens, whose purpose was to continue fighting the Persian Empire after the Greek victory in the Battle of Plataea at the end of the Second Persian...
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# Delian League ## Background thumb\|upright=1.5\|Athenian Empire in 445 BC, according to the Tribute Lists. The islands of Lesbos, Chios and Samos (shaded on the map) did not pay tribute. The Greco-Persian Wars had their roots in the conquest of the Greek cities of Asia Minor, and particularly Ionia, by the Achaemen...
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# Delian League ## Formation After Byzantion, Sparta was eager to end its involvement in the war. The Spartans greatly feared the rise of the Athenians as a challenge to their power. Additionally, the Spartans were of the view that, with the liberation of mainland Greece, and the Greek cities of Asia Minor, the war\'...
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# Delian League ## Members The Delian League, also known as the Athenian Empire, was a collection of Greek city-states largely based around the Aegean Sea which operated under the hegemony of Athens. This alliance initially served the purpose of coordinating a united Greek front against a perceived looming Persian th...
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# Delian League ## Policies of the League {#policies_of_the_league} In 461 BC, Cimon was ostracized and was succeeded in his influence by democrats such as Ephialtes and Pericles. This signaled a complete change in Athenian foreign policy, neglecting the alliance with the Spartans and instead allying with her enemies...
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# Delian League ## Wars in Greece {#wars_in_greece} Soon, war with the Peloponnesians broke out. In 458 BC, the Athenians blockaded the island of Aegina, and simultaneously defended Megara from the Corinthians by sending out an army composed of those too young or old for regular military service. The following year, ...
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# Delian League ## Athenian Empire (454--404 BC) {#athenian_empire_454404_bc} By 454 BC, the Delian League could be fairly characterised as an Athenian Empire; a key event of 454 BC was the moving of the treasury of the Delian League from Delos to Athens. This is often seen as a key marker of the transition from alli...
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# August Horch **August Horch** (12 October 1868 -- 3 February 1951) was a German engineer and automobile pioneer, the founder of the manufacturing giant that eventually became Audi. ## Beginnings Horch was born in Winningen, Rhenish Prussia. His initial trade was as a blacksmith, and then was educated at *italic=no...
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# Avionics **Avionics** (a portmanteau of *aviation* and *electronics*) are the electronic systems used on aircraft. Avionic systems include communications, navigation, the display and management of multiple systems, and the hundreds of systems that are fitted to aircraft to perform individual functions. These can be ...
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# Avionics ## Aircraft avionics {#aircraft_avionics} The cockpit or, in larger aircraft, under the cockpit of an aircraft or in a movable nosecone, is a typical location for avionic bay equipment, including control, monitoring, communication, navigation, weather, and anti-collision systems. The majority of aircraft p...
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# Avionics ## Aircraft avionics {#aircraft_avionics} ### Monitoring The first hints of glass cockpits emerged in the 1970s when flight-worthy cathode-ray tube (CRT) screens began to replace electromechanical displays, gauges and instruments. A \"glass\" cockpit refers to the use of computer monitors instead of gauges...
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# Avionics ## Mission or tactical avionics {#mission_or_tactical_avionics} Military aircraft have been designed either to deliver a weapon or to be the eyes and ears of other weapon systems. The vast array of sensors available to the military is used for whatever tactical means required. As with aircraft management, ...
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# Ares **Ares** (`{{IPAc-en|ˈ|ɛər|iː|z}}`{=mediawiki}; *Ἄρης*, *Árēs* `{{IPA|el|árɛːs|}}`{=mediawiki}) is the Greek god of war and courage. He is one of the Twelve Olympians, and the son of Zeus and Hera. The Greeks were ambivalent towards him. He embodies the physical valor necessary for success in war but can also p...
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# Ares ## Cult In mainland Greece and the Peloponnese, only a few places are known to have had a formal temple and cult of Ares.`{{refn|group=n|Burkert lists temples at or near Troizen, Geronthrai and Halicarnassus. The Oxford Classical Dictionary adds Argos, Megalopolis, Therapne and Tegea in the Peloponnese, Athens...
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# Ares ## Cult ### Sacrifices thumb\|upright=1.3\|Ares (right) with Demeter, Dionysus and Hermes on the frieze of the Parthenon, ca. 447--433 BC, British Museum. Like most Greek deities, Ares was given animal sacrifice; in Sparta, after battle, he was given an ox for a victory by stratagem, or a rooster for victory ...
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# Ares ## Cult ### Aksum In Africa, Maḥrem, the principal god of the kings of Aksum prior to the 4th century AD, was invoked as Ares in Greek inscriptions. The anonymous king who commissioned the Monumentum Adulitanum in the late 2nd or early 3rd century refers to \"my greatest god, Ares, who also begat me, through w...
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# Ares ## Characterisation Ares was one of the Twelve Olympians in the archaic tradition represented by the *Iliad* and *Odyssey.* In Greek literature, Ares often represents the physical or violent and untamed aspect of war and is the personification of sheer brutality and bloodlust (\"overwhelming, insatiable in bat...
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# Ares ## Mythology ### Birth He is one of the Twelve Olympians, and the son of Zeus and Hera. ### Argonautica In the *Argonautica*, the *Golden Fleece* hangs in a grove sacred to Ares, until its theft by Jason. The Birds of Ares (*Ornithes Areioi*) drop feather darts in defense of the Amazons\' shrine to Ares, as...
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# Ares ## Mythology ### Attendants Deimos (\"Terror\" or \"Dread\") and Phobos (\"Fear\") are Ares\' companions in war, and according to Hesiod, are also his children by Aphrodite. Eris, the goddess of discord, or Enyo, the goddess of war, bloodshed, and violence, was considered the sister and companion of the violen...
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# Ares ## Renaissance and later depictions {#renaissance_and_later_depictions} In Renaissance and Neoclassical works of art, Ares\'s symbols are a spear and helmet, his animal is a dog, and his bird is the vulture. In literary works of these eras, Ares is replaced by the Roman Mars, a romantic emblem of manly valor r...
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# Alpha compositing In computer graphics, **alpha compositing** or **alpha blending** is the process of combining one image with a background to create the appearance of partial or full transparency. It is often useful to render picture elements (pixels) in separate passes or layers and then combine the resulting 2D i...
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# Alpha compositing ## Description In a 2D image a color combination is stored for each picture element (pixel), often a combination of red, green and blue (RGB). When alpha compositing is in use, each pixel has an additional numeric value stored in its **alpha channel**, with a value ranging from 0 to 1. A value of ...
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# Alpha compositing ## Straight versus premultiplied {#straight_versus_premultiplied} If an alpha channel is used in an image, there are two common representations that are available: straight (unassociated) alpha and premultiplied (associated) alpha. - With **straight alpha**, the RGB components represent the col...
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# Alpha compositing ## Image formats supporting alpha channels {#image_formats_supporting_alpha_channels} The most popular image formats that support the alpha channel are PNG and TIFF. GIF supports alpha channels, but is considered to be inefficient when it comes to file size. Support for alpha channels is present i...
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# Alpha compositing ## Gamma correction {#gamma_correction} The RGB values of typical digital images do not directly correspond to the physical light intensities, but are rather compressed by a gamma correction function: : $C_\text{encoded} = C_\text{linear}^{1/\gamma}$ This transformation better utilizes the lim...
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# Array (data structure) In computer science, an **array** is a data structure consisting of a collection of *elements* (values or variables), of same memory size, each identified by at least one *array index* or *key*, a collection of which may be a tuple, known as an index tuple. An array is stored such that the pos...
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# Array (data structure) ## Element identifier and addressing formulas {#element_identifier_and_addressing_formulas} When data objects are stored in an array, individual objects are selected by an index that is usually a non-negative scalar integer. Indexes are also called subscripts. An index *maps* the array value ...
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# Array (data structure) ## Element identifier and addressing formulas {#element_identifier_and_addressing_formulas} ### Compact layouts {#compact_layouts} Often the coefficients are chosen so that the elements occupy a contiguous area of memory. However, that is not necessary. Even if arrays are always created with ...
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# Array (data structure) ## Efficiency Both *store* and *select* take (deterministic worst case) constant time. Arrays take linear (O(*n*)) space in the number of elements *n* that they hold. In an array with element size *k* and on a machine with a cache line size of B bytes, iterating through an array of *n* eleme...
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# Array (data structure) ## Dimension The *dimension* of an array is the number of indices needed to select an element. Thus, if the array is seen as a function on a set of possible index combinations, it is the dimension of the space of which its domain is a discrete subset. Thus a one-dimensional array is a list of...
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# Aristide Maillol **Aristide Joseph Bonaventure Maillol** (`{{IPA|fr|mɑjɔl|lang}}`{=mediawiki}; December 8, 1861 -- September 27, 1944) was a French sculptor, painter, and printmaker. He began his career as a painter and developed an early interest in the decorative arts. He became primarily interested in sculpture ...
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# Aristide Maillol ## Works - Mme Henry Clemens van de Velde (c. 1899) - *The Mediterranean (1902--05)* - *Action in Chains* (1905) - *Flora, Nude* (1910) - *L\'Été sans bras* (1911) - *Bathing Woman with Raised Arms* (1921) - *Nymph* (1930) - *The Mountain* (1937) - *L\'Air* (1938) - *The River* ...
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# Antonio Canova **Antonio Canova** (`{{IPA|it|anˈtɔːnjo kaˈnɔːva}}`{=mediawiki}; 1 November 1757 -- 13 October 1822) was an Italian Neoclassical sculptor, famous for his marble sculptures. Often regarded as the greatest of the Neoclassical artists, his sculpture was inspired by the Baroque and the classical revival, ...
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# Antonio Canova ## Life ### Rome Canova arrived in Rome, on 28 December 1780. Prior to his departure, his friends had applied to the Venetian Senate for a pension. Successful in the application, the stipend allotted amounted to three hundred ducats, limited to three years. While in Rome, Canova spent time studying ...
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# Antonio Canova ## Life ### France and England {#france_and_england} By 1800, Canova was the most celebrated artist in Europe. He systematically promoted his reputation by publishing engravings of his works and having marble versions of plaster casts made in his workshop. He became so successful that he had acquired...
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# Antonio Canova ## Life ### Returning to Italy {#returning_to_italy} In 1816, Canova returned to Rome with some of the art Napoleon had taken. He was rewarded with several marks of distinction: he was appointed President of the Accademia di San Luca, inscribed into the \"Golden Book of Roman Nobles\" by the Pope\'s ...
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# Antonio Canova ## Works Among Canova\'s most notable works are: ### *Psyche Revived by Cupid\'s Kiss* (1787) {#psyche_revived_by_cupids_kiss_1787} *Main article: Psyche Revived by Cupid\'s Kiss* *Psyche Revived by Cupid\'s Kiss* was commissioned in 1787 by Colonel John Campbell. It is regarded as a masterpiece o...
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# Antonio Canova ## Artistic process {#artistic_process} Canova had a distinct, signature style in which he combined Greek and Roman art practices with early stirrings of romanticism to delve into a new path of Neoclassicism. Canova\'s sculptures fall into three categories: Heroic compositions, compositions of grace,...
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# Antonio Canova ## Gallery Antonio Canova from the studio if Canova c.1813.jpg\|Antonio Canova from the studio of Canova, c. 1813 Amor-Psyche-Canova-JBU02.JPG\|*Psyche Revived by Cupid\'s Kiss*, 1787--1793, Louvre Amor-Psyche-Canova-JBU04.JPG\|*Psyche Revived by Cupid\'s Kiss*, 1787--1793, Louvre (detail) Venus Ital...
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# Act of Settlement 1701 The **Act of Settlement** (12 & 13 Will. 3. c. 2) is an act of the Parliament of England that settled the succession to the English and Irish crowns to only Protestants, which passed in 1701. More specifically, anyone who became a Roman Catholic, or who married one, became disqualified to inhe...
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# Act of Settlement 1701 ## Background Following the Glorious Revolution, the line of succession to the English throne was governed by the Bill of Rights 1689, which declared that the flight of James II from England to France during the revolution amounted to an abdication of the throne and that James\'s daughter Mar...
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# Act of Settlement 1701 ## The act {#the_act} The Act of Settlement provided that the throne would pass to the Electress Sophia of Hanover -- a granddaughter of James VI and I and a niece of King Charles I -- and her descendants, but it excluded \"for ever\" \"all and every Person and Persons who \... is are or shal...
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# Act of Settlement 1701 ## Legacy For different reasons, various constitutionalists have praised the Act of Settlement: Henry Hallam called the Act \"the seal of our constitutional laws\" and David Lindsay Keir placed its importance above the Bill of Rights of 1689. Naamani Tarkow wrote: \"If one is to make sweeping...
