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The opah is the only fish known to exhibit whole body endothermy where all the internal organs are kept at a higher temperature than the surrounding water. This feature allows opahs to maintain an active lifestyle in the cold waters they inhabit. Unlike birds and mammals, the opah is not a homeotherm despite being an endotherm: while its body temperature is raised above the surrounding water temperature, it still varies with the external temperature and is not held constant. In addition to whole body endothermy, the opah also exhibits regional endothermy by raising the temperature of its brain and eyes above that of the rest of the body. Regional endothermy also arose by convergent evolution in tuna, lamnid sharks and billfishes where the swimming muscles and cranial organs are maintained at an elevated temperature compared with the surrounding water.
The large muscles powering the pectoral fins generate most of the heat in the opah. In addition to the heat they generate while moving, these muscles have special regions that can generate additional heat without contracting. The opah has a thick layer of fat that insulates its internal organs and cranium from the surrounding water. However, fat alone is insufficient to retain heat within a fish's body. The gills are the main point of heat loss in fishes as this is where blood from the entire body must continuously be brought in close contact with the surrounding water. Opahs prevent heat loss through their gills using a special structure in the gill blood vessels called the rete mirabile. The rete mirabile is a dense network of blood vessels where the warm blood flowing from the heart to the gills transfers its heat to the cold blood returning from the gills. Hence, the rete mirabile prevents warm blood from coming in contact with the cold water (and losing its heat) and also ensures that the blood returning to the internal organs is warmed up to body temperature. Within the rete, the warm and cold blood flow past each other in opposite directions through thin vessels to maximise the heat transferred. This mechanism is called a counter-current heat exchanger. | Opah | Wikipedia | 437 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals |
In addition to the rete mirabile in its gills, the opah also has a rete in the blood supply to its brain and eyes. This helps to trap heat in the cranium and further raise its temperature above the rest of the body. While the rete mirabile in the gills is unique to the opah, the cranial rete mirabile has also evolved independently in other fishes. Unlike in billfish which have a specialised noncontractile tissue that functions as a brain heater, the opah cranium is heated by the contractions of the large eye muscles.
Behavior
Almost nothing is known of opah biology and ecology. They are presumed to live out their entire lives in the open ocean, at mesopelagic depths of 50 to 500 m, with possible forays into the bathypelagic zone. They are apparently solitary, but are known to school with tuna and other scombrids. The fish propel themselves by a lift-based labriform mode of swimming, that is, by flapping their pectoral fins. This, together with their forked caudal fins and depressible median fins, indicates they swim at constantly high speeds like tuna.
Lampris guttatus are able to maintain their eyes and brain at 2 °C warmer than their bodies, a phenomenon called cranial endothermy and one they share with sharks in the family Lamnidae, billfishes, and some tunas. This may allow their eyes and brains to continue functioning during deep dives into water below 4 °C.
Squid and euphausiids (krill) make up the bulk of the opah diet; small fish are also taken. Pop-up archival transmitting tagging operations have indicated that, aside from humans, large pelagic sharks, such as great white sharks and mako sharks, are primary predators of opah. The tetraphyllidean tapeworm, Pelichnibothrium speciosum, has been found in L. guttatus, which may be an intermediate or paratenic host. | Opah | Wikipedia | 430 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals |
The planktonic opah larvae initially resemble those of certain ribbonfishes (Trachipteridae), but are distinguished by the former's lack of dorsal and pelvic fin ornamentation. The slender hatchlings later undergo a marked and rapid transformation from a slender to deep-bodied form; this transformation is complete by 10.6 mm standard length in L. guttatus. Opahs are believed to have a low population resilience. | Opah | Wikipedia | 94 | 1490711 | https://en.wikipedia.org/wiki/Opah | Biology and health sciences | Acanthomorpha | Animals |
Tridacna gigas, the giant clam, is the best-known species of the giant clam genus Tridacna. Giant clams are the largest living bivalve mollusks. Several other species of "giant clam" in the genus Tridacna are often misidentified as Tridacna gigas.
These clams were known to indigenous peoples of East Asia for thousands of years and the Venetian scholar and explorer Antonio Pigafetta documented them in a journal as early as 1521. One of a number of large clam species native to the shallow coral reefs of the South Pacific and Indian oceans, they may weigh more than , measure as much as across, and have an average lifespan in the wild of more than 100 years. They also are found off the shores of the Philippines and in the South China Sea in the coral reefs of Malaysia.
The giant clam lives in flat coral sand or broken coral and may be found at depths of as great as 20 m (66 ft). Its range covers the Indo-Pacific, but populations are diminishing quickly and the giant clam has become extinct in many areas where it was once common. The maxima clam has the largest geographical distribution among giant clam species; it may be found off high- or low-elevation islands, in lagoons or fringing reefs. Its rapid growth rate is likely due to its ability to cultivate algae in its body tissue.
Although larval clams are planktonic, they become sessile in adulthood. The creature's mantle tissues act as a habitat for the symbiotic single-celled dinoflagellate algae (zooxanthellae) from which the adult clams get most of their nutrition. By day, the clam opens its shell and extends its mantle tissue so that the algae receive the sunlight they need to photosynthesise. This method of algal farming is under study as a model for highly efficient bioreactors. | Giant clam | Wikipedia | 407 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
Anatomy
Young T. gigas are difficult to distinguish from other species of Tridacninae. Adult T. gigas are the only giant clams unable to close their shells completely, allowing part of the brownish-yellow mantle to remain visible. Tridacna gigas has four or five vertical folds in its shell, which serves as the main characteristic differentiating it from the similar T. derasa that has six or seven vertical folds. Similar to coral matrices composed of calcium carbonate, giant clams grow their shells through the process of biomineralization, which is very sensitive to seasonal temperature. The isotopic ratio of oxygen in carbonate and the ratio between Strontium and Calcium together may be used to determine historical sea surface temperature.
The mantle border itself is covered in several hundred to several thousand pinhole eyespots approximately in diameter. Each one consists of a small cavity containing a pupil-like aperture and a base of 100 or more photoreceptors sensitive to three different ranges of light, including UV, which may be unique among molluscs. These receptors allow T. gigas to partially close their shells in response to dimming of light, change in the direction of light, or the movement of an object. The optical system forms an image by sequential, local dimming of some eyes using pigment from the aperture.
Largest specimens
The largest known T. gigas specimen measured , and it weighed 230 kg (510 lb) dead and was estimated to be 250 kg (550 lb) alive. It was discovered around 1817 on the north western coast of Sumatra, Indonesia, and its shells are now on display in a museum in Northern Ireland.
A heavier giant clam was found in 1956 off the Japanese island of Ishigaki. The shell's length was , and it weighed dead and estimated alive.
Ecology
Feeding
Giant clams are filter-feeders, yet 65-70 percent of their nutritional needs are supplied by zooxanthellae. This enables giant clams to grow as large as one meter in length even in nutrient-poor coral-reef waters. The clams cultivate algae in a special circulatory system that enables them to keep a substantially higher number of symbionts per unit of volume. The mantle's edges are packed with symbiotic zooxanthellae, which presumably use carbon dioxide, phosphates, and nitrates supplied by the clam. | Giant clam | Wikipedia | 492 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
In very small clams— dry tissue weight—filter feeding provides approximately 65% of total carbon needed for respiration and growth; comparatively larger clams () acquire only 34% of carbon from this source. A single species of zooxenthellae may be symbionts of both giant clams and nearby reef–building (hermatypic) corals.
Reproduction
Tridacna gigas reproduce sexually and are hermaphrodites (producing both eggs and sperm by one clam). While self-fertilization is not possible, having both characteristics does allow them to reproduce with any other member of the species as well as hermaphrodically. As with all other forms of sexual reproduction, hermaphroditism ensures that new gene combinations be passed to further generations. This flexibility in reproduction reduces the burden of finding a compatible mate, while simultaneously doubling the number of offspring produced.
Since giant clams cannot move themselves, they adopt broadcast spawning, releasing sperm and eggs into the water. A transmitter substance called spawning induced substance (SIS) helps synchronize the release of sperm and eggs to ensure fertilization. The substance is released through a syphonal outlet. Other clams can detect SIS immediately. Incoming water passes chemoreceptors situated close to the incurrent syphon that transmit the information directly to the cerebral ganglia, a simple form of brain.
Detection of SIS stimulates the giant clam to swell its mantle in the central region and to contract its adductor muscle. Each clam then fills its water chambers and closes the incurrent syphon. The shell contracts vigorously with the adductor's help, so the excurrent chamber's contents flows through the excurrent syphon. After a few contractions containing only water, eggs and sperm appear in the excurrent chamber and then pass through the excurrent syphon into the water. Female eggs have a diameter of . Egg release initiates the reproductive process. An adult T. gigas can release more than 500 million eggs at a time.
Spawning seems to coincide with incoming tides near the second (full), third, and fourth (new) quarters of the moon phase. Spawning contractions occur every two or three minutes, with intense spawning ranging from thirty minutes to two and a half hours. Clams that do not respond to the spawning of neighboring clams may be reproductively inactive.
Development | Giant clam | Wikipedia | 499 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
The fertilized egg floats in the sea for approximately 12 hours until eventually a larva (trochophore) hatches. It then starts to produce a calcium carbonate shell. Two days after fertilization it measures . Soon it develops a "foot," which is used to move on the ground. Larvae also can swim to search for appropriate habitat.
At roughly one week of age, the clam settles on the ground, although it changes location frequently within the first few weeks. The larva does not yet have symbiotic algae, so it depends completely on plankton. Also, free-floating zooxanthellae are captured while filtering food. Eventually the front adductor muscle disappears and the rear muscle moves into the clam's center. Many small clams die at this stage. The clam is considered a juvenile when it reaches a length of . It is difficult to observe the growth rate of T. gigas in the wild, but laboratory-reared giant clams have been observed to grow a year.
The ability for Tridacna to grow to such large sizes with fleshy mantles that extend beyond the edges of their shells is considered to be the result of total reorganization of bivalve development and morphology. Historically, two evolutionary explanations have been suggested for this process. Sir Yonge suggested and maintained for many years that the visceral-pedal ganglia complex rotate 180 degrees relative to the shell, requiring that they develop and evolve independently. Stasek proposed instead that the growth occurs primarily in a posterior direction instead of the more typical direction of ventral in most bivalves, which is reflected in the transitional stages of alternative ways of growing that juveniles undergo.
Human relevance
The main reason that giant clams are becoming endangered is likely to be intensive exploitation by bivalve fishers. Mainly large adults are killed because they are the most profitable.
The giant clam is considered a delicacy in Japan (known as himejako), France, Southeast Asia, and many Pacific Islands. Some Asian foods include the meat from the muscles of clams. Large amounts of money are paid for the adductor muscle, which Chinese people believe to have aphrodisiac powers.
On the black market, giant clam shells are sold as decorative accoutrements.
Legend
As is often the case historically with uncharacteristically large species, the giant clam has been misunderstood. | Giant clam | Wikipedia | 495 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
Even in countries where giant clams are easily seen, stories incorrectly depict giant clams as aggressive beings. For instance, although the clams are unable to close their shells completely, a Polynesian folk tale relates that a monkey's hand was bitten off by one, and even though once past larval stage, the clams are sessile, a Maori legend relates a supposed attack on a canoe by a giant clam. Starting from the eighteenth century, claims of danger had been related to the western world. In the 1920s, a reputable science magazine Popular Mechanics once claimed that the great mollusc had caused deaths. Versions of the U.S. Navy Diving Manual even gave detailed instructions for releasing oneself from its grasp by severing the adductor muscles used to close its shell. In an account of the discovery of the Pearl of Lao Tzu, Wilburn Cobb said he was told that a Dyak diver was drowned when the Tridacna closed its shell on his arm. In reality, the slow speed of their abductor muscle contraction and the need to force water out of their shells while closing, prevents them from trapping a human.
Other myths focus on the huge size of giant clams being associated with long age. While giant clams do live a long time and may serve as a bio-metric for historic climatic conditions, their large size is more likely associated with rapid growth.
Aquaculture
Mass culture of giant clams began at the Micronesian Mariculture Demonstration Center in Palau (Belau). A large Australian government-funded project from 1985 to 1992 mass-cultured giant clams, particularly T. gigas at James Cook University's Orpheus Island Research Station, and supported the development of hatcheries in the Pacific Islands and the Philippines. Seven of the ten known species of giant clams in the world are found in the coral reefs of the South China Sea.
Conservation status
There is concern among conservationists about whether those who use the species as a source of livelihood are overexploiting it. The numbers in the wild have been greatly reduced by extensive harvesting for food and the aquarium trade. The species is listed in Appendix II of the Convention on International Trade in Endangered Species (CITES) meaning international trade (including in parts and derivatives) is regulated. | Giant clam | Wikipedia | 467 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
T. gigas has been reported as locally extinct in peninsular Malaysia, while T. derasa and Hippopus porcellanus are restricted to Eastern Malaysia. These recent local extinctions have motivated the introduction of giant clams to Hawaii and Micronesia following maricultural advancements. Restocked individuals in the Philippines have successfully dispersed their own spawned larvae to at least several hundred meters away after only ten years. | Giant clam | Wikipedia | 83 | 1491594 | https://en.wikipedia.org/wiki/Giant%20clam | Biology and health sciences | Bivalvia | Animals |
An electric stove, electric cooker or electric range is a stove with an integrated electrical heating device to cook and bake. Electric stoves became popular as replacements for solid-fuel (wood or coal) stoves which required more labor to operate and maintain. Some modern stoves come in a unit with built-in extractor hoods.
The stove's one or more "burners" (heating elements) may be controlled by a rotary switch with a finite number of positions; or may have an "infinite switch" called a simmerstat that allows constant variability between minimum and maximum heat settings. Some stove burners and controls incorporate thermostats.
History
Early patents
On September 20, 1859, George B. Simpson was awarded US patent #25532 for an 'electro-heater' surface heated by a platinum-wire coil powered by batteries. In his words, useful to "warm rooms, boil water, cook victuals...".
Canadian inventor Thomas Ahearn filed patent #39916 in 1892 for an "Electric Oven," a device he probably employed in preparing a meal for an Ottawa hotel that year. Ahearn and Warren Y. Soper were owners of Ottawa's Chaudiere Electric Light and Power Company. The electric stove was showcased at the Chicago World's Fair in 1893, where an electrified model kitchen was shown. Unlike the gas stove, the electrical stove was slow to catch on, partly due to the unfamiliar technology, and the need for cities and towns to be electrified.
In 1897, William Hadaway was granted US patent # 574537 for an "Automatically Controlled Electric Oven".
Kalgoorlie Stove
In November 1905, David Curle Smith, the Municipal Electrical Engineer of Kalgoorlie, Western Australia, applied for a patent (Aust Patent No 4699/05) for a device that adopted (following the design of gas stoves) what later became the configuration for most electric stoves: an oven surmounted by a hotplate with a grill tray between them. Curle Smith's stove did not have a thermostat; heat was controlled by the number of the appliance's nine elements that were switched on. | Electric stove | Wikipedia | 454 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null |
After the patent was granted in 1906, manufacturing of Curle Smith's design commenced in October that year. The entire production run was acquired by the electricity supply department of Kalgoorlie Municipality, which hired out the stoves to residents. About 50 appliances were produced before cost overruns became a factor in Council politics and the project was suspended. This was the first time household electric stoves were produced with the express purpose of bringing "cooking by electricity ... within the reach of anyone". There are no extant examples of this stove, many of which were salvaged for their copper content during World War I.
To promote the stove, David Curle Smith's wife, H. Nora Curle Smith (née Helen Nora Murdoch, and a member of the Murdoch family prominent in Australian public life), wrote a cookbook containing operating instructions and 161 recipes. Thermo-Electrical Cooking Made Easy, published in March 1907, is therefore the world's first cookbook for electric stoves.
Since 1908
Three companies, in the United States, began selling electric stoves in 1908. However, sales and public acceptance were slow to develop. Early electric stoves were unsatisfactory due to the cost of electricity (compared with wood, coal, or city gas), limited power available from the electrical supply company, poor temperature regulation, and short life of heating elements. The invention of nichrome alloy for resistance wires improved the cost and durability of heating elements. As late as the 1920s, an electric stove was still considered a novelty.
By the 1930s, the maturing of the technology, the decreased cost of electric power and modernized styling of electric stoves had greatly increased their acceptance. The electrical stove slowly began to replace the gas stove, especially in household kitchens.
Electric stoves and other household appliances were marketed by electrical utilities to build demand for electric power. During the expansion of rural electrification, demonstrations of cooking on an electric stove were popular.
Variants
Early electric stoves had resistive heating coils which heated iron hotplates, on top of which the pots were placed. Eventually, composite heating elements were introduced, with the resistive wires encased in hollow metal tubes packed with magnesite. These tubes, arranged in a spiral, support the cookware directly. | Electric stove | Wikipedia | 464 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null |
In the 1970s, glass-ceramic cooktops started to appear. Glass-ceramic has very low thermal conductivity and a near-zero coefficient of thermal expansion, but lets infrared radiation pass very well. Electrical heating coils or halogen lamps are used as heating elements. Because of its physical characteristics,
A third technology is the induction stove, which also has a smooth glass-ceramic surface. Only ferromagnetic cookware works with induction stoves, which heat by dint of electromagnetic induction.
