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In biology, the gonadosomatic index ( GSI ) is the calculation of the gonad mass as a proportion of the total body mass . It is represented by the formula: [ 1 ]
GSI = [gonad weight / total tissue weight] × 100
It is a tool for measuring the sexual maturity of animals in correlation to ovary development and testicle development. The index is frequently used as reporting point in OECD test guideline, which may be used as indication or evidence of potential endocrine disruption effect of chemicals in regulatory framework (EFSA and ECHA, 2017). [ 2 ] [ 3 ]
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it .
This zoology –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gonadosomatic_index |
The gonadotropin receptors are a group of receptors that bind a group of pituitary hormones called gonadotropins . They include the:
This transmembrane receptor -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gonadotropin_receptor |
Gondwana ( / ɡ ɒ n d ˈ w ɑː n ə / gond-WAHN-ə ; [ 1 ] Sanskrit: [goːɳɖɐʋɐnɐ] ) was a large landmass, sometimes referred to as a supercontinent . The remnants of Gondwana make up around two-thirds of today's continental area, including South America , Africa , Antarctica , Australia , Zealandia , Arabia , and the Indian subcontinent .
Gondwana was formed by the accretion of several cratons (large stable blocks of the Earth's crust), beginning c. 800 to 650 Ma with the East African Orogeny , the collision of India and Madagascar with East Africa, and culminating in c. 600 to 530 Ma with the overlapping Brasiliano and Kuunga orogenies, the collision of South America with Africa, and the addition of Australia and Antarctica, respectively. [ 2 ] Eventually, Gondwana became the largest piece of continental crust of the Paleozoic Era, covering an area of some 100,000,000 km 2 (39,000,000 sq mi), [ 3 ] about one-fifth of the Earth's surface. It fused with Laurasia during the Carboniferous to form Pangaea . It began to separate from northern Pangea ( Laurasia ) during the Triassic , and started to fragment during the Early Jurassic (around 180 million years ago). The final stages of break-up saw the fragmentation of the Antarctic land bridge (involving the separation of Antarctica from South America and Australia, forming the Drake and Tasmanian Passages ), which occurred during the Paleogene (from around 66 to 23 million years ago (Ma)). Gondwana was not considered a supercontinent by the earliest definition, since the landmasses of Baltica , Laurentia , and Siberia were separated from it. [ 4 ] To differentiate it from the Indian region of the same name (see § Name ), it is also commonly called Gondwanaland . [ 5 ]
Regions that were part of Gondwana shared floral and faunal elements that persist to the present day.
The continent of Gondwana was named by the Austrian scientist Eduard Suess after the Indian region of the same name , which is derived from Sanskrit गोण्डवन goṇḍavana ('forest of the Gonds '). [ 6 ] The name had been previously used in a geological context, first by H. B. Medlicott in 1872, [ 7 ] from which the Gondwana sedimentary sequences ( Permian - Triassic ) are also described.
Some scientists prefer the term "Gondwanaland" for the supercontinent to make a clear distinction between the region and the supercontinent. [ 8 ]
The assembly of Gondwana was a protracted process during the Neoproterozoic and Paleozoic , which remains incompletely understood because of the lack of paleo-magnetic data. Several orogenies , collectively known as the Pan-African orogeny , caused the continental fragments of a much older supercontinent, Rodinia , to amalgamate. One of those orogenic belts, the Mozambique Belt , formed 800 to 650 Ma and was originally interpreted as the suture between East (India, Madagascar, Antarctica, Australia) and West Gondwana (Africa and South America). Three orogenies were recognised during the 1990s as a result of data sets compiled on behalf of oil and mining companies: [ 10 ] the East African Orogeny ( 650 to 800 Ma ) and Kuunga orogeny (including the Malagasy orogeny in southern Madagascar) ( 550 Ma ), the collision between East Gondwana and East Africa in two steps, and the Brasiliano orogeny ( 660 to 530 Ma ), the successive collision between South American and African cratons . [ 11 ]
The last stages of Gondwanan assembly overlapped with the opening of the Iapetus Ocean between Laurentia and western Gondwana. [ 12 ] During this interval, the Cambrian explosion occurred. Laurentia was docked against the western shores of a united Gondwana for a brief period near the Precambrian and Cambrian boundary, forming the short-lived and still disputed supercontinent Pannotia . [ 13 ]
The Mozambique Ocean separated the Congo – Tanzania – Bangweulu Block of central Africa from Neoproterozoic India (India, the Antongil Block in far eastern Madagascar, the Seychelles , and the Napier and Rayner Complexes in East Antarctica ). The Azania continent [ 14 ] (much of central Madagascar , the Horn of Africa and parts of Yemen and Arabia) was an island in the Mozambique Ocean.
The continents of Australia and East Antarctica were still separated from India, eastern Africa, and Kalahari by c. 600 Ma , when most of western Gondwana had already been amalgamated. By c. 550 Ma, India had reached its Gondwanan position, which initiated the Kuunga orogeny (also known as the Pinjarra orogeny). Meanwhile, on the other side of the newly forming Africa, Kalahari collided with Congo and Rio de la Plata which closed the Adamastor Ocean . c. 540–530 Ma, the closure of the Mozambique Ocean brought India next to Australia–East Antarctica, and both North China and South China were in proximity to Australia. [ 15 ]
As the rest of Gondwana formed, a complex series of orogenic events assembled the eastern parts of Gondwana (eastern Africa, Arabian-Nubian Shield, Seychelles, Madagascar, India, Sri Lanka, East Antarctica, Australia) c. 750 to 530 Ma . First, the Arabian-Nubian Shield collided with eastern Africa (in the Kenya-Tanzania region) in the East African Orogeny c. 750 to 620 Ma . Then Australia and East Antarctica were merged with the remaining Gondwana c. 570 to 530 Ma in the Kuunga Orogeny. [ 16 ]
The later Malagasy orogeny at about 550–515 Mya affected Madagascar, eastern East Africa and southern India. In it, Neoproterozoic India collided with the already combined Azania and Congo–Tanzania–Bangweulu Block, suturing along the Mozambique Belt. [ 17 ]
The 18,000 km-long (11,000 mi) Terra Australis Orogen developed along Gondwana's western, southern, and eastern margins. [ 18 ] Proto-Gondwanan Cambrian arc belts from this margin have been found in eastern Australia, Tasmania, New Zealand, and Antarctica. Though these belts formed a continuous arc chain, the direction of subduction was different between the Australian-Tasmanian and New Zealand-Antarctica arc segments. [ 19 ]
Many terranes were accreted to Eurasia during Gondwana's existence, but the Cambrian or Precambrian origin of many of these terranes remains uncertain. For example, some Paleozoic terranes and microcontinents that now make up Central Asia, often called the "Kazakh" and "Mongolian terranes", were progressively amalgamated into the continent Kazakhstania in the late Silurian . Whether these blocks originated on the shores of Gondwana is not known. [ 20 ]
In the Early Paleozoic, the Armorican terrane , which today form large parts of France, was part of Peri-Gondwana; the Rheic Ocean closed in front of it and the Paleo-Tethys Ocean opened behind it. Precambrian rocks from the Iberian Peninsula suggest that it, too, formed part of core Gondwana before its detachment as an orocline in the Variscan orogeny close to the Carboniferous–Permian boundary. [ 21 ]
South-east Asia was made of Gondwanan and Cathaysian continental fragments that were assembled during the Mid-Paleozoic and Cenozoic. This process can be divided into three phases of rifting along Gondwana's northern margin: first, in the Devonian, North and South China , together with Tarim and Quidam (north-western China) rifted, