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The mineral olivine () is a magnesium iron silicate with the chemical formula . It is a type of nesosilicate or orthosilicate. The primary component of the Earth's upper mantle, it is a common mineral in Earth's subsurface, but weathers quickly on the surface. Olivine has many uses, such as the gemstone peridot (or chrysolite), as well as industrial applications like metalworking processes.
The ratio of magnesium to iron varies between the two endmembers of the solid solution series: forsterite (Mg-endmember: ) and fayalite (Fe-endmember: ). Compositions of olivine are commonly expressed as molar percentages of forsterite (Fo) and/or fayalite (Fa) (e.g., Fo70Fa30, or just Fo70 with Fa30 implied). Forsterite's melting temperature is unusually high at atmospheric pressure, almost , while fayalite's is much lower – about . Melting temperature varies smoothly between the two endmembers, as do other properties. Olivine incorporates only minor amounts of elements other than oxygen (O), silicon (Si), magnesium (Mg) and iron (Fe). Manganese (Mn) and nickel (Ni) commonly are the additional elements present in highest concentrations.
Olivine gives its name to the group of minerals with a related structure (the olivine group) – which includes tephroite (Mn2SiO4), monticellite (CaMgSiO4), larnite (Ca2SiO4) and kirschsteinite (CaFeSiO4) (commonly also spelled kirschteinite).
Olivine's crystal structure incorporates aspects of the orthorhombic P Bravais lattice, which arise from each silica (SiO4) unit being joined by metal divalent cations with each oxygen in SiO4 bound to three metal ions. It has a spinel-like structure similar to magnetite but uses one quadrivalent and two divalent cations M22+ M4+O4 instead of two trivalent and one divalent cations.
Identification and paragenesis
Olivine is named for its typically olive-green color, though it may alter to a reddish color from the oxidation of iron. | Olivine | Wikipedia | 497 | 45159 | https://en.wikipedia.org/wiki/Olivine | Physical sciences | Silicate minerals | Earth science |
Translucent olivine is sometimes used as a gemstone called peridot (péridot, the French word for olivine). It is also called chrysolite (or chrysolithe, from the Greek words for gold and stone), though this name is now rarely used in the English language. Some of the finest gem-quality olivine has been obtained from a body of mantle rocks on Zabargad Island in the Red Sea.
Olivine occurs in both mafic and ultramafic igneous rocks and as a primary mineral in certain metamorphic rocks. Mg-rich olivine crystallizes from magma that is rich in magnesium and low in silica. That magma crystallizes to mafic rocks such as gabbro and basalt. Ultramafic rocks usually contain substantial olivine, and those with an olivine content of over 40% are described as peridotites. Dunite has an olivine content of over 90% and is likely a cumulate formed by olivine crystallizing and settling out of magma or a vein mineral lining magma conduits. Olivine and high pressure structural variants constitute over 50% of the Earth's upper mantle, and olivine is one of the Earth's most common minerals by volume. The metamorphism of impure dolomite or other sedimentary rocks with high magnesium and low silica content also produces Mg-rich olivine, or forsterite.
Fe-rich olivine fayalite is relatively much less common, but it occurs in igneous rocks in small amounts in rare granites and rhyolites, and extremely Fe-rich olivine can exist stably with quartz and tridymite. In contrast, Mg-rich olivine does not occur stably with silica minerals, as it would react with them to form orthopyroxene (). | Olivine | Wikipedia | 398 | 45159 | https://en.wikipedia.org/wiki/Olivine | Physical sciences | Silicate minerals | Earth science |
Mg-rich olivine is stable to pressures equivalent to a depth of about within Earth. Because it is thought to be the most abundant mineral in Earth's mantle at shallower depths, the properties of olivine have a dominant influence upon the rheology of that part of Earth and hence upon the solid flow that drives plate tectonics. Experiments have documented that olivine at high pressures (12 GPa, the pressure at depths of about ) can contain at least as much as about 8900 parts per million (weight) of water, and that such water content drastically reduces the resistance of olivine to solid flow. Moreover, because olivine is so abundant, more water may be dissolved in olivine of the mantle than is contained in Earth's oceans.
Olivine pine forest (a plant community) is unique to Norway. It is rare and found on dry olivine ridges in the fjord districts of Sunnmøre and Nordfjord.
Extraterrestrial occurrences
Mg-rich olivine has also been discovered in meteorites, on the Moon and Mars, falling into infant stars, as well as on asteroid 25143 Itokawa. Such meteorites include chondrites, collections of debris from the early Solar System; and pallasites, mixes of iron-nickel and olivine. The rare A-type asteroids are suspected to have a surface dominated by olivine.
The spectral signature of olivine has been seen in the dust disks around young stars. The tails of comets (which formed from the dust disk around the young Sun) often have the spectral signature of olivine, and the presence of olivine was verified in samples of a comet from the Stardust spacecraft in 2006. Comet-like (magnesium-rich) olivine has also been detected in the planetesimal belt around the star Beta Pictoris.
Crystal structure
Minerals in the olivine group crystallize in the orthorhombic system (space group Pbnm) with isolated silicate tetrahedra, meaning that olivine is a nesosilicate. The structure can be described as a hexagonal, close-packed array of oxygen ions with half of the octahedral sites occupied with magnesium or iron ions and one-eighth of the tetrahedral sites occupied by silicon ions. | Olivine | Wikipedia | 484 | 45159 | https://en.wikipedia.org/wiki/Olivine | Physical sciences | Silicate minerals | Earth science |
There are three distinct oxygen sites (marked O1, O2 and O3 in figure 1), two distinct metal sites (M1 and M2) and only one distinct silicon site. O1, O2, M2 and Si all lie on mirror planes, while M1 exists on an inversion center. O3 lies in a general position.
High-pressure polymorphs
At the high temperatures and pressures found at depth within the Earth the olivine structure is no longer stable. Below depths of about olivine undergoes an exothermic phase transition to the sorosilicate, wadsleyite and, at about depth, wadsleyite transforms exothermically into ringwoodite, which has the spinel structure. At a depth of about , ringwoodite decomposes into silicate perovskite () and ferropericlase () in an endothermic reaction. These phase transitions lead to a discontinuous increase in the density of the Earth's mantle that can be observed by seismic methods. They are also thought to influence the dynamics of mantle convection in that the exothermic transitions reinforce flow across the phase boundary, whereas the endothermic reaction hampers it.
The pressure at which these phase transitions occur depends on temperature and iron content. At , the pure magnesium end member, forsterite, transforms to wadsleyite at and to ringwoodite at pressures above . Increasing the iron content decreases the pressure of the phase transition and narrows the wadsleyite stability field. At about 0.8 mole fraction fayalite, olivine transforms directly to ringwoodite over the pressure range . Fayalite transforms to spinel at pressures below . Increasing the temperature increases the pressure of these phase transitions.
Weathering
Olivine is one of the less stable common minerals on the surface according to the Goldich dissolution series. It alters into iddingsite (a combination of clay minerals, iron oxides and ferrihydrite) readily in the presence of water. Artificially increasing the weathering rate of olivine, e.g. by dispersing fine-grained olivine on beaches, has been proposed as a cheap way to sequester CO2. The presence of iddingsite on Mars would suggest that liquid water once existed there, and might enable scientists to determine when there was last liquid water on the planet. | Olivine | Wikipedia | 498 | 45159 | https://en.wikipedia.org/wiki/Olivine | Physical sciences | Silicate minerals | Earth science |
Because of its rapid weathering, olivine is rarely found in sedimentary rock.
Mining
Norway
Norway is the main source of olivine in Europe, particularly in an area stretching from Åheim to Tafjord, and from Hornindal to Flemsøy in the Sunnmøre district. There is also olivine in Eid municipality. About 50% of the world's olivine for industrial use is produced in Norway. At Svarthammaren in Norddal olivine was mined from around 1920 to 1979, with a daily output up to 600 metric tons. Olivine was also obtained from the construction site of the hydro power stations in Tafjord. At Robbervika in Norddal municipality an open-pit mine has been in operation since 1984. The characteristic red color is reflected in several local names with "red" such as Raudbergvik (Red rock bay) or Raudnakken (Red ridge).
Hans Strøm in 1766 described the olivine's typical red color on the surface and the blue color within. Strøm wrote that in Norddal district large quantities of olivine were broken from the bedrock and used as sharpening stones.
Kallskaret near Tafjord is a nature reserve with olivine.
Applications
Olivine is used as a substitute for dolomite in steel works.
The aluminium foundry industry uses olivine sand to cast objects in aluminium. Olivine sand requires less water than silica sands while still holding the mold together during handling and pouring of the metal. Less water means less gas (steam) to vent from the mold as metal is poured into the mold.
In Finland, olivine is marketed as an ideal rock for sauna stoves because of its comparatively high density and resistance to weathering under repeated heating and cooling.
Gem-quality olivine is used as a gemstone called peridot.
Experimental uses
Removal of atmospheric CO2 via reaction with crushed olivine has been considered. The end-products of the very slow reaction are silicon dioxide, magnesium carbonate, and iron oxides. A nonprofit, Project Vesta, is investigating this approach on beaches which increase the agitation and surface area of crushed olivine through wave action. | Olivine | Wikipedia | 466 | 45159 | https://en.wikipedia.org/wiki/Olivine | Physical sciences | Silicate minerals | Earth science |
Peridot ( ), sometimes called chrysolite, is a yellow-green transparent variety of olivine. Peridot is one of the few gemstones that occur in only one color.
Peridot can be found in mafic and ultramafic rocks occurring in lava and peridotite xenoliths of the mantle. The gem occurs in silica-deficient rocks such as volcanic basalt and pallasitic meteorites. Along with diamonds, peridot is one of only two gems observed to be formed not in Earth's crust, but in the molten rock of the upper mantle. Gem-quality peridot is rare on Earth's surface due to its susceptibility to alteration during its movement from deep within the mantle and weathering at the surface. Peridot has a chemical formula of .
Peridot is one of the birthstones for the month of August.
Etymology
The origin of the name peridot is uncertain. The Oxford English Dictionary suggests an alteration of Anglo–Norman (classical Latin -), a kind of opal, rather than the Arabic word , meaning "gemstone".
The Middle English Dictionarys entry on peridot includes several variations: , , and — other variants substitute y for letter i used here.
The earliest use of the word in English is possibly in the 1705 register of the St. Albans Abbey: The dual entry is in Latin with the translation to English listed as peridot. It records that on his death in 1245, Bishop John bequeathed various items, including peridot gems, to the Abbey.
Appearance
Peridot is one of the few gemstones that occur in only one color: an olive-green. The intensity and tint of the green, however, depends on the percentage of iron in the crystal structure, so the color of individual peridot gems can vary from yellow, to olive, to brownish-green. In rare cases, peridot may have a medium-dark toned, pure green with no secondary yellow hue or brown mask. Lighter-colored gems are due to lower iron concentrations.
Mineral properties
Crystal structure | Peridot | Wikipedia | 436 | 45162 | https://en.wikipedia.org/wiki/Peridot | Physical sciences | Silicate minerals | Earth science |
The molecular structure of peridot consists of isomorphic olivine, silicate, magnesium and iron in an orthorhombic crystal system. In an alternative view, the atomic structure can be described as a hexagonal, close-packed array of oxygen ions with half of the octahedral sites occupied by magnesium or iron ions and one-eighth of the tetrahedral sites occupied by silicon ions.
Surface property
Oxidation of peridot does not occur at natural surface temperature and pressure but begins to occur slowly at with rates increasing with temperature. The oxidation of the olivine occurs by an initial breakdown of the fayalite component, and subsequent reaction with the forsterite component, to give magnetite and orthopyroxene.
Occurrence
Geologically
Olivine, of which peridot is a type, is a common mineral in mafic and ultramafic rocks, often found in lava and in peridotite xenoliths of the mantle, which lava carries to the surface; however, gem-quality peridot occurs in only a fraction of these settings. Peridots can also be found in meteorites.
Peridots can be differentiated by size and composition. A peridot formed as a result of volcanic activity tends to contain higher concentrations of lithium, nickel and zinc than those found in meteorites.
Olivine is an abundant mineral, but gem-quality peridot is rather rare due to its chemical instability on Earth's surface. Olivine is usually found as small grains and tends to exist in a heavily weathered state, unsuitable for decorative use. Large crystals of forsterite, the variety most often used to cut peridot gems, are rare; as a result, peridot is considered to be precious.
In the ancient world, the mining of peridot was called topazios then, on St. John's Island, in the Red Sea began about 300 .
The principal source of peridot olivine today is the San Carlos Apache Indian Reservation in Arizona.
It is also mined at another location in Arizona, and in Arkansas, Hawaii, Nevada, and New Mexico at Kilbourne Hole, in the US; and in Australia, Brazil, China, Egypt, Kenya, Mexico, Myanmar (Burma), Norway, Pakistan, Saudi Arabia, South Africa, Sri Lanka, and Tanzania.
In meteorites | Peridot | Wikipedia | 488 | 45162 | https://en.wikipedia.org/wiki/Peridot | Physical sciences | Silicate minerals | Earth science |
Peridot crystals have been collected from some pallasite meteorites. The most commonly studied pallasitic peridot belongs to the Indonesian Jeppara meteorite, but others exist such as the Brenham, Esquel, Fukang, and Imilac meteorites.
Pallasitic (extraterrestrial) peridot differs chemically from its earthbound counterpart, in that pallasitic peridot lacks nickel.
Gemology
Orthorhombic minerals, like peridot, have biaxial birefringence defined by three principal axes: , and . Refractive index readings of faceted gems can range around = 1.651, = 1.668, and = 1.689, with a biaxial positive birefringence of 0.037–0.038. With decreasing magnesium and increasing iron concentration, the specific gravity, color darkness and refractive indices increase, and the shifts toward the index. Increasing iron concentration ultimately forms the iron-rich end-member of the olivine solid solution series fayalite.
A study of Chinese peridot gem samples determined the hydro-static specific gravity to be 3.36 . The visible-light spectroscopy of the same Chinese peridot samples showed light bands between 493.0–481.0 nm, the strongest absorption at 492.0 nm.
The largest cut peridot olivine is a specimen in the gem collection of the Smithsonian Museum in Washington, D.C.
Inclusions are common in peridot crystals but their presence depends on the location where it was found and the geological conditions that led to its crystallization.
Primary negative crystals – rounded gas bubbles – form in situ with peridot, and are common in Hawaiian peridots.
Secondary negative crystals form in peridot fractures.
"Lily pad" cleavages are often seen in San Carlos peridots, and are a type of secondary negative crystal. They can easily be seen under reflected light as circular discs surrounding a negative crystal.
Silky and rod-like inclusions are common in Pakistani peridots.
The most common mineral inclusion in peridot is the chromium-rich mineral chromite.
Magnesium-rich minerals also can exist in the form of pyrope and magnesiochromite. These two types of mineral inclusions are typically surrounded "lily-pad" cleavages.
Biotite flakes appear flat, brown, translucent, and tabular. | Peridot | Wikipedia | 509 | 45162 | https://en.wikipedia.org/wiki/Peridot | Physical sciences | Silicate minerals | Earth science |
Cultural history
Peridot has been prized since the earliest civilizations for its claimed protective powers to drive away fears and nightmares, according to superstitions. There is a superstition that it carries the gift of "inner radiance", sharpening the mind and opening it to new levels of awareness and growth, helping one to recognize and realize one's destiny and spiritual purpose. (There is no scientific evidence for any such claims.)
Peridot olivine is the birthstone for the month of August.
Peridot has often been mistaken for emerald beryl and other green gems. Noted gemologist G.F. Kunz discussed the confusion between beryl and peridot in many church treasures, most notably the "Three Magi treasure" in the Dom of Cologne, Germany.
Gallery | Peridot | Wikipedia | 164 | 45162 | https://en.wikipedia.org/wiki/Peridot | Physical sciences | Silicate minerals | Earth science |
Orthoclase, or orthoclase feldspar (endmember formula KAlSi3O8), is an important tectosilicate mineral which forms igneous rock. The name is from the Ancient Greek for "straight fracture", because its two cleavage planes are at right angles to each other. It is a type of potassium feldspar, also known as K-feldspar. The gem known as moonstone (see below) is largely composed of orthoclase.
Formation and subtypes
Orthoclase is a common constituent of most granites and other felsic igneous rocks and often forms huge crystals and masses in pegmatite.
Typically, the pure potassium endmember of orthoclase forms a solid solution with albite, the sodium endmember (NaAlSi3O8), of plagioclase. While slowly cooling within the earth, sodium-rich albite lamellae form by exsolution, enriching the remaining orthoclase with potassium. The resulting intergrowth of the two feldspars is called perthite.
The higher-temperature polymorph of KAlSi3O8 is sanidine. Sanidine is common in rapidly cooled volcanic rocks such as obsidian and felsic pyroclastic rocks, and is notably found in trachytes of the Drachenfels, Germany. The lower-temperature polymorph of KAlSi3O8 is microcline.
Adularia is a low temperature form of either microcline or orthoclase originally reported from the low temperature hydrothermal deposits in the Adula Alps of Switzerland. It was first described by Ermenegildo Pini in 1781. The optical effect of adularescence in moonstone is typically due to adularia.
The largest documented single crystal of orthoclase was found in the Ural Mountains in Russia. It measured around and weighed around .
Applications
Together with the other potassium feldspars, orthoclase is a common raw material for the manufacture of some glasses and some ceramics such as porcelain, and as a constituent of scouring powder. | Orthoclase | Wikipedia | 453 | 45165 | https://en.wikipedia.org/wiki/Orthoclase | Physical sciences | Silicate minerals | Earth science |
Some intergrowths of orthoclase and albite have an attractive pale luster and are called moonstone when used in jewelry. Most moonstones are translucent and white, although grey and peach-colored varieties also occur. In gemology, their luster is called adularescence and is typically described as creamy or silvery white with a "billowy" quality. It is the state gem of Florida.
The gemstone commonly called rainbow moonstone is more properly a colorless form of labradorite and can be distinguished from "true" moonstone by its greater transparency and play of color, although their value and durability do not greatly differ.
Orthoclase is one of the ten defining minerals of the Mohs scale of mineral hardness, on which it is listed as having a hardness of 6.
NASA's Curiosity rover discovery of high levels of orthoclase in Martian sandstones suggested that some Martian rocks may have experienced complex geological processing, such as repeated melting. | Orthoclase | Wikipedia | 201 | 45165 | https://en.wikipedia.org/wiki/Orthoclase | Physical sciences | Silicate minerals | Earth science |
Microcline (KAlSi3O8) is an important igneous rock-forming tectosilicate mineral. It is a potassium-rich alkali feldspar. Microcline typically contains minor amounts of sodium. It is common in granite and pegmatites. Microcline forms during slow cooling of orthoclase; it is more stable at lower temperatures than orthoclase. Sanidine is a polymorph of alkali feldspar stable at yet higher temperature. Microcline may be clear, white, pale-yellow, brick-red, or green; it is generally characterized by cross-hatch twinning that forms as a result of the transformation of monoclinic orthoclase into triclinic microcline.
The chemical compound name is potassium aluminium silicate, and it is known as E number reference E555.