Electricity consumption
Typical electricity consumption of one heating element depending on size is 1–3 kW. | Electric stove | Wikipedia | 119 | 1491814 | https://en.wikipedia.org/wiki/Electric%20stove | Technology | Household appliances | null |
The eastern mole or common mole (Scalopus aquaticus) is a medium-sized North American mole. It is the only species in the genus Scalopus. It is found in forested and open areas with moist sandy soils in northern Mexico, the eastern United States and the southwestern corner of Ontario in Canada.
The eastern mole has grey-brown fur with silver-grey underparts, a pointed nose and a short tail. It is about in length including a long tail and weighs about . Its front paws are broad and spade-shaped, specialized for digging. It has 36 teeth. Its eyes are covered by fur and its ears are not visible.
The eastern mole spends most of its time underground, foraging in shallow burrows for earthworms, grubs, beetles, insect larvae and some plant matter. It is active year-round. It is mainly solitary except during mating in early spring. The female has a litter of two to five young in a deep burrow.
Subspecies
A majority of the moles throughout their range are Scalopus aquaticus aquaticus. All the other subspecies exist in small pocket ranges. | Eastern mole | Wikipedia | 226 | 1493798 | https://en.wikipedia.org/wiki/Eastern%20mole | Biology and health sciences | Eulipotyphla | Animals |
Glacial landforms are landforms created by the action of glaciers. Most of today's glacial landforms were created by the movement of large ice sheets during the Quaternary glaciations. Some areas, like Fennoscandia and the southern Andes, have extensive occurrences of glacial landforms; other areas, such as the Sahara, display rare and very old fossil glacial landforms.
Erosional landforms
As the glaciers expand, due to their accumulating weight of snow and ice they crush, abrade, and scour surfaces such as rocks and bedrock. The resulting erosional landforms include striations, cirques, glacial horns, arêtes, trim lines, U-shaped valleys, roches moutonnées, overdeepenings and hanging valleys.
Striations: grooves and indentations in rock outcrops, formed by the scraping of small sediments on the bottom of a glacier across the Earth's surface. The direction of striations display the direction the glacier was moving.
Cirque: Starting location for mountain glaciers, leaving behind a bowl shaped indentation in the mountain side once the small glacier has melted.(add geology book citation already in the article)
Cirque stairway: a sequence of cirques
U-shaped, or trough, valley: U-shaped valleys are created by mountain glaciers. When filled with ocean water so as to create an inlet, these valleys are called fjords.
Arête: spiky high land between two glaciers. If the glacial action erodes through, a spillway (or col) forms
Horn: a sharp peak connecting multiple glacier intersections, made up of multiple arêtes.
Valley step: an abrupt change in the longitudinal slope of a glacial valley
Hanging Valleys: Formed by glacial meltwater eroding the land partially, often accompanied by a waterfall.
Roche moutonnée
Nunatak
Depositional landforms | Glacial landform | Wikipedia | 387 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null |
Later, when the glaciers retreated leaving behind their freight of crushed rock and sand (glacial drift), they created characteristic depositional landforms. Depositional landforms are often made of glacial till, which is composed of unsorted sediments (some quite large, others small) that were eroded, carried, and deposited by the glacier some distance away from their original rock source. Examples include glacial moraines, eskers, and kames. Drumlins and ribbed moraines are also landforms left behind by retreating glaciers. Many depositional landforms result from sediment deposited or reshaped by meltwater and are referred to as fluvioglacial landforms. Fluvioglacial deposits differ from glacial till in that they were deposited by means of water, rather than the glacial itself, and the sediments are thus also more size sorted than glacial till is. The stone walls of New England contain many glacial erratics, rocks that were dragged by a glacier many miles from their bedrock origin.
Esker: Built up bed of a subglacial stream, forming small, string-like mounds left behind as a glacier retreats.
Kame: Irregularly shaped mound of sediments previously deposited by falling into an opening of glacial ice.
Moraine: Built up mound of glacial till along a spot on the glacier. Feature can be terminal (at the end of a glacier, showing how far the glacier extended), lateral (along the sides of a glacier), or medial (formed by the merger of lateral moraines from contributory glaciers). Types: Pulju, Rogen, Sevetti, terminal, Veiki
Outwash fan: Braided stream flowing from the front end of a glacier into a more flat, lower elevation plain of sediments.
Glacial lakes and ponds
Lakes and ponds may also be caused by glacial movement. Kettle lakes form when a retreating glacier leaves behind an underground or surface chunk of ice that later melts to form a depression containing water. Moraine-dammed lakes occur when glacial debris dam a stream (or snow runoff). Jackson Lake and Jenny Lake in Grand Teton National Park are examples of moraine-dammed lakes, though Jackson Lake is enhanced by a man-made dam. | Glacial landform | Wikipedia | 449 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null |
Kettle lake: Depression, formed by a block of ice separated from the main glacier, in which the lake forms
Tarn: A lake formed in a cirque by overdeepening
Paternoster lake: A series of lakes in a glacial valley, formed when a stream is dammed by successive recessional moraines left by an advancing or retreating glacier
Glacial lake: A lake that formed between the front of a glacier and the last recessional moraine
Ice features
Apart from the landforms left behind by glaciers, glaciers themselves are striking features of the terrain, particularly in the polar regions of Earth.
Notable examples include valley glaciers where glacial flow is restricted by the valley walls, crevasses in the upper section of glacial ice, and icefalls—the ice equivalent of waterfalls.
Disputed origin
The glacial origin of some landforms has been questioned:
Erling Lindström has advanced the thesis that roches moutonnées may not be entirely glacial landforms, and may have already had most of their shape before glaciation. Jointing that contributes to their shape typically predates glaciation, and roche moutonnée-like forms can be found in tropical areas such as East Africa and Australia. Further, at Ivö Lake in Sweden, rock surfaces exposed by kaolin mining and then weathered resemble roche moutonnée.
The idea of elevated flat surfaces being shaped by glaciation—the glacial buzzsaw effect—has been rejected by various scholars. In the case of Norway the elevated paleic surface has been proposed to have been shaped by the glacial buzzsaw effect. However, this proposal is difficult to reconcile with the fact that the paleic surfaces consist of a series of steps at different levels. Glacial cirques, that in the buzzsaw hypothesis contribute to leveling the landscape, are not associated with any paleosurface levels of the composite paleic surface, nor does the modern equilibrium line altitude (ELA) or the Last Glacial Maximum ELA match any given level of the paleic surface. The elevated plains of West Greenland are also unrelated to any glacial buzzsaw effect.
The Gulf of Bothnia and Hudson Bay, two large depressions at the centre of former ice sheets, are known to be more the result of tectonics than of any weak glacial erosion. | Glacial landform | Wikipedia | 469 | 1494235 | https://en.wikipedia.org/wiki/Glacial%20landform | Physical sciences | Glacial landforms | null |
Google Maps is a web mapping platform and consumer application offered by Google. It offers satellite imagery, aerial photography, street maps, 360° interactive panoramic views of streets (Street View), real-time traffic conditions, and route planning for traveling by foot, car, bike, air (in beta) and public transportation. , Google Maps was being used by over one billion people every month around the world.
Google Maps began as a C++ desktop program developed by brothers Lars and Jens Rasmussen in Australia at Where 2 Technologies. In October 2004, the company was acquired by Google, which converted it into a web application. After additional acquisitions of a geospatial data visualization company and a real-time traffic analyzer, Google Maps was launched in February 2005. The service's front end utilizes JavaScript, XML, and Ajax. Google Maps offers an API that allows maps to be embedded on third-party websites, and offers a locator for businesses and other organizations in numerous countries around the world. Google Map Maker allowed users to collaboratively expand and update the service's mapping worldwide but was discontinued from March 2017. However, crowdsourced contributions to Google Maps were not discontinued as the company announced those features would be transferred to the Google Local Guides program, although users that are not Local Guides can still contribute.
Google Maps' satellite view is a "top-down" or bird's-eye view; most of the high-resolution imagery of cities is aerial photography taken from aircraft flying at , while most other imagery is from satellites. Much of the available satellite imagery is no more than three years old and is updated on a regular basis, according to a 2011 report. Google Maps previously used a variant of the Mercator projection, and therefore could not accurately show areas around the poles. In August 2018, the desktop version of Google Maps was updated to show a 3D globe. It is still possible to switch back to the 2D map in the settings.
Google Maps for mobile devices were first released in 2006; the latest versions feature GPS turn-by-turn navigation along with dedicated parking assistance features. By 2013, it was found to be the world's most popular smartphone app, with over 54% of global smartphone owners using it. In 2017, the app was reported to have two billion users on Android, along with several other Google services including YouTube, Chrome, Gmail, Search, and Google Play.
History | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Acquisitions
Google Maps first started as a C++ program designed by two Danish brothers, Lars and Jens Eilstrup Rasmussen, and Noel Gordon and Stephen Ma, at the Sydney-based company Where 2 Technologies, which was founded in early 2003. The program was initially designed to be separately downloaded by users, but the company later pitched the idea for a purely Web-based product to Google management, changing the method of distribution. In October 2004, the company was acquired by Google Inc. where it transformed into the web application Google Maps. The Rasmussen brothers, Gordon and Ma joined Google at that time.
In the same month, Google acquired Keyhole, a geospatial data visualization company (with investment from the CIA), whose marquee application suite, Earth Viewer, emerged as the Google Earth application in 2005 while other aspects of its core technology were integrated into Google Maps. In September 2004, Google acquired ZipDash, a company that provided real-time traffic analysis.
2005–2010
The launch of Google Maps was first announced on the Google Blog on February 8, 2005.
In September 2005, in the aftermath of Hurricane Katrina, Google Maps quickly updated its satellite imagery of New Orleans to allow users to view the extent of the flooding in various parts of that city.
As of 2007, Google Maps was equipped with a miniature view with a draggable rectangle that denotes the area shown in the main viewport, and "Info windows" for previewing details about locations on maps. As of 2024, this feature had been removed (likely several years prior).
On November 28, 2007, Google Maps for Mobile 2.0 was released. It featured a beta version of a "My Location" feature, which uses the GPS / Assisted GPS location of the mobile device, if available, supplemented by determining the nearest wireless networks and cell sites. The software looks up the location of the cell site using a database of known wireless networks and sites. By triangulating the different signal strengths from cell transmitters and then using their location property (retrieved from the database), My Location determines the user's current location.
On September 23, 2008, coinciding with the announcement of the first commercial Android device, Google announced that a Google Maps app had been released for its Android operating system.
In October 2009, Google replaced Tele Atlas as their primary supplier of geospatial data in the US version of Maps and used their own data. | Google Maps | Wikipedia | 497 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
2011–2015
On April 19, 2011, Map Maker was added to the American version of Google Maps, allowing any viewer to edit and add changes to Google Maps. This provides Google with local map updates almost in real-time instead of waiting for digital map data companies to release more infrequent updates.
On January 31, 2012, Google, due to offering its Maps for free, was found guilty of abusing the dominant position of its Google Maps application and ordered by a court to pay a fine and damages to Bottin Cartographer, a French mapping company. This ruling was overturned on appeal.
In June 2012, Google started mapping the UK's rivers and canals in partnership with the Canal and River Trust. The company has stated that "it would update the program during the year to allow users to plan trips which include locks, bridges and towpaths along the 2,000 miles of river paths in the UK."
In December 2012, the Google Maps application was separately made available in the App Store, after Apple removed it from its default installation of the mobile operating system version iOS 6 in September 2012.
On January 29, 2013, Google Maps was updated to include a map of North Korea. , Google Maps recognizes Palestine as a country, instead of redirecting to the Palestinian territories.
In August 2013, Google Maps removed the Wikipedia Layer, which provided links to Wikipedia content about locations shown in Google Maps using Wikipedia geocodes.
On April 12, 2014, Google Maps was updated to reflect the annexation of Ukrainian Crimea by Russia. Crimea is shown as the Republic of Crimea in Russia and as the Autonomous Republic of Crimea in Ukraine. All other versions show a dotted disputed border.
In April 2015, on a map near the Pakistani city of Rawalpindi, the imagery of the Android logo urinating on the Apple logo was added via Map Maker and appeared on Google Maps. The vandalism was soon removed and Google publicly apologized. However, as a result, Google disabled user moderation on Map Maker, and on May 12, disabled editing worldwide until it could devise a new policy for approving edits and avoiding vandalism.
On April 29, 2015, users of the classic Google Maps were forwarded to the new Google Maps with the option to be removed from the interface.
On July 14, 2015, the Chinese name for Scarborough Shoal was removed after a petition from the Philippines was posted on Change.org.
2016–2018 | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
On June 27, 2016, Google rolled out new satellite imagery worldwide sourced from Landsat 8, comprising over 700 trillion pixels of new data. In September 2016, Google Maps acquired mapping analytics startup Urban Engines.
In 2016, the Government of South Korea offered Google conditional access to the country's geographic database – access that already allows indigenous Korean mapping providers high-detail maps. Google declined the offer, as it was unwilling to accept restrictions on reducing the quality around locations the South Korean Government felt were sensitive (see restrictions on geographic data in South Korea).
On October 16, 2017, Google Maps was updated with accessible imagery of several planets and moons such as Titan, Mercury, and Venus, as well as direct access to imagery of the Moon and Mars.
In May 2018, Google announced major changes to the API structure starting June 11, 2018. This change consolidated the 18 different endpoints into three services and merged the basic and premium plans into one pay-as-you-go plan. This meant a 1400% price raise for users on the basic plan, with only six weeks of notice. This caused a harsh reaction within the developers community. In June, Google postponed the change date to July 16, 2018.
In August 2018, Google Maps designed its overall view (when zoomed out completely) into a 3D globe dropping the Mercator projection that projected the planet onto a flat surface.
2019–present
In January 2019, Google Maps added speed trap and speed camera alerts as reported by other users.
On October 17, 2019, Google Maps was updated to include incident reporting, resembling a functionality in Waze which was acquired by Google in 2013.
In December 2019, Incognito mode was added, allowing users to enter destinations without saving entries to their Google accounts.
In February 2020, Maps received a 15th anniversary redesign. It notably added a brand-new app icon, which now resembles the original icon in 2005.
On September 23, 2020, Google announced a COVID-19 Layer update for Google maps, which is designed to offer a seven-day average data of the total COVID-19-positive cases per 100,000 people in the area selected on the map. It also features a label indicating the rise and fall in the number of cases.
In January 2021, Google announced that it would be launching a new feature displaying COVID-19 vaccination sites. | Google Maps | Wikipedia | 478 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
In January 2021, Google announced updates to the route planner that would accommodate drivers of electric vehicles. Routing would take into account the type of vehicle, vehicle status including current charge, and the locations of charging stations.
In June 2022, Google Maps added a layer displaying air quality for certain countries.
In September 2022, Google removed the COVID-19 Layer from Google Maps due to lack of usage of the feature.
Functionality
Directions and transit
Google Maps provides a route planner, allowing users to find available directions through driving, public transportation, walking, or biking. Google has partnered globally with over 800 public transportation providers to adopt GTFS (General Transit Feed Specification), making the data available to third parties. The app can indicate users' transit route, thanks to an October 2019 update. The incognito mode, eyes-free walking navigation features were released earlier. A July 2020 update provided bike share routes.
In February 2024, Google Maps started rolling out glanceable directions for its Android and iOS apps. The feature allows users to track their journey from their device's lock screen.
Traffic conditions
In 2007, Google began offering traffic data as a colored overlay on top of roads and motorways to represent the speed of vehicles on particular roads. Crowdsourcing is used to obtain the GPS-determined locations of a large number of cellphone users, from which live traffic maps are produced.
Google has stated that the speed and location information it collects to calculate traffic conditions is anonymous. Options available in each phone's settings allow users not to share information about their location with Google Maps. Google stated, "Once you disable or opt out of My Location, Maps will not continue to send radio information back to Google servers to determine your handset's approximate location".
Street View
On May 25, 2007, Google released Google Street View, a feature of Google Maps providing 360° panoramic street-level views of various locations. On the date of release, the feature only included five cities in the U.S. It has since expanded to thousands of locations around the world. In July 2009, Google began mapping college campuses and surrounding paths and trails.
Street View garnered much controversy after its release because of privacy concerns about the uncensored nature of the panoramic photographs, although the views are only taken on public streets. Since then, Google has blurred faces and license plates through automated facial recognition. | Google Maps | Wikipedia | 485 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
In late 2014, Google launched Google Underwater Street View, including of the Australian Great Barrier Reef in 3D. The images are taken by special cameras which turn 360 degrees and take shots every 3 seconds.
In 2017, in both Google Maps and Google Earth, Street View navigation of the International Space Station interior spaces became available.
3D imagery
Google Maps has incorporated 3D models of hundreds of cities in over 40 countries from Google Earth into its satellite view. The models were developed using aerial photogrammetry techniques.
Immersive View
At the I/O 2022 event, Google announced Immersive View, a feature of Google Maps which would involve composite 3D images generated from Street View and aerial images of locations using AI, complete with synchronous information. It was to be initially in five cities worldwide, with plans to add it to other cities later on. The feature was previewed in September 2022 with 250 photorealistic aerial 3D images of landmarks, and was full launched in February 2023. An expansion of Immersive View to routes was announced at Google I/O 2023, and was launched in October 2023 for 15 cities globally.
The feature uses predictive modelling and neural radiance fields to scan Street View and aerial images to generate composite 3D imagery of locations, including both exteriors and interiors, and routes, including driving, walking or cycling, as well as generate synchronous information and forecasts up to a month ahead from historical and environmental data about both such as weather, traffic and busyness.
Immersive View has been available in the following locations:
Landmark Icons
Google added icons of city attractions, in a similar style to Apple Maps, on October 3, 2019. In the first stage, such icons were added to 9 cities.
45° imagery
In December 2009, Google introduced a new view consisting of 45° angle aerial imagery, offering a "bird's-eye view" of cities. The first cities available were San Jose and San Diego. This feature was initially available only to developers via the Google Maps API. In February 2010, it was introduced as an experimental feature in Google Maps Labs. In July 2010, 45° imagery was made available in Google Maps in select cities in South Africa, the United States, Germany and Italy.
Weather
In February 2024, Google Maps incorporated a small weather icon on the top left corner of the Android and iOS mobile apps, giving access to weather and air quality index details. | Google Maps | Wikipedia | 493 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Lens in Maps
Previously called Search with Live View, Lens In Maps identifies shops, restaurants, transit stations and other street features with a phone's camera and places relevant information and a category pin on top, like closing/opening times, current busyness, pricing and reviews using AI and augmented reality. The feature, if available on the device, can be accessed through tapping the Lens icon in the search bar. It was expanded to 50 new cities in October 2023 in its biggest expansion yet, after initially being released in late 2022 in Los Angeles, San Francisco, New York, London, and Paris. Lens in Maps shares features with Live View, which also displays information relating to street features while guiding a user to a selected destination with virtual arrows, signs and guidance.
Business listings
Google collates business listings from multiple on-line and off-line sources. To reduce duplication in the index, Google's algorithm combines listings automatically based on address, phone number, or geocode, but sometimes information for separate businesses will be inadvertently merged with each other, resulting in listings inaccurately incorporating elements from multiple businesses. Google allows business owners to create and verify their own business data through Google Business Profile (GBP), formerly Google My Business (GMB). Owners are encouraged to provide Google with business information including address, phone number, business category, and photos. Google has staff in India who check and correct listings remotely as well as support businesses with issues. Google also has teams on the ground in most countries that validate physical addresses in person. In May 2024, Google announced it would discontinue the chat feature in Google Business Profile. Starting July 15, 2024, new chat conversations would be disabled, and by July 31, 2024, all chat functionalities would end.
Google Maps can be manipulated by businesses that are not physically located in the area in which they record a listing. There are cases of people abusing Google Maps to overtake their competition by placing unverified listings on online directory sites, knowing the information will roll across to Google (duplicate sites). The people who update these listings do not use a registered business name. They place keywords and location details on their Google Maps business title, which can overtake credible business listings. In Australia in particular, genuine companies and businesses are noticing a trend of fake business listings in a variety of industries. | Google Maps | Wikipedia | 481 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Genuine business owners can also optimize their business listings to gain greater visibility in Google Maps, through a type of search engine marketing called local search engine optimization.
Indoor maps
In March 2011, indoor maps were added to Google Maps, giving users the ability to navigate themselves within buildings such as airports, museums, shopping malls, big-box stores, universities, transit stations, and other public spaces (including underground facilities). Google encourages owners of public facilities to submit floor plans of their buildings in order to add them to the service. Map users can view different floors of a building or subway station by clicking on a level selector that is displayed near any structures which are mapped on multiple levels.
My Maps
My Maps is a feature in Google Maps launched in April 2007 that enables users to create custom maps for personal use or sharing. Users can add points, lines, shapes, notes and images on top of Google Maps using a WYSIWYG editor. An Android app for My Maps, initially released in March 2013 under the name Google Maps Engine Lite, was available until its removal from the Play Store in October 2021.
Google Local Guides
Google Local Guides is a volunteer program launched by Google Maps to enable users to contribute to Google Maps when registered. It sometimes provides them additional perks and benefits for their collaboration. Users can achieve Level 1 to 10, and be awarded with badges. The program is partially a successor to Google Map Maker as features from the former program became integrated into the website and app.
The program consists of adding reviews, photos, basic information, and videos; and correcting information such as wheelchair accessibility. Adding reviews, photos, videos, new places, new roads or providing useful information gives points to the users. The level of users is upgraded when they get a certain amount of points. Starting with Level 4, a star is shown near the avatar of the user.
Timelapse
Earth Timelapse, released in April 2021, is a program in which users can see how the earth has been changed in the last 37 years. They combined the 15 million satellite images (roughly ten quadrillion pixels) to create the 35 global cloud-free Images for this program.
Timeline
If a user shares their location with Google, Timeline summarises this location for each day on a Timeline map. Timeline estimates the mode of travel used to move between places and will also show photos taken at that location. In June 2024, Google started progressively removing access to the timeline on web browsers, with the information instead being stored on a local device. | Google Maps | Wikipedia | 512 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Implementation
As the user drags the map, the grid squares are downloaded from the server and inserted into the page. When a user searches for a business, the results are downloaded in the background for insertion into the side panel and map; the page is not reloaded. A hidden iframe with form submission is used because it preserves browser history. Like many other Google web applications, Google Maps uses JavaScript extensively. The site also uses protocol buffers for data transfer rather than JSON, for performance reasons.
The version of Google Street View for classic Google Maps required Adobe Flash. In October 2011, Google announced MapsGL, a WebGL version of Maps with better renderings and smoother transitions. Indoor maps use JPG, .PNG, .PDF, .BMP, or .GIF, for floor plans.
Users who are logged into a Google Account can save locations so that they are overlaid on the map with various colored "pins" whenever they browse the application. These "Saved places" can be organized into default groups or user named groups and shared with other users. "Starred places" is one default group example. It previously automatically created a record within the now-discontinued product Google Bookmarks.
Map data and imagery
The Google Maps terms and conditions state that usage of material from Google Maps is regulated by Google Terms of Service and some additional restrictions. Google has either purchased local map data from established companies, or has entered into lease agreements to use copyrighted map data. The owner of the copyright is listed at the bottom of zoomed maps. For example, street maps in Japan are leased from Zenrin. Street maps in China are leased from AutoNavi. Russian street maps are leased from Geocentre Consulting and Tele Atlas. Data for North Korea is sourced from the companion project Google Map Maker.
Street map overlays, in some areas, may not match up precisely with the corresponding satellite images. The street data may be entirely erroneous, or simply out of date: "The biggest challenge is the currency of data, the authenticity of data," said Google Earth representative Brian McClendon. As a result, in March 2008 Google added a feature to edit the locations of houses and businesses.
Restrictions have been placed on Google Maps through the apparent censoring of locations deemed potential security threats. In some cases the area of redaction is for specific buildings, but in other cases, such as Washington, D.C., the restriction is to use outdated imagery. | Google Maps | Wikipedia | 507 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Google Maps API
Google Maps API, now called Google Maps Platform, hosts about 17 different APIs, which are themed under the following categories: Maps, Places and Routes.
After the success of reverse-engineered mashups such as chicagocrime.org and housingmaps.com, Google launched the Google Maps API in June 2005 to allow developers to integrate Google Maps into their websites. It was a free service that did not require an API key until June 2018 (changes went into effect on July 16), when it was announced that an API key linked to a Google Cloud account with billing enabled would be required to access the API. The API does not contain ads, but Google states in their terms of use that they reserve the right to display ads in the future.
By using the Google Maps API, it is possible to embed Google Maps into an external website, onto which site-specific data can be overlaid. Although initially only a JavaScript API, the Maps API was expanded to include an API for Adobe Flash applications (but this has been deprecated), a service for retrieving static map images, and web services for performing geocoding, generating driving directions, and obtaining elevation profiles. Over 1,000,000 web sites use the Google Maps API, making it the most heavily used web application development API. In September 2011, Google announced it would deprecate the Google Maps API for Flash.
The Google Maps API was free for commercial use, provided that the site on which it is being used is publicly accessible and did not charge for access, and was not generating more than 25,000 map accesses a day. Sites that did not meet these requirements could purchase the Google Maps API for Business.
As of June 21, 2018, Google increased the prices of the Maps API and requires a billing profile.
Google Maps in China
Due to restrictions on geographic data in China, Google Maps must partner with a Chinese digital map provider in order to legally show Chinese map data. Since 2006, this partner has been AutoNavi. | Google Maps | Wikipedia | 413 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Within China, the State Council mandates that all maps of China use the GCJ-02 coordinate system, which is offset from the WGS-84 system used in most of the world. google.cn/maps (formerly Google Ditu) uses the GCJ-02 system for both its street maps and satellite imagery. google.com/maps also uses GCJ-02 data for the street map, but uses WGS-84 coordinates for satellite imagery, causing the so-called China GPS shift problem.
Frontier alignments also present some differences between google.cn/maps and google.com/maps. On the latter, sections of the Chinese border with India and Pakistan are shown with dotted lines, indicating areas or frontiers in dispute. However, google.cn shows the Chinese frontier strictly according to Chinese claims with no dotted lines indicating the border with India and Pakistan. For example, the South Tibet region claimed by China but administered by India as a large part of Arunachal Pradesh is shown inside the Chinese frontier by google.cn, with Indian highways ending abruptly at the Chinese claim line. Google.cn also shows Taiwan and the South China Sea Islands as part of China. Google Ditu's street map coverage of Taiwan no longer omits major state organs, such as the Presidential Palace, the five Yuans, and the Supreme Court.
Feature-wise, google.cn/maps does not feature My Maps. On the other hand, while google.cn displays virtually all text in Chinese, google.com/maps displays most text (user-selectable real text as well as those on map) in English. This behavior of displaying English text is not consistent but intermittent – sometimes it is in English, sometimes it is in Chinese. The criteria for choosing which language is displayed are not known publicly.
Criticism and controversies
Incorrect location naming
There are cases where Google Maps had added out-of-date neighborhood monikers. Thus, in Los Angeles, the name "Brooklyn Heights" was revived from its 1870s usage and "Silver Lake Heights" from its 1920s usage, or mistakenly renamed areas (in Detroit, the neighborhood "Fiskhorn" became "Fishkorn"). Because many companies utilize Google Maps data, these previously obscure or incorrect names then gain traction; the names are often used by realtors, hotels, food delivery sites, dating sites, and news organizations. | Google Maps | Wikipedia | 481 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Google has said it created its maps from third-party data, public sources, satellites, and users, but many names used have not been connected to any official record. According to a former Google Maps employee (who was not authorized to speak publicly), users can submit changes to Google Maps, but some submissions are ruled upon by people with little local knowledge of a place, such as contractors in India. Critics maintain that names likes "BoCoCa" (for the area in Brooklyn between Boerum Hill, Cobble Hill and Carroll Gardens), are "just plain puzzling" or simply made up. Some names used by Google have been traced to non-professionally made maps with typographical errors that survived on Google Maps.
Potential misuse
In 2005 the Australian Nuclear Science and Technology Organisation (ANSTO) complained about the potential for terrorists to use the satellite images in planning attacks, with specific reference to the Lucas Heights nuclear reactor; however, the Australian Federal government did not support the organization's concern. At the time of the ANSTO complaint, Google had colored over some areas for security (mostly in the U.S.), such as the rooftop of the White House and several other Washington, D.C. buildings.
In October 2010, Nicaraguan military commander Edén Pastora stationed Nicaraguan troops on the Isla Calero (in the delta of the San Juan River), justifying his action on the border delineation given by Google Maps. Google has since updated its data which it found to be incorrect.
On January 27, 2014, documents leaked by Edward Snowden revealed that the NSA and the GCHQ intercepted Google Maps queries made on smartphones, and used them to locate the users making these queries. One leaked document, dating to 2008, stated that "[i]t effectively means that anyone using Google Maps on a smartphone is working in support of a GCHQ system."
In May 2015, searches on Google Maps for offensive racial epithets for African Americans such as "nigger", "nigger king", and "nigger house" pointed the user to the White House; Google apologized for the incident.
In December 2015, 3 Japanese netizens were charged with vandalism after they were found to have added an unrelated law firm's name as well as indecent names to locations such as "Nuclear test site" to the Atomic Bomb Dome and "Izumo Satya" to the Izumo Taisha. | Google Maps | Wikipedia | 504 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
In February 2020, the artist Simon Weckert used 99 cell phones to fake a Google Maps traffic jam.
In September 2024, several schools in Taiwan and Hong Kong were altered to incorrect labels, such as "psychiatric hospitals" or "prisons". Initially, it was believed to be the result of hacker attacks. However, police later revealed that local students had carried out the prank. Google quickly corrected the mislabeled entries. Education officials in Taiwan and Hong Kong expressed concern over the incident.
Misdirection incidents
Australia
In August 2023, a woman driving from Alice Springs to the Harts Range Racecourse was stranded in the Central Australian desert for a night after following directions provided by Google Maps. She later discovered that Google Maps was providing directions for the actual Harts Range instead of the rodeo. Google said it was looking into the naming of the two locations and consulting with "local and authoritative sources" to solve the issue.
In February 2024, two German tourists were stranded for a week after Google Maps directed them to follow a dirt track through Oyala Thumotang National Park and their vehicle became trapped in mud. Queensland Parks and Wildlife Service ranger Roger James said, "People should not trust Google Maps when they're travelling in remote regions of Queensland, and they need to follow the signs, use official maps or other navigational devices."
North America
In June 2019, Google Maps provided nearly 100 Colorado drivers an alternative route that led to a dirt road after a crash occurred on Peña Boulevard. The road had been turned to mud by rain, resulting in nearly 100 vehicles being trapped. Google said in a statement, "While we always work to provide the best directions, issues can arise due to unforeseen circumstances such as weather. We encourage all drivers to follow local laws, stay attentive, and use their best judgment while driving."
In September 2023, Google was sued by a North Carolina resident who alleged that Google Maps had directed her husband over the Snow Creek Bridge in Hickory the year prior, resulting in him drowning. According to the lawsuit, multiple people had notified Google about the state of the bridge, which collapsed in 2013, but Google had not updated the route information and continued to direct users over the bridge. At the time of the man's death, the barriers placed to block access to the bridge had been vandalized. | Google Maps | Wikipedia | 478 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
In November 2023, a hiker was rescued by helicopter on the backside of Mount Fromme in Vancouver. North Shore Rescue stated on its Facebook page that the hiker had followed a non-existent hiking trail on Google Maps. This was also the second hiker in two months to require rescuing after following the same trail. The fake trail has since been removed from the app.
Also in November 2023, Google apologized after users were directed through desert roads after parts of Interstate 15 were closed due to a dust storm. Drivers became stranded after following the suggested detour route, which was a "bumpy dirt trail". Following the incident, Google stated that Google Maps would "no longer route drivers traveling between Las Vegas and Barstow down through those roads."
Russia
In 2020, a teenage motorist was found frozen to death while his passenger was still alive but suffered from severe frostbite after using Google Maps, which had led them to a shorter but abandoned section of the R504 Kolyma Highway, where their Toyota Chaser became disabled.
India
In 2024, three men from Uttar Pradesh died after their car fell from an under-construction bridge. They were using Google Maps for driving which misdirected them and the car fell into the Ramganga river.
Discontinued features
Google Latitude
Google Latitude was a feature that let users share their physical locations with other people. This service was based on Google Maps, specifically on mobile devices. There was an iGoogle widget for desktops and laptops as well. Some concerns were expressed about the privacy issues raised by the use of the service. On August 9, 2013, this service was discontinued, and on March 22, 2017, Google incorporated the features from Latitude into the Google Maps app.
Google Map Maker
In areas where Google Map Maker was available, for example, much of Asia, Africa, Latin America and Europe as well as the United States and Canada, anyone who logged into their Google account could directly improve the map by fixing incorrect driving directions, adding biking trails, or adding a missing building or road. General map errors in Australia, Austria, Belgium, Denmark, France, Liechtenstein, Netherlands, New Zealand, Norway, South Africa, Switzerland, and the United States could be reported using the Report a Problem link in Google Maps and would be updated by Google. For areas where Google used Tele Atlas data, map errors could be reported using Tele Atlas map insight. | Google Maps | Wikipedia | 492 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
If imagery was missing, outdated, misaligned, or generally incorrect, one could notify Google through their contact request form.
In November 2016, Google announced the discontinuation of Google Map Maker as of March 2017.
Mobile app
Google Maps is available as a mobile app for the Android and iOS mobile operating systems. The first mobile version of Google Maps (then known as Google Local for Mobile) was launched in beta in November 2005 for mobile platforms supporting J2ME. It was released as Google Maps for Mobile in 2006. In 2007 it came preloaded on the first iPhone in a deal with Apple. A version specifically for Windows Mobile was released in February 2007 and the Symbian app was released in November 2007.
Version 2.0 of Google Maps Mobile was announced at the end of 2007, with a stand out My Location feature to find the user's location using the cell towers, without needing GPS. In September 2008, Google Maps was released for and preloaded on Google's own new platform Android.
Up until iOS 6, the built-in maps application on the iOS operating system was powered by Google Maps. However, with the announcement of iOS 6 in June 2012, Apple announced that they had created their own Apple Maps mapping service, which officially replaced Google Maps when iOS 6 was released on September 19, 2012. However, at launch, Apple Maps received significant criticism from users due to inaccuracies, errors and bugs. One day later, The Guardian reported that Google was preparing its own Google Maps app, which was released on December 12, 2012. Within two days, the application had been downloaded over ten million times.
Features
The Google Maps apps for iOS and Android have many of the same features, including turn-by-turn navigation, street view, and public transit information. Turn-by-turn navigation was originally announced by Google as a separate beta testing app exclusive to Android 2.0 devices in October 2009. The original standalone iOS version did not support the iPad, but tablet support was added with version 2.0 in July 2013. An update in June 2012 for Android devices added support for offline access to downloaded maps of certain regions, a feature that was eventually released for iOS devices, and made more robust on Android, in May 2014. | Google Maps | Wikipedia | 459 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
At the end of 2015 Google Maps announced its new offline functionality, but with various limitations – downloaded area cannot exceed 120,000 square kilometers and require a considerable amount of storage space. In January 2017, Google added a feature exclusively to Android that will, in some U.S. cities, indicate the level of difficulty in finding available parking spots, and on both Android and iOS, the app can, as of an April 2017 update, remember where users parked. In August 2017, Google Maps for Android was updated with new functionality to actively help the user in finding parking lots and garages close to a destination. In December 2017, Google added a new two-wheeler mode to its Android app, designed for users in India, allowing for more accessibility in traffic conditions. In 2019 the Android version introduced the new feature called live view that allows to view directions directly on the road thanks to augmented reality. Google Maps won the 2020 Webby Award for Best User Interface in the category Apps, Mobile & Voice. In March 2021, Google added a feature in which users can draw missing roads. In June 2022, Google implemented support for toll calculation. Both iOS and Android apps report how much the user has to pay in tolls when a route that includes toll roads is input. The feature is available for roads in the US, India, Japan and Indonesia with further expansion planned. As per reports the total number of toll roads covered in this phase is around 2000. | Google Maps | Wikipedia | 292 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Reception
USA Today welcomed the application back to iOS, saying: "The reemergence in the middle of the night of a Google Maps app for the iPhone is like the return of an old friend. Only your friend, who'd gone missing for three months, comes back looking better than ever." Jason Parker of CNET, calling it "the king of maps", said, "With its iOS Maps app, Google sets the standard for what mobile navigation should be and more." Bree Fowler of the Associated Press compared Google's and Apple's map applications, saying: "The one clear advantage that Apple has is style. Like Apple devices, the maps are clean and clear and have a fun, pretty element to them, especially in 3-D. But when it comes down to depth and information, Google still reigns superior and will no doubt be welcomed back by its fans." Gizmodo gave it a ranking of 4.5 stars, stating: "Maps Done Right". According to The New York Times, Google "admits that it's [iOS app is] even better than Google Maps for Android phones, which has accommodated its evolving feature set mainly by piling on menus".
Google Maps' location tracking is regarded by some as a threat to users' privacy, with Dylan Tweney of VentureBeat writing in August 2014 that "Google is probably logging your location, step by step, via Google Maps", and linked users to Google's location history map, which "lets you see the path you've traced for any given day that your smartphone has been running Google Maps". Tweney then provided instructions on how to disable location history. The history tracking was also noticed, and recommended disabled, by editors at CNET and TechCrunch. Additionally, Quartz reported in April 2014 that a "sneaky new privacy change" would have an effect on the majority of iOS users. The privacy change, an update to the Gmail iOS app that "now supports sign-in across Google iOS apps, including Maps, Drive, YouTube and Chrome", meant that Google would be able to identify users' actions across its different apps.
The Android version of the app surpassed five billion installations in March 2019. By November 2021, the Android app had surpassed 10 billion installations.
Go version
Google Maps Go, a version of the app designed for lower-end devices, was released in beta in January 2018. By September 2018, the app had over 10 million installations. | Google Maps | Wikipedia | 509 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
Artistic and literary uses
The German "geo-novel" Senghor on the Rocks (2008) presents its story as a series of spreads showing a Google Maps location on the left and the story's text on the right. Annika Richterich explains that the "satellite pictures in Senghor on the Rocks illustrate the main character's travel through the West-African state of Senegal".
Artists have used Google Street View in a range of ways. Emilio Vavarella's The Google Trilogy includes glitchy images and unintended portraits of the drivers of the Street View cars. The Japanese band group inou used Google Street View backgrounds to make a music video for their song EYE. The Canadian band Arcade Fire made a customized music video that used Street View to show the viewer their own childhood home. | Google Maps | Wikipedia | 164 | 1494648 | https://en.wikipedia.org/wiki/Google%20Maps | Technology | Utility | null |
A Faraday cage or Faraday shield is an enclosure used to block some electromagnetic fields. A Faraday shield may be formed by a continuous covering of conductive material, or in the case of a Faraday cage, by a mesh of such materials. Faraday cages are named after scientist Michael Faraday, who first constructed one in 1836.
Faraday cages work because an external electrical field will cause the electric charges within the cage's conducting material to be distributed in a way that cancels out the field's effect inside the cage. This phenomenon can be used to protect sensitive electronic equipment (for example RF receivers) from external radio frequency interference (RFI) often during testing or alignment of the device. Faraday cages are also used to protect people and equipment against electric currents such as lightning strikes and electrostatic discharges, because the cage conducts electrical current around the outside of the enclosed space and none passes through the interior.
Faraday cages cannot block stable or slowly varying magnetic fields, such as the Earth's magnetic field (a compass will still work inside one). To a large degree, however, they shield the interior from external electromagnetic radiation if the conductor is thick enough and any holes are significantly smaller than the wavelength of the radiation. For example, certain computer forensic test procedures of electronic systems that require an environment free of electromagnetic interference can be carried out within a screened room. These rooms are spaces that are completely enclosed by one or more layers of a fine metal mesh or perforated sheet metal. The metal layers are grounded to dissipate any electric currents generated from external or internal electromagnetic fields, and thus they block a large amount of the electromagnetic interference (see also electromagnetic shielding). They provide less attenuation of outgoing transmissions than incoming: they can block electromagnetic pulse (EMP) waves from natural phenomena very effectively, but especially in upper frequencies, a tracking device may be able to penetrate from within the cage (e.g., some cell phones operate at various radio frequencies so while one frequency may not work, another one will). | Faraday cage | Wikipedia | 415 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
The reception or transmission of radio waves, a form of electromagnetic radiation, to or from an antenna within a Faraday cage is heavily attenuated or blocked by the cage; however, a Faraday cage has varied attenuation depending on wave form, frequency, or the distance from receiver or transmitter, and receiver or transmitter power. Near-field, high-powered frequency transmissions like HF RFID are more likely to penetrate. Solid cages generally attenuate fields over a broader range of frequencies than mesh cages.
History
In 1754, Jean-Antoine Nollet published an account of the cage effect in his Leçons de physique expérimentale.
In 1755, Benjamin Franklin observed the effect by lowering an uncharged cork ball suspended on a silk thread through an opening in an electrically charged metal can. The behavior is that of a Faraday cage or shield.
In 1836, Michael Faraday observed that the excess charge on a charged conductor resided only on its exterior and had no influence on anything enclosed within it. To demonstrate this, he built a room coated with metal foil and allowed high-voltage discharges from an electrostatic generator to strike the outside of the room. He used an electroscope to show that there was no electric charge present on the inside of the room walls.
Operation
Continuous
A continuous Faraday shield is a hollow conductor. Externally or internally applied electromagnetic fields produce forces on the charge carriers (usually electrons) within the conductor; the charges are redistributed accordingly due to electrostatic induction. The redistributed charges greatly reduce the voltage within the surface, to an extent depending on the capacitance; however, full cancellation does not occur. | Faraday cage | Wikipedia | 339 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
Interior charges
If charge +Q is placed inside an ungrounded Faraday shield without touching the walls, the internal face of the shield becomes charged with −Q, leading to field lines originating at the charge and extending to charges inside the inner surface of the metal. The field line paths in this inside space (to the endpoint negative charges) are dependent on the shape of the inner containment walls. Simultaneously +Q accumulates on the outer face of the shield. The spread of charges on the outer face is not affected by the position of the internal charge inside the enclosure, but rather determined by the shape of outer face. So for all intents and purposes, the Faraday shield generates the same static electric field on the outside that it would generate if the metal were simply charged with +Q. See Faraday's ice pail experiment, for example, for more details on electric field lines and the decoupling of the outside from the inside. Note that electromagnetic waves are not static charges.
If the cage is grounded, the excess charges will be neutralized as the ground connection creates an equipotential bonding between the outside of the cage and the environment, so there is no voltage between them and therefore also no field. The inner face and the inner charge will remain the same so the field is kept inside.
Exterior fields
Effectiveness of the shielding of a static electric field is largely independent of the geometry of the conductive material; however, the static magnetic fields can penetrate the shield completely.
In the case of varying electromagnetic fields, the faster the variations are (i.e., the higher the frequencies), the better the material resists magnetic field penetration. In this case the shielding also depends on the electrical conductivity, the magnetic properties of the conductive materials used in the cages, as well as their thicknesses.
A good example of the effectiveness of a Faraday shield can be obtained from considerations of skin depth. With skin depth, the current flowing is mostly in the surface, and decays exponentially with depth through the material. Because a Faraday shield has finite thickness, this determines how well the shield works; a thicker shield can attenuate electromagnetic fields better, and to a lower frequency. | Faraday cage | Wikipedia | 447 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
Faraday cage
Faraday cages are Faraday shields that have holes in them and are therefore more complex to analyze. Whereas continuous shields essentially attenuate all wavelengths whose skin depth in the hull material is less than the thickness of the hull, the holes in a cage may permit shorter wavelengths to pass through or set up "evanescent fields" (oscillating fields that do not propagate as EM waves) just beyond the surface. The shorter the wavelength, the better it passes through a mesh of given size. Thus, to work well at short wavelengths (i.e., high frequencies), the holes in the cage must be smaller than the wavelength of the incident wave.
Examples | Faraday cage | Wikipedia | 141 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
Faraday cages are routinely used in analytical chemistry to reduce noise while making sensitive measurements.
Faraday cages, more specifically dual paired seam Faraday bags, are often used in digital forensics to prevent remote wiping and alteration of criminal digital evidence.
Faraday bags are portable containers fabricated with metallic materials that are used to contain devices in order to protect them from electromagnetic transmissions for a wide range of applications, from enhancing digital privacy of cell telephones to protecting credit cards from RFID skimming.
The U.S. and NATO Tempest standards, and similar standards in other countries, include Faraday cages as part of a broader effort to provide emission security for computers.
Automobile and airplane passenger compartments are essentially Faraday cages, protecting passengers from electric charges, such as lightning.
Electronic components in automobiles and aircraft use Faraday cages to protect signals from interference. Sensitive components may include wireless door locks, navigation/GPS systems, and lane departure warning systems. Faraday cages and shields are also critical to vehicle infotainment systems (e.g. radio, Wi-Fi, and GPS display units), which may be designed with the capability to function as critical circuits in emergency situations.
A booster bag (shopping bag lined with aluminum foil) acts as a Faraday cage. It is often used by shoplifters to steal RFID-tagged items. Similar containers are used to resist RFID skimming.
Elevators and other rooms with metallic conducting frames and walls simulate a Faraday cage effect, leading to a loss of signal and "dead zones" for users of cellular phones, radios, and other electronic devices that require external electromagnetic signals. During training, firefighters and other first responders are cautioned that their two-way radios will probably not work inside elevators and to make allowances for that.
Small, physical Faraday cages are used by electronics engineers during equipment testing to simulate such an environment to make sure that the device gracefully handles these conditions.
Properly designed conductive clothing can also form a protective Faraday cage. Some electrical linemen wear Faraday suits, which allow them to work on live, high-voltage power lines without risk of electrocution. The suit prevents electric current from flowing through the body and it has no theoretical voltage limit. Linemen have successfully worked even the highest voltage (Kazakhstan's Ekibastuz–Kokshetau line 1150 kV) lines safely. | Faraday cage | Wikipedia | 478 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
The scan room of a magnetic resonance imaging (MRI) machine is designed as a Faraday cage. This prevents external RF (radio frequency) signals from being added to data collected from the patient, which would affect the resulting image. Technologists are trained to identify the characteristic artifacts created on images should the Faraday cage be damaged, such as during a thunderstorm.
A microwave oven uses a partial Faraday shield (on five of its interior six sides) and a partial Faraday cage, consisting of a wire mesh, on the sixth side (the transparent window), to contain the electromagnetic energy within the oven and to protect the user from exposure to microwave radiation.
Plastic bags that are impregnated with metal are used to enclose electronic toll collection devices whenever tolls should not be charged to those devices, such as during transit or when the user is paying cash.
The shield of a screened cable, such as USB cables or the coaxial cable used for cable television, protects the internal conductors from external electrical noise and prevents the RF signals from leaking out.
Electronic components in some music instruments, such as in an electric guitar, are protected by Faraday cages made from copper or aluminum foils that protect the instrument's electromagnetic pickups from interference from speakers, amplifiers, stage lights, and other musical equipment.
Some buildings, such as prisons, are constructed as a Faraday cage because they have reasons to block both incoming and outgoing cell phone calls by prisoners. The exhibit hall of the Green Bank Observatory is a Faraday cage to prevent interference with the operations of their radio telescopes. | Faraday cage | Wikipedia | 320 | 151590 | https://en.wikipedia.org/wiki/Faraday%20cage | Technology | Signal transmission | null |
Fomalhaut (, ) is the brightest star in the southern constellation of Piscis Austrinus, the Southern Fish, and one of the brightest stars in the night sky. It has the Bayer designation Alpha Piscis Austrini, which is an alternative form of α Piscis Austrini, and is abbreviated Alpha PsA or α PsA. This is a class A star on the main sequence approximately from the Sun as measured by the Hipparcos astrometry satellite. Since 1943, the spectrum of this star has served as one of the stable anchor points by which other stars are classified.
It is classified as a Vega-like star that emits excess infrared radiation, indicating it is surrounded by a circumstellar disk. Fomalhaut, K-type main-sequence star TW Piscis Austrini, and M-type, red dwarf star LP 876-10 constitute a triple system, even though the companions are separated by approximately 8 degrees.
Fomalhaut was the first stellar system with an extrasolar planet candidate imaged at visible wavelengths, designated Fomalhaut b. However, analyses in 2019 and 2023 of existing and new observations indicate that Fomalhaut b is not a planet, but rather an expanding region of debris from a massive planetesimal collision.
Nomenclature
α Piscis Austrini, or Alpha Piscis Austrini, is the system's Bayer designation. It also bears the Flamsteed designation of 24 Piscis Austrini. The classical astronomer Ptolemy included it in the constellation of Aquarius, along with the rest of Piscis Austrinus. In the 17th century, Johann Bayer firmly planted it in the primary position of Piscis Austrinus. Following Ptolemy, John Flamsteed in 1725 additionally denoted it 79 Aquarii. The current designation reflects modern consensus on Bayer's decision, that the star belongs in Piscis Austrinus. Under the rules for naming objects in multiple-star systems, the three components – Fomalhaut, TW Piscis Austrini and LP 876-10 – are designated A, B and C, respectively. | Fomalhaut | Wikipedia | 458 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
The star's traditional name derives from Fom al-Haut from scientific Arabic "the mouth of the [Southern] Fish" (literally, "mouth of the whale"), a translation of how Ptolemy labeled it.
Fam in Arabic means "mouth", al "the", and ḥūt "fish"
or "whale".
In 2016, the International Astronomical Union organized a Working Group on Star Names (WGSN) to catalog and standardize proper names for stars. The WGSN's first bulletin of July 2016 included a table of the first two batches of names approved by the WGSN, which included the name "Fomalhaut" for this star.
In July 2014, the International Astronomical Union (IAU) launched NameExoWorlds, a process for giving proper names to certain exoplanets. The process involved public nomination and voting for the new names. In December 2015, the IAU announced "Dagon" as the winning name for Fomalhaut b. The winning name was proposed by Todd Vaccaro and forwarded by the St. Cloud State University Planetarium of St. Cloud, Minnesota, United States of America, to the IAU for consideration. Dagon was a Semitic deity, often represented as half-man, half-fish.
Fomalhaut A
At a declination of −29.6°, Fomalhaut is located south of the celestial equator, and hence is best viewed from the Southern Hemisphere. However, its southerly declination is not as great as that of stars such as Acrux, Alpha Centauri and Canopus, meaning that, unlike them, Fomalhaut is visible from a large part of the Northern Hemisphere as well, being best seen in autumn. Its declination is greater than that of Sirius and similar to that of Antares. At 40°N, Fomalhaut rises above the horizon for eight hours and reaches only 20° above the horizon, while Capella, which rises at approximately the same time, will stay above the horizon for twenty hours. Fomalhaut can be located in northern latitudes by the fact that the western (right-hand) side of the Square of Pegasus points to it. Continuing the line from Beta to Alpha Pegasi towards the southern horizon, Fomalhaut is about 45˚ south of Alpha Pegasi, with no bright stars in between. | Fomalhaut | Wikipedia | 505 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
Properties
Fomalhaut is a young star, for many years thought to be only 100 to 300 million years old, with a potential lifespan of a billion years. A 2012 study gave a slightly higher age of . The surface temperature of the star is around . Fomalhaut's mass is about 1.92 times that of the Sun, its luminosity is about 16.6 times greater, and its diameter is roughly 1.84 times as large.
Fomalhaut is slightly metal-deficient compared to the Sun, which means it is composed of a smaller percentage of elements other than hydrogen and helium. The metallicity is typically determined by measuring the abundance of iron in the photosphere relative to the abundance of hydrogen. A 1997 spectroscopic study measured a value equal to 93% of the Sun's abundance of iron. A second 1997 study deduced a value of 78%, by assuming Fomalhaut has the same metallicity as the neighboring star TW Piscis Austrini, which has since been argued to be a physical companion. In 2004, a stellar evolutionary model of Fomalhaut yielded a metallicity of 79%. Finally, in 2008, a spectroscopic measurement gave a significantly lower value of 46%.
Fomalhaut has been claimed to be one of approximately 16 stars belonging to the Castor Moving Group. This is an association of stars which share a common motion through space, and have been claimed to be physically associated. Other members of this group include Castor and Vega. The moving group has an estimated age of and originated from the same location. More recent work has found that purported members of the Castor Moving Group appear to not only have a wide range of ages, but their velocities are too different to have been possibly associated with one another in the distant past. Hence, "membership" in this dynamical group has no bearing on the age of the Fomalhaut system.
Debris disks and suspected planets
Fomalhaut is surrounded by several debris disks.
The inner disk is a high-carbon small-grain (10–300 nm) ash disk, clustering at 0.1 AU from the star. Next is a disk of larger particles, with inner edge 0.4-1 AU of the star. The innermost disk is unexplained as yet. | Fomalhaut | Wikipedia | 489 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
The outermost disk is at a radial distance of , in a toroidal shape with a very sharp inner edge, all inclined 24 degrees from edge-on. The dust is distributed in a belt about 25 AU wide. The geometric center of the disk is offset by about from Fomalhaut. The disk is sometimes referred to as "Fomalhaut's Kuiper belt". Fomalhaut's dusty disk is believed to be protoplanetary, and emits considerable infrared radiation. Measurements of Fomalhaut's rotation indicate that the disk is located in the star's equatorial plane, as expected from theories of star and planet formation.
Herschel Space Observatory images of Fomalhaut, analysed in 2012, reveal that a large amount of fluffy micrometer-sized dust is present in the outer dust belt. Because such dust is expected to be blown out of the system by stellar radiation pressure on short timescales, its presence indicates a constant replenishment by collisions of planetesimals. The fluffy morphology of the grains suggests a cometary origin. The collision rate is estimated to be approximately 2000 kilometre-sized comets per day. Observations of this outer dust ring by the Atacama Large Millimeter Array also suggested the possible existence of two planets in the system. If there are additional planets from 4 to 10 AU, they must be under ; if from 2.5 outward, then . | Fomalhaut | Wikipedia | 296 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
On November 13, 2008, astronomers announced an extrasolar planet candidate, orbiting just inside the outer debris ring. This was the first extrasolar orbiting object candidate to be directly imaged in visible light, captured by the Hubble Space Telescope. The mass of the tentative planet, Fomalhaut b, was estimated to be less than three times the mass of Jupiter, and at least the mass of Neptune. However, M-band images taken from the MMT Observatory put strong limits on the existence of gas giants within 40 AU of the star, and Spitzer Space Telescope imaging suggested that the object Fomalhaut b was more likely to be a dust cloud. A later 2019 synthesis of new and existing direct observations of the object confirmed that it is expanding, losing brightness, has not enough mass to detectably perturb the outer ring while crossing it, and is probably a dispersing cloud of debris from a massive planetesimal collision on a hyperbolic orbit destined to leave the Fomalhaut A system. Further 2022 observations with the James Webb Space Telescope in mid-infrared failed to resolve the object in the MIRI wideband filter wavelength range, reported by the same team to be consistent with the previous result.
The same 2022 JWST imaging data discovered another apparent feature in the outer disk, dubbed the "Great Dust Cloud". However, another team's analysis, which included other existing data, preferred its interpretation as a coincident background object, not part of the outer ring. Another 2023 study detected 10 point sources around Fomalhaut; all but one of these are background objects, including the "Great Dust Cloud", but the nature of the last is unclear. It may be a background object, or a planetary companion to Fomalhaut.
|-
| Outer hot disk
| colspan="4"| 0.21–0.62 AU or 0.88–1.08 AU
| —
| —
Fomalhaut B (TW Piscis Austrini) | Fomalhaut | Wikipedia | 420 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
Fomalhaut forms a binary star with the K4-type star TW Piscis Austrini (TW PsA), which lies away from Fomalhaut, and its space velocity agrees with that of Fomalhaut within , consistent with being a bound companion. A recent age estimate for TW PsA () agrees very well with the isochronal age for Fomalhaut (), further arguing for the two stars forming a physical binary.
The designation TW Piscis Austrini is astronomical nomenclature for a variable star. Fomalhaut B is a flare star of the type known as a BY Draconis variable. It varies slightly in apparent magnitude, ranging from 6.44 to 6.49 over a 10.3 day period. While smaller than the Sun, it is relatively large for a flare star. Most flare stars are red M-type dwarfs.
In 2019, a team of researchers analyzing the astrometry, radial velocity measurements, and images of Fomalhaut B suggested the existence of a planet orbiting the star with a mass of Jupiter masses, and a poorly defined orbital period with an estimate loosely centering around 25 years.
Fomalhaut C (LP 876-10) | Fomalhaut | Wikipedia | 263 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
LP 876-10 is also associated with the Fomalhaut system, making it a trinary star. In October 2013, Eric Mamajek and collaborators from the RECONS consortium announced that the previously known high-proper-motion star LP 876-10 had a distance, velocity, and color-magnitude position consistent with being another member of the Fomalhaut system. LP 876-10 was originally catalogued as a high-proper-motion star by Willem Luyten in his 1979 NLTT catalogue; however, a precise trigonometric parallax and radial velocity was only measured quite recently. LP 876-10 is a red dwarf of spectral type M4V, and located even farther from Fomalhaut A than TW PsA—about 5.7° away from Fomalhaut A in the sky, in the neighbouring constellation Aquarius, whereas both Fomalhaut A and TW PsA are located in constellation Piscis Austrinus. Its current separation from Fomalhaut A is about , and it is currently located away from TW PsA (Fomalhaut B). LP 876-10 is located well within the tidal radius of the Fomalhaut system, which is . Although LP 876-10 is itself catalogued as a binary star in the Washington Double Star Catalog (called "WSI 138"), there was no sign of a close-in stellar companion in the imaging, spectral, or astrometric data in the Mamajek et al. study. In December 2013, Kennedy et al. reported the discovery of a cold dusty debris disk associated with Fomalhaut C, using infrared images from the Herschel Space Observatory. Multiple-star systems hosting multiple debris disks are exceedingly rare. | Fomalhaut | Wikipedia | 375 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
Etymology and cultural significance
Fomalhaut has had various names ascribed to it through time, and has been recognized by many cultures of the northern hemisphere, including the Arabs, Persians, and Chinese. It marked the solstice in 2500 BC. It was also a marker for the worship of Demeter in Eleusis.
It is considered to be one of the four "royal stars" of the Persians.
The Latin names are "the mouth of the Southern Fish".
A folk name among the early Arabs was Difdi' al Awwal ( ) "the first frog" (the second frog is Beta Ceti).
The Chinese name (Mandarin: Běiluòshīmén), meaning North Gate of the Military Camp, because this star is marking itself and stands alone in North Gate of the Military Camp asterism, Encampment mansion (see: Chinese constellations). (Běiluòshīmén), westernized into Pi Lo Sze Mun by R.H. Allen.
To the Moporr Aboriginal people of South Australia, it is a male being called Buunjill. The Wardaman people of the Northern Territory called Fomalhaut Menggen —white cockatoo.
Fomalhaut-Earthwork B, in Mounds State Park near Anderson, Indiana, lines up with the rising of the star Fomalhaut in the fall months, according to the Indiana Department of Natural Resources. In 1980, astronomer Jack Robinson proposed that the rising azimuth of Fomalhaut was marked by cairn placements at both the Bighorn medicine wheel in Wyoming, USA, and the Moose Mountain medicine wheel in Saskatchewan, Canada.
New Scientist magazine termed it the "Great Eye of Sauron", comparing its shape and debris ring to the aforementioned "eye" in the Peter Jackson Lord of the Rings films.
USS Fomalhaut (AK-22) was a United States navy amphibious cargo ship. | Fomalhaut | Wikipedia | 398 | 151651 | https://en.wikipedia.org/wiki/Fomalhaut | Physical sciences | Notable stars | Astronomy |
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
One way of stating the approximation involves the logarithm of the factorial:
where the big O notation means that, for all sufficiently large values of , the difference between and will be at most proportional to the logarithm of . In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form
The error term in either base can be expressed more precisely as , corresponding to an approximate formula for the factorial itself,
Here the sign means that the two quantities are asymptotic, that is, their ratio tends to 1 as tends to infinity.
Derivation
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
with an integral:
The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating , one considers its natural logarithm, as this is a slowly varying function:
The right-hand side of this equation minus
is the approximation by the trapezoid rule of the integral
and the error in this approximation is given by the Euler–Maclaurin formula:
where is a Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that
Denote this limit as . Because the remainder in the Euler–Maclaurin formula satisfies
where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:
Taking the exponential of both sides and choosing any positive integer , one obtains a formula involving an unknown quantity . For , the formula is
The quantity can be found by taking the limit on both sides as tends to infinity and using Wallis' product, which shows that . Therefore, one obtains Stirling's formula:
Alternative derivations
An alternative formula for using the gamma function is
(as can be seen by repeated integration by parts). Rewriting and changing variables , one obtains
Applying Laplace's method one has
which recovers Stirling's formula: | Stirling's approximation | Wikipedia | 485 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null |
Higher orders
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that , so we "peel off" this dominant term, then perform two changes of variables, to obtain:To verify this: .
Now the function is unimodal, with maximum value zero. Locally around zero, it looks like , which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by . This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us . Now plug back to the equation to obtainnotice how we don't need to actually find , since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in , which can be obtained programmatically.
Thus we get Stirling's formula to two orders:
Complex-analytic version
A complex-analysis version of this method is to consider as a Taylor coefficient of the exponential function , computed by Cauchy's integral formula as
This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius . The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.
Using the Central Limit Theorem and the Poisson distribution
An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem.
Since the Poisson distribution with parameter converges to a normal distribution with mean and variance , their density functions will be approximately the same:
Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to:
Taking logs then results in:
which can easily be rearranged to give:
Evaluating at gives the usual, more precise form of Stirling's approximation.
Speed of convergence and error estimates
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series): | Stirling's approximation | Wikipedia | 428 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null |
An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the On-Line Encyclopedia of Integer Sequences as and . The first graph in this section shows the relative error vs. , for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients:which shows that it grows superexponentially, and that by the ratio test the radius of convergence is zero.
As , the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion. It is not a convergent series; for any particular value of there are only so many terms of the series that improve accuracy, after which accuracy worsens. This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let be the Stirling series to terms evaluated at . The graphs show
which, when small, is essentially the relative error.
Writing Stirling's series in the form
it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term.
Other bounds, due to Robbins, valid for all positive integers are
This upper bound corresponds to stopping the above series for after the term. The lower bound is weaker than that obtained by stopping the series after the term. A looser version of this bound is that for all .
Stirling's formula for the gamma function
For all positive integers,
where denotes the gamma function.
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If , then
Repeated integration by parts gives
where is the th Bernoulli number (note that the limit of the sum as is not convergent, so this formula is just an asymptotic expansion). The formula is valid for large enough in absolute value, when , where is positive, with an error term of . The corresponding approximation may now be written:
where the expansion is identical to that of Stirling's series above for , except that is replaced with .
A further application of this asymptotic expansion is for complex argument with constant . See for example the Stirling formula applied in of the Riemann–Siegel theta function on the straight line . | Stirling's approximation | Wikipedia | 502 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null |
Error bounds
For any positive integer , the following notation is introduced:
and
Then
For further information and other error bounds, see the cited papers.
A convergent version of Stirling's formula
Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series. Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula:
One way to do this is by means of a convergent series of inverted rising factorials. If
then
where
where denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's series
which converges when .
Stirling's formula may also be given in convergent form as
where
Versions suitable for calculators
The approximation
and its equivalent form
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function. This approximation is good to more than 8 decimal digits for with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:
or equivalently,
An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is
for . The equivalent approximation for has an asymptotic error of and is given by
The approximation may be made precise by giving paired upper and lower bounds; one such inequality is
History
The formula was first discovered by Abraham de Moivre in the form
De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely . | Stirling's approximation | Wikipedia | 386 | 151783 | https://en.wikipedia.org/wiki/Stirling%27s%20approximation | Mathematics | Specific functions | null |
In medicine, a side effect is an effect of the use of a medicinal drug or other treatment, usually adverse but sometimes beneficial, that is unintended. Herbal and traditional medicines also have side effects.
A drug or procedure usually used for a specific effect may be used specifically because of a beneficial side-effect; this is termed "off-label use" until such use is approved. For instance, X-rays have long been used as an imaging technique; the discovery of their oncolytic capability led to their use in radiotherapy for ablation of malignant tumours.
Frequency of side effects
The World Health Organization and other health organisations characterise the probability of experiencing side effects as:
Very common, ≥ 1⁄10
Common (frequent), 1⁄10 to 1⁄100
Uncommon (infrequent), 1⁄100 to 1⁄1000
Rare, 1⁄1000 to 1⁄10000
Very rare, < 1⁄10000
The European Commission recommends that the list should contain only effects where there is "at least a reasonable possibility" that they are caused by the drug and the frequency "should represent crude incidence rates (and not differences or relative risks calculated against placebo or other comparator)". The frequency describes how often symptoms appear after taking the drug, without assuming that they were necessarily caused by the drug. Both healthcare providers and lay people misinterpret the frequency of side effects as describing the increase in frequency caused by the drug.
Examples of therapeutic side effects | Side effect | Wikipedia | 305 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health |
Most drugs and procedures have a multitude of reported adverse side effects; the information leaflets provided with virtually all drugs list possible side effects. Beneficial side effects are less common; some examples, in many cases of side-effects that ultimately gained regulatory approval as intended effects, are:
Bevacizumab (Avastin), used to slow the growth of blood vessels, has been used against dry age-related macular degeneration, as well as macular edema from diseases such as diabetic retinopathy and central retinal vein occlusion.
Buprenorphine has been shown experimentally (1982–1995) to be effective against severe, refractory depression.
Bupropion (Wellbutrin), an anti-depressant, also helps smoking cessation; this indication was later approved, and the name of the drug as sold for smoking cessation is Zyban. Bupropion branded as Zyban may be sold at a higher price than as Wellbutrin, so some physicians prescribe Wellbutrin for smoking cessation.
Carbamazepine is an approved treatment for bipolar disorder and epileptic seizures, but it has side effects useful in treating attention-deficit hyperactivity disorder (ADHD), schizophrenia, phantom limb syndrome, paroxysmal extreme pain disorder, neuromyotonia, and post-traumatic stress disorder.
Dexamethasone and betamethasone in premature labor, to enhance pulmonary maturation of the fetus.
Doxepin has been used to treat angioedema and severe allergic reactions due to its strong antihistamine properties.
Gabapentin, approved for treatment of seizures and postherpetic neuralgia in adults, has side effects which are useful in treating bipolar disorder, essential tremor, hot flashes, migraine prophylaxis, neuropathic pain syndromes, phantom limb syndrome, and restless leg syndrome.
Hydroxyzine, an antihistamine, is also used as an anxiolytic.
Magnesium sulfate in obstetrics for premature labor and preeclampsia.
Methotrexate (MTX), approved for the treatment of choriocarcinoma, is frequently used for the medical treatment of an unruptured ectopic pregnancy.
The SSRI medication sertraline is approved as an antidepressant but delays sexual climax in men, and can be used to treat premature ejaculation. | Side effect | Wikipedia | 512 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health |
Sildenafil was originally intended for pulmonary hypertension; subsequently, it was discovered that it also produces erections, for which it was later approved.
Terazosin, an α1-adrenergic antagonist approved to treat benign prostatic hyperplasia (enlarged prostate) and hypertension, is (one of several drugs) used off-label to treat drug induced diaphoresis and hyperhidrosis (excessive sweating).
Thalidomide, a drug sold over the counter from 1957 to 1961 as a tranquiliser and treatment for morning sickness of pregnancy, became notorious for causing tens of thousands of babies to be born without limbs and with other conditions, or stillborn. The drug, though still subject to other adverse side-effects, is now used to treat cancers and skin disorders, and is on the World Health Organization's List of Essential Medicines. | Side effect | Wikipedia | 178 | 151828 | https://en.wikipedia.org/wiki/Side%20effect | Biology and health sciences | General concepts_2 | Health |
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region".
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to Green's theorem.
Explanation using liquid flow
Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Consider an imaginary closed surface S inside a body of liquid, enclosing a volume of liquid. The flux of liquid out of the volume at any time is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface.
Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no sources or sinks inside the volume then the flux of liquid out of S is zero. If the liquid is moving, it may flow into the volume at some points on the surface S and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the net flux of liquid out of the volume is zero. | Divergence theorem | Wikipedia | 432 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
However if a source of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface S. The flux outward through S equals the volume rate of flow of fluid into S from the pipe. Similarly if there is a sink or drain inside S, such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The volume rate of flow of liquid inward through the surface S equals the rate of liquid removed by the sink.
If there are multiple sources and sinks of liquid inside S, the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks. The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the divergence of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by S equals the volume rate of flux through S. This is the divergence theorem.
The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Mathematical statement
Suppose is a subset of (in the case of represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then:
The left side is a volume integral over the volume , and the right side is the surface integral over the boundary of the volume . The closed, measurable set is oriented by outward-pointing normals, and is the outward pointing unit normal at almost each point on the boundary . ( may be used as a shorthand for .) In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume , and the right-hand side represents the total flow across the boundary . | Divergence theorem | Wikipedia | 462 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
Informal derivation
The divergence theorem follows from the fact that if a volume is partitioned into separate parts, the flux out of the original volume is equal to the algebraic sum of the flux out of each component volume. This is true despite the fact that the new subvolumes have surfaces that were not part of the original volume's surface, because these surfaces are just partitions between two of the subvolumes and the flux through them just passes from one volume to the other and so cancels out when the flux out of the subvolumes is summed.
See the diagram. A closed, bounded volume is divided into two volumes and by a surface (green). The flux out of each component region is equal to the sum of the flux through its two faces, so the sum of the flux out of the two parts is
where and are the flux out of surfaces and , is the flux through out of volume 1, and is the flux through out of volume 2. The point is that surface is part of the surface of both volumes. The "outward" direction of the normal vector is opposite for each volume, so the flux out of one through is equal to the negative of the flux out of the other so these two fluxes cancel in the sum.
Therefore:
Since the union of surfaces and is
This principle applies to a volume divided into any number of parts, as shown in the diagram. Since the integral over each internal partition (green surfaces) appears with opposite signs in the flux of the two adjacent volumes they cancel out, and the only contribution to the flux is the integral over the external surfaces (grey). Since the external surfaces of all the component volumes equal the original surface.
The flux out of each volume is the surface integral of the vector field over the surface
The goal is to divide the original volume into infinitely many infinitesimal volumes. As the volume is divided into smaller and smaller parts, the surface integral on the right, the flux out of each subvolume, approaches zero because the surface area approaches zero. However, from the definition of divergence, the ratio of flux to volume, , the part in parentheses below, does not in general vanish but approaches the divergence as the volume approaches zero.
As long as the vector field has continuous derivatives, the sum above holds even in the limit when the volume is divided into infinitely small increments
As approaches zero volume, it becomes the infinitesimal , the part in parentheses becomes the divergence, and the sum becomes a volume integral over | Divergence theorem | Wikipedia | 512 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
Since this derivation is coordinate free, it shows that the divergence does not depend on the coordinates used.
Proofs
For bounded open subsets of Euclidean space
We are going to prove the following:
Proof of Theorem.
For compact Riemannian manifolds with boundary
We are going to prove the following:
Proof of Theorem.
We use the Einstein summation convention. By using a partition of unity, we may assume that and have compact support in a coordinate patch . First consider the case where the patch is disjoint from . Then is identified with an open subset of and integration by parts produces no boundary terms:
In the last equality we used the Voss-Weyl coordinate formula for the divergence, although the preceding identity could be used to define as the formal adjoint of . Now suppose intersects . Then is identified with an open set in . We zero extend and to and perform integration by parts to obtain
where .
By a variant of the straightening theorem for vector fields, we may choose so that is the inward unit normal at . In this case is the volume element on and the above formula reads
This completes the proof.
Corollaries
By replacing in the divergence theorem with specific forms, other useful identities can be derived (cf. vector identities).
With for a scalar function and a vector field ,
A special case of this is , in which case the theorem is the basis for Green's identities.
With for two vector fields and , where denotes a cross product,
With for two vector fields and , where denotes a dot product,
With for a scalar function and vector field c:
The last term on the right vanishes for constant or any divergence free (solenoidal) vector field, e.g. Incompressible flows without sources or sinks such as phase change or chemical reactions etc. In particular, taking to be constant:
With for vector field and constant vector c:
By reordering the triple product on the right hand side and taking out the constant vector of the integral,
Hence,
Example
Suppose we wish to evaluate
where is the unit sphere defined by
and is the vector field
The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
where is the unit ball:
Since the function is positive in one hemisphere of and negative in the other, in an equal and opposite way, its total integral over is zero. The same is true for :
Therefore, | Divergence theorem | Wikipedia | 508 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
because the unit ball has volume .
Applications
Differential and integral forms of physical laws
As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
Continuity equations
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamics, electromagnetism, quantum mechanics, relativity theory, and a number of other fields, there are continuity equations that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of sources or sinks of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
Inverse-square laws
Any inverse-square law can instead be written in a Gauss's law-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss's law-type equation from the inverse-square formulation or vice versa is exactly the same in both cases; see either of those articles for details.
History
Joseph-Louis Lagrange introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his Mécanique Analytique. Lagrange employed surface integrals in his work on fluid mechanics. He discovered the divergence theorem in 1762. | Divergence theorem | Wikipedia | 428 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem. He proved additional special cases in 1833 and 1839. But it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow. Special cases were proven by George Green in 1828 in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Siméon Denis Poisson in 1824 in a paper on elasticity, and Frédéric Sarrus in 1828 in his work on floating bodies.
Worked examples
Example 1
To verify the planar variant of the divergence theorem for a region :
and the vector field:
The boundary of is the unit circle, , that can be represented parametrically by:
such that where units is the length arc from the point to the point on . Then a vector equation of is
At a point on :
Therefore,
Because , we can evaluate and because . Thus
Example 2
Let's say we wanted to evaluate the flux of the following vector field defined by bounded by the following inequalities:
By the divergence theorem,
We now need to determine the divergence of . If is a three-dimensional vector field, then the divergence of is given by .
Thus, we can set up the following flux integral
as follows:
Now that we have set up the integral, we can evaluate it.
Generalizations
Multiple dimensions
One can use the generalised Stokes' theorem to equate the -dimensional volume integral of the divergence of a vector field over a region to the -dimensional surface integral of over the boundary of :
This equation is also known as the divergence theorem.
When , this is equivalent to Green's theorem.
When , it reduces to the fundamental theorem of calculus, part 2.
Tensor fields
Writing the theorem in Einstein notation:
suggestively, replacing the vector field with a rank- tensor field , this can be generalized to:
where on each side, tensor contraction occurs for at least one index. This form of the theorem is still in 3d, each index takes values 1, 2, and 3. It can be generalized further still to higher (or lower) dimensions (for example to 4d spacetime in general relativity). | Divergence theorem | Wikipedia | 470 | 151864 | https://en.wikipedia.org/wiki/Divergence%20theorem | Mathematics | Multivariable and vector calculus | null |
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
Gradient:
Divergence:
Curl:
Definition
In the Cartesian coordinate system with coordinates and standard basis , del is a vector operator whose components are the partial derivative operators ; that is,
Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system with coordinates and standard basis or unit vectors of axes , del is written as
As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.
More specifically, for any scalar field and any vector field , if one defines
then using the above definition of , one may write
and
and
Example:
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates. | Del | Wikipedia | 473 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null |
Notational uses
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.
Gradient
The vector derivative of a scalar field is called the gradient, and it can be represented as:
It always points in the direction of greatest increase of , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane , the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
However, the rules for dot products do not turn out to be simple, as illustrated by:
Divergence
The divergence of a vector field
is a scalar field that can be represented as:
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.
The power of the del notation is shown by the following product rule:
The formula for the vector product is slightly less intuitive, because this product is not commutative:
Curl
The curl of a vector field is a vector function that can be represented as:
The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.
The vector product operation can be visualized as a pseudo-determinant:
Again the power of the notation is shown by the product rule:
The rule for the vector product does not turn out to be simple:
Directional derivative
The directional derivative of a scalar field in the direction
is defined as:
Which is equal to the following when the gradient exists
This gives the rate of change of a field in the direction of , scaled by the magnitude of . In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid. | Del | Wikipedia | 481 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null |
Note that is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
Laplacian
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
and the definition for more general coordinate systems is given in vector Laplacian.
The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.
Hessian matrix
While usually represents the Laplacian, sometimes also represents the Hessian matrix. The former refers to the inner product of , while the latter refers to the dyadic product of :
.
So whether refers to a Laplacian or a Hessian matrix depends on the context.
Tensor derivative
Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as , where represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.
For a small displacement , the change in the vector field is given by:
Product rules
For vector calculus:
For matrix calculus (for which can be written ):
Another relation of interest (see e.g. Euler equations) is the following, where is the outer product tensor:
Second derivatives
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved ( in most cases), two of them are always zero:
Two of them are always equal: | Del | Wikipedia | 508 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null |
The 3 remaining vector derivatives are related by the equation:
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
Precautions
Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.
Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.
A counterexample that demonstrates the divergence () and the advection operator () are not commutative:
A counterexample that relies on del's differential properties:
Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.
For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule. | Del | Wikipedia | 255 | 151925 | https://en.wikipedia.org/wiki/Del | Mathematics | Calculus and analysis | null |
Vaccinium vitis-idaea is a small evergreen shrub in the heath family, Ericaceae. It is known colloquially as the lingonberry, partridgeberry, foxberry, mountain cranberry, or cowberry. It is native to boreal forest and Arctic tundra throughout the Northern Hemisphere. Commercially cultivated in the United States Pacific Northwest and the Netherlands, the edible berries are also picked in the wild and used in various dishes, especially in Nordic cuisine.
Description
Vaccinium vitis-idaea spreads by underground stems to form dense clonal colonies. Slender and brittle roots grow from the underground stems. The stems are rounded in cross-section and grow from in height. Leaves grow alternately and are oval, long, with a slightly wavy margin, and sometimes with a notched tip.
The flowers are bell-shaped, white to pale pink, long. V. vitis-idaea begins to produce flowers from five to ten years of age. They are pollinated by multiple insect species, including Andrena lapponica and several species of bumblebee.
The fruit is a red berry across, with an acidic taste, ripening in late summer to autumn. While bitter early in the season, they sweeten if left on the branch through winter. Cytology is 2n = 24.
Related species
Vaccinium vitis-idaea differs from the related cranberries in having white flowers with petals partially enclosing the stamens and stigma, rather than pink flowers with petals reflexed backwards, and rounder, less pear-shaped berries. Vaccinium oxycoccos is similar.
Hybrids between Vaccinium vitis-idaea and V. myrtillus, named Vaccinium × intermedium Ruthe, are occasionally found in Europe.
Taxonomy
Varieties
There are two regional varieties or subspecies of V. vitis-idaea, one in Eurasia and one in North America, differing in leaf size:
V. vitis-idaea var. vitis-idaea L.—syn. V. vitis-idaea subsp. vitis-idaea.cowberry. Eurasia. Leaves are long.
V. vitis-idaea var. minus Lodd.—syn. V. vitis-idaea subsp. minus (Lodd.) Hultén.lingonberry. North America. Leaves are long. | Vaccinium vitis-idaea | Wikipedia | 485 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants |
Etymology
Vaccinium vitis-idaea is most commonly known in English as 'lingonberry' or 'cowberry'. The name 'lingonberry' originates from the Swedish name () for the species deriving from Old Norse lyngr, a cognate (thus also a doublet) to 'ling'.
The genus name Vaccinium is a classical Latin name for a plant, possibly the bilberry or hyacinth, and may be derived from the Latin , 'berry'. The specific name is derived from Latin ('vine') and , the feminine form of (literally 'from Mount Ida', used in reference to raspberries Rubus idaeus).
Worldwide, Vaccinium vitis-idaea is known by at least 25 other common English names, including:
bearberry
beaverberry
cougarberry
foxberry
lowbush cranberry
mountain bilberry
mountain cranberry
partridgeberry (in Newfoundland and Cape Breton Island)
quailberry
red whortleberry
redberry (in Labrador and the Lower North Shore of Quebec)
Distribution and habitat
It is native to boreal forest and Arctic tundra throughout the Northern Hemisphere, including Eurasia and North America.
Ecology
Vaccinium vitis-idaea keeps its leaves all winter even in the coldest years, unusual for a broad-leaved plant, though in its natural habitat it is usually protected from severe cold by snow cover. It is extremely hardy, tolerating temperatures as low as −50 °F (−45 °C) or lower, but grows poorly where summers are hot. It prefers some shade (as from a forest canopy) and constantly moist, acidic soil. Nutrient-poor soils are tolerated but not alkaline soils.
Conservation
The plant is endangered in Michigan. The minus subspecies is listed as a species of special concern and believed extirpated in Connecticut.
Cultivation
Lingonberry has been commercially cultivated in the Netherlands and other countries since the 1960s.
Some cultivars are grown for their ornamental rather than culinary value. In the United Kingdom, the Koralle Group has gained the Royal Horticultural Society's Award of Garden Merit.
Uses
Culinary
Raw lingonberries are 86% water, 13% carbohydrates, 1% protein, and contain negligible fat. In a reference amount, lingonberries supply 54 kcal, and are low-to-moderate sources of vitamin C, B vitamins, and dietary minerals. | Vaccinium vitis-idaea | Wikipedia | 503 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants |
The berries collected in the wild are a popular fruit in northern, central and eastern Europe, notably in the Nordic countries, the Baltic states, central and northern Europe. In some areas, they can be picked legally on both public and private lands in accordance with the freedom to roam.
The berries are quite tart, so they are often cooked and sweetened before eating in the form of lingonberry jam, compote, juice, smoothie or syrup. The raw fruits are also frequently simply mashed with sugar, which preserves most of their nutrients and taste. This mix can be stored at room temperature in closed but not necessarily sealed containers, but in this condition, they are best preserved frozen. Fruit served this way or as compote often accompanies game and liver dishes.
In Sweden the traditional Swedish meatballs are served with lingonberry jam alongside boiled or mashed potatoes and gravy sauce. In Sweden, Finland and Norway, reindeer and elk steaks are traditionally served with gravy and lingonberry sauce. Preserved fruit is commonly eaten with meatballs, as well as potato pancakes. A traditional Swedish dessert is (literally 'lingonberry pears'), consisting of fresh pears which are peeled, boiled and preserved in (lingonberry juice) and is commonly eaten during Christmas. This was very common in old times, because it was an easy and tasty way to preserve pears. In Sweden and Russia, when sugar was still a luxury item, the berries were usually preserved simply by putting them whole into bottles of water. This was known as (watered lingonberries); the procedure preserved them until next season. This was also a home remedy against scurvy.
This traditional Russian soft drink, known as "lingonberry water", is mentioned by Alexander Pushkin in Eugene Onegin. In Russian folk medicine, lingonberry water was used as a mild laxative. A traditional Finnish dish is sautéed reindeer () with mashed potatoes and lingonberries on the side, either raw, thawed or as a jam. In Finland, whipped semolina pudding flavored with lingonberry () is also popular. In Poland, the berries are often mixed with pears to create a sauce served with poultry or game. The berries can also be used to replace redcurrants when creating Cumberland sauce. | Vaccinium vitis-idaea | Wikipedia | 481 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants |
The berries are also popular as a wild picked fruit in Eastern Canada, for example in Newfoundland and Labrador and Cape Breton, where they are locally known as partridgeberries or redberries, and on the mainland of Nova Scotia, where they are known as foxberries. In this region they are incorporated into jams, syrups, and baked goods, such as pies, scones, and muffins.
In Sweden lingonberries are often sold as jam and juice, and as a key ingredient in dishes. They are used to make Lillehammer berry liqueur; and, in East European countries, lingonberry vodka is sold, and vodka with lingonberry juice or mors is a cocktail.
The berries are an important food for bears and foxes, and many fruit-eating birds. Caterpillars of the case-bearer moths Coleophora glitzella, Coleophora idaeella and Coleophora vitisella are obligate feeders on V. vitis-idaea leaves.
Indigenous North American cuisine
Alaska natives mix the berries with rose hip pulp and sugar to make jam, cook the berries as a sauce, and store the berries for future use. The Dakelh use the berries to make jam. The Koyukon freeze the berries for winter use. Inuit dilute and sweeten the juice to make a beverage, freeze and store the berries for spring, and use the berries to make jams and jellies. The Iñupiat use the berries to make two different desserts, one in which the berries are whipped with frozen fish eggs and eaten, and one in which raw berries are mashed with canned milk and seal oil. They also make a dish of the berries cooked with fish eggs, fish (whitefish, sheefish or pike) and blubber.
The Upper Tanana boil the berries with sugar and flour to thicken; eat the raw berries, either plain or mixed with sugar, grease or a combination of the two; fry them in grease with sugar or dried fish eggs; or make them into pies, jam, and jelly. They also preserve the berries alone or in grease and store them in a birchbark basket in an underground cache, or freeze them. | Vaccinium vitis-idaea | Wikipedia | 455 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants |
Use of the minus subspecies
The Anticosti people use the fruit to make jams and jellies. The Nihithawak Cree store the berries by freezing them outside during the winter, mix the berries with boiled fish eggs, livers, air bladders and fat and eat them, eat the berries raw as a snack food, or stew them with fish or meat. The Iñupiat of Nelson Island eat the berries, as do the Iñupiat of the Northern Bering Sea and Arctic regions of Alaska, as well as the Inuvialuit. The Haida people, Hesquiaht First Nation, Wuikinuxv and Tsimshian all use the berries as food.
Traditional medicine
In traditional medicine, V. vitis-idaea was used as an apéritif and astringent. The Upper Tanana ate the berries or used their juice to treat minor respiratory disorders.
Other uses
The Nihithawak Cree use the berries of the minus subspecies to color porcupine quills, and put the firm, ripe berries on a string to wear as a necklace. The Western Canadian Inuit use the minus subspecies as a tobacco additive or substitute.
Explanatory notes | Vaccinium vitis-idaea | Wikipedia | 248 | 151942 | https://en.wikipedia.org/wiki/Vaccinium%20vitis-idaea | Biology and health sciences | Berries | Plants |
Vernalization () is the induction of a plant's flowering process by exposure to the prolonged cold of winter, or by an artificial equivalent. After vernalization, plants have acquired the ability to flower, but they may require additional seasonal cues or weeks of growth before they will actually do so. The term is sometimes used to refer to the need of herbal (non-woody) plants for a period of cold dormancy in order to produce new shoots and leaves, but this usage is discouraged.
Many plants grown in temperate climates require vernalization and must experience a period of low winter temperature to initiate or accelerate the flowering process. This ensures that reproductive development and seed production occurs in spring and winters, rather than in autumn. The needed cold is often expressed in chill hours. Typical vernalization temperatures are between 1 and 7 degrees Celsius (34 and 45 degrees Fahrenheit).
For many perennial plants, such as fruit tree species, a period of cold is needed first to induce dormancy and then later, after the requisite period, re-emerge from that dormancy prior to flowering. Many monocarpic winter annuals and biennials, including some ecotypes of Arabidopsis thaliana and winter cereals such as wheat, must go through a prolonged period of cold before flowering occurs.
History of vernalization research
In the history of agriculture, farmers observed a traditional distinction between "winter cereals", whose seeds require chilling (to trigger their subsequent emergence and growth), and "spring cereals", whose seeds can be sown in spring, and germinate, and then flower soon thereafter. Scientists in the early 19th century had discussed how some plants needed cold temperatures to flower. In 1857 an American agriculturist John Hancock Klippart, Secretary of the Ohio Board of Agriculture, reported the importance and effect of winter temperature on the germination of wheat. One of the most significant works was by a German plant physiologist Gustav Gassner who made a detailed discussion in his 1918 paper. Gassner was the first to systematically differentiate the specific requirements of winter plants from those of summer plants, and also that early swollen germinating seeds of winter cereals are sensitive to cold. | Vernalization | Wikipedia | 456 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology |
In 1928, the Soviet agronomist Trofim Lysenko published his works on the effects of cold on cereal seeds, and coined the term "яровизация" (yarovizatsiya : "jarovization") to describe a chilling process he used to make the seeds of winter cereals behave like spring cereals (from яровой : yarvoy, Tatar root ярый : yaryiy meaning ardent, fiery, associated with the god of spring). Lysenko himself translated the term into "vernalization" (from the Latin vernum meaning Spring). After Lysenko the term was used to explain the ability of flowering in some plants after a period of chilling due to physiological changes and external factors. The formal definition was given in 1960 by a French botanist P. Chouard, as "the acquisition or acceleration of the ability to flower by a chilling treatment."
Lysenko's 1928 paper on vernalization and plant physiology drew wide attention due to its practical consequences for Russian agriculture. Severe cold and lack of winter snow had destroyed many early winter wheat seedlings. By treating wheat seeds with moisture as well as cold, Lysenko induced them to bear a crop when planted in spring. Later however, according to Richard Amasino, Lysenko inaccurately asserted that the vernalized state could be inherited, i.e. the offspring of a vernalized plant would behave as if they themselves had also been vernalized and would not require vernalization in order to flower quickly. Opposing this view and supporting Lysenko's claim, Xiuju Li and Yongsheng Liu have detailed experimental evidence from the USSR, Hungary, Bulgaria and China that shows the conversion between spring wheat and winter wheat, positing that "it is not unreasonable to postulate epigenetic mechanisms that could plausibly result in the conversion of spring to winter wheat or vice versa."
Early research on vernalization focused on plant physiology; the increasing availability of molecular biology has made it possible to unravel its underlying mechanisms. For example, a lengthening daylight period (longer days), as well as cold temperatures are required for winter wheat plants to go from the vegetative to the reproductive state; the three interacting genes are called VRN1, VRN2, and FT (VRN3).
In Arabidopsis thaliana | Vernalization | Wikipedia | 502 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology |
Arabidopsis thaliana ("thale cress") is a much-studied model for vernalization. Some ecotypes (varieties), called "winter annuals", have delayed flowering without vernalization; others ("summer annuals") do not. The genes that underlie this difference in plant physiology have been intensively studied.
The reproductive phase change of A. thaliana occurs by a sequence of two related events: first, the bolting transition (flower stalk elongates), then the floral transition (first flower appears). Bolting is a robust predictor of flower formation, and hence a good indicator for vernalization research.
In winter annual Arabidopsis, vernalization of the meristem appears to confer competence to respond to floral inductive signals. A vernalized meristem retains competence for as long as 300 days in the absence of an inductive signal.
At the molecular level, flowering is repressed by the protein Flowering Locus C (FLC), which binds to and represses genes that promote flowering, thus blocking flowering. Winter annual ecotypes of Arabidopsis have an active copy of the gene FRIGIDA (FRI), which promotes FLC expression, thus repression of flowering. Prolonged exposure to cold (vernalization) induces expression of VERNALIZATION INSENSTIVE3, which interacts with the VERNALIZATION2 (VRN2) polycomb-like complex to reduce FLC expression through chromatin remodeling. Levels of VRN2 protein increase during long-term cold exposure as a result of inhibition of VRN2 turnover via its N-degron. The events of histone deacetylation at Lysine 9 and 14 followed by methylation at Lys 9 and 27 is associated with the vernalization response. The epigenetic silencing of FLC by chromatin remodeling is also thought to involve the cold-induced expression of antisense FLC COOLAIR or COLDAIR transcripts. Vernalization is registered by the plant by the stable silencing of individual FLC loci. The removal of silent chromatin marks at FLC during embryogenesis prevents the inheritance of the vernalized state. | Vernalization | Wikipedia | 471 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology |
Since vernalization also occurs in flc mutants (lacking FLC), vernalization must also activate a non-FLC pathway. A day-length mechanism is also important. Vernalization response works in concert with the photo-periodic genes CO, FT, PHYA, CRY2 to induce flowering.
Devernalization
It is possible to devernalize a plant by
exposure to sometimes low and high temperatures subsequent to vernalization. For example, commercial onion growers store sets at low temperatures, but devernalize them before planting, because they want the plant's energy to go into enlarging its bulb (underground stem), not making flowers. | Vernalization | Wikipedia | 139 | 151965 | https://en.wikipedia.org/wiki/Vernalization | Biology and health sciences | Plant reproduction | Biology |
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .
Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing.
Intuition
Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least of them), identified with subsets of the set of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis.
In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model of the set theory, which is itself a set in the "real universe" . By the Löwenheim–Skolem theorem, can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in ) of that are not in . Specifically, there is an ordinal that "plays the role of the cardinal " in , but is actually countable in . Working in , it should be easy to find one distinct subset of per each element of . (For simplicity, this family of subsets can be characterized with a single subset .) | Forcing (mathematics) | Wikipedia | 379 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
However, in some sense, it may be desirable to "construct the expanded model within ". This would help ensure that "resembles" in certain aspects, such as being the same as (more generally, that cardinal collapse does not occur), and allow fine control over the properties of . More precisely, every member of should be given a (non-unique) name in . The name can be thought as an expression in terms of , just like in a simple field extension every element of can be expressed in terms of . A major component of forcing is manipulating those names within , so sometimes it may help to directly think of as "the universe", knowing that the theory of forcing guarantees that will correspond to an actual model.
A subtle point of forcing is that, if is taken to be an arbitrary "missing subset" of some set in , then the constructed "within " may not even be a model. This is because may encode "special" information about that is invisible within (e.g. the countability of ), and thus prove the existence of sets that are "too complex for to describe".
Forcing avoids such problems by requiring the newly introduced set to be a generic set relative to . Some statements are "forced" to hold for any generic : For example, a generic is "forced" to be infinite. Furthermore, any property (describable in ) of a generic set is "forced" to hold under some forcing condition. The concept of "forcing" can be defined within , and it gives enough reasoning power to prove that is indeed a model that satisfies the desired properties.
Cohen's original technique, now called ramified forcing, is slightly different from the unramified forcing expounded here. Forcing is also equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.
The role of the model
In order for the above approach to work smoothly, must in fact be a standard transitive model in , so that membership and other elementary notions can be handled intuitively in both and . A standard transitive model can be obtained from any standard model through the Mostowski collapse lemma, but the existence of any standard model of (or any variant thereof) is in itself a stronger assumption than the consistency of . | Forcing (mathematics) | Wikipedia | 473 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
To get around this issue, a standard technique is to let be a standard transitive model of an arbitrary finite subset of (any axiomatization of has at least one axiom schema, and thus an infinite number of axioms), the existence of which is guaranteed by the reflection principle. As the goal of a forcing argument is to prove consistency results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.
Forcing conditions and forcing posets
Each forcing condition can be regarded as a finite piece of information regarding the object adjoined to the model. There are many different ways of providing information about an object, which give rise to different forcing notions. A general approach to formalizing forcing notions is to regard forcing conditions as abstract objects with a poset structure.
A forcing poset is an ordered triple, , where is a preorder on , and is the largest element. Members of are the forcing conditions (or just conditions). The order relation means " is stronger than ". (Intuitively, the "smaller" condition provides "more" information, just as the smaller interval provides more information about the number than the interval does.) Furthermore, the preorder must be atomless, meaning that it must satisfy the splitting condition:
For each , there are such that , with no such that .
In other words, it must be possible to strengthen any forcing condition in at least two incompatible directions. Intuitively, this is because is only a finite piece of information, whereas an infinite piece of information is needed to determine .
There are various conventions in use. Some authors require to also be antisymmetric, so that the relation is a partial order. Some use the term partial order anyway, conflicting with standard terminology, while some use the term preorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by Saharon Shelah and his co-authors.
Examples
Let be any infinite set (such as ), and let the generic object in question be a new subset . In Cohen's original formulation of forcing, each forcing condition is a finite set of sentences, either of the form or , that are self-consistent (i.e. and for the same value of do not appear in the same condition). This forcing notion is usually called Cohen forcing. | Forcing (mathematics) | Wikipedia | 496 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
The forcing poset for Cohen forcing can be formally written as , the finite partial functions from to under reverse inclusion. Cohen forcing satisfies the splitting condition because given any condition , one can always find an element not mentioned in , and add either the sentence or to to get two new forcing conditions, incompatible with each other.
Another instructive example of a forcing poset is , where and is the collection of Borel subsets of having non-zero Lebesgue measure. The generic object associated with this forcing poset is a random real number . It can be shown that falls in every Borel subset of with measure 1, provided that the Borel subset is "described" in the original unexpanded universe (this can be formalized with the concept of Borel codes). Each forcing condition can be regarded as a random event with probability equal to its measure. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.
Generic filters
Even though each individual forcing condition cannot fully determine the generic object , the set of all true forcing conditions does determine . In fact, without loss of generality, is commonly considered to be the generic object adjoined to , so the expanded model is called . It is usually easy enough to show that the originally desired object is indeed in the model .
Under this convention, the concept of "generic object" can be described in a general way. Specifically, the set should be a generic filter on relative to . The "filter" condition means that it makes sense that is a set of all true forcing conditions:
if , then
if , then there exists an such that
For to be "generic relative to " means:
If is a "dense" subset of (that is, for each , there exists a such that ), then .
Given that is a countable model, the existence of a generic filter follows from the Rasiowa–Sikorski lemma. In fact, slightly more is true: Given a condition , one can find a generic filter such that . Due to the splitting condition on , if is a filter, then is dense. If , then because is a model of . For this reason, a generic filter is never in .
P-names and interpretations
Associated with a forcing poset is the class of -names. A -name is a set of the form
Given any filter on , the interpretation or valuation map from -names is given by | Forcing (mathematics) | Wikipedia | 500 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
The -names are, in fact, an expansion of the universe. Given , one defines to be the -name
Since , it follows that . In a sense, is a "name for " that does not depend on the specific choice of .
This also allows defining a "name for " without explicitly referring to :
so that .
Rigorous definitions
The concepts of -names, interpretations, and may be defined by transfinite recursion. With the empty set, the successor ordinal to ordinal , the power-set operator, and a limit ordinal, define the following hierarchy:
Then the class of -names is defined as
The interpretation map and the map can similarly be defined with a hierarchical construction.
Forcing
Given a generic filter , one proceeds as follows. The subclass of -names in is denoted . Let
To reduce the study of the set theory of to that of , one works with the "forcing language", which is built up like ordinary first-order logic, with membership as the binary relation and all the -names as constants.
Define (to be read as " forces in the model with poset "), where is a condition, is a formula in the forcing language, and the 's are -names, to mean that if is a generic filter containing , then . The special case is often written as "" or simply "". Such statements are true in , no matter what is.
What is important is that this external definition of the forcing relation is equivalent to an internal definition within , defined by transfinite induction (specifically -induction) over the -names on instances of and , and then by ordinary induction over the complexity of formulae. This has the effect that all the properties of are really properties of , and the verification of in becomes straightforward. This is usually summarized as the following three key properties:
Truth: if and only if it is forced by , that is, for some condition , we have .
Definability: The statement "" is definable in .
Coherence: .
Internal definition
There are many different but equivalent ways to define the forcing relation in . One way to simplify the definition is to first define a modified forcing relation that is strictly stronger than . The modified relation still satisfies the three key properties of forcing, but and are not necessarily equivalent even if the first-order formulae and are equivalent. The unmodified forcing relation can then be defined as
In fact, Cohen's original concept of forcing is essentially rather than . | Forcing (mathematics) | Wikipedia | 511 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
The modified forcing relation can be defined recursively as follows:
means
means
means
means
means
Other symbols of the forcing language can be defined in terms of these symbols: For example, means , means , etc. Cases 1 and 2 depend on each other and on case 3, but the recursion always refer to -names with lesser ranks, so transfinite induction allows the definition to go through.
By construction, (and thus ) automatically satisfies Definability. The proof that also satisfies Truth and Coherence is by inductively inspecting each of the five cases above. Cases 4 and 5 are trivial (thanks to the choice of and as the elementary symbols), cases 1 and 2 relies only on the assumption that is a filter, and only case 3 requires to be a generic filter.
Formally, an internal definition of the forcing relation (such as the one presented above) is actually a transformation of an arbitrary formula to another formula where and are additional variables. The model does not explicitly appear in the transformation (note that within , just means " is a -name"), and indeed one may take this transformation as a "syntactic" definition of the forcing relation in the universe of all sets regardless of any countable transitive model. However, if one wants to force over some countable transitive model , then the latter formula should be interpreted under (i.e. with all quantifiers ranging only over ), in which case it is equivalent to the external "semantic" definition of described at the top of this section:
For any formula there is a theorem of the theory (for example conjunction of finite number of axioms) such that for any countable transitive model such that and any atomless partial order and any -generic filter over
This the sense under which the forcing relation is indeed "definable in ".
Consistency
The discussion above can be summarized by the fundamental consistency result that, given a forcing poset , we may assume the existence of a generic filter , not belonging to the universe , such that is again a set-theoretic universe that models . Furthermore, all truths in may be reduced to truths in involving the forcing relation. | Forcing (mathematics) | Wikipedia | 445 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
Both styles, adjoining to either a countable transitive model or the whole universe , are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.
Cohen forcing
The simplest nontrivial forcing poset is , the finite partial functions from to under reverse inclusion. That is, a condition is essentially two disjoint finite subsets and of , to be thought of as the "yes" and "no" parts of with no information provided on values outside the domain of . " is stronger than " means that , in other words, the "yes" and "no" parts of are supersets of the "yes" and "no" parts of , and in that sense, provide more information.
Let be a generic filter for this poset. If and are both in , then is a condition because is a filter. This means that is a well-defined partial function from to because any two conditions in agree on their common domain.
In fact, is a total function. Given , let . Then is dense. (Given any , if is not in 's domain, adjoin a value for —the result is in .) A condition has in its domain, and since , we find that is defined.
Let , the set of all "yes" members of the generic conditions. It is possible to give a name for directly. Let
Then Now suppose that in . We claim that . Let
Then is dense. (Given any , find that is not in its domain, and adjoin a value for contrary to the status of "".) Then any witnesses . To summarize, is a "new" subset of , necessarily infinite.
Replacing with , that is, consider instead finite partial functions whose inputs are of the form , with and , and whose outputs are or , one gets new subsets of . They are all distinct, by a density argument: Given , let
then each is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the th new set. | Forcing (mathematics) | Wikipedia | 455 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map onto , or onto . For example, if one considers instead , finite partial functions from to , the first uncountable ordinal, one gets in a bijection from to . In other words, has collapsed, and in the forcing extension, is a countable ordinal.
The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals. For this, a sufficient combinatorial property is that all of the antichains of the forcing poset are countable.
The countable chain condition
An (strong) antichain of is a subset such that if and , then and are incompatible (written ), meaning there is no in such that and . In the example on Borel sets, incompatibility means that has zero measure. In the example on finite partial functions, incompatibility means that is not a function, in other words, and assign different values to some domain input.
satisfies the countable chain condition (c.c.c.) if and only if every antichain in is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)
It is easy to see that satisfies the c.c.c. because the measures add up to at most . Also, satisfies the c.c.c., but the proof is more difficult.
Given an uncountable subfamily , shrink to an uncountable subfamily of sets of size , for some . If for uncountably many , shrink this to an uncountable subfamily and repeat, getting a finite set and an uncountable family of incompatible conditions of size such that every is in for at most countable many . Now, pick an arbitrary , and pick from any that is not one of the countably many members that have a domain member in common with . Then and are compatible, so is not an antichain. In other words, -antichains are countable. | Forcing (mathematics) | Wikipedia | 463 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal antichain is one that cannot be extended to a larger antichain. This means that every element is compatible with some member of . The existence of a maximal antichain follows from Zorn's Lemma. Given a maximal antichain , let
Then is dense, and if and only if . Conversely, given a dense set , Zorn's Lemma shows that there exists a maximal antichain , and then if and only if .
Assume that satisfies the c.c.c. Given , with a function in , one can approximate inside as follows. Let be a name for (by the definition of ) and let be a condition that forces to be a function from to . Define a function , by
By the definability of forcing, this definition makes sense within . By the coherence of forcing, a different come from an incompatible . By c.c.c., is countable.
In summary, is unknown in as it depends on , but it is not wildly unknown for a c.c.c.-forcing. One can identify a countable set of guesses for what the value of is at any input, independent of .
This has the following very important consequence. If in , is a surjection from one infinite ordinal onto another, then there is a surjection in , and consequently, a surjection in . In particular, cardinals cannot collapse. The conclusion is that in .
Easton forcing
The exact value of the continuum in the above Cohen model, and variants like for cardinals in general, was worked out by Robert M. Solovay, who also worked out how to violate (the generalized continuum hypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if holds in , then holds in .
William B. Easton worked out the proper class version of violating the for regular cardinals, basically showing that the known restrictions, (monotonicity, Cantor's Theorem and König's Theorem), were the only -provable restrictions (see Easton's Theorem). | Forcing (mathematics) | Wikipedia | 454 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of . For example, forcing with , where is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing with introduces a countable enumeration of the ordinals. In both cases, the resulting is visibly not a model of .
At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of singular cardinals. However, this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in and with the forcing models depending on the consistency of various large-cardinal properties. Many open problems remain.
Random reals
Random forcing can be defined as forcing over the set of all compact subsets of of positive measure ordered by relation (smaller set in context of inclusion is smaller set in ordering and represents condition with more information). There are two types of important dense sets:
For any positive integer the set is dense, where is diameter of the set .
For any Borel subset of measure 1, the set is dense.
For any filter and for any finitely many elements there is such that holds . In case of this ordering, this means that any filter is set of compact sets with finite intersection property. For this reason, intersection of all elements of any filter is nonempty. If is a filter intersecting the dense set for any positive integer , then the filter contains conditions of arbitrarily small positive diameter. Therefore, the intersection of all conditions from has diameter 0. But the only nonempty sets of diameter 0 are singletons. So there is exactly one real number such that .
Let be any Borel set of measure 1. If intersects , then . | Forcing (mathematics) | Wikipedia | 379 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
However, a generic filter over a countable transitive model is not in . The real defined by is provably not an element of . The problem is that if , then " is compact", but from the viewpoint of some larger universe , can be non-compact and the intersection of all conditions from the generic filter is actually empty. For this reason, we consider the set of topological closures of conditions from G (i.e., ). Because of and the finite intersection property of , the set also has the finite intersection property. Elements of the set are bounded closed sets as closures of bounded sets. Therefore,
is a set of compact sets with the finite intersection property and thus has nonempty intersection. Since and the ground model inherits a metric from the universe , the set has elements of arbitrarily small diameter. Finally, there is exactly one real that belongs to all members of the set . The generic filter can be reconstructed from as .
If is name of , and for holds " is Borel set of measure 1", then holds
for some . There is name such that for any generic filter holds
Then
holds for any condition .
Every Borel set can, non-uniquely, be built up, starting from intervals with rational endpoints and applying the operations of complement and countable unions, a countable number of times. The record of such a construction is called a Borel code. Given a Borel set in , one recovers a Borel code, and then applies the same construction sequence in , getting a Borel set . It can be proven that one gets the same set independent of the construction of , and that basic properties are preserved. For example, if , then . If has measure zero, then has measure zero. This mapping is injective.
For any set such that and " is a Borel set of measure 1" holds .
This means that is "infinite random sequence of 0s and 1s" from the viewpoint of , which means that it satisfies all statistical tests from the ground model .
So given , a random real, one can show that
Because of the mutual inter-definability between and , one generally writes for . | Forcing (mathematics) | Wikipedia | 446 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
A different interpretation of reals in was provided by Dana Scott. Rational numbers in have names that correspond to countably-many distinct rational values assigned to a maximal antichain of Borel sets – in other words, a certain rational-valued function on . Real numbers in then correspond to Dedekind cuts of such functions, that is, measurable functions.
Boolean-valued models
Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement is assigned a truth value from some complete atomless Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model that contains this ultrafilter, which can be understood as a new model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get a model that has the desired property. In it, only statements that must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).
Meta-mathematical explanation
In forcing, we usually seek to show that some sentence is consistent with (or optionally some extension of ). One way to interpret the argument is to assume that is consistent and then prove that combined with the new sentence is also consistent.
Each "condition" is a finite piece of information – the idea is that only finite pieces are relevant for consistency, since, by the compactness theorem, a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then we can pick an infinite set of consistent conditions to extend our model. Therefore, assuming the consistency of , we prove the consistency of extended by this infinite set.
Logical explanation
By Gödel's second incompleteness theorem, one cannot prove the consistency of any sufficiently strong formal theory, such as , using only the axioms of the theory itself, unless the theory is inconsistent. Consequently, mathematicians do not attempt to prove the consistency of using only the axioms of , or to prove that is consistent for any hypothesis using only . For this reason, the aim of a consistency proof is to prove the consistency of relative to the consistency of . Such problems are known as problems of relative consistency, one of which proves | Forcing (mathematics) | Wikipedia | 496 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:
For any given proof, can verify the validity of this proof. This is provable by induction on the length of the proof.
Then resolve
By proving the following
it can be concluded that
which is equivalent to
which gives (*). The core of the relative consistency proof is proving (**). A proof of can be constructed for any given finite subset of the axioms (by instruments of course). (No universal proof of of course.)
In , it is provable that for any condition , the set of formulas (evaluated by names) forced by is deductively closed. Furthermore, for any axiom, proves that this axiom is forced by . Then it suffices to prove that there is at least one condition that forces .
In the case of Boolean-valued forcing, the procedure is similar: proving that the Boolean value of is not .
Another approach uses the Reflection Theorem. For any given finite set of axioms, there is a proof that this set of axioms has a countable transitive model. For any given finite set of axioms, there is a finite set of axioms such that proves that if a countable transitive model satisfies , then satisfies . By proving that there is finite set of axioms such that if a countable transitive model satisfies , then satisfies the hypothesis . Then, for any given finite set of axioms, proves .
Sometimes in (**), a stronger theory than is used for proving . Then we have proof of the consistency of relative to the consistency of . Note that , where is (the axiom of constructibility). | Forcing (mathematics) | Wikipedia | 363 | 152205 | https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | Mathematics | Set theory | null |
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.
The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.
The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.
History
Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.
Applications
The compactness theorem has many applications in model theory; a few typical results are sketched here.
Robinson's principle
The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.
Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant such that the sentence holds for every field of characteristic larger than This can be seen as follows: suppose is a sentence that holds in every field of characteristic zero. Then its negation together with the field axioms and the infinite sequence of sentences | Compactness theorem | Wikipedia | 387 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null |
is not satisfiable (because there is no field of characteristic 0 in which holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset of these sentences that is not satisfiable. must contain because otherwise it would be satisfiable. Because adding more sentences to does not change unsatisfiability, we can assume that contains the field axioms and, for some the first sentences of the form Let contain all the sentences of except Then any field with a characteristic greater than is a model of and together with is not satisfiable. This means that must hold in every model of which means precisely that holds in every field of characteristic greater than This completes the proof.
The Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence in the language of rings is true in (or equivalently, in ) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes for which is true in algebraically closed field of characteristic in which case is true in algebraically closed fields of sufficiently large non-0 characteristic
One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials are surjective (indeed, it can even be shown that its inverse will also be a polynomial). In fact, the surjectivity conclusion remains true for any injective polynomial where is a finite field or the algebraic closure of such a field.
Upward Löwenheim–Skolem theorem | Compactness theorem | Wikipedia | 335 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null |
A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'. To achieve this, let be the initial theory and let be any cardinal number. Add to the language of one constant symbol for every element of Then add to a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of sentences). Since every subset of this new theory is satisfiable by a sufficiently large finite model of or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least .
Non-standard analysis
A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol to the language and adjoining to the axiom and the axioms for all positive integers Clearly, the standard real numbers are a model for every finite subset of these axioms, because the real numbers satisfy everything in and, by suitable choice of can be made to satisfy any finite subset of the axioms about By the compactness theorem, there is a model that satisfies and also contains an infinitesimal element
A similar argument, this time adjoining the axioms etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization of the reals.
It can be shown that the hyperreal numbers satisfy the transfer principle: a first-order sentence is true of if and only if it is true of
Proofs | Compactness theorem | Wikipedia | 417 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null |
One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.
Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found; that is, proofs that refer to but not to . One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:
Proof:
Fix a first-order language and let be a collection of -sentences such that every finite subcollection of -sentences, of it has a model Also let be the direct product of the structures and be the collection of finite subsets of For each let
The family of all of these sets generates a proper filter, so there is an ultrafilter containing all sets of the form
Now for any sentence in
the set is in
whenever then hence holds in
the set of all with the property that holds in is a superset of hence also in
Łoś's theorem now implies that holds in the ultraproduct So this ultraproduct satisfies all formulas in | Compactness theorem | Wikipedia | 305 | 152207 | https://en.wikipedia.org/wiki/Compactness%20theorem | Mathematics | Model theory | null |
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets and there is a new set containing exactly and . Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). | Zermelo–Fraenkel set theory | Wikipedia | 465 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
History
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes.
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number aleph-omega () and the set where is any infinite set and is the power set operation. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
Formal language | Zermelo–Fraenkel set theory | Wikipedia | 407 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
Formally, ZFC is a one-sorted theory in first-order logic. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The signature has a single predicate symbol, usually denoted , which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a set membership relation. For example, the formula means that is an element of the set (also read as is a member of ).
There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as functional completeness. This section attempts to strike a balance between simplicity and intuitiveness.
The language's alphabet consists of:
A countably infinite amount of variables used for representing sets
The logical connectives , ,
The quantifier symbols ,
The equality symbol
The set membership symbol
Brackets ( )
With this alphabet, the recursive rules for forming well-formed formulae (wff) are as follows:
Let and be metavariables for any variables. These are the two ways to build atomic formulae (the simplest wffs):
Let and be metavariables for any wff, and be a metavariable for any variable. These are valid wff constructions:
A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes and have exactly two child nodes, while nodes , and have exactly one. There are countably infinitely many wffs, however, each wff has a finite number of nodes.
Axioms
There are many equivalent formulations of the ZFC axioms. The following particular axiom set is from . The axioms in order below are expressed in a mixture of first order logic and high-level abbreviations.
Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following , we use the equivalent well-ordering theorem in place of the axiom of choice for axiom 9. | Zermelo–Fraenkel set theory | Wikipedia | 448 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, although he notes that he does so only "for emphasis". Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, . Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that some set exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic, in which it is not provable from logic alone that something exists, the axiom of infinity asserts that an infinite set exists. This implies that a set exists, and so, once again, it is superfluous to include an axiom asserting as much.
Axiom of extensionality
Two sets are equal (are the same set) if they have the same elements.
The converse of this axiom follows from the substitution property of equality. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which you are constructing set theory does not include equality "", may be defined as an abbreviation for the following formula:
In this case, the axiom of extensionality can be reformulated as
which says that if and have the same elements, then they belong to the same sets.
Axiom of regularity (also called the axiom of foundation)
Every non-empty set contains a member such that and are disjoint sets.
or in modern notation:
This (along with the axioms of pairing and union) implies, for example, that no set is an element of itself and that every set has an ordinal rank.
Axiom schema of specification (or of separation, or of restricted comprehension)
Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset of the integers satisfying the congruence modulo predicate : | Zermelo–Fraenkel set theory | Wikipedia | 505 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
In general, the subset of a set obeying a formula with one free variable may be written as:
The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one axiom for each ). Formally, let be any formula in the language of ZFC with all free variables among ( is not free in ). Then:
Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:
This restriction is necessary to avoid Russell's paradox (let then ) and its variants that accompany naive set theory with unrestricted comprehension (since under this restriction only refers to sets within that don't belong to themselves, and has not been established, even though is the case, so stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build a on the basis of a formula , we need to previously restrict the sets will regard within a set that leaves outside so can't refer to itself; or, in other words, sets shouldn't refer to themselves).
In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement and the axiom of the empty set.
On the other hand, the axiom schema of specification can be used to prove the existence of the empty set, denoted , once at least one set is known to exist. One way to do this is to use a property which no set has. For example, if is any existing set, the empty set can be constructed as
Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on ). It is common to make a definitional extension that adds the symbol "" to the language of ZFC.
Axiom of pairing
If and are sets, then there exists a set which contains and as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}} | Zermelo–Fraenkel set theory | Wikipedia | 465 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the axiom of infinity, or by the and the axiom of the power set applied twice to any set.
Axiom of union
The union over the elements of a set exists. For example, the union over the elements of the set is
The axiom of union states that for any set of sets , there is a set containing every element that is a member of some member of :
Although this formula doesn't directly assert the existence of , the set can be constructed from in the above using the axiom schema of specification:
Axiom schema of replacement
The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.
Formally, let be any formula in the language of ZFC whose free variables are among so that in particular is not free in . Then:
(The unique existential quantifier denotes the existence of exactly one element such that it follows a given statement.)
In other words, if the relation represents a definable function , represents its domain, and is a set for every then the range of is a subset of some set . The form stated here, in which may be larger than strictly necessary, is sometimes called the axiom schema of collection.
Axiom of infinity
Let abbreviate where is some set. (We can see that is a valid set by applying the axiom of pairing with so that the set is ). Then there exists a set such that the empty set , defined axiomatically, is a member of and, whenever a set is a member of then is also a member of .
or in modern notation:
More colloquially, there exists a set having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set satisfying the axiom of infinity is the von Neumann ordinal which can also be thought of as the set of natural numbers
Axiom of power set | Zermelo–Fraenkel set theory | Wikipedia | 500 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
By definition, a set is a subset of a set if and only if every element of is also an element of :
The Axiom of power set states that for any set , there is a set that contains every subset of :
The axiom schema of specification is then used to define the power set as the subset of such a containing the subsets of exactly:
Axioms 1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in . Some ZF axiomatizations include an axiom asserting that the empty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set whose existence is being asserted are just those sets which the axiom asserts must contain.
The following axiom is added to turn ZF into ZFC:
Axiom of well-ordering (choice)
The last axiom, commonly known as the axiom of choice, is presented here as a property about well-orders, as in .
For any set , there exists a binary relation which well-orders . This means is a linear order on such that every nonempty subset of has a least element under the order .
Given axioms 1 – 8, many statements are equivalent to axiom 9. The most common of these goes as follows. Let be a set whose members are all nonempty. Then there exists a function from to the union of the members of , called a "choice function", such that for all one has . A third version of the axiom, also equivalent, is Zorn's lemma.
Since the existence of a choice function when is a finite set is easily proved from axioms 1–8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed".
Motivation via the cumulative hierarchy | Zermelo–Fraenkel set theory | Wikipedia | 399 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann. In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
It is provable that a set is in V if and only if the set is pure and well-founded. And V satisfies all the axioms of ZFC if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then, every subset of x is also added at (or before) stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at (or before) stage α, and that the powerset of x will be added at the next stage after α.
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.
It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy, which gives the constructible universe L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional "axiom of constructibility".
Metamathematics | Zermelo–Fraenkel set theory | Wikipedia | 512 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
Virtual classes
Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC).
An alternative to proper classes while staying within ZF and ZFC is the virtual class notational construct introduced by , where the entire construct y ∈ { x | Fx } is simply defined as Fy. This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach of . Virtual classes are also used in , , and in the Metamath implementation of ZFC.
Finite axiomatization
The axiom schemata of replacement and separation each contain infinitely many instances. included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes proper classes as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem not mentioning classes and provable in one theory can be proved in the other. | Zermelo–Fraenkel set theory | Wikipedia | 307 | 152214 | https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | Mathematics | Axiomatic systems | null |
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