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# Act of Settlement 1701 ## Amendment proposals {#amendment_proposals} Challenges have been made against the Act of Settlement, especially its provisions regarding Roman Catholics and preference for males. However, changing the act is a complex process, since the act governs the shared succession of all the Commonwea...
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# Act of Settlement 1701 ## Amendment proposals {#amendment_proposals} ### Canada In Canada, where the Act of Settlement (*Acte d\'établissement*) is now a part of Canadian constitutional law, Tony O\'Donohue, a Canadian civic politician, took issue with the provisions that exclude Roman Catholics from the throne, an...
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# Act of Settlement 1701 ## Amendment proposals {#amendment_proposals} ### United Kingdom {#united_kingdom} From time to time there has been debate over repealing the clause that prevents Roman Catholics, or those who marry one, from ascending to the British throne. Proponents of repeal argue that the clause is a big...
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# Act of Settlement 1701 ## Amendment proposals {#amendment_proposals} ### Across the realms {#across_the_realms} In early 2011 Keith Vaz, a Labour Member of Parliament, introduced to the House of Commons at Westminster a private member\'s bill which proposed that the Act of Settlement be amended to remove the provis...
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# A Fire Upon the Deep ***A Fire Upon the Deep*** is a 1992 science fiction novel by American writer Vernor Vinge. It is a space opera involving superhuman intelligences, aliens, variable physics, space battles, love, betrayal, genocide, and a communication medium resembling Usenet. *A Fire Upon the Deep* won the Hugo...
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# A Fire Upon the Deep ## Plot An expedition from Straumli Realm, a young human civilization in the high Beyond, investigates a newly discovered five-billion-year-old data archive in the low Transcend that offers the possibility of unimaginable riches. The expedition\'s facility, High Lab, is gradually and secretly c...
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# A Fire Upon the Deep ## Intelligent species {#intelligent_species} ### Aprahanti A race of humanoids with colorful butterfly-like wings who attempt to use the chaos wrought by the Blight to reestablish their waning hegemony. Despite their attractive, delicate appearance, the Aprahanti are an extremely fearsome and...
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# Abimelech **Abimelech** (also spelled **Abimelek** or **Avimelech**; `{{Hebrew Name 2|אֲבִימֶלֶךְ|אֲבִימָלֶךְ|ʼAvīméleḵ|ʼAvīmáleḵ|ʼAḇīmeleḵ|ʼAḇīmāleḵ|"my father is a king"/"my father reigns"}}`{=mediawiki}) was the generic name given to all Philistine kings in the Hebrew Bible from the time of Abraham through King D...
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# Abimelech ## Abimelech son of Jerubbaal/Gideon {#abimelech_son_of_jerubbaalgideon} The Book of Judges mentions Abimelech, son of judge Gideon (also known as Jerubbaal). According to the biblical narrative, Abimelech was an extremely conniving and evil person. He persuaded his mother\'s brothers to encourage the peo...
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# Applesoft BASIC **Applesoft BASIC** is a dialect of Microsoft BASIC, developed by Marc McDonald and Ric Weiland, supplied with Apple II computers. It supersedes Integer BASIC and is the BASIC in ROM in all Apple II series computers after the original Apple II model. It is also referred to as **FP BASIC** (from float...
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# Applesoft BASIC ## Features Applesoft is similar to Commodore\'s BASIC 2.0 aside from features inherited from Integer BASIC. There are a few minor differences such as Applesoft\'s lack of bitwise operators; otherwise most BASIC programs that do not use hardware-dependent features will run on both BASICs. The `{{mo...
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# Applesoft BASIC ## Features ### Bugs A deficiency with error-trapping via `ONERR` means that the system stack is not reset if an error-handling routine does not invoke `RESUME`, potentially leading to a crash. The built-in pseudorandom number generator function `RND` is capable of producing a predictable series of ...
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# Applesoft BASIC ## Sample code {#sample_code} Hello, World! in Applesoft BASIC can be entered as the following: ``` cbmbas 10TEXT:HOME 20?"HELLO WORLD" ``` Multiple commands can be included on the same line of code if separated by a colon (`:`). The `?` can be used in Applesoft BASIC (and almost all versions of M...
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# Adalbert of Prague **Adalbert of Prague** (*Sanctus Adalbertus*, *svatý Vojtěch*, *svätý Vojtech*, *święty Wojciech*, *Szent Adalbert (Béla)*; c. 956`{{snd}}`{=mediawiki}23 April 997), known in the Czech Republic, Poland and Slovakia by his birth name **Vojtěch** (*Voitecus*), was a Czech missionary and Christian s...
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# Adalbert of Prague ## Life ### Mission and martyrdom in Prussia {#mission_and_martyrdom_in_prussia} Adalbert again relinquished his diocese, namely that of Gniezno, and set out as a missionary to preach to the inhabitants near Prussia. Bolesław I, Duke (and, later, King) of Poland, sent soldiers with Adalbert on hi...
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# Adalbert of Prague ## Veneration and relics {#veneration_and_relics} A few years after his martyrdom, Adalbert was canonized as **Saint Adalbert of Prague.** His life was written in *Vita Sancti Adalberti Pragensis* by various authors, the earliest being traced to imperial Aachen and the Bishop of Liège, Notger von...
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# Adalbert of Prague ## Feast day {#feast_day} - 25 January -- commemoration of translation of relics to Church of Saint Roch, - 22 April -- commemoration in Diocese of Innsbruck, - 22 April -- commemoration in Catholic Church in England and Wales, - 23 April -- commemoration of death anniversary, - 14 May ...
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# Ælfheah of Canterbury **Ælfheah** (c. 953 -- 19 April 1012), more commonly known today as **Alphege**, was an Anglo-Saxon Bishop of Winchester, later Archbishop of Canterbury. He became an anchorite before being elected abbot of Bath Abbey. His reputation for piety and sanctity led to his promotion to the episcopate...
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# Ælfheah of Canterbury ## Veneration thumb\|upright=.8\|An 1868 statue on the West Front of Salisbury Cathedral by James Redfern, showing Ælfheah holding the stones used in his martyrdom. Pope Gregory VII canonised Ælfheah in 1078, with a feast day of 19 April. Lanfranc, the first post-Conquest archbishop, was dubio...
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# Associative algebra In mathematics, an **associative algebra** *A* over a commutative ring (often a field) *K* is a ring *A* together with a ring homomorphism from *K* into the center of *A*. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by th...
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# Associative algebra ## Examples The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics. ### Algebra - Any rin...
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# Associative algebra ## Constructions Subalgebras : A subalgebra of an *R*-algebra *A* is a subset of *A* which is both a subring and a submodule of *A*. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of *A*.\ Quotient algebras : Let *A...
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# Associative algebra ## Dual of an associative algebra {#dual_of_an_associative_algebra} Let *A* be an associative algebra over a commutative ring *R*. Since *A* is in particular a module, we can take the dual module *A*^\*^ of *A*. A priori, the dual *A*^\*^ need not have a structure of an associative algebra. Howe...
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# Associative algebra ## Finite-dimensional algebra {#finite_dimensional_algebra} Let *A* be a finite-dimensional algebra over a field *k*. Then *A* is an Artinian ring. ### Commutative case {#commutative_case} As *A* is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose resid...
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# Associative algebra ## Representations A representation of an algebra *A* is an algebra homomorphism `{{nowrap|''ρ'' : ''A'' → End(''V'')}}`{=mediawiki} from *A* to the endomorphism algebra of some vector space (or module) *V*. The property of *ρ* being an algebra homomorphism means that *ρ* preserves the multiplic...
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# Associative algebra ## Non-unital algebras {#non_unital_algebras} Some authors use the term \"associative algebra\" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non-unital associative algebra is g...
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# Axiom of regularity In mathematics, the **axiom of regularity** (also known as the **axiom of foundation**) is an axiom of Zermelo--Fraenkel set theory that states that every non-empty set *A* contains an element that is disjoint from *A*. In first-order logic, the axiom reads: $\forall x\,(x \neq \varnothing \righ...
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# Axiom of regularity ## Elementary implications of regularity {#elementary_implications_of_regularity} ### No set is an element of itself {#no_set_is_an_element_of_itself} Let $A$ be a set, and apply the axiom of regularity to $\{A\}$, which is a set by the axiom of pairing. By this axiom, there must be an element ...
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# Axiom of regularity ## Regularity and the rest of ZF(C) axioms {#regularity_and_the_rest_of_zfc_axioms} Regularity was shown to be relatively consistent with the rest of ZF by Skolem and von Neumann, meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. The axiom of reg...
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