Geology
Microcline may be chemically the same as monoclinic orthoclase, but because it belongs to the triclinic crystal system, the prism angle is slightly less than right angles; hence the name "microcline" from the Greek "small slope". It is a fully ordered triclinic modification of potassium feldspar and is dimorphous with orthoclase. Microcline is identical to orthoclase in many physical properties, and can be distinguished by x-ray or optical examination. When viewed under a polarizing microscope, microcline exhibits a minute multiple twinning which forms a grating-like structure that is unmistakable.
Perthite is either microcline or orthoclase with thin lamellae of exsolved albite.
Amazon stone, or amazonite, is a green variety of microcline. It is not found anywhere in the Amazon Basin, however. The Spanish explorers who named it apparently confused it with another green mineral from that region.
The largest documented single crystals of microcline were found in Devil's Hole Beryl Mine, Colorado, US and measured ~50 × 36 × 14 m. This could be one of the largest crystals of any material found so far.
Microcline is exceptionally active ice-nucleating agent in the atmosphere. Recently it has been possible to understand how water binds to the microcline surface. | Microcline | Wikipedia | 477 | 45166 | https://en.wikipedia.org/wiki/Microcline | Physical sciences | Silicate minerals | Earth science |
As food additive
The chemical compound name is potassium aluminium silicate, and it is known as E number reference E555. It was the subject in 2018 of a Call for technical and toxicological data from the EFSA.
In 2008, it (along with other Aluminum compounds) was the subject of
a Scientific Opinion of the Panel on Food Additives, Flavourings, Processing Aids and Food Contact Materials from the EFSA. | Microcline | Wikipedia | 88 | 45166 | https://en.wikipedia.org/wiki/Microcline | Physical sciences | Silicate minerals | Earth science |
Plagioclase ( ) is a series of tectosilicate (framework silicate) minerals within the feldspar group. Rather than referring to a particular mineral with a specific chemical composition, plagioclase is a continuous solid solution series, more properly known as the plagioclase feldspar series. This was first shown by the German mineralogist Johann Friedrich Christian Hessel (1796–1872) in 1826. The series ranges from albite to anorthite endmembers (with respective compositions NaAlSi3O8 to CaAl2Si2O8), where sodium and calcium atoms can substitute for each other in the mineral's crystal lattice structure. Plagioclase in hand samples is often identified by its polysynthetic crystal twinning or "record-groove" effect.
Plagioclase is a major constituent mineral in Earth's crust and is consequently an important diagnostic tool in petrology for identifying the composition, origin and evolution of igneous rocks. Plagioclase is also a major constituent of rock in the highlands of the Moon. Analysis of thermal emission spectra from the surface of Mars suggests that plagioclase is the most abundant mineral in the crust of Mars.
Its name comes , in reference to its two cleavage angles.
Properties
Plagioclase is the most common and abundant mineral group in the Earth's crust. Part of the feldspar family of minerals, it is abundant in igneous and metamorphic rock, and it is also common as a detrital mineral in sedimentary rock. It is not a single mineral, but is a solid solution of two end members, albite or sodium feldspar () and anorthite or calcium feldspar (). These can be present in plagioclase in any proportion from pure anorthite to pure albite. The composition of plagioclase can thus be written as where x ranges from 0 for pure albite to 1 for pure anorthite. This solid solution series is known as the plagioclase series. The composition of a particular sample of plagioclase is customarily expressed as the mol% of anorthite in the sample. For example, plagioclase that is 40 mol% anorthite would be described as An40 plagioclase. | Plagioclase | Wikipedia | 482 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
The ability of albite and anorthite to form solid solutions in any proportions at elevated temperature reflects the ease with which calcium and aluminium can substitute for sodium and silicon in the plagioclase crystal structure. Although a calcium ion has a charge of +2, versus +1 for a sodium ion, the two ions have very nearly the same effective radius. The difference in charge is accommodated by the coupled substitution of aluminium (charge +3) for silicon (charge +4), both of which can occupy tetrahedral sites (surrounded by four oxygen ions). This contrasts with potassium, which has the same charge as sodium, but is a significantly larger ion. As a result of the size and charge difference between potassium and calcium, there is a very wide miscibility gap between anorthite and potassium feldspar, (), the third common rock-forming feldspar end member. Potassium feldspar does form a solid solution series with albite, due to the identical charges of sodium and potassium ions, which is known as the alkali feldspar series. Thus, almost all feldspar found on Earth is either plagioclase or alkali feldspar, with the two series overlapping for pure albite. When a plagioclase composition is described by its anorthite mol% (such as An40 in the previous example) it is assumed that the remainder is albite, with only a minor component of potassium feldspar.
Plagioclase of any composition shares many basic physical characteristics, while other characteristics vary smoothly with composition. The Mohs hardness of all plagioclase species is 6 to 6.5, and cleavage is perfect on [001] and good on [010], with the cleavage planes meeting at an angle of 93 to 94 degrees. It is from this slightly oblique cleavage angle that plagioclase gets its name, Ancient Greek ( 'oblique') + ( 'fracture'). The name was introduced by August Breithaupt in 1847. There is also a poor cleavage on [110] rarely seen in hand samples. | Plagioclase | Wikipedia | 433 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
The luster is vitreous to pearly and the diaphaneity is transparent to translucent. The tenacity is brittle, and the fracture is uneven or conchoidal, but the fracture is rarely observed due to the strong tendency of the mineral to cleave instead. At low temperature, the crystal structure belongs to the triclinic system, space group P Well-formed crystals are rare and are most commonly sodic in composition. Well-shaped samples are instead typically cleavage fragments. Well-formed crystals are typically bladed or tabular parallel to [010].
Plagioclase is usually white to greyish-white in color, with a slight tendency for more calcium-rich samples to be darker. Impurities can infrequently tint the mineral greenish, yellowish, or flesh-red. Ferric iron (Fe3+) gives a pale yellow color in plagioclase feldspar from Lake County, Oregon. The specific gravity increases smoothly with calcium content, from 2.62 for pure albite to 2.76 for pure anorthite, and this can provide a useful estimate of composition if measured accurately. The index of refraction likewise varies smoothly from 1.53 to 1.58, and, if measured carefully, this also gives a useful composition estimate.
Plagioclase almost universally shows a characteristic polysynthetic twinning that produces twinning striations on [010]. These striations allow plagioclase to be distinguished from alkali feldspar. Plagioclase often also displays Carlsbad, Baveno, and Manebach Law twinning.
Plagioclase series members
The composition of a plagioclase feldspar is typically denoted by its overall fraction of anorthite (%An) or albite (%Ab). There are several named plagioclase feldspars that fall between albite and anorthite in the series. The following table shows their compositions in terms of constituent anorthite and albite percentages. | Plagioclase | Wikipedia | 424 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
The distinction between these minerals cannot easily be made in the field. The composition can be roughly determined by specific gravity, but accurate measurement requires chemical or optical tests. The composition in a crushed grain mount can be obtained by the Tsuboi method, which yields an accurate measurement of the minimum refractive index that in turn gives an accurate composition. In thin section, the composition can be determined by either the Michel Lévy or Carlsbad-albite methods. The former relies on accurate measure of minimum index of refraction, while the latter relies on measuring the extinction angle under a polarizing microscope. The extinction angle is an optical characteristic and varies with the albite fraction (%Ab).
Endmembers
Anorthite was named by Gustav Rose in 1823 from Greek ('not') + ('straight'), literally 'oblique', referring to its triclinic crystallization. Anorthite is a comparatively rare mineral but occurs in the basic plutonic rocks of some orogenic calc-alkaline suites.
Albite is named from the Latin , in reference to its unusually pure white color. The name was first applied by Johan Gottlieb Gahn and Jöns Jacob Berzelius in 1815. It is a relatively common and important rock-making mineral associated with the more silica-rich rock types, in hydrothermal veins, with greenschist facies metamorphic rocks, and in pegmatite dikes, often as the variety cleavelandite and associated with rarer minerals like tourmaline and beryl.
Intermediate members
The intermediate members of the plagioclase group are very similar to each other and normally cannot be distinguished except by their optical properties. The specific gravity in each member (albite 2.62) increases 0.02 per 10% increase in anorthite (2.75).
Bytownite, named after the former name for Ottawa, Ontario, Canada—Bytown— is a rare mineral occasionally found in more basic rocks. | Plagioclase | Wikipedia | 412 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
Labradorite is the characteristic feldspar of the more basic rock types such as gabbro or basalt. Labradorite frequently shows an iridescent display of colors due to light refracting within the lamellae of the crystal. It is named after Labrador, where it is a constituent of the intrusive igneous rock anorthosite which is composed almost entirely of plagioclase. A variety of labradorite known as spectrolite is found in Finland.
Andesine is a characteristic mineral of rocks such as diorite which contain a moderate amount of silica and related volcanics such as andesite.
Oligoclase is common in granite and monzonite. The name oligoclase is derived from the Greek ('small, slight') + ('fracture'), in reference to the fact that its cleavage angle differs significantly from 90°. The term was first used by Breithaupt in 1826. Sunstone is mainly oligoclase (sometimes albite) with flakes of hematite.
Petrogenesis
Plagioclase is the primary aluminium-bearing mineral in mafic rocks formed at low pressure. It is normally the first and most abundant feldspar to crystallize from a cooling primitive magma. Anorthite has a much higher melting point than albite, and, as a result, calcium-rich plagioclase is the first to crystallize. The plagioclase becomes more enriched in sodium as the temperature drops, forming Bowen's continuous reaction series. However, the composition with which plagioclase crystallizes also depends on the other components of the melt, so it is not by itself a reliable thermometer. | Plagioclase | Wikipedia | 353 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
The liquidus of plagioclase (the temperature at which the plagioclase first begins to crystallize) is about for olivine basalt, with a composition of 50.5 wt% silica; in andesite with a silica content of 60.7 wt%; and in dacite with a silica content of 69.9 wt%. These values are for dry magma. The liquidus is greatly lowered by the addition of water, and much more for plagioclase than for mafic minerals. The eutectic (minimum melting mixture) for a mixture of anorthite and diopside shifts from 40 wt% anorthite to 78 wt% anorthite as the water vapor pressure goes from 1 bar to 10 kbar. The presence of water also shifts the composition of the crystallizing plagioclase towards anorthite. The eutectic for this wet mixture drops to about .
Crystallizing plagioclase is always richer in anorthite than the melt from which it crystallizes. This plagioclase effect causes the residual melt to be enriched in sodium and silicon and depleted in aluminium and calcium. However, the simultaneous crystallization of mafic minerals not containing aluminium can partially offset the depletion in aluminium. In volcanic rock, the crystallized plagioclase incorporates most of the potassium in the melt as a trace element.
New plagioclase crystals nucleate only with difficulty, and diffusion is very slow within the solid crystals. As a result, as a magma cools, increasingly sodium-rich plagioclase is usually crystallized onto the rims of existing plagioclase crystals, which retain their more calcium-rich cores. This results in compositional zoning of plagioclase in igneous rocks. In rare cases, plagioclase shows reverse zoning, with a more calcium-rich rim on a more sodium-rich core. Plagioclase also sometimes shows oscillatory zoning, with the zones fluctuating between sodium-rich and calcium-rich compositions, though this is usually superimposed on an overall normal zoning trend. | Plagioclase | Wikipedia | 446 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
Classification of igneous rocks
Plagioclase is very important for the classification of crystalline igneous rocks. Generally, the more silica is present in the rock, the fewer the mafic minerals, and the more sodium-rich the plagioclase. Alkali feldspar appears as the silica content becomes high. Under the QAPF classification, plagioclase is one of the three key minerals, along with quartz and alkali feldspar, used to make the initial classification of the rock type. Low-silica igneous rocks are further divided into dioritic rocks having sodium-rich plagioclase (An<50) and gabbroic rocks having calcium-rich plagioclase (An>50). Anorthosite is an intrusive rock composed of at least 90% plagioclase.
Albite is an end member of both the alkali and plagioclase series. However, it is included in the alkali feldspar fraction of the rock in the QAPF classification.
In metamorphic rocks
Plagioclase is also common in metamorphic rock. Plagioclase tends to be albite in low-grade metamorphic rock, while oligoclase to andesine are more common in medium- to high-grade metamorphic rock. Metacarbonate rock sometimes contains fairly pure anorthite.
In sedimentary rocks
Feldspar makes up between 10 and 20 percent of the framework grains in typical sandstones. Alkali feldspar is usually more abundant than plagioclase in sandstone because Alkali feldspars are more resistant to chemical weathering and more stable, but sandstone derived from volcanic rock contains more plagioclase. Plagioclase weathers relatively rapidly to clay minerals such as smectite.
At the Mohorovičić discontinuity
The Mohorovičić discontinuity, which defines the boundary between the Earth's crust and the upper mantle, is thought to be the depth where feldspar disappears from the rock. While plagioclase is the most important aluminium-bearing mineral in the crust, it breaks down at the high pressure of the upper mantle, with the aluminium tending to be incorporated into clinopyroxene as Tschermak's molecule () or in jadeite . At still higher pressure, the aluminium is incorporated into garnet.
Exsolution | Plagioclase | Wikipedia | 496 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
At very high temperatures, plagioclase forms a solid solution with potassium feldspar, but this becomes highly unstable on cooling. The plagioclase separates from the potassium feldspar, a process called exsolution. The resulting rock, in which fine streaks of plagioclase (lamellae) are present in potassium feldspar, is called perthite.
The solid solution between anorthite and albite remains stable to lower temperatures, but ultimately becomes unstable as the rock approaches ambient surface temperatures. The resulting exsolution results in very fine lamellar and other intergrowths, normally detected only by sophisticated means. However, exsolution in the andesine to labradorite compositional range sometimes produces lamellae with thicknesses comparable to the wavelength of visible light. This acts like a diffraction grating, causing the labradorite to show the beautiful play of colors known as chatoyance.
Uses
In addition to its importance to geologists in classifying igneous rocks, plagioclase finds practical use as construction aggregate, as dimension stone, and in powdered form as a filler in paint, plastics, and rubber. Sodium-rich plagioclase finds use in the manufacture of glass and ceramics.
Anorthosite could someday be important as a source of aluminium. | Plagioclase | Wikipedia | 268 | 45168 | https://en.wikipedia.org/wiki/Plagioclase | Physical sciences | Silicate minerals | Earth science |
In computing, a process is the instance of a computer program that is being executed by one or many threads. There are many different process models, some of which are light weight, but almost all processes (even entire virtual machines) are rooted in an operating system (OS) process which comprises the program code, assigned system resources, physical and logical access permissions, and data structures to initiate, control and coordinate execution activity. Depending on the OS, a process may be made up of multiple threads of execution that execute instructions concurrently.
While a computer program is a passive collection of instructions typically stored in a file on disk, a process is the execution of those instructions after being loaded from the disk into memory. Several processes may be associated with the same program; for example, opening up several instances of the same program often results in more than one process being executed.
Multitasking is a method to allow multiple processes to share processors (CPUs) and other system resources. Each CPU (core) executes a single process at a time. However, multitasking allows each processor to switch between tasks that are being executed without having to wait for each task to finish (preemption). Depending on the operating system implementation, switches could be performed when tasks initiate and wait for completion of input/output operations, when a task voluntarily yields the CPU, on hardware interrupts, and when the operating system scheduler decides that a process has expired its fair share of CPU time (e.g, by the Completely Fair Scheduler of the Linux kernel).
A common form of multitasking is provided by CPU's time-sharing that is a method for interleaving the execution of users' processes and threads, and even of independent kernel tasks – although the latter feature is feasible only in preemptive kernels such as Linux. Preemption has an important side effect for interactive processes that are given higher priority with respect to CPU bound processes, therefore users are immediately assigned computing resources at the simple pressing of a key or when moving a mouse. Furthermore, applications like video and music reproduction are given some kind of real-time priority, preempting any other lower priority process. In time-sharing systems, context switches are performed rapidly, which makes it seem like multiple processes are being executed simultaneously on the same processor. This seemingly-simultaneous execution of multiple processes is called concurrency.
For security and reliability, most modern operating systems prevent direct communication between independent processes, providing strictly mediated and controlled inter-process communication.
Representation | Process (computing) | Wikipedia | 507 | 45178 | https://en.wikipedia.org/wiki/Process%20%28computing%29 | Technology | Operating systems | null |
In general, a computer system process consists of (or is said to own) the following resources:
An image of the executable machine code associated with a program.
Memory (typically some region of virtual memory); which includes the executable code, process-specific data (input and output), a call stack (to keep track of active subroutines and/or other events), and a heap to hold intermediate computation data generated during run time.
Operating system descriptors of resources that are allocated to the process, such as file descriptors (Unix terminology) or handles (Windows), and data sources and sinks.
Security attributes, such as the process owner and the process' set of permissions (allowable operations).
Processor state (context), such as the content of registers and physical memory addressing. The state is typically stored in computer registers when the process is executing, and in memory otherwise.
The operating system holds most of this information about active processes in data structures called process control blocks. Any subset of the resources, typically at least the processor state, may be associated with each of the process' threads in operating systems that support threads or child processes.
The operating system keeps its processes separate and allocates the resources they need, so that they are less likely to interfere with each other and cause system failures (e.g., deadlock or thrashing). The operating system may also provide mechanisms for inter-process communication to enable processes to interact in safe and predictable ways.
Multitasking and process management
A multitasking operating system may just switch between processes to give the appearance of many processes executing simultaneously (that is, in parallel), though in fact only one process can be executing at any one time on a single CPU (unless the CPU has multiple cores, then multithreading or other similar technologies can be used).
It is usual to associate a single process with a main program, and child processes with any spin-off, parallel processes, which behave like asynchronous subroutines. A process is said to own resources, of which an image of its program (in memory) is one such resource. However, in multiprocessing systems many processes may run off of, or share, the same reentrant program at the same location in memory, but each process is said to own its own image of the program. | Process (computing) | Wikipedia | 488 | 45178 | https://en.wikipedia.org/wiki/Process%20%28computing%29 | Technology | Operating systems | null |
Processes are often called "tasks" in embedded operating systems. The sense of "process" (or task) is "something that takes up time", as opposed to "memory", which is "something that takes up space".
The above description applies to both processes managed by an operating system, and processes as defined by process calculi.
If a process requests something for which it must wait, it will be blocked. When the process is in the blocked state, it is eligible for swapping to disk, but this is transparent in a virtual memory system, where regions of a process's memory may be really on disk and not in main memory at any time. Even portions of active processes/tasks (executing programs) are eligible for swapping to disk, if the portions have not been used recently. Not all parts of an executing program and its data have to be in physical memory for the associated process to be active.
Process states
An operating system kernel that allows multitasking needs processes to have certain states. Names for these states are not standardised, but they have similar functionality.
First, the process is "created" by being loaded from a secondary storage device (hard disk drive, CD-ROM, etc.) into main memory. After that the process scheduler assigns it the "waiting" state.
While the process is "waiting", it waits for the scheduler to do a so-called context switch. The context switch loads the process into the processor and changes the state to "running" while the previously "running" process is stored in a "waiting" state.
If a process in the "running" state needs to wait for a resource (wait for user input or file to open, for example), it is assigned the "blocked" state. The process state is changed back to "waiting" when the process no longer needs to wait (in a blocked state).
Once the process finishes execution, or is terminated by the operating system, it is no longer needed. The process is removed instantly or is moved to the "terminated" state. When removed, it just waits to be removed from main memory.
Inter-process communication | Process (computing) | Wikipedia | 439 | 45178 | https://en.wikipedia.org/wiki/Process%20%28computing%29 | Technology | Operating systems | null |
When processes need to communicate with each other they must share parts of their address spaces or use other forms of inter-process communication (IPC).
For instance in a shell pipeline, the output of the first process needs to pass to the second one, and so on. Another example is a task that has been decomposed into cooperating but partially independent processes which can run simultaneously (i.e., using concurrency, or true parallelism – the latter model is a particular case of concurrent execution and is feasible whenever multiple CPU cores are available for the processes that are ready to run).
It is even possible for two or more processes to be running on different machines that may run different operating system (OS), therefore some mechanisms for communication and synchronization (called communications protocols for distributed computing) are needed (e.g., the Message Passing Interface {MPI}).
History
By the early 1960s, computer control software had evolved from monitor control software, for example IBSYS, to executive control software. Over time, computers got faster while computer time was still neither cheap nor fully utilized; such an environment made multiprogramming possible and necessary. Multiprogramming means that several programs run concurrently. At first, more than one program ran on a single processor, as a result of underlying uniprocessor computer architecture, and they shared scarce and limited hardware resources; consequently, the concurrency was of a serial nature. On later systems with multiple processors, multiple programs may run concurrently in parallel.
Programs consist of sequences of instructions for processors. A single processor can run only one instruction at a time: it is impossible to run more programs at the same time. A program might need some resource, such as an input device, which has a large delay, or a program might start some slow operation, such as sending output to a printer. This would lead to processor being "idle" (unused). To keep the processor busy at all times, the execution of such a program is halted and the operating system switches the processor to run another program. To the user, it will appear that the programs run at the same time (hence the term "parallel"). | Process (computing) | Wikipedia | 443 | 45178 | https://en.wikipedia.org/wiki/Process%20%28computing%29 | Technology | Operating systems | null |
Shortly thereafter, the notion of a "program" was expanded to the notion of an "executing program and its context". The concept of a process was born, which also became necessary with the invention of re-entrant code. Threads came somewhat later. However, with the advent of concepts such as time-sharing, computer networks, and multiple-CPU shared memory computers, the old "multiprogramming" gave way to true multitasking, multiprocessing and, later, multithreading. | Process (computing) | Wikipedia | 106 | 45178 | https://en.wikipedia.org/wiki/Process%20%28computing%29 | Technology | Operating systems | null |
{{DISPLAYTITLE:Lp space}}
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz .
spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.
Preliminaries
The -norm in finite dimensions
The Euclidean length of a vector in the -dimensional real vector space is given by the Euclidean norm:
The Euclidean distance between two points and is the length of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of -norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.
For a real number the -norm or -norm of is defined by
The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets.
The Euclidean norm from above falls into this class and is the -norm, and the -norm is the norm that corresponds to the rectilinear distance.
The -norm or maximum norm (or uniform norm) is the limit of the -norms for , given by: | Lp space | Wikipedia | 394 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
For all the -norms and maximum norm satisfy the properties of a "length function" (or norm), that is:
only the zero vector has zero length,
the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).
Abstractly speaking, this means that together with the -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space.
Relations between -norms
The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
This fact generalizes to -norms in that the -norm of any given vector does not grow with :
For the opposite direction, the following relation between the -norm and the -norm is known:
This inequality depends on the dimension of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.
In general, for vectors in where
This is a consequence of Hölder's inequality.
When
In for the formula
defines an absolutely homogeneous function for however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the formula
defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm, though, which is homogeneous of degree
Hence, the function
defines a metric. The metric space is denoted by
Although the -unit ball around the origin in this metric is "concave", the topology defined on by the metric is the usual vector space topology of hence is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of is to denote by the smallest constant such that the scalar multiple of the -unit ball contains the convex hull of which is equal to The fact that for fixed we have
shows that the infinite-dimensional sequence space defined below, is no longer locally convex.
When
There is one norm and another function called the "norm" (with quotation marks). | Lp space | Wikipedia | 483 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
The mathematical definition of the norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm on the product metric:
The -normed space is studied in functional analysis, probability theory, and harmonic analysis.
Another function was called the "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector Many authors abuse terminology by omitting the quotation marks. Defining the zero "norm" of is equal to
This is not a norm because it is not homogeneous. For example, scaling the vector by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.
spaces and sequence spaces
The -norm can be extended to vectors that have an infinite number of components (sequences), which yields the space This contains as special cases:
the space of sequences whose series are absolutely convergent,
the space of square-summable sequences, which is a Hilbert space, and
the space of bounded sequences.
The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by:
Define the -norm:
Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, will have an infinite -norm for The space is then defined as the set of all infinite sequences of real (or complex) numbers such that the -norm is finite.
One can check that as increases, the set grows larger. For example, the sequence
is not in but it is in for as the series
diverges for (the harmonic series), but is convergent for
One also defines the -norm using the supremum:
and the corresponding space of all bounded sequences. It turns out that
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider spaces for
The -norm thus defined on is indeed a norm, and together with this norm is a Banach space. | Lp space | Wikipedia | 512 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
General ℓp-space
In complete analogy to the preceding definition one can define the space over a general index set (and ) as
where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence).
With the norm
the space becomes a Banach space.
In the case where is finite with elements, this construction yields with the -norm defined above.
If is countably infinite, this is exactly the sequence space defined above.
For uncountable sets this is a non-separable Banach space which can be seen as the locally convex direct limit of -sequence spaces.
For the -norm is even induced by a canonical inner product called the , which means that holds for all vectors This inner product can expressed in terms of the norm by using the polarization identity.
On it can be defined by
Now consider the case Define
where for all
The index set can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space is just a special case of the more general -space (defined below).
Lp spaces and Lebesgue integrals
An space may be defined as a space of measurable functions for which the -th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let be a measure space and
When , consider the set of all measurable functions from to or whose absolute value raised to the -th power has a finite integral, or in symbols:
To define the set for recall that two functions and defined on are said to be , written , if the set is measurable and has measure zero.
Similarly, a measurable function (and its absolute value) is (or ) by a real number written , if the (necessarily) measurable set has measure zero.
The space is the set of all measurable functions that are bounded almost everywhere (by some real ) and is defined as the infimum of these bounds:
When then this is the same as the essential supremum of the absolute value of :
For example, if is a measurable function that is equal to almost everywhere then for every and thus for all | Lp space | Wikipedia | 447 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
For every positive the value under of a measurable function and its absolute value are always the same (that is, for all ) and so a measurable function belongs to if and only if its absolute value does. Because of this, many formulas involving -norms are stated only for non-negative real-valued functions. Consider for example the identity which holds whenever is measurable, is real, and (here when ). The non-negativity requirement can be removed by substituting in for which gives
Note in particular that when is finite then the formula relates the -norm to the -norm.
Seminormed space of -th power integrable functions
Each set of functions forms a vector space when addition and scalar multiplication are defined pointwise.
That the sum of two -th power integrable functions and is again -th power integrable follows from
although it is also a consequence of Minkowski's inequality
which establishes that satisfies the triangle inequality for (the triangle inequality does not hold for ).
That is closed under scalar multiplication is due to being absolutely homogeneous, which means that for every scalar and every function
Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm.
Thus is a seminorm and the set of -th power integrable functions together with the function defines a seminormed vector space. In general, the seminorm is not a norm because there might exist measurable functions that satisfy but are not equal to ( is a norm if and only if no such exists).
Zero sets of -seminorms
If is measurable and equals a.e. then for all positive
On the other hand, if is a measurable function for which there exists some such that then almost everywhere. When is finite then this follows from the case and the formula mentioned above.
Thus if is positive and is any measurable function, then if and only if almost everywhere. Since the right hand side ( a.e.) does not mention it follows that all have the same zero set (it does not depend on ). So denote this common set by
This set is a vector subspace of for every positive
Quotient vector space
Like every seminorm, the seminorm induces a norm (defined shortly) on the canonical quotient vector space of by its vector subspace | Lp space | Wikipedia | 486 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
This normed quotient space is called and it is the subject of this article. We begin by defining the quotient vector space.
Given any the coset consists of all measurable functions that are equal to almost everywhere.
The set of all cosets, typically denoted by
forms a vector space with origin when vector addition and scalar multiplication are defined by and
This particular quotient vector space will be denoted by
Two cosets are equal if and only if (or equivalently, ), which happens if and only if almost everywhere; if this is the case then and are identified in the quotient space. Hence, strictly speaking consists of equivalence classes of functions.
Given any the value of the seminorm on the coset is constant and equal to , that is:
The map is a norm on called the .
The value of a coset is independent of the particular function that was chosen to represent the coset, meaning that if is any coset then for every (since for every ).
The Lebesgue space
The normed vector space is called or the of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the [[Riesz–Fischer theorem#Completeness of Lp, 0 < p ≤ ∞|Riesz–Fischer theorem]]).
When the underlying measure space is understood then is often abbreviated or even just
Depending on the author, the subscript notation might denote either or
If the seminorm on happens to be a norm (which happens if and only if ) then the normed space will be linearly isometrically isomorphic to the normed quotient space via the canonical map (since ); in other words, they will be, up to a linear isometry, the same normed space and so they may both be called " space".
The above definitions generalize to Bochner spaces.
In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of in For however, there is a theory of lifts enabling such recovery. | Lp space | Wikipedia | 442 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
Special cases
For the spaces are a special case of spaces; when are the natural numbers and is the counting measure. More generally, if one considers any set with the counting measure, the resulting space is denoted For example, is the space of all sequences indexed by the integers, and when defining the -norm on such a space, one sums over all the integers. The space where is the set with elements, is with its -norm as defined above.
Similar to spaces, is the only Hilbert space among spaces. In the complex case, the inner product on is defined by
Functions in are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral .
As any Hilbert space, every space is linearly isometric to a suitable where the cardinality of the set is the cardinality of an arbitrary basis for this particular
If we use complex-valued functions, the space is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of defines a bounded operator on any space by multiplication.
When
If then can be defined as above, that is:
In this case, however, the -norm does not satisfy the triangle inequality and defines only a quasi-norm. The inequality valid for implies that
and so the function
is a metric on The resulting metric space is complete.
In this setting satisfies a reverse Minkowski inequality, that is for
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces for .
The space for is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in or every open convex set containing the function is unbounded for the -quasi-norm; therefore, the vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space contains an infinite family of disjoint measurable sets of finite positive measure. | Lp space | Wikipedia | 484 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
The only nonempty convex open set in is the entire space. Consequently, there are no nonzero continuous linear functionals on the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (i.e. ), the bounded linear functionals on are exactly those that are bounded on , i.e., those given by sequences in Although does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.
Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on rather than work with for it is common to work with the Hardy space whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn–Banach theorem still fails in for .
Properties
Hölder's inequality
Suppose satisfy . If and then and
This inequality, called Hölder's inequality, is in some sense optimal since if and is a measurable function such that
where the supremum is taken over the closed unit ball of then and
Atomic decomposition
If then every non-negative has an , meaning that there exist a sequence of non-negative real numbers and a sequence of non-negative functions called , whose supports are pairwise disjoint sets of measure such that
and for every integer
and
and where moreover, the sequence of functions depends only on (it is independent of ).
These inequalities guarantee that for all integers while the supports of being pairwise disjoint implies
Dual spaces
The dual space of for has a natural isomorphism with where is such that . This isomorphism associates with the functional defined by
for every
is a well defined continuous linear mapping which is an isometry by the extremal case of Hölder's inequality. If is a -finite measure space one can use the Radon–Nikodym theorem to show that any can be expressed this way, i.e., is an isometric isomorphism of Banach spaces. Hence, it is usual to say simply that is the continuous dual space of
For the space is reflexive. Let be as above and let be the corresponding linear isometry. Consider the map from to obtained by composing with the transpose (or adjoint) of the inverse of | Lp space | Wikipedia | 475 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
This map coincides with the canonical embedding of into its bidual. Moreover, the map is onto, as composition of two onto isometries, and this proves reflexivity.
If the measure on is sigma-finite, then the dual of is isometrically isomorphic to (more precisely, the map corresponding to is an isometry from onto
The dual of is subtler. Elements of can be identified with bounded signed finitely additive measures on that are absolutely continuous with respect to See ba space for more details. If we assume the axiom of choice, this space is much bigger than except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of is
Embeddings
Colloquially, if then contains functions that are more locally singular, while elements of can be more spread out. Consider the Lebesgue measure on the half line A continuous function in might blow up near but must decay sufficiently fast toward infinity. On the other hand, continuous functions in need not decay at all but no blow-up is allowed. More formally, suppose that , then:
if and only if does not contain sets of finite but arbitrarily large measure (e.g. any finite measure).
if and only if does not contain sets of non-zero but arbitrarily small measure (e.g. the counting measure).
Neither condition holds for the Lebesgue measure on the real line while both conditions holds for the counting measure on any finite set. As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator is a bounded linear map from to in the first case and to in the second. Indeed, if the domain has finite measure, one can make the following explicit calculation using Hölder's inequality
leading to
The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity is precisely
the case of equality being achieved exactly when -almost-everywhere.
Dense subspaces
Let and be a measure space and consider an integrable simple function on given by
where are scalars, has finite measure and is the indicator function of the set for By construction of the integral, the vector space of integrable simple functions is dense in
More can be said when is a normal topological space and its Borel –algebra. | Lp space | Wikipedia | 511 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
Suppose is an open set with Then for every Borel set contained in there exist a closed set and an open set such that
for every . Subsequently, there exists a Urysohn function on that is on and on with
If can be covered by an increasing sequence of open sets that have finite measure, then the space of –integrable continuous functions is dense in More precisely, one can use bounded continuous functions that vanish outside one of the open sets
This applies in particular when and when is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the space of integrable step functions are dense in .
Closed subspaces
Suppose . If is a probability space and is a closed subspace of then is finite-dimensional.
It is crucial that the vector space be a subset of since it is possible to construct an infinite-dimensional closed vector subspace of which lies in ; taking the Lebesgue measure on the circle group divided by as the probability measure.
Applications
Statistics
In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems.
In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero). Elastic net regularization uses a penalty term that is a combination of the norm and the squared norm of the parameter vector.
Hausdorff–Young inequality
The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps to (or to ) respectively, where and This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.
By contrast, if the Fourier transform does not map into | Lp space | Wikipedia | 432 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
Hilbert spaces
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces and are both Hilbert spaces. In fact, by choosing a Hilbert basis i.e., a maximal orthonormal subset of or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to (same as above), i.e., a Hilbert space of type
Generalizations and extensions
Weak
Let be a measure space, and a measurable function with real or complex values on The distribution function of is defined for by
If is in for some with then by Markov's inequality,
A function is said to be in the space weak , or if there is a constant such that, for all
The best constant for this inequality is the -norm of and is denoted by
The weak coincide with the Lorentz spaces so this notation is also used to denote them.
The -norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for in
and in particular
In fact, one has
and raising to power and taking the supremum in one has
Under the convention that two functions are equal if they are equal almost everywhere, then the spaces are complete .
For any the expression
is comparable to the -norm. Further in the case this expression defines a norm if Hence for the weak spaces are Banach spaces .
A major result that uses the -spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.
Weighted spaces
As before, consider a measure space Let be a measurable function. The -weighted space is defined as where means the measure defined by
or, in terms of the Radon–Nikodym derivative, the norm for is explicitly
As -spaces, the weighted spaces have nothing special, since is equal to But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem: for the classical Hilbert transform is defined on where denotes the unit circle and the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on Muckenhoupt's theorem describes weights such that the Hilbert transform remains bounded on and the maximal operator on
spaces on manifolds
One may also define spaces on a manifold, called the intrinsic spaces of the manifold, using densities.
Vector-valued spaces | Lp space | Wikipedia | 482 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
Given a measure space and a locally convex space (here assumed to be complete), it is possible to define spaces of -integrable -valued functions on in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies that are (each in their own way) a natural generalization of the usual topology. Another way involves topological tensor products of with Element of the vector space are finite sums of simple tensors where each simple tensor may be identified with the function that sends This tensor product is then endowed with a locally convex topology that turns it into a topological tensor product, the most common of which are the projective tensor product, denoted by and the injective tensor product, denoted by In general, neither of these space are complete so their completions are constructed, which are respectively denoted by and (this is analogous to how the space of scalar-valued simple functions on when seminormed by any is not complete so a completion is constructed which, after being quotiented by is isometrically isomorphic to the Banach space ). Alexander Grothendieck showed that when is a nuclear space (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.
space of measurable functions
The vector space of (equivalence classes of) measurable functions on is denoted . By definition, it contains all the and is equipped with the topology of convergence in measure. When is a probability measure (i.e., ), this mode of convergence is named convergence in probability. The space is always a topological abelian group but is only a topological vector space if This is because scalar multiplication is continuous if and only if If is -finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure for a suitable choice of probability measure
The description is easier when is finite. If is a finite measure on the function admits for the convergence in measure the following fundamental system of neighborhoods
The topology can be defined by any metric of the form | Lp space | Wikipedia | 480 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
where is bounded continuous concave and non-decreasing on with and when (for example, Such a metric is called Lévy-metric for Under this metric the space is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if . To see this, consider the Lebesgue measurable function defined by . Then clearly . The space is in general not locally bounded, and not locally convex.
For the infinite Lebesgue measure on the definition of the fundamental system of neighborhoods could be modified as follows
The resulting space , with the topology of local convergence in measure, is isomorphic to the space for any positive –integrable density | Lp space | Wikipedia | 134 | 45194 | https://en.wikipedia.org/wiki/Lp%20space | Mathematics | Mathematical analysis | null |
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.
A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details.
A function that is not injective is sometimes called many-to-one.
Definition
Let be a function whose domain is a set The function is said to be injective provided that for all and in if then ; that is, implies Equivalently, if then in the contrapositive statement.
Symbolically,
which is logically equivalent to the contrapositive,An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, or ), although some authors specifically reserve ↪ for an inclusion map. | Injective function | Wikipedia | 323 | 45196 | https://en.wikipedia.org/wiki/Injective%20function | Mathematics | Functions: General | null |
Examples
For visual examples, readers are directed to the gallery section.
For any set and any subset the inclusion map (which sends any element to itself) is injective. In particular, the identity function is always injective (and in fact bijective).
If the domain of a function is the empty set, then the function is the empty function, which is injective.
If the domain of a function has one element (that is, it is a singleton set), then the function is always injective.
The function defined by is injective.
The function defined by is injective, because (for example) However, if is redefined so that its domain is the non-negative real numbers [0,+∞), then is injective.
The exponential function defined by is injective (but not surjective, as no real value maps to a negative number).
The natural logarithm function defined by is injective.
The function defined by is not injective, since, for example,
More generally, when and are both the real line then an injective function is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the .
Injections can be undone
Functions with left inverses are always injections. That is, given if there is a function such that for every , , then is injective. In this case, is called a retraction of Conversely, is called a section of
Conversely, every injection with a non-empty domain has a left inverse . It can be defined by choosing an element in the domain of and setting to the unique element of the pre-image (if it is non-empty) or to (otherwise).
The left inverse is not necessarily an inverse of because the composition in the other order, may differ from the identity on In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.
Injections may be made invertible
In fact, to turn an injective function into a bijective (hence invertible) function, it suffices to replace its codomain by its actual image That is, let such that for all ; then is bijective. Indeed, can be factored as where is the inclusion function from into | Injective function | Wikipedia | 498 | 45196 | https://en.wikipedia.org/wiki/Injective%20function | Mathematics | Functions: General | null |
More generally, injective partial functions are called partial bijections.
Other properties
If and are both injective then is injective.
If is injective, then is injective (but need not be).
is injective if and only if, given any functions whenever then In other words, injective functions are precisely the monomorphisms in the category Set of sets.
If is injective and is a subset of then Thus, can be recovered from its image
If is injective and and are both subsets of then
Every function can be decomposed as for a suitable injection and surjection This decomposition is unique up to isomorphism, and may be thought of as the inclusion function of the range of as a subset of the codomain of
If is an injective function, then has at least as many elements as in the sense of cardinal numbers. In particular, if, in addition, there is an injection from to then and have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
If both and are finite with the same number of elements, then is injective if and only if is surjective (in which case is bijective).
An injective function which is a homomorphism between two algebraic structures is an embedding.
Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function is injective can be decided by only considering the graph (and not the codomain) of
Proving that functions are injective
A proof that a function is injective depends on how the function is presented and what properties the function holds.
For functions that are given by some formula there is a basic idea.
We use the definition of injectivity, namely that if then
Here is an example:
Proof: Let Suppose So implies which implies Therefore, it follows from the definition that is injective. | Injective function | Wikipedia | 424 | 45196 | https://en.wikipedia.org/wiki/Injective%20function | Mathematics | Functions: General | null |
There are multiple other methods of proving that a function is injective. For example, in calculus if is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if is a linear transformation it is sufficient to show that the kernel of contains only the zero vector. If is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.
A graphical approach for a real-valued function of a real variable is the horizontal line test. If every horizontal line intersects the curve of in at most one point, then is injective or one-to-one.
Gallery | Injective function | Wikipedia | 152 | 45196 | https://en.wikipedia.org/wiki/Injective%20function | Mathematics | Functions: General | null |
A submarine communications cable is a cable laid on the seabed between land-based stations to carry telecommunication signals across stretches of ocean and sea. The first submarine communications cables were laid beginning in the 1850s and carried telegraphy traffic, establishing the first instant telecommunications links between continents, such as the first transatlantic telegraph cable which became operational on 16 August 1858.
Submarine cables first connected all the world's continents (except Antarctica) when Java was connected to Darwin, Northern Territory, Australia, in 1871 in anticipation of the completion of the Australian Overland Telegraph Line in 1872 connecting to Adelaide, South Australia and thence to the rest of Australia.
Subsequent generations of cables carried telephone traffic, then data communications traffic. These early cables used copper wires in their cores, but modern cables use optical fiber technology to carry digital data, which includes telephone, Internet and private data traffic. Modern cables are typically about in diameter and weigh around for the deep-sea sections which comprise the majority of the run, although larger and heavier cables are used for shallow-water sections near shore.
Early history: telegraph and coaxial cables
First successful trials
After William Cooke and Charles Wheatstone had introduced their working telegraph in 1839, the idea of a submarine line across the Atlantic Ocean began to be thought of as a possible triumph of the future. Samuel Morse proclaimed his faith in it as early as 1840, and in 1842, he submerged a wire, insulated with tarred hemp and India rubber, in the water of New York Harbor, and telegraphed through it. The following autumn, Wheatstone performed a similar experiment in Swansea Bay. A good insulator to cover the wire and prevent the electric current from leaking into the water was necessary for the success of a long submarine line. India rubber had been tried by Moritz von Jacobi, the Prussian electrical engineer, as far back as the early 19th century. | Submarine communications cable | Wikipedia | 372 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Another insulating gum which could be melted by heat and readily applied to wire made its appearance in 1842. Gutta-percha, the adhesive juice of the Palaquium gutta tree, was introduced to Europe by William Montgomerie, a Scottish surgeon in the service of the British East India Company. Twenty years earlier, Montgomerie had seen whips made of gutta-percha in Singapore, and he believed that it would be useful in the fabrication of surgical apparatus. Michael Faraday and Wheatstone soon discovered the merits of gutta-percha as an insulator, and in 1845, the latter suggested that it should be employed to cover the wire which was proposed to be laid from Dover to Calais. In 1847 William Siemens, then an officer in the army of Prussia, laid the first successful underwater cable using gutta percha insulation, across the Rhine between Deutz and Cologne. In 1849, Charles Vincent Walker, electrician to the South Eastern Railway, submerged of wire coated with gutta-percha off the coast from Folkestone, which was tested successfully.
First commercial cables
In August 1850, having earlier obtained a concession from the French government, John Watkins Brett's English Channel Submarine Telegraph Company laid the first line across the English Channel, using the converted tugboat Goliath. It was simply a copper wire coated with gutta-percha, without any other protection, and was not successful. However, the experiment served to secure renewal of the concession, and in September 1851, a protected core, or true, cable was laid by the reconstituted Submarine Telegraph Company from a government hulk, Blazer, which was towed across the Channel.
In 1853, more successful cables were laid, linking Great Britain with Ireland, Belgium, and the Netherlands, and crossing The Belts in Denmark. The British & Irish Magnetic Telegraph Company completed the first successful Irish link on May 23 between Portpatrick and Donaghadee using the collier William Hutt. The same ship was used for the link from Dover to Ostend in Belgium, by the Submarine Telegraph Company. Meanwhile, the Electric & International Telegraph Company completed two cables across the North Sea, from Orford Ness to Scheveningen, the Netherlands. These cables were laid by Monarch, a paddle steamer which later became the first vessel with permanent cable-laying equipment. | Submarine communications cable | Wikipedia | 476 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
In 1858, the steamship Elba was used to lay a telegraph cable from Jersey to Guernsey, on to Alderney and then to Weymouth, the cable being completed successfully in September of that year. Problems soon developed with eleven breaks occurring by 1860 due to storms, tidal and sand movements, and wear on rocks. A report to the Institution of Civil Engineers in 1860 set out the problems to assist in future cable-laying operations.
Crimean War (1853–1856)
In the Crimean War various forms of telegraphy played a major role; this was a first. At the start of the campaign there was a telegraph link at Bucharest connected to London. In the winter of 1854 the French extended the telegraph link to the Black Sea coast. In April 1855 the British laid an underwater cable from Varna to the Crimean peninsula so that news of the Crimean War could reach London in a handful of hours.
Transatlantic telegraph cable
The first attempt at laying a transatlantic telegraph cable was promoted by Cyrus West Field, who persuaded British industrialists to fund and lay one in 1858. However, the technology of the day was not capable of supporting the project; it was plagued with problems from the outset, and was in operation for only a month. Subsequent attempts in 1865 and 1866 with the world's largest steamship, the SS Great Eastern, used a more advanced technology and produced the first successful transatlantic cable. Great Eastern later went on to lay the first cable reaching to India from Aden, Yemen, in 1870.
British dominance of early cable
From the 1850s until 1911, British submarine cable systems dominated the most important market, the North Atlantic Ocean. The British had both supply side and demand side advantages. In terms of supply, Britain had entrepreneurs willing to put forth enormous amounts of capital necessary to build, lay and maintain these cables. In terms of demand, Britain's vast colonial empire led to business for the cable companies from news agencies, trading and shipping companies, and the British government. Many of Britain's colonies had significant populations of European settlers, making news about them of interest to the general public in the home country. | Submarine communications cable | Wikipedia | 421 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
British officials believed that depending on telegraph lines that passed through non-British territory posed a security risk, as lines could be cut and messages could be interrupted during wartime. They sought the creation of a worldwide network within the empire, which became known as the All Red Line, and conversely prepared strategies to quickly interrupt enemy communications. Britain's very first action after declaring war on Germany in World War I was to have the cable ship Alert (not the CS Telconia as frequently reported) cut the five cables linking Germany with France, Spain and the Azores, and through them, North America. Thereafter, the only way Germany could communicate was by wireless, and that meant that Room 40 could listen in.
The submarine cables were an economic benefit to trading companies, because owners of ships could communicate with captains when they reached their destination and give directions as to where to go next to pick up cargo based on reported pricing and supply information. The British government had obvious uses for the cables in maintaining administrative communications with governors throughout its empire, as well as in engaging other nations diplomatically and communicating with its military units in wartime. The geographic location of British territory was also an advantage as it included both Ireland on the east side of the Atlantic Ocean and Newfoundland in North America on the west side, making for the shortest route across the ocean, which reduced costs significantly.
A few facts put this dominance of the industry in perspective. In 1896, there were 30 cable-laying ships in the world, 24 of which were owned by British companies. In 1892, British companies owned and operated two-thirds of the world's cables and by 1923, their share was still 42.7 percent. During World War I, Britain's telegraph communications were almost completely uninterrupted, while it was able to quickly cut Germany's cables worldwide.
Cable to India, Singapore, East Asia and Australia | Submarine communications cable | Wikipedia | 376 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Throughout the 1860s and 1870s, British cable expanded eastward, into the Mediterranean Sea and the Indian Ocean. An 1863 cable to Bombay (now Mumbai), India, provided a crucial link to Saudi Arabia. In 1870, Bombay was linked to London via submarine cable in a combined operation by four cable companies, at the behest of the British Government. In 1872, these four companies were combined to form the mammoth globe-spanning Eastern Telegraph Company, owned by John Pender. A spin-off from Eastern Telegraph Company was a second sister company, the Eastern Extension, China and Australasia Telegraph Company, commonly known simply as "the Extension." In 1872, Australia was linked by cable to Bombay via Singapore and China and in 1876, the cable linked the British Empire from London to New Zealand.
Submarine cables across the Pacific, 1902-1991
The first trans-Pacific cables providing telegraph service were completed in 1902 and 1903, linking the US mainland to Hawaii in 1902 and Guam to the Philippines in 1903. Canada, Australia, New Zealand and Fiji were also linked in 1902 with the trans-Pacific segment of the All Red Line. Japan was connected into the system in 1906. Service beyond Midway Atoll was abandoned in 1941 due to World War II, but the remainder stayed in operation until 1951 when the FCC gave permission to cease operations.
The first trans-Pacific telephone cable was laid from Hawaii to Japan in 1964, with an extension from Guam to The Philippines. Also in 1964, the Commonwealth Pacific Cable System (COMPAC), with 80 telephone channel capacity, opened for traffic from Sydney to Vancouver, and in 1967, the South East Asia Commonwealth (SEACOM) system, with 160 telephone channel capacity, opened for traffic. This system used microwave radio from Sydney to Cairns (Queensland), cable running from Cairns to Madang (Papua New Guinea), Guam, Hong Kong, Kota Kinabalu (capital of Sabah, Malaysia), Singapore, then overland by microwave radio to Kuala Lumpur. In 1991, the North Pacific Cable system was the first regenerative system (i.e., with repeaters) to completely cross the Pacific from the US mainland to Japan. The US portion of NPC was manufactured in Portland, Oregon, from 1989 to 1991 at STC Submarine Systems, and later Alcatel Submarine Networks. The system was laid by Cable & Wireless Marine on the CS Cable Venture.
Construction, 19-20th century | Submarine communications cable | Wikipedia | 487 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Transatlantic cables of the 19th century consisted of an outer layer of iron and later steel wire, wrapping India rubber, wrapping gutta-percha, which surrounded a multi-stranded copper wire at the core. The portions closest to each shore landing had additional protective armour wires. Gutta-percha, a natural polymer similar to rubber, had nearly ideal properties for insulating submarine cables, with the exception of a rather high dielectric constant which made cable capacitance high. William Thomas Henley had developed a machine in 1837 for covering wires with silk or cotton thread that he developed into a wire wrapping capability for submarine cable with a factory in 1857 that became W.T. Henley's Telegraph Works Co., Ltd. The India Rubber, Gutta Percha and Telegraph Works Company, established by the Silver family and giving that name to a section of London, furnished cores to Henley's as well as eventually making and laying finished cable. In 1870 William Hooper established Hooper's Telegraph Works to manufacture his patented vulcanized rubber core, at first to furnish other makers of finished cable, that began to compete with the gutta-percha cores. The company later expanded into complete cable manufacture and cable laying, including the building of the first cable ship specifically designed to lay transatlantic cables.
Gutta-percha and rubber were not replaced as a cable insulation until polyethylene was introduced in the 1930s. Even then, the material was only available to the military and the first submarine cable using it was not laid until 1945 during World War II across the English Channel. In the 1920s, the American military experimented with rubber-insulated cables as an alternative to gutta-percha, since American interests controlled significant supplies of rubber but did not have easy access to gutta-percha manufacturers. The 1926 development by John T. Blake of deproteinized rubber improved the impermeability of cables to water. | Submarine communications cable | Wikipedia | 388 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Many early cables suffered from attack by sea life. The insulation could be eaten, for instance, by species of Teredo (shipworm) and Xylophaga. Hemp laid between the steel wire armouring gave pests a route to eat their way in. Damaged armouring, which was not uncommon, also provided an entrance. Cases of sharks biting cables and attacks by sawfish have been recorded. In one case in 1873, a whale damaged the Persian Gulf Cable between Karachi and Gwadar. The whale was apparently attempting to use the cable to clean off barnacles at a point where the cable descended over a steep drop. The unfortunate whale got its tail entangled in loops of cable and drowned. The cable repair ship Amber Witch was only able to winch up the cable with difficulty, weighed down as it was with the dead whale's body.
Bandwidth problems
Early long-distance submarine telegraph cables exhibited formidable electrical problems. Unlike modern cables, the technology of the 19th century did not allow for in-line repeater amplifiers in the cable. Large voltages were used to attempt to overcome the electrical resistance of their tremendous length but the cables' distributed capacitance and inductance combined to distort the telegraph pulses in the line, reducing the cable's bandwidth, severely limiting the data rate for telegraph operation to 10–12 words per minute.
As early as 1816, Francis Ronalds had observed that electric signals were slowed in passing through an insulated wire or core laid underground, and outlined the cause to be induction, using the analogy of a long Leyden jar. The same effect was noticed by Latimer Clark (1853) on cores immersed in water, and particularly on the lengthy cable between England and The Hague. Michael Faraday showed that the effect was caused by capacitance between the wire and the earth (or water) surrounding it. Faraday had noticed that when a wire is charged from a battery (for example when pressing a telegraph key), the electric charge in the wire induces an opposite charge in the water as it travels along. In 1831, Faraday described this effect in what is now referred to as Faraday's law of induction. As the two charges attract each other, the exciting charge is retarded. The core acts as a capacitor distributed along the length of the cable which, coupled with the resistance and inductance of the cable, limits the speed at which a signal travels through the conductor of the cable. | Submarine communications cable | Wikipedia | 506 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Early cable designs failed to analyse these effects correctly. Famously, E.O.W. Whitehouse had dismissed the problems and insisted that a transatlantic cable was feasible. When he subsequently became chief electrician of the Atlantic Telegraph Company, he became involved in a public dispute with William Thomson. Whitehouse believed that, with enough voltage, any cable could be driven. Thomson believed that his law of squares showed that retardation could not be overcome by a higher voltage. His recommendation was a larger cable. Because of the excessive voltages recommended by Whitehouse, Cyrus West Field's first transatlantic cable never worked reliably, and eventually short circuited to the ocean when Whitehouse increased the voltage beyond the cable design limit.
Thomson designed a complex electric-field generator that minimized current by resonating the cable, and a sensitive light-beam mirror galvanometer for detecting the faint telegraph signals. Thomson became wealthy on the royalties of these, and several related inventions. Thomson was elevated to Lord Kelvin for his contributions in this area, chiefly an accurate mathematical model of the cable, which permitted design of the equipment for accurate telegraphy. The effects of atmospheric electricity and the geomagnetic field on submarine cables also motivated many of the early polar expeditions.
Thomson had produced a mathematical analysis of propagation of electrical signals into telegraph cables based on their capacitance and resistance, but since long submarine cables operated at slow rates, he did not include the effects of inductance. By the 1890s, Oliver Heaviside had produced the modern general form of the telegrapher's equations, which included the effects of inductance and which were essential to extending the theory of transmission lines to the higher frequencies required for high-speed data and voice.
Transatlantic telephony
While laying a transatlantic telephone cable was seriously considered from the 1920s, the technology required for economically feasible telecommunications was not developed until the 1940s. A first attempt to lay a "pupinized" telephone cable—one with loading coils added at regular intervals—failed in the early 1930s due to the Great Depression.
TAT-1 (Transatlantic No. 1) was the first transatlantic telephone cable system. Between 1955 and 1956, cable was laid between Gallanach Bay, near Oban, Scotland and Clarenville, Newfoundland and Labrador, in Canada. It was inaugurated on September 25, 1956, initially carrying 36 telephone channels. | Submarine communications cable | Wikipedia | 482 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
In the 1960s, transoceanic cables were coaxial cables that transmitted frequency-multiplexed voiceband signals. A high-voltage direct current on the inner conductor powered repeaters (two-way amplifiers placed at intervals along the cable). The first-generation repeaters remain among the most reliable vacuum tube amplifiers ever designed. Later ones were transistorized. Many of these cables are still usable, but have been abandoned because their capacity is too small to be commercially viable. Some have been used as scientific instruments to measure earthquake waves and other geomagnetic events.
Other uses
In 1942, Siemens Brothers of New Charlton, London, in conjunction with the United Kingdom National Physical Laboratory, adapted submarine communications cable technology to create the world's first submarine oil pipeline in Operation Pluto during World War II.
Active fiber-optic cables may be useful in detecting seismic events which alter cable polarization.
Modern history
Optical telecommunications cables
In the 1980s, fiber-optic cables were developed. The first transatlantic telephone cable to use optical fiber was TAT-8, which went into operation in 1988. A fiber-optic cable comprises multiple pairs of fibers. Each pair has one fiber in each direction. TAT-8 had two operational pairs and one backup pair. Except for very short lines, fiber-optic submarine cables include repeaters at regular intervals.
Modern optical fiber repeaters use a solid-state optical amplifier, usually an erbium-doped fiber amplifier (EDFA). Each repeater contains separate equipment for each fiber. These comprise signal reforming, error measurement and controls. A solid-state laser dispatches the signal into the next length of fiber. The solid-state laser excites a short length of doped fiber that itself acts as a laser amplifier. As the light passes through the fiber, it is amplified. This system also permits wavelength-division multiplexing, which dramatically increases the capacity of the fiber. EDFA amplifiers were first used in submarine cables in 1995. | Submarine communications cable | Wikipedia | 399 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Repeaters are powered by a constant direct current passed down the conductor near the centre of the cable, so all repeaters in a cable are in series. Power feed equipment (PFE) is installed at the terminal stations. Typically both ends share the current generation with one end providing a positive voltage and the other a negative voltage. A virtual earth point exists roughly halfway along the cable under normal operation. The amplifiers or repeaters derive their power from the potential difference across them. The voltage passed down the cable is often anywhere from 3000 to 15,000VDC at a current of up to 1,100mA, with the current increasing with decreasing voltage; the current at 10,000VDC is up to 1,650mA. Hence the total amount of power sent into the cable is often up to 16.5 kW.
The optic fiber used in undersea cables is chosen for its exceptional clarity, permitting runs of more than between repeaters to minimize the number of amplifiers and the distortion they cause. Unrepeated cables are cheaper than repeated cables and their maximum transmission distance is limited, although this has increased over the years; in 2014 unrepeated cables of up to in length were in service; however these require unpowered repeaters to be positioned every 100 km.
The rising demand for these fiber-optic cables outpaced the capacity of providers such as AT&T. Having to shift traffic to satellites resulted in lower-quality signals. To address this issue, AT&T had to improve its cable-laying abilities. It invested $100 million in producing two specialized fiber-optic cable laying vessels. These included laboratories in the ships for splicing cable and testing its electrical properties. Such field monitoring is important because the glass of fiber-optic cable is less malleable than the copper cable that had been formerly used. The ships are equipped with thrusters that increase maneuverability. This capability is important because fiber-optic cable must be laid straight from the stern, which was another factor that copper-cable-laying ships did not have to contend with. | Submarine communications cable | Wikipedia | 417 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Originally, submarine cables were simple point-to-point connections. With the development of submarine branching units (SBUs), more than one destination could be served by a single cable system. Modern cable systems now usually have their fibers arranged in a self-healing ring to increase their redundancy, with the submarine sections following different paths on the ocean floor. One reason for this development was that the capacity of cable systems had become so large that it was not possible to completely back up a cable system with satellite capacity, so it became necessary to provide sufficient terrestrial backup capability. Not all telecommunications organizations wish to take advantage of this capability, so modern cable systems may have dual landing points in some countries (where back-up capability is required) and only single landing points in other countries where back-up capability is either not required, the capacity to the country is small enough to be backed up by other means, or having backup is regarded as too expensive.
A further redundant-path development over and above the self-healing rings approach is the mesh network whereby fast switching equipment is used to transfer services between network paths with little to no effect on higher-level protocols if a path becomes inoperable. As more paths become available to use between two points, it is less likely that one or two simultaneous failures will prevent end-to-end service.
As of 2012, operators had "successfully demonstrated long-term, error-free transmission at 100 Gbps across Atlantic Ocean" routes of up to , meaning a typical cable can move tens of terabits per second overseas. Speeds improved rapidly in the previous few years, with 40 Gbit/s having been offered on that route only three years earlier in August 2009. | Submarine communications cable | Wikipedia | 344 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Switching and all-by-sea routing commonly increases the distance and thus the round trip latency by more than 50%. For example, the round trip delay (RTD) or latency of the fastest transatlantic connections is under 60 ms, close to the theoretical optimum for an all-sea route. While in theory, a great circle route (GCP) between London and New York City is only , this requires several land masses (Ireland, Newfoundland, Prince Edward Island and the isthmus connecting New Brunswick to Nova Scotia) to be traversed, as well as the extremely tidal Bay of Fundy and a land route along Massachusetts' north shore from Gloucester to Boston and through fairly built up areas to Manhattan itself. In theory, using this partial land route could result in round trip times below 40 ms (which is the speed of light minimum time), and not counting switching. Along routes with less land in the way, round trip times can approach speed of light minimums in the long term.
The type of optical fiber used in unrepeated and very long cables is often PCSF (pure silica core) due to its low loss of 0.172dB per kilometer when carrying a 1550 nm wavelength laser light. The large chromatic dispersion of PCSF means that its use requires transmission and receiving equipment designed with this in mind; this property can also be used to reduce interference when transmitting multiple channels through a single fiber using wavelength division multiplexing (WDM), which allows for multiple optical carrier channels to be transmitted through a single fiber, each carrying its own information. WDM is limited by the optical bandwidth of the amplifiers used to transmit data through the cable and by the spacing between the frequencies of the optical carriers; however this minimum spacing is also limited, with the minimum spacing often being 50 GHz (0.4 nm). The use of WDM can reduce the maximum length of the cable although this can be overcome by designing equipment with this in mind. | Submarine communications cable | Wikipedia | 403 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Optical post amplifiers, used to increase the strength of the signal generated by the optical transmitter often use a diode-pumped erbium-doped fiber laser. The diode is often a high power 980 or 1480 nm laser diode. This setup allows for an amplification of up to +24dBm in an affordable manner. Using an erbium-ytterbium doped fiber instead allows for a gain of +33dBm, however again the amount of power that can be fed into the fiber is limited. In single carrier configurations the dominating limitation is self phase modulation induced by the Kerr effect which limits the amplification to +18 dBm per fiber. In WDM configurations the limitation due to crossphase modulation becomes predominant instead. Optical pre-amplifiers are often used to negate the thermal noise of the receiver. Pumping the pre-amplifier with a 980 nm laser leads to a noise of at most 3.5 dB, with a noise of 5 dB usually obtained with a 1480 nm laser. The noise has to be filtered using optical filters.
Raman amplification can be used to extend the reach or the capacity of an unrepeatered cable, by launching 2 frequencies into a single fiber; one carrying data signals at 1550 nm, and the other pumping them at 1450 nm. Launching a pump frequency (pump laser light) at a power of just one watt leads to an increase in reach of 45 km or a 6-fold increase in capacity.
Another way to increase the reach of a cable is by using unpowered repeaters called remote optical pre-amplifiers (ROPAs); these still make a cable count as unrepeatered since the repeaters do not require electrical power but they do require a pump laser light to be transmitted alongside the data carried by the cable; the pump light and the data are often transmitted in physically separate fibers. The ROPA contains a doped fiber that uses the pump light (often a 1480 nm laser light) to amplify the data signals carried on the rest of the fibers. | Submarine communications cable | Wikipedia | 423 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
WDM or wavelength division multiplexing was first implemented in submarine fiber optic cables from the 1990s to the 2000s, followed by DWDM or dense wavelength division mulltiplexing around 2007. Each fiber can carry 30 wavelengths at a time. SDM or spatial division multiplexing submarine cables have at least 12 fiber pairs which is an increase from the maximum of 8 pairs found in conventional submarine cables, and submarine cables with up to 24 fiber pairs have been deployed. The type of modulation employed in a submarine cable can have a major impact in its capacity. SDM is combined with DWDM to improve capacity.
Transponders are used to send data through the cable. The open cable concept allows for the design of a submarine cable independently of the transponders that will be used to transmit data through the cable. SLTE (Submarine Line Terminal Equipment) has transponders and a ROADM (Reconfigurable optical add-drop multiplexer) used for handling the signals in the cable via software control. The ROADM is used to improve the reliability of the cable by allowing it to operate even if it has faults. This equipment is located inside a cable landing station (CLS). C-OTDR (Coherent Optical Time Domain Reflectometry) is used in submarine cables to detect the location of cable faults. The wet plant of a submarine cable comprises the cable itself, branching units, repeaters and possibly OADMs (Optical add-drop multiplexers).
Investment and finances
A typical multi-terabit, transoceanic submarine cable system costs several hundred million dollars to construct. Almost all fiber-optic cables from TAT-8 in 1988 until approximately 1997 were constructed by consortia of operators. For example, TAT-8 counted 35 participants including most major international carriers at the time such as AT&T Corporation. Two privately financed, non-consortium cables were constructed in the late 1990s, which preceded a massive, speculative rush to construct privately financed cables that peaked in more than $22 billion worth of investment between 1999 and 2001. This was followed by the bankruptcy and reorganization of cable operators such as Global Crossing, 360networks, FLAG, Worldcom, and Asia Global Crossing. Tata Communications' Global Network (TGN) is the only wholly owned fiber network circling the planet. | Submarine communications cable | Wikipedia | 469 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Most cables in the 20th century crossed the Atlantic Ocean, to connect the United States and Europe. However, capacity in the Pacific Ocean was much expanded starting in the 1990s. For example, between 1998 and 2003, approximately 70% of undersea fiber-optic cable was laid in the Pacific. This is in part a response to the emerging significance of Asian markets in the global economy.
After decades of heavy investment in already developed markets such as the transatlantic and transpacific routes, efforts increased in the 21st century to expand the submarine cable network to serve the developing world. For instance, in July 2009, an underwater fiber-optic cable line plugged East Africa into the broader Internet. The company that provided this new cable was SEACOM, which is 75% owned by East African and South African investors. The project was delayed by a month due to increased piracy along the coast.
Investments in cables present a commercial risk because cables cover 6,200 km of ocean floor, cross submarine mountain ranges and rifts. Because of this most companies only purchase capacity after the cable is finished.
Antarctica
Antarctica is the only continent not yet reached by a submarine telecommunications cable. Phone, video, and e-mail traffic must be relayed to the rest of the world via satellite links that have limited availability and capacity. Bases on the continent itself are able to communicate with one another via radio, but this is only a local network. To be a viable alternative, a fiber-optic cable would have to be able to withstand temperatures of as well as massive strain from ice flowing up to per year. Thus, plugging into the larger Internet backbone with the high bandwidth afforded by fiber-optic cable is still an as-yet infeasible economic and technical challenge in the Antarctic.
Arctic
The climate change induced melting of Arctic ice has provided the opportunity to lay new cable networks, linking continents and remote regions. Several projects are underway in the Arctic including 12,650 km "Polar Express" and 14,500 km Far North Fiber. However, scholars have raised environmental concerns about the laying of submarine cables in the region and the general lack of a nuanced regulatory framework. Environmental concerns pertain both to ice-related hazards damaging the cables, and cable installation disturbing the seabed or electromagnetic fields and thermal radiation of the cables impacting sensitive organisms. | Submarine communications cable | Wikipedia | 465 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Importance of submarine cables
Submarine cables, while often perceived as ‘insignificant’ parts of communication infrastructure as they lay “hidden” in the seabed, are an essential infrastructure in the digital era, carrying 99% of the data traffic across the oceans. This data includes all internet traffic, military transmissions, and financial transactions.
The total carrying capacity of a submarine cable is in the terabits per second, while a satellite typically offers only 1 gigabit per second, a ratio of more than 1000 to 1. Satellites handle less than 5% - to an estimate of even 0.5% - of global data transmission, and are less efficient, slower, and more expensive. Therefore, satellites are often exclusively considered for remote areas with challenging conditions for laying submarine cables. Submarine cables are thus the essential technical infrastructure for all internet communication.
National security
As a result of these cables' cost and usefulness, they are highly valued not only by the corporations building and operating them for profit, but also by national governments. For instance, the Australian government considers its submarine cable systems to be "vital to the national economy". Accordingly, the Australian Communications and Media Authority (ACMA) has created protection zones that restrict activities that could potentially damage cables linking Australia to the rest of the world. The ACMA also regulates all projects to install new submarine cables.
Due to their critical role, disruptions to these cables can lead to communication blackouts and, thus, extensive economic losses. The impact of such disruptions is often exemplified by the 2022 Tonga volcanic eruption that severed the island's only submarine cable and thus connectivity to the rest of the world for several days. The cable break was declared a “national crisis,” and repairs took several weeks, leaving Tonga largely isolated during a crucial period for disaster response.
Submarine cable infrastructure may even have additional technical advantages, such as carrying SMART environmental sensors supporting national disaster early warning systems. Furthermore, the cables are predicted to become even more critical with growing demands from 5G networks, the ‘Internet of Things’ (IoT), and artificial intelligence on large data transfers.
International security
Submarine communication cables are a critical infrastructure within the context of international security. Transmitting massive amounts of sensitive data every day, they are essential for both state operations and private enterprises. One of the catalysts for the amount and sensitivity of data flowing through these cables has been the global rise of cloud computing. | Submarine communications cable | Wikipedia | 483 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
The U.S military, for example, uses the submarine cable network for data transfer from conflict zones to command staff in the United States (U.S.). Interruption of the cable network during intense operations could have direct consequences for the military on the ground.
The criticality of cable services makes their geopolitical influence profound. Scholars argue that state dominance in cable networks can exert political pressure, or shape global internet governance.
An example of such state dominance in the global cable infrastructure is China’s ‘Digital Silk Road’ strategy funding the expansion of Chinese cable networks, with the Chinese company HMN Technologies often criticised for providing networks for other states, holding up to 10% of the global market share. Some critiques argue that Chinese investments in critical cable infrastructure, being involvement in approximately 25% of global submarine cables, such as the PEACE cable linking Eastafrica and Europe, may enable China to reroute data traffic through its own networks, and thus apply political pressure. The strategy is countered by the U.S., supporting alternative projects.
Vulnerabilities of submarine cables to organized crime
Submarine cables are exposed to a variety of potential threats. Many of these threats are accidental, such as by fishing trawlers, ship anchors, earthquakes, turbidity currents, and even shark bites.
Based on surveying breaks in the Atlantic Ocean and the Caribbean Sea, it was found that between 1959 and 1996, fewer than 9% were due to natural events. In response to this threat to the communications network, the practice of cable burial has developed. The average incidence of cable faults was 3.7 per per year from 1959 to 1979. That rate was reduced to 0.44 faults per 1,000 km per year after 1985, due to widespread burial of cable starting in 1980.
Still, cable breaks are by no means a thing of the past, with more than 50 repairs a year in the Atlantic Ocean alone, and significant breaks in 2006, 2008, 2009 and 2011.
Several vulnerabilities of submarine communication cables make them attractive targets for organized crime. The following section explores these vulnerabilities and currently proposed counter measures to organized crime from different perspectives.
Technical perspective
Technical vulnerabilities
The remoteness of these cables in international waters, poses significant challenges for continuous monitoring and increases their attractiveness as targets of physical tampering, data theft, and service disruptions. | Submarine communications cable | Wikipedia | 483 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
The cables' vulnerability is further compounded by technological advancements, such as the development of Unmanned Underwater Vehicles (UUVs), which enable covert cable damage while avoiding detection. However, even low-tech attacks can impact the cable's security significantly, as demonstrated in 2013, when three divers were arrested for severing the main cable linking Egypt with Europe, drastically lowering Egypt's internet speed.
Even in shallow waters, cables remain exposed to risks, as illustrated in the context of the Korea Strait. Such sea passages are often marked as ‘maritime choke points’ where several nations have conflicting interests, increasing the risk of harm from shipping activities and disputes.
Further, most cable locations are publicly available, making them an easy target for criminal acts such as disrupting services or stealing cable materials, which potentially lead to substantial communication blackouts. The stealing of submarine cable has been reported in Vietnam, where more than 11 km of cables went missing in 2007 and were later presumed to be found on fishing boats, attributed to their incentives to sell them, according to media reports.
Technical countermeasures
Typically, cables are buried in waters with a depth of less than 2,000 meters, but increasingly, they are buried in deeper seabed as a means of protection against high seas fishing and bottom trawling. However, this may also be advantageous against physical attacks from organized crime.
Further technical solutions are advanced protective casings, and monitoring them with, e.g., UUVs. Such technical solutions, however, can be challenging to implement and are limited in the remote areas of the high sea. Other proposed solutions include spatial modelling through protective or safety zones and penalties, increasing resources for surveillance, and a more collaborative approach between states and the private sector. However, how to implement and enforce these solutions remains to be determined. The cables' remoteness thus complicates both physical attacks and their protection.
Cable repair
Shore stations can locate a break in a cable by electrical measurements, such as through spread-spectrum time-domain reflectometry (SSTDR), a type of time-domain reflectometry that can be used in live environments very quickly. Presently, SSTDR can collect a complete data set in 20ms. Spread spectrum signals are sent down the wire and then the reflected signal is observed. It is then correlated with the copy of the sent signal and algorithms are applied to the shape and timing of the signals to locate the break. | Submarine communications cable | Wikipedia | 497 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
A cable repair ship will be sent to the location to drop a marker buoy near the break. Several types of grapples are used depending on the situation. If the sea bed in question is sandy, a grapple with rigid prongs is used to plough under the surface and catch the cable. If the cable is on a rocky sea surface, the grapple is more flexible, with hooks along its length so that it can adjust to the changing surface. In especially deep water, the cable may not be strong enough to lift as a single unit, so a special grapple that cuts the cable soon after it has been hooked is used and only one length of cable is brought to the surface at a time, whereupon a new section is spliced in. The repaired cable is longer than the original, so the excess is deliberately laid in a "U" shape on the seabed. A submersible can be used to repair cables that lie in shallower waters.
A number of ports near important cable routes became homes to specialized cable repair ships. Halifax, Nova Scotia, was home to a half dozen such vessels for most of the 20th century including long-lived vessels such as the CS Cyrus West Field, CS Minia and CS Mackay-Bennett. The latter two were contracted to recover victims from the sinking of the RMS Titanic. The crews of these vessels developed many new techniques and devices to repair and improve cable laying, such as the "plough".
Cybersecurity perspective
Cyber vulnerabilities
Increasingly, sophisticated cyber-attacks threaten the data traffic on the cables, with incentives ranging from financial gain, espionage, or extortion by either state actors or non-state actors. Further, hybrid warfare tactics can interfere with or even weaponize the data transferred by the cables. For example, low-intensity cyber-attacks can be employed for ransomware, data manipulation and theft, opening up new a new opportunity for the use of cybercrime and grey-zone tactics in interstate disputes.
The lack of binding international cybersecurity standards may create a gap in dealing with cyber-enabled sabotage, that can be used by organized crime. However, attributing an incident to a specific actor or motivation of such actor can be challenging, specifically in cyberspace. | Submarine communications cable | Wikipedia | 463 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Cyber espionage and Intelligence-gathering
The rising sophistication of cyberattacks underscores the vulnerability of submarine cables to cyberespionage, ultimately complicating their security. Techniques like cable tapping, hacking into network management systems, and targeting cable landing stations enable covert data access by intelligence agencies, with Russia, the U.S., and the United Kingdom (U.K.) noted as primary players.
These activities are driven by both strategic and economic motives, with advancements in technology making interception and data manipulation more effective and difficult to detect. Recent technological advancements increasing the vulnerability include the use of remote access portals and remote network management systems centralizing control over components, enabling attackers to monitor traffic and potentially disrupt data flows.
Intelligence-gathering techniques have been deployed since the late 19th century. Frequently at the beginning of wars, nations have cut the cables of the other sides to redirect the information flow into cables that were being monitored. The most ambitious efforts occurred in World War I, when British and German forces systematically attempted to destroy the others' worldwide communications systems by cutting their cables with surface ships or submarines.
During the Cold War, the United States Navy and National Security Agency (NSA) succeeded in placing wire taps on Soviet underwater communication lines in Operation Ivy Bells.
These historical intelligence-gathering techniques were eventually countered with technological advancements like the widespread use of end-to-end encryption minimizing the threat of wire tapping.
Cybersecurity countermeasures
Cybersecurity strategies for submarine cables, such as encryption, access controls, and continuous monitoring, primarily focus on preventing unauthorized data access but do not adequately address the physical protection of cables in vulnerable, remote, high-sea areas as stated above.
As a result, while cybersecurity protocols are effective near coastal landing points, their enforcement across vast stretches of the open ocean becomes a challenge. To address these limitations, experts suggest a broader, multi-layered approach that integrates physical security measures with international cooperation and legal frameworks, especially given the jurisdictional ambiguities in international waters.
Multilateral agreements to establish cybersecurity standards specific to submarine cables are highlighted as critical. These agreements can help bridge the jurisdictional ambiguities and often resulting enforcement gaps in international waters, which ultimately hinder effective protection and are frequently exploited by organized crime. | Submarine communications cable | Wikipedia | 476 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Some scholars advocate for heightened European Union (E.U.) coordination, recommending improvements in surveillance and response capabilities across various agencies, such as the Coast Guard and specific telecommunication regulators. Given the central role of private companies in cable ownership, some experts also underscore the need for stronger collaboration between governments and tech firms to pool resources and develop more innovative security measures tailored to this critical infrastructure.
Geopolitical perspective
Geopolitical vulnerabilities
Fishing vessels are the leading cause of accidental damage to submarine communication cables. However, some of the academic discussions and recent incidents point to geopolitical tactics influencing the cable's security more than previously expected. These tactics include the ease and potential with which fishing vessels can blend into regular maritime traffic and implement their attacks.
The propensity for fishing trawler nets to cause cable faults may well have been exploited during the Cold War. For example, in February 1959, a series of 12 breaks occurred in five American trans-Atlantic communications cables. In response, a U.S. naval vessel, the USS Roy O. Hale, detained and investigated the Soviet trawler Novorosiysk. A review of the ship's log indicated it had been in the region of each of the cables when they broke. Broken sections of cable were also found on the deck of the Novorosiysk. It appeared that the cables had been dragged along by the ship's nets, and then cut once they were pulled up onto the deck to release the nets. The Soviet Union's stance on the investigation was that it was unjustified, but the U.S. cited the Convention for the Protection of Submarine Telegraph Cables of 1884 to which Russia had signed (prior to the formation of the Soviet Union) as evidence of violation of international protocol.
Several media outlets and organizations indicate that Russian fishing vessels, particularly in 2022, passed over a damaged submarine cable up to 20 times, suggesting potential political motives and the possibility of hybrid warfare tactics used from Russia's side. Russian naval activities near submarine cables are often linked to increased hybrid warfare strategies targeting submarine cables, where sabotage is argued to serve as a tool to disrupt communication networks during conflict and destabilise adversaries. | Submarine communications cable | Wikipedia | 448 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
These tactics elevate cable security to a significant geopolitical issue. Criminal actors may further target cables as a means of economic warfare, aiming to destabilize economies or convey political messages. The disruption of submarine communication cables in highly politicised maritime areas thus has a significant political component that is receiving increased attention.
After two cable breaks in the Baltic Sea in November 2024, one between Lithuania and Sweden and the other between Finland and Germany, Defence Minister Boris Pistorius argued:
“No one believes that these cables were cut accidentally. I also don't want to believe in versions that these were ship anchors that accidentally caused the damage. Therefore, we have to state, without knowing specifically who it came from, that it is a 'hybrid' action. And we also have to assume, without knowing it yet, that it is sabotage."
This statement underlines the current discourse to recognize cable disruptions as threats to national securiy, which ultimately leads to their securitization in the international context.
Geopolitical risks and countermeasures
Submarine cables are inherently vulnerable to transnational threats like organized crime. International collaboration to address these threats tends to fall to existing organizations with a cable specific focus - such as the International Cable Protection Committee (ICPC) - which represent key submarine stakeholders, and play a vital role in promoting cooperation and information sharing among stakeholders. Such organizations are argued to be crucial to develop and implement a comprehensive and coordinated global strategy for cable security.
As of 2025, a tense U.S.-China relationship complicates this task especially in the South China Sea where there are territorial disputes. China has increasing control and influence over global cables networks, while both it and the USA financially supports allied-owned cable projects and exerts diplomatic pressure and regulatory action, e.g. against Vietnam.
In light of Nord Stream pipelines sabotage in the Baltic Sea, where subsea infrastructure vital to Germany and Russia was physically destroyed, and other incidents there, NATO has increased patrols and monitoring operations.
Legal perspective
Legal vulnerabilities
Submarine cables are internationally regulated within the framework of the United Nations Convention on the Law of the Sea (UNCLOS), in particular through the provisions of Articles 112 and 97, 112 and 115, which mandate operational freedom to lay cables in international waters and beyond the continental shelf and reward measures to protect against shipping accidents. | Submarine communications cable | Wikipedia | 483 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
However, submarine cables face significant legal challenges and lack specific legal protection in UNCLOS and enforcement mechanisms against emerging threats, particular in international waters. This is further complicated by the non-ratification of the treaty by key states such as the U.S. and Turkey. Many countries lack explicit legal provisions to criminalize the destruction or theft of undersea cables, creating jurisdictional ambiguities that organized crime can exploit. Other legal frameworks, such as the 1884 Convention for the Protection of Submarine Telegraph Cables are outdated and fail to address modern threats like cyberattacks and hybrid warfare tactics. The unclear jurisdiction and weak enforcement mechanisms, demonstrate the difficulty to protect submarine cables from organized crime.
The Arctic Ocean in particular exemplifies the challenges associated with surveillance and enforcement in vast and remote areas, leaving a legal vacuum that criminals may exploit. In the Arctic, the absence of a central international authority to oversee submarine cable protection and the reliance on military organizations like NATO hinders general coordinated global responses.
Organizations such as the ICPC thus highlight the need for updated and more comprehensive legal frameworks to ensure the security of submarine cables.
Legal countermeasures
The legal challenges of protecting submarine cables from organized crime have resulted in recommendations ranging from treaty amendments to domestic law reforms and multi-level governance models.
Some scholars argue that UNCLOS should be updated to protect cables extensively, including cooperative monitoring and enforcement protocols. Additionally, principles from the law of the sea, state responsibility, and the laws on the use of force could be creatively applied to strengthen protections for cables. Enforcement issues could be tackled by aligning domestic laws with UNCLOS, implementing national response protocols, and creating streamlined points of contact for cable incidents. Given the increased involvement of organizations like NATO, others recommend to clarify the roles of military and non-military actors in cable security and enhanced multi-level governance models.
While these proposed legal solutions seem promising, their practical implementation still remains a challenge due to the complexity of international treaties, the need for international cooperation, the lack of domestic criminalization of cable damage, and the evolving nature of technological threats. Additionally, while UNCLOS's ambiguous jurisdiction in international waters hinders effective enforcement, limited political interests seems to hamper treaty development.
Environmental impact
The presence of cables in the oceans can be a danger to marine life. With the proliferation of cable installations and the increasing demand for inter-connectivity that today's society demands, the environmental impact is increasing.
Submarine cables can impact marine life in a number of ways. | Submarine communications cable | Wikipedia | 509 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Alteration of the seabed
Seabed ecosystems can be disturbed by the installation and maintenance of cables. The effects of cable installation are generally limited to specific areas. The intensity of disturbance depends on the installation method.
Cables are often laid in the so-called benthic zone of the seabed. The benthic zone is the ecological region at the bottom of the sea where benthos, clams and crabs live, and where the surface sediments, which are deposits of matter and particles in the water that provide a habitat for marine species, are located.
Sediment can be damaged by cable installation by trenching with water jets or ploughing. This can lead to reworking of the sediments, altering the substrate of which they are composed.
According to several studies, the biota of the benthic zone is only slightly affected by the presence of cables. However, the presence of cables can trigger behavioral disturbances in living organisms. The main observation is that the presence of cables provides a hard substrate for anemones attachment. These organisms are found in large number around cables that run through soft sediments, which are not normally suitable for these organisms. This is also the case for flatfish. Although little observed, the presence of cables can also change the water temperature and therefore disturb the surrounding natural habitat.
However, these disturbances are not very persistent over time, and can stabilize within a few days. Cable operators are trying to implement measures to route cables in such a way as to avoid areas with sensitive and vulnerable ecosystems.
Entanglement
Entanglement of marine animals in cables is one of the main causes of cable damage. Whales and sperm whales are the main animals that entangle themselves in cables and damage them. The encounter between these animals and cables can cause injury and sometimes death. Studies carried out between 1877 and 1955 reported 16 cable ruptures caused by whale entanglement, 13 of them by sperm whales. Between 1907 and 2006, 39 such events were recorded. Cable burial techniques are gradually being introduced to prevent such incidents.
The risk of fishing
Although submarine cables are located on the seabed, fishing activity can damage the cables. Fishermen using fishing techniques that involve scraping the seabed, or dragging equipment such as trawls or cages, can damage the cables, resulting in the loss of liquids and the chemical and toxic materials that make up the cables. | Submarine communications cable | Wikipedia | 469 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
Areas with a high density of submarine cables have the advantage of being safer from fishing. At the expense of benthic and sedimentary zones, marine fauna is better protected in these maritime regions, thanks to limitations and bans. Studies have shown a positive effect on the fauna surrounding cable installation zones.
Pollution
Submarine cables are made of copper or optical fibers, surrounded by several protective layers of plastic, wire or synthetic materials. Cables can also be composed of dielectric fluids or hydrocarbon fluids, which act as electrical insulators. These substances can be harmful to marine life.
Fishing, aging cables and marine species that collide with or become entangled in cables can damage cables and spread toxic and harmful substances into the sea. However, the impact of submarine cables is limited compared with other sources of ocean pollution.
There is also a risk of releasing pollutants buried in sediments. When sediments are re-suspended due to the installation of cables, toxic substances such as hydrocarbons may be released.
Preliminary analyses can assess the level of sediment toxicity and select a cable route that avoids the remobilization and dispersion of sediment pollutants. And new, more modern techniques will make it possible to use less polluting materials for cable construction.
Sound waves and electromagnetic waves
The installation and maintenance of cables requires the use of machinery and equipment that can trigger sound waves or electromagnetic waves that can disturb animals that use waves to find their bearings in space or to communicate. Underwater sound waves depend on the equipment used, the characteristics of the seabed area where the cables are located, and the relief of the area.
Underwater noise and waves can modify the behavior of certain underwater species, such as migratory behavior, disrupting communication or reproduction. Available information is that underwater noise generated by submarine cable engineering operations has limited acoustic footprint and limited duration. | Submarine communications cable | Wikipedia | 366 | 45206 | https://en.wikipedia.org/wiki/Submarine%20communications%20cable | Technology | Telecommunications | null |
A communications satellite is an artificial satellite that relays and amplifies radio telecommunication signals via a transponder; it creates a communication channel between a source transmitter and a receiver at different locations on Earth. Communications satellites are used for television, telephone, radio, internet, and military applications. Many communications satellites are in geostationary orbit above the equator, so that the satellite appears stationary at the same point in the sky; therefore the satellite dish antennas of ground stations can be aimed permanently at that spot and do not have to move to track the satellite. Others form satellite constellations in low Earth orbit, where antennas on the ground have to follow the position of the satellites and switch between satellites frequently.
The radio waves used for telecommunications links travel by line of sight and so are obstructed by the curve of the Earth. The purpose of communications satellites is to relay the signal around the curve of the Earth allowing communication between widely separated geographical points. Communications satellites use a wide range of radio and microwave frequencies. To avoid signal interference, international organizations have regulations for which frequency ranges or "bands" certain organizations are allowed to use. This allocation of bands minimizes the risk of signal interference.
History
Origins
In October 1945, Arthur C. Clarke published an article titled "Extraterrestrial Relays" in the British magazine Wireless World. The article described the fundamentals behind the deployment of artificial satellites in geostationary orbits to relay radio signals. Because of this, Arthur C. Clarke is often quoted as being the inventor of the concept of the communications satellite, and the term 'Clarke Belt' is employed as a description of the orbit.
The first artificial Earth satellite was Sputnik 1, which was put into orbit by the Soviet Union on 4 October 1957. It was developed by Mikhail Tikhonravov and Sergey Korolev, building on work by Konstantin Tsiolkovsky. Sputnik 1 was equipped with an on-board radio transmitter that worked on two frequencies of 20.005 and 40.002 MHz, or 7 and 15 meters wavelength. The satellite was not placed in orbit to send data from one point on Earth to another, but the radio transmitter was meant to study the properties of radio wave distribution throughout the ionosphere. The launch of Sputnik 1 was a major step in the exploration of space and rocket development, and marks the beginning of the Space Age. | Communications satellite | Wikipedia | 482 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Early active and passive satellite experiments
There are two major classes of communications satellites, passive and active. Passive satellites only reflect the signal coming from the source, toward the direction of the receiver. With passive satellites, the reflected signal is not amplified at the satellite, and only a small amount of the transmitted energy actually reaches the receiver. Since the satellite is so far above Earth, the radio signal is attenuated due to free-space path loss, so the signal received on Earth is very weak. Active satellites, on the other hand, amplify the received signal before retransmitting it to the receiver on the ground. Passive satellites were the first communications satellites, but are little used now.
Work that was begun in the field of electrical intelligence gathering at the United States Naval Research Laboratory in 1951 led to a project named Communication Moon Relay. Military planners had long shown considerable interest in secure and reliable communications lines as a tactical necessity, and the ultimate goal of this project was the creation of the longest communications circuit in human history, with the Moon, Earth's natural satellite, acting as a passive relay. After achieving the first transoceanic communication between Washington, D.C., and Hawaii on 23 January 1956, this system was publicly inaugurated and put into formal production in January 1960.
The first satellite purpose-built to actively relay communications was Project SCORE, led by Advanced Research Projects Agency (ARPA) and launched on 18 December 1958, which used a tape recorder to carry a stored voice message, as well as to receive, store, and retransmit messages. It was used to send a Christmas greeting to the world from U.S. President Dwight D. Eisenhower. The satellite also executed several realtime transmissions before the non-rechargeable batteries failed on 30 December 1958 after eight hours of actual operation.
The direct successor to SCORE was another ARPA-led project called Courier. Courier 1B was launched on 4 October 1960 to explore whether it would be possible to establish a global military communications network by using "delayed repeater" satellites, which receive and store information until commanded to rebroadcast them. After 17 days, a command system failure ended communications from the satellite. | Communications satellite | Wikipedia | 444 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
NASA's satellite applications program launched the first artificial satellite used for passive relay communications in Echo 1 on 12 August 1960. Echo 1 was an aluminized balloon satellite acting as a passive reflector of microwave signals. Communication signals were bounced off the satellite from one point on Earth to another. This experiment sought to establish the feasibility of worldwide broadcasts of telephone, radio, and television signals.
More firsts and further experiments
Telstar was the first active, direct relay communications commercial satellite and marked the first transatlantic transmission of television signals. Belonging to AT&T as part of a multi-national agreement between AT&T, Bell Telephone Laboratories, NASA, the British General Post Office, and the French National PTT (Post Office) to develop satellite communications, it was launched by NASA from Cape Canaveral on 10 July 1962, in the first privately sponsored space launch.
Another passive relay experiment primarily intended for military communications purposes was Project West Ford, which was led by Massachusetts Institute of Technology's Lincoln Laboratory. After an initial failure in 1961, a launch on 9 May 1963 dispersed 350 million copper needle dipoles to create a passive reflecting belt. Even though only about half of the dipoles properly separated from each other, the project was able to successfully experiment and communicate using frequencies in the SHF X band spectrum.
An immediate antecedent of the geostationary satellites was the Hughes Aircraft Company's Syncom 2, launched on 26 July 1963. Syncom 2 was the first communications satellite in a geosynchronous orbit. It revolved around the Earth once per day at constant speed, but because it still had north–south motion, special equipment was needed to track it. Its successor, Syncom 3, launched on 19 July 1964, was the first geostationary communications satellite. Syncom 3 obtained a geosynchronous orbit, without a north–south motion, making it appear from the ground as a stationary object in the sky.
A direct extension of the passive experiments of Project West Ford was the Lincoln Experimental Satellite program, also conducted by the Lincoln Laboratory on behalf of the United States Department of Defense. The LES-1 active communications satellite was launched on 11 February 1965 to explore the feasibility of active solid-state X band long-range military communications. A total of nine satellites were launched between 1965 and 1976 as part of this series.
International commercial satellite projects | Communications satellite | Wikipedia | 479 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
In the United States, 1962 saw the creation of the Communications Satellite Corporation (COMSAT) private corporation, which was subject to instruction by the US Government on matters of national policy. Over the next two years, international negotiations led to the Intelsat Agreements, which in turn led to the launch of Intelsat 1, also known as Early Bird, on 6 April 1965, and which was the first commercial communications satellite to be placed in geosynchronous orbit. Subsequent Intelsat launches in the 1960s provided multi-destination service and video, audio, and data service to ships at sea (Intelsat 2 in 1966–67), and the completion of a fully global network with Intelsat 3 in 1969–70. By the 1980s, with significant expansions in commercial satellite capacity, Intelsat was on its way to become part of the competitive private telecommunications industry, and had started to get competition from the likes of PanAmSat in the United States, which, ironically, was then bought by its archrival in 2005.
When Intelsat was launched, the United States was the only launch source outside of the Soviet Union, who did not participate in the Intelsat agreements. The Soviet Union launched its first communications satellite on 23 April 1965 as part of the Molniya program. This program was also unique at the time for its use of what then became known as the Molniya orbit, which describes a highly elliptical orbit, with two high apogees daily over the northern hemisphere. This orbit provides a long dwell time over Russian territory as well as over Canada at higher latitudes than geostationary orbits over the equator.
In the 2020s, the popularity of low Earth orbit satellite internet constellations providing relatively low-cost internet services led to reducing demand for new geostationary orbit communications satellites.
Satellite orbits | Communications satellite | Wikipedia | 364 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Communications satellites usually have one of three primary types of orbit, while other orbital classifications are used to further specify orbital details. MEO and LEO are non-geostationary orbit (NGSO).
Geostationary satellites have a geostationary orbit (GEO), which is from Earth's surface. This orbit has the special characteristic that the apparent position of the satellite in the sky when viewed by a ground observer does not change, the satellite appears to "stand still" in the sky. This is because the satellite's orbital period is the same as the rotation rate of the Earth. The advantage of this orbit is that ground antennas do not have to track the satellite across the sky, they can be fixed to point at the location in the sky the satellite appears.
Medium Earth orbit (MEO) satellites are closer to Earth. Orbital altitudes range from above Earth.
The region below medium orbits is referred to as low Earth orbit (LEO), and is about above Earth.
As satellites in MEO and LEO orbit the Earth faster, they do not remain visible in the sky to a fixed point on Earth continually like a geostationary satellite, but appear to a ground observer to cross the sky and "set" when they go behind the Earth beyond the visible horizon. Therefore, to provide continuous communications capability with these lower orbits requires a larger number of satellites, so that one of these satellites will always be visible in the sky for transmission of communication signals. However, due to their closer distance to the Earth, LEO or MEO satellites can communicate to ground with reduced latency and at lower power than would be required from a geosynchronous orbit.
Low Earth orbit (LEO)
A low Earth orbit (LEO) typically is a circular orbit about above the Earth's surface and, correspondingly, a period (time to revolve around the Earth) of about 90 minutes.
Because of their low altitude, these satellites are only visible from within a radius of roughly from the sub-satellite point. In addition, satellites in low Earth orbit change their position relative to the ground position quickly. So even for local applications, many satellites are needed if the mission requires uninterrupted connectivity. | Communications satellite | Wikipedia | 445 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Low-Earth-orbiting satellites are less expensive to launch into orbit than geostationary satellites and, due to proximity to the ground, do not require as high signal strength (signal strength falls off as the square of the distance from the source, so the effect is considerable). Thus there is a trade off between the number of satellites and their cost.
In addition, there are important differences in the onboard and ground equipment needed to support the two types of missions.
Satellite constellation
A group of satellites working in concert is known as a satellite constellation. Two such constellations, intended to provide satellite phone and low-speed data services, primarily to remote areas, are the Iridium and Globalstar systems. The Iridium system has 66 satellites, which orbital inclination of 86.4° and inter-satellite links provide service availability over the entire surface of Earth. Starlink is a satellite internet constellation operated by SpaceX, that aims for global satellite Internet access coverage.
It is also possible to offer discontinuous coverage using a low-Earth-orbit satellite capable of storing data received while passing over one part of Earth and transmitting it later while passing over another part. This will be the case with the CASCADE system of Canada's CASSIOPE communications satellite. Another system using this store and forward method is Orbcomm.
Medium Earth orbit (MEO)
A medium Earth orbit is a satellite in orbit somewhere between above the Earth's surface. MEO satellites are similar to LEO satellites in functionality. MEO satellites are visible for much longer periods of time than LEO satellites, usually between 2 and 8 hours. MEO satellites have a larger coverage area than LEO satellites. A MEO satellite's longer duration of visibility and wider footprint means fewer satellites are needed in a MEO network than a LEO network. One disadvantage is that a MEO satellite's distance gives it a longer time delay and weaker signal than a LEO satellite, although these limitations are not as severe as those of a GEO satellite.
Like LEOs, these satellites do not maintain a stationary distance from the Earth. This is in contrast to the geostationary orbit, where satellites are always from Earth.
Typically the orbit of a medium Earth orbit satellite is about above Earth. In various patterns, these satellites make the trip around Earth in anywhere from 2 to 8 hours. | Communications satellite | Wikipedia | 469 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Examples of MEO
In 1962, the communications satellite, Telstar, was launched. It was a medium Earth orbit satellite designed to help facilitate high-speed telephone signals. Although it was the first practical way to transmit signals over the horizon, its major drawback was soon realised. Because its orbital period of about 2.5 hours did not match the Earth's rotational period of 24 hours, continuous coverage was impossible. It was apparent that multiple MEOs needed to be used in order to provide continuous coverage.
In 2013, the first four of a constellation of 20 MEO satellites was launched. The O3b satellites provide broadband internet services, in particular to remote locations and maritime and in-flight use, and orbit at an altitude of ).
Geostationary orbit (GEO)
To an observer on Earth, a satellite in a gestationary orbit appears motionless, in a fixed position in the sky. This is because it revolves around the Earth at Earth's own angular velocity (one revolution per sidereal day, in an equatorial orbit).
A geostationary orbit is useful for communications because ground antennas can be aimed at the satellite without their having to track the satellite's motion. This is relatively inexpensive.
In applications that require many ground antennas, such as DirecTV distribution, the savings in ground equipment can more than outweigh the cost and complexity of placing a satellite into orbit. | Communications satellite | Wikipedia | 281 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Examples of GEO
The first geostationary satellite was Syncom 3, launched on 19 August 1964, and used for communication across the Pacific starting with television coverage of the 1964 Summer Olympics. Shortly after Syncom 3, Intelsat I, aka Early Bird, was launched on 6 April 1965 and placed in orbit at 28° west longitude. It was the first geostationary satellite for telecommunications over the Atlantic Ocean.
On 9 November 1972, Canada's first geostationary satellite serving the continent, Anik A1, was launched by Telesat Canada, with the United States following suit with the launch of Westar 1 by Western Union on 13 April 1974.
On 30 May 1974, the first geostationary communications satellite in the world to be three-axis stabilized was launched: the experimental satellite ATS-6 built for NASA.
After the launches of the Telstar through Westar 1 satellites, RCA Americom (later GE Americom, now SES) launched Satcom 1 in 1975. It was Satcom 1 that was instrumental in helping early cable TV channels such as WTBS (now TBS), HBO, CBN (now Freeform) and The Weather Channel become successful, because these channels distributed their programming to all of the local cable TV headends using the satellite. Additionally, it was the first satellite used by broadcast television networks in the United States, like ABC, NBC, and CBS, to distribute programming to their local affiliate stations. Satcom 1 was widely used because it had twice the communications capacity of the competing Westar 1 in America (24 transponders as opposed to the 12 of Westar 1), resulting in lower transponder-usage costs. Satellites in later decades tended to have even higher transponder numbers.
By 2000, Hughes Space and Communications (now Boeing Satellite Development Center) had built nearly 40 percent of the more than one hundred satellites in service worldwide. Other major satellite manufacturers include Space Systems/Loral, Orbital Sciences Corporation with the Star Bus series, Indian Space Research Organisation, Lockheed Martin (owns the former RCA Astro Electronics/GE Astro Space business), Northrop Grumman, Alcatel Space, now Thales Alenia Space, with the Spacebus series, and Astrium.
Molniya orbit | Communications satellite | Wikipedia | 460 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Geostationary satellites must operate above the equator and therefore appear lower on the horizon as the receiver gets farther from the equator. This will cause problems for extreme northerly latitudes, affecting connectivity and causing multipath interference (caused by signals reflecting off the ground and into the ground antenna).
Thus, for areas close to the North (and South) Pole, a geostationary satellite may appear below the horizon. Therefore, Molniya orbit satellites have been launched, mainly in Russia, to alleviate this problem.
Molniya orbits can be an appealing alternative in such cases. The Molniya orbit is highly inclined, guaranteeing good elevation over selected positions during the northern portion of the orbit. (Elevation is the extent of the satellite's position above the horizon. Thus, a satellite at the horizon has zero elevation and a satellite directly overhead has elevation of 90 degrees.)
The Molniya orbit is designed so that the satellite spends the great majority of its time over the far northern latitudes, during which its ground footprint moves only slightly. Its period is one half day, so that the satellite is available for operation over the targeted region for six to nine hours every second revolution. In this way a constellation of three Molniya satellites (plus in-orbit spares) can provide uninterrupted coverage.
The first satellite of the Molniya series was launched on 23 April 1965 and was used for experimental transmission of TV signals from a Moscow uplink station to downlink stations located in Siberia and the Russian Far East, in Norilsk, Khabarovsk, Magadan and Vladivostok. In November 1967 Soviet engineers created a unique system of national TV network of satellite television, called Orbita, that was based on Molniya satellites.
Polar orbit
In the United States, the National Polar-orbiting Operational Environmental Satellite System (NPOESS) was established in 1994 to consolidate the polar satellite operations of
NASA (National Aeronautics and Space Administration)
NOAA (National Oceanic and Atmospheric Administration). NPOESS manages a number of satellites for various purposes; for example, METSAT for meteorological satellite, EUMETSAT for the European branch of the program, and METOP for meteorological operations. | Communications satellite | Wikipedia | 448 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
These orbits are Sun synchronous, meaning that they cross the equator at the same local time each day. For example, the satellites in the NPOESS (civilian) orbit will cross the equator, going from south to north, at times 1:30 P.M., 5:30 P.M., and 9:30 P.M.
Beyond geostationary orbit
There are plans and initiatives to bring dedicated communications satellite beyond geostationary orbits.
NASA proposed LunaNet as a data network aiming to provide a "Lunar Internet" for cis-lunar spacecraft and Installations.
The Moonlight Initiative is an equivalent ESA project that is stated to be compatible and providing navigational services for the lunar surface. Both programmes are satellite constellations of several satellites in various orbits around the Moon.
Other orbits are also planned to be used. Positions in the Earth-Moon-Libration points are also proposed for communication satellites covering the Moon alike communication satellites in geosynchronous orbit cover the Earth. Also, dedicated communication satellites in orbits around Mars supporting different missions on surface and other orbits are considered, such as the Mars Telecommunications Orbiter.
Structure
Communications Satellites are usually composed of the following subsystems:
Communication Payload, normally composed of transponders, antennas, amplifiers and switching systems
Engines used to bring the satellite to its desired orbit
A station keeping tracking and stabilization subsystem used to keep the satellite in the right orbit, with its antennas pointed in the right direction, and its power system pointed towards the Sun
Power subsystem, used to power the Satellite systems, normally composed of solar cells, and batteries that maintain power during solar eclipse
Command and Control subsystem, which maintains communications with ground control stations. The ground control Earth stations monitor the satellite performance and control its functionality during various phases of its life-cycle.
The bandwidth available from a satellite depends upon the number of transponders provided by the satellite. Each service (TV, Voice, Internet, radio) requires a different amount of bandwidth for transmission. This is typically known as link budgeting and a network simulator can be used to arrive at the exact value. | Communications satellite | Wikipedia | 431 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Frequency allocation for satellite systems
Allocating frequencies to satellite services is a complicated process which requires international coordination and planning. This is carried out under the auspices of the International Telecommunication Union (ITU).
To facilitate frequency planning, the world is divided into three regions:
Region 1: Europe, Africa, the Middle East, what was formerly the Soviet Union, and Mongolia
Region 2: North and South America and Greenland
Region 3: Asia (excluding region 1 areas), Australia, and the southwest Pacific
Within these regions, frequency bands are allocated to various satellite services, although a given service may be allocated different frequency bands in different regions. Some of the services provided by satellites are:
Fixed satellite service (FSS)
Broadcasting satellite service (BSS)
Mobile-satellite service
Radionavigation-satellite service
Meteorological-satellite service
Applications
Telephony
The first and historically most important application for communication satellites was in intercontinental long distance telephony. The fixed Public Switched Telephone Network relays telephone calls from land line telephones to an Earth station, where they are then transmitted to a geostationary satellite. The downlink follows an analogous path. Improvements in submarine communications cables through the use of fiber-optics caused some decline in the use of satellites for fixed telephony in the late 20th century.
Satellite communications are still used in many applications today. Remote islands such as Ascension Island, Saint Helena, Diego Garcia, and Easter Island, where no submarine cables are in service, need satellite telephones. There are also regions of some continents and countries where landline telecommunications are rare to non existent, for example large regions of South America, Africa, Canada, China, Russia, and Australia. Satellite communications also provide connection to the edges of Antarctica and Greenland. Other land use for satellite phones are rigs at sea, a backup for hospitals, military, and recreation. Ships at sea, as well as planes, often use satellite phones.
Satellite phone systems can be accomplished by a number of means. On a large scale, often there will be a local telephone system in an isolated area with a link to the telephone system in a main land area. There are also services that will patch a radio signal to a telephone system. In this example, almost any type of satellite can be used. Satellite phones connect directly to a constellation of either geostationary or low-Earth-orbit satellites. Calls are then forwarded to a satellite teleport connected to the Public Switched Telephone Network .
Television | Communications satellite | Wikipedia | 494 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
As television became the main market, its demand for simultaneous delivery of relatively few signals of large bandwidth to many receivers being a more precise match for the capabilities of geosynchronous comsats. Two satellite types are used for North American television and radio: Direct broadcast satellite (DBS), and Fixed Service Satellite (FSS).
The definitions of FSS and DBS satellites outside of North America, especially in Europe, are a bit more ambiguous. Most satellites used for direct-to-home television in Europe have the same high power output as DBS-class satellites in North America, but use the same linear polarization as FSS-class satellites. Examples of these are the Astra, Eutelsat, and Hotbird spacecraft in orbit over the European continent. Because of this, the terms FSS and DBS are more so used throughout the North American continent, and are uncommon in Europe.
Fixed Service Satellites use the C band, and the lower portions of the Ku band. They are normally used for broadcast feeds to and from television networks and local affiliate stations (such as program feeds for network and syndicated programming, live shots, and backhauls), as well as being used for distance learning by schools and universities, business television (BTV), Videoconferencing, and general commercial telecommunications. FSS satellites are also used to distribute national cable channels to cable television headends.
Free-to-air satellite TV channels are also usually distributed on FSS satellites in the Ku band. The Intelsat Americas 5, Galaxy 10R and AMC 3 satellites over North America provide a quite large amount of FTA channels on their Ku band transponders.
The American Dish Network DBS service has also recently used FSS technology as well for their programming packages requiring their SuperDish antenna, due to Dish Network needing more capacity to carry local television stations per the FCC's "must-carry" regulations, and for more bandwidth to carry HDTV channels. | Communications satellite | Wikipedia | 402 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
A direct broadcast satellite is a communications satellite that transmits to small DBS satellite dishes (usually 18 to 24 inches or 45 to 60 cm in diameter). Direct broadcast satellites generally operate in the upper portion of the microwave Ku band. DBS technology is used for DTH-oriented (Direct-To-Home) satellite TV services, such as DirecTV, DISH Network and Orby TV in the United States, Bell Satellite TV and Shaw Direct in Canada, Freesat and Sky in the UK, Ireland, and New Zealand and DSTV in South Africa.
Operating at lower frequency and lower power than DBS, FSS satellites require a much larger dish for reception (3 to 8 feet (1 to 2.5 m) in diameter for Ku band, and 12 feet (3.6 m) or larger for C band). They use linear polarization for each of the transponders' RF input and output (as opposed to circular polarization used by DBS satellites), but this is a minor technical difference that users do not notice. FSS satellite technology was also originally used for DTH satellite TV from the late 1970s to the early 1990s in the United States in the form of TVRO (Television Receive Only) receivers and dishes. It was also used in its Ku band form for the now-defunct Primestar satellite TV service.
Some satellites have been launched that have transponders in the Ka band, such as DirecTV's SPACEWAY-1 satellite, and Anik F2. NASA and ISRO have also launched experimental satellites carrying Ka band beacons recently.
Some manufacturers have also introduced special antennas for mobile reception of DBS television. Using Global Positioning System (GPS) technology as a reference, these antennas automatically re-aim to the satellite no matter where or how the vehicle (on which the antenna is mounted) is situated. These mobile satellite antennas are popular with some recreational vehicle owners. Such mobile DBS antennas are also used by JetBlue Airways for DirecTV (supplied by LiveTV, a subsidiary of JetBlue), which passengers can view on-board on LCD screens mounted in the seats.
Radio broadcasting
Satellite radio offers audio broadcast services in some countries, notably the United States. Mobile services allow listeners to roam a continent, listening to the same audio programming anywhere.
A satellite radio or subscription radio (SR) is a digital radio signal that is broadcast by a communications satellite, which covers a much wider geographical range than terrestrial radio signals.
Amateur radio | Communications satellite | Wikipedia | 503 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
Amateur radio operators have access to amateur satellites, which have been designed specifically to carry amateur radio traffic. Most such satellites operate as spaceborne repeaters, and are generally accessed by amateurs equipped with UHF or VHF radio equipment and highly directional antennas such as Yagis or dish antennas. Due to launch costs, most current amateur satellites are launched into fairly low Earth orbits, and are designed to deal with only a limited number of brief contacts at any given time. Some satellites also provide data-forwarding services using the X.25 or similar protocols.
Internet access
After the 1990s, satellite communication technology has been used as a means to connect to the Internet via broadband data connections. This can be very useful for users who are located in remote areas, and cannot access a broadband connection, or require high availability of services.
Military
Communications satellites are used for military communications applications, such as Global Command and Control Systems. Examples of military systems that use communication satellites are the MILSTAR, the DSCS, and the FLTSATCOM of the United States, NATO satellites, United Kingdom satellites (for instance Skynet), and satellites of the former Soviet Union. India has launched its first Military Communication satellite GSAT-7, its transponders operate in UHF, F, C and bands. Typically military satellites operate in the UHF, SHF (also known as X-band) or EHF (also known as Ka band) frequency bands.
Data collection
Near-ground in situ environmental monitoring equipment (such as tide gauges, weather stations, weather buoys, and radiosondes), may use satellites for one-way data transmission or two-way telemetry and telecontrol. It may be based on a secondary payload of a weather satellite (as in the case of GOES and METEOSAT and others in the Argos system) or in dedicated satellites (such as SCD). The data rate is typically much lower than in satellite Internet access. | Communications satellite | Wikipedia | 393 | 45207 | https://en.wikipedia.org/wiki/Communications%20satellite | Technology | Media and communication | null |
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.
The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.
For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for rings.
Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is isomorphic to the quotient by the kernel.
The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a congruence relation.
This article is a survey for some important types of kernels in algebraic structures.
Survey of examples
Linear maps
Let V and W be vector spaces over a field (or more generally, modules over a ring) and let T be a linear map from V to W. If 0W is the zero vector of W, then the kernel of T is the preimage of the zero subspace {0W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0W. The kernel is usually denoted as , or some variation thereof:
Since a linear map preserves zero vectors, the zero vector 0V of V must belong to the kernel. The transformation T is injective if and only if its kernel is reduced to the zero subspace. | Kernel (algebra) | Wikipedia | 494 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
The kernel ker T is always a linear subspace of V. Thus, it makes sense to speak of the quotient space . The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T (which is a subspace of W). As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image.
If V and W are finite-dimensional and bases have been chosen, then T can be described by a matrix M, and the kernel can be computed by solving the homogeneous system of linear equations . In this case, the kernel of T may be identified to the kernel of the matrix M, also called "null space" of M. The dimension of the null space, called the nullity of M, is given by the number of columns of M minus the rank of M, as a consequence of the rank–nullity theorem.
Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators.
For instance, in order to find all twice-differentiable functions f from the real line to itself such that
let V be the space of all twice differentiable functions, let W be the space of all functions, and define a linear operator T from V to W by
for f in V and x an arbitrary real number.
Then all solutions to the differential equation are in .
One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory).
Group homomorphisms
Let G and H be groups and let f be a group homomorphism from G to H. If eH is the identity element of H, then the kernel of f is the preimage of the singleton set {eH}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element eH.
The kernel is usually denoted (or a variation). In symbols:
Since a group homomorphism preserves identity elements, the identity element eG of G must belong to the kernel. | Kernel (algebra) | Wikipedia | 447 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
The homomorphism f is injective if and only if its kernel is only the singleton set {eG}. If f were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist such that and . Thus . f is a group homomorphism, so inverses and group operations are preserved, giving ; in other words, , and ker f would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element , then , thus f would not be injective.
is a subgroup of G and further it is a normal subgroup. Thus, there is a corresponding quotient group . This is isomorphic to f(G), the image of G under f (which is a subgroup of H also), by the first isomorphism theorem for groups.
In the special case of abelian groups, there is no deviation from the previous section.
Example
Let G be the cyclic group on 6 elements with modular addition, H be the cyclic on 2 elements with modular addition, and f the homomorphism that maps each element g in G to the element g modulo 2 in H. Then , since all these elements are mapped to 0H. The quotient group has two elements: and . It is indeed isomorphic to H.
Ring homomorphisms
Let R and S be rings (assumed unital) and let f be a ring homomorphism from R to S.
If 0S is the zero element of S, then the kernel of f is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the zero ideal , which is, the subset of R consisting of all those elements of R that are mapped by f to the element 0S.
The kernel is usually denoted (or a variation).
In symbols:
Since a ring homomorphism preserves zero elements, the zero element 0R of R must belong to the kernel.
The homomorphism f is injective if and only if its kernel is only the singleton set .
This is always the case if R is a field, and S is not the zero ring. | Kernel (algebra) | Wikipedia | 450 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
Since ker f contains the multiplicative identity only when S is the zero ring, it turns out that the kernel is generally not a subring of R. The kernel is a subrng, and, more precisely, a two-sided ideal of R.
Thus, it makes sense to speak of the quotient ring .
The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S). (Note that rings need not be unital for the kernel definition).
To some extent, this can be thought of as a special case of the situation for modules, since these are all bimodules over a ring R:
R itself;
any two-sided ideal of R (such as ker f);
any quotient ring of R (such as ); and
the codomain of any ring homomorphism whose domain is R (such as S, the codomain of f).
However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.
This example captures the essence of kernels in general Mal'cev algebras.
Monoid homomorphisms
Let M and N be monoids and let f be a monoid homomorphism from M to N. Then the kernel of f is the subset of the direct product consisting of all those ordered pairs of elements of M whose components are both mapped by f to the same element in N. The kernel is usually denoted (or a variation thereof). In symbols:
Since f is a function, the elements of the form must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the diagonal set .
It turns out that is an equivalence relation on M, and in fact a congruence relation. Thus, it makes sense to speak of the quotient monoid . The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N; for the congruence relation).
This is very different in flavour from the above examples. In particular, the preimage of the identity element of N is not enough to determine the kernel of f.
Universal algebra
All the above cases may be unified and generalized in universal algebra. | Kernel (algebra) | Wikipedia | 496 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
General case
Let A and B be algebraic structures of a given type and let f be a homomorphism of that type from A to B.
Then the kernel of f is the subset of the direct product consisting of all those ordered pairs of elements of A whose components are both mapped by f to the same element in B.
The kernel is usually denoted (or a variation).
In symbols:
Since f is a function, the elements of the form must belong to the kernel.
The homomorphism f is injective if and only if its kernel is exactly the diagonal set .
It is easy to see that ker f is an equivalence relation on A, and in fact a congruence relation.
Thus, it makes sense to speak of the quotient algebra .
The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B).
Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept.
For more on this general concept, outside of abstract algebra, see kernel of a function.
Malcev algebras
In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element.
To be specific, let A and B be Malcev algebraic structures of a given type and let f be a homomorphism of that type from A to B. If eB is the neutral element of B, then the kernel of f is the preimage of the singleton set {eB}; that is, the subset of A consisting of all those elements of A that are mapped by f to the element eB.
The kernel is usually denoted (or a variation). In symbols:
Since a Malcev algebra homomorphism preserves neutral elements, the identity element eA of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set . | Kernel (algebra) | Wikipedia | 486 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
The notion of ideal generalises to any Malcev algebra (as linear subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules).
It turns out that ker f is not a subalgebra of A, but it is an ideal.
Then it makes sense to speak of the quotient algebra .
The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B).
The connection between this and the congruence relation for more general types of algebras is as follows.
First, the kernel-as-an-ideal is the equivalence class of the neutral element eA under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings).
Using this, elements a and b of A are equivalent under the kernel-as-a-congruence if and only if their quotient a/b is an element of the kernel-as-an-ideal.
Algebras with nonalgebraic structure
Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations.
For example, one may consider topological groups or topological vector spaces, which are equipped with a topology.
In this case, we would expect the homomorphism f to preserve this additional structure; in the topological examples, we would want f to be a continuous map.
The process may run into a snag with the quotient algebras, which may not be well-behaved.
In the topological examples, we can avoid problems by requiring that topological algebraic structures be Hausdorff (as is usually done); then the kernel (however it is constructed) will be a closed set and the quotient space will work fine (and also be Hausdorff).
Kernels in category theory
The notion of kernel in category theory is a generalisation of the kernels of abelian algebras; see Kernel (category theory).
The categorical generalisation of the kernel as a congruence relation is the kernel pair.
(There is also the notion of difference kernel, or binary equaliser.) | Kernel (algebra) | Wikipedia | 502 | 45240 | https://en.wikipedia.org/wiki/Kernel%20%28algebra%29 | Mathematics | Abstract algebra | null |
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
History
The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether.
Three years later, B.L. van der Waerden published his influential Moderne Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.
Groups
We first present the isomorphism theorems of the groups.
Theorem A (groups)
Let G and H be groups, and let f : G → H be a homomorphism. Then:
The kernel of f is a normal subgroup of G,
The image of f is a subgroup of H, and
The image of f is isomorphic to the quotient group G / ker(f).
In particular, if f is surjective then H is isomorphic to G / ker(f).
This theorem is usually called the first isomorphism theorem.
Theorem B (groups) | Isomorphism theorems | Wikipedia | 426 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
Let be a group. Let be a subgroup of , and let be a normal subgroup of . Then the following hold:
The product is a subgroup of ,
The subgroup is a normal subgroup of ,
The intersection is a normal subgroup of , and
The quotient groups and are isomorphic.
Technically, it is not necessary for to be a normal subgroup, as long as is a subgroup of the normalizer of in . In this case, is not a normal subgroup of , but is still a normal subgroup of the product .
This theorem is sometimes called the second isomorphism theorem, diamond theorem or the parallelogram theorem.
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting , the group of invertible 2 × 2 complex matrices, , the subgroup of determinant 1 matrices, and the normal subgroup of scalar matrices , we have , where is the identity matrix, and . Then the second isomorphism theorem states that:
Theorem C (groups)
Let be a group, and a normal subgroup of .
Then
If is a subgroup of such that , then has a subgroup isomorphic to .
Every subgroup of is of the form for some subgroup of such that .
If is a normal subgroup of such that , then has a normal subgroup isomorphic to .
Every normal subgroup of is of the form for some normal subgroup of such that .
If is a normal subgroup of such that , then the quotient group is isomorphic to .
The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem.
Theorem D (groups)
Let be a group, and a normal subgroup of .
The canonical projection homomorphism defines a bijective correspondence
between the set of subgroups of containing and the set of (all) subgroups of . Under this correspondence normal subgroups correspond to normal subgroups.
This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth isomorphism theorem.
The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.
Discussion | Isomorphism theorems | Wikipedia | 452 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism . The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into , where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object and a monomorphism (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from to and .
If the sequence is right split (i.e., there is a morphism σ that maps to a -preimage of itself), then G is the semidirect product of the normal subgroup and the subgroup . If it is left split (i.e., there exists some such that ), then it must also be right split, and is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence .
In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.
The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects. | Isomorphism theorems | Wikipedia | 447 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
Note on numbers and names
Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.
It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, as one of isomorphism theorems, but when included, it is the last one.
Rings
The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.
Theorem A (rings)
Let and be rings, and let be a ring homomorphism. Then:
The kernel of is an ideal of ,
The image of is a subring of , and
The image of is isomorphic to the quotient ring .
In particular, if is surjective then is isomorphic to .
Theorem B (rings)
Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:
The sum S + I = {s + i | s ∈ S, i ∈ I } is a subring of R,
The intersection S ∩ I is an ideal of S, and
The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.
Theorem C (rings)
Let R be a ring, and I an ideal of R. Then
If is a subring of such that , then is a subring of .
Every subring of is of the form for some subring of such that .
If is an ideal of such that , then is an ideal of .
Every ideal of is of the form for some ideal of such that .
If is an ideal of such that , then the quotient ring is isomorphic to .
Theorem D (rings)
Let be an ideal of . The correspondence is an inclusion-preserving bijection between the set of subrings of that contain and the set of subrings of . Furthermore, (a subring containing ) is an ideal of if and only if is an ideal of . | Isomorphism theorems | Wikipedia | 466 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
Modules
The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem.
In the following, "module" will mean "R-module" for some fixed ring R.
Theorem A (modules)
Let M and N be modules, and let φ : M → N be a module homomorphism. Then:
The kernel of φ is a submodule of M,
The image of φ is a submodule of N, and
The image of φ is isomorphic to the quotient module M / ker(φ).
In particular, if φ is surjective then N is isomorphic to M / ker(φ).
Theorem B (modules)
Let M be a module, and let S and T be submodules of M. Then:
The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M,
The intersection S ∩ T is a submodule of M, and
The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.
Theorem C (modules)
Let M be a module, T a submodule of M.
If is a submodule of such that , then is a submodule of .
Every submodule of is of the form for some submodule of such that .
If is a submodule of such that , then the quotient module is isomorphic to .
Theorem D (modules)
Let be a module, a submodule of . There is a bijection between the submodules of that contain and the submodules of . The correspondence is given by for all . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of and the lattice of submodules of that contain ).
Universal algebra
To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations. | Isomorphism theorems | Wikipedia | 492 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
A congruence on an algebra is an equivalence relation that forms a subalgebra of considered as an algebra with componentwise operations. One can make the set of equivalence classes into an algebra of the same type by defining the operations via representatives; this will be well-defined since is a subalgebra of . The resulting structure is the quotient algebra.
Theorem A (universal algebra)
Let be an algebra homomorphism. Then the image of is a subalgebra of , the relation given by (i.e. the kernel of ) is a congruence on , and the algebras and are isomorphic. (Note that in the case of a group, iff , so one recovers the notion of kernel used in group theory in this case.)
Theorem B (universal algebra)
Given an algebra , a subalgebra of , and a congruence on , let be the trace of in and the collection of equivalence classes that intersect . Then
is a congruence on ,
is a subalgebra of , and
the algebra is isomorphic to the algebra .
Theorem C (universal algebra)
Let be an algebra and two congruence relations on such that . Then is a congruence on , and is isomorphic to
Theorem D (universal algebra)
Let be an algebra and denote the set of all congruences on . The set
is a complete lattice ordered by inclusion.
If is a congruence and we denote by the set of all congruences that contain (i.e. is a principal filter in , moreover it is a sublattice), then
the map is a lattice isomorphism. | Isomorphism theorems | Wikipedia | 339 | 45241 | https://en.wikipedia.org/wiki/Isomorphism%20theorems | Mathematics | Abstract algebra | null |
In the industrial design field of human–computer interaction, a user interface (UI) is the space where interactions between humans and machines occur. The goal of this interaction is to allow effective operation and control of the machine from the human end, while the machine simultaneously feeds back information that aids the operators' decision-making process. Examples of this broad concept of user interfaces include the interactive aspects of computer operating systems, hand tools, heavy machinery operator controls and process controls. The design considerations applicable when creating user interfaces are related to, or involve such disciplines as, ergonomics and psychology.
Generally, the goal of user interface design is to produce a user interface that makes it easy, efficient, and enjoyable (user-friendly) to operate a machine in the way which produces the desired result (i.e. maximum usability). This generally means that the operator needs to provide minimal input to achieve the desired output, and also that the machine minimizes undesired outputs to the user.
User interfaces are composed of one or more layers, including a human–machine interface (HMI) that typically interfaces machines with physical input hardware (such as keyboards, mice, or game pads) and output hardware (such as computer monitors, speakers, and printers). A device that implements an HMI is called a human interface device (HID). User interfaces that dispense with the physical movement of body parts as an intermediary step between the brain and the machine use no input or output devices except electrodes alone; they are called brain–computer interfaces (BCIs) or brain–machine interfaces (BMIs).
Other terms for human–machine interfaces are man–machine interface (MMI) and, when the machine in question is a computer, human–computer interface. Additional UI layers may interact with one or more human senses, including: tactile UI (touch), visual UI (sight), auditory UI (sound), olfactory UI (smell), equilibria UI (balance), and gustatory UI (taste). | User interface | Wikipedia | 417 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
Composite user interfaces (CUIs) are UIs that interact with two or more senses. The most common CUI is a graphical user interface (GUI), which is composed of a tactile UI and a visual UI capable of displaying graphics. When sound is added to a GUI, it becomes a multimedia user interface (MUI). There are three broad categories of CUI: standard, virtual and augmented. Standard CUI use standard human interface devices like keyboards, mice, and computer monitors. When the CUI blocks out the real world to create a virtual reality, the CUI is virtual and uses a virtual reality interface. When the CUI does not block out the real world and creates augmented reality, the CUI is augmented and uses an augmented reality interface. When a UI interacts with all human senses, it is called a qualia interface, named after the theory of qualia. CUI may also be classified by how many senses they interact with as either an X-sense virtual reality interface or X-sense augmented reality interface, where X is the number of senses interfaced with. For example, a Smell-O-Vision is a 3-sense (3S) Standard CUI with visual display, sound and smells; when virtual reality interfaces interface with smells and touch it is said to be a 4-sense (4S) virtual reality interface; and when augmented reality interfaces interface with smells and touch it is said to be a 4-sense (4S) augmented reality interface.
Overview
The user interface or human–machine interface is the part of the machine that handles the human–machine interaction. Membrane switches, rubber keypads and touchscreens are examples of the physical part of the Human Machine Interface which we can see and touch.
In complex systems, the human–machine interface is typically computerized. The term human–computer interface refers to this kind of system. In the context of computing, the term typically extends as well to the software dedicated to control the physical elements used for human–computer interaction.
The engineering of human–machine interfaces is enhanced by considering ergonomics (human factors). The corresponding disciplines are human factors engineering (HFE) and usability engineering (UE) which is part of systems engineering. | User interface | Wikipedia | 457 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
Tools used for incorporating human factors in the interface design are developed based on knowledge of computer science, such as computer graphics, operating systems, programming languages. Nowadays, we use the expression graphical user interface for human–machine interface on computers, as nearly all of them are now using graphics.
Multimodal interfaces allow users to interact using more than one modality of user input.
Terminology
There is a difference between a user interface and an operator interface or a human–machine interface (HMI).
The term "user interface" is often used in the context of (personal) computer systems and electronic devices.
Where a network of equipment or computers are interlinked through an MES (Manufacturing Execution System)-or Host to display information.
A human–machine interface (HMI) is typically local to one machine or piece of equipment, and is the interface method between the human and the equipment/machine. An operator interface is the interface method by which multiple pieces of equipment, linked by a host control system, are accessed or controlled.
The system may expose several user interfaces to serve different kinds of users. For example, a computerized library database might provide two user interfaces, one for library patrons (limited set of functions, optimized for ease of use) and the other for library personnel (wide set of functions, optimized for efficiency).
The user interface of a mechanical system, a vehicle or an industrial installation is sometimes referred to as the human–machine interface (HMI). HMI is a modification of the original term MMI (man–machine interface). In practice, the abbreviation MMI is still frequently used although some may claim that MMI stands for something different now. Another abbreviation is HCI, but is more commonly used for human–computer interaction. Other terms used are operator interface console (OIC) and operator interface terminal (OIT). However it is abbreviated, the terms refer to the 'layer' that separates a human that is operating a machine from the machine itself. Without a clean and usable interface, humans would not be able to interact with information systems.
In science fiction, HMI is sometimes used to refer to what is better described as a direct neural interface. However, this latter usage is seeing increasing application in the real-life use of (medical) prostheses—the artificial extension that replaces a missing body part (e.g., cochlear implants). | User interface | Wikipedia | 495 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
In some circumstances, computers might observe the user and react according to their actions without specific commands. A means of tracking parts of the body is required, and sensors noting the position of the head, direction of gaze and so on have been used experimentally. This is particularly relevant to immersive interfaces.
History
The history of user interfaces can be divided into the following phases according to the dominant type of user interface:
1945–1968: Batch interface
In the batch era, computing power was extremely scarce and expensive. User interfaces were rudimentary. Users had to accommodate computers rather than the other way around; user interfaces were considered overhead, and software was designed to keep the processor at maximum utilization with as little overhead as possible.
The input side of the user interfaces for batch machines was mainly punched cards or equivalent media like paper tape. The output side added line printers to these media. With the limited exception of the system operator's console, human beings did not interact with batch machines in real time at all.
Submitting a job to a batch machine involved first preparing a deck of punched cards that described a program and its dataset. The program cards were not punched on the computer itself but on keypunches, specialized, typewriter-like machines that were notoriously bulky, unforgiving, and prone to mechanical failure. The software interface was similarly unforgiving, with very strict syntaxes designed to be parsed by the smallest possible compilers and interpreters.
Once the cards were punched, one would drop them in a job queue and wait. Eventually, operators would feed the deck to the computer, perhaps mounting magnetic tapes to supply another dataset or helper software. The job would generate a printout, containing final results or an abort notice with an attached error log. Successful runs might also write a result on magnetic tape or generate some data cards to be used in a later computation.
The turnaround time for a single job often spanned entire days. If one was very lucky, it might be hours; there was no real-time response. But there were worse fates than the card queue; some computers required an even more tedious and error-prone process of toggling in programs in binary code using console switches. The very earliest machines had to be partly rewired to incorporate program logic into themselves, using devices known as plugboards. | User interface | Wikipedia | 476 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
Early batch systems gave the currently running job the entire computer; program decks and tapes had to include what we would now think of as operating system code to talk to I/O devices and do whatever other housekeeping was needed. Midway through the batch period, after 1957, various groups began to experiment with so-called "load-and-go" systems. These used a monitor program which was always resident on the computer. Programs could call the monitor for services. Another function of the monitor was to do better error checking on submitted jobs, catching errors earlier and more intelligently and generating more useful feedback to the users. Thus, monitors represented the first step towards both operating systems and explicitly designed user interfaces.
1969–present: Command-line user interface
Command-line interfaces (CLIs) evolved from batch monitors connected to the system console. Their interaction model was a series of request-response transactions, with requests expressed as textual commands in a specialized vocabulary. Latency was far lower than for batch systems, dropping from days or hours to seconds. Accordingly, command-line systems allowed the user to change their mind about later stages of the transaction in response to real-time or near-real-time feedback on earlier results. Software could be exploratory and interactive in ways not possible before. But these interfaces still placed a relatively heavy mnemonic load on the user, requiring a serious investment of effort and learning time to master.
The earliest command-line systems combined teleprinters with computers, adapting a mature technology that had proven effective for mediating the transfer of information over wires between human beings. Teleprinters had originally been invented as devices for automatic telegraph transmission and reception; they had a history going back to 1902 and had already become well-established in newsrooms and elsewhere by 1920. In reusing them, economy was certainly a consideration, but psychology and the rule of least surprise mattered as well; teleprinters provided a point of interface with the system that was familiar to many engineers and users. | User interface | Wikipedia | 407 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
The widespread adoption of video-display terminals (VDTs) in the mid-1970s ushered in the second phase of command-line systems. These cut latency further, because characters could be thrown on the phosphor dots of a screen more quickly than a printer head or carriage can move. They helped quell conservative resistance to interactive programming by cutting ink and paper consumables out of the cost picture, and were to the first TV generation of the late 1950s and 60s even more iconic and comfortable than teleprinters had been to the computer pioneers of the 1940s.
Just as importantly, the existence of an accessible screen—a two-dimensional display of text that could be rapidly and reversibly modified—made it economical for software designers to deploy interfaces that could be described as visual rather than textual. The pioneering applications of this kind were computer games and text editors; close descendants of some of the earliest specimens, such as rogue(6), and vi(1), are still a live part of Unix tradition.
1985: SAA user interface or text-based user interface
In 1985, with the beginning of Microsoft Windows and other graphical user interfaces, IBM created what is called the Systems Application Architecture (SAA) standard which include the Common User Access (CUA) derivative. CUA successfully created what we know and use today in Windows, and most of the more recent DOS or Windows Console Applications will use that standard as well.
This defined that a pulldown menu system should be at the top of the screen, status bar at the bottom, shortcut keys should stay the same for all common functionality (F2 to Open for example would work in all applications that followed the SAA standard). This greatly helped the speed at which users could learn an application so it caught on quick and became an industry standard.
1968–present: Graphical user interface | User interface | Wikipedia | 377 | 45249 | https://en.wikipedia.org/wiki/User%20interface | Technology | User interface | null |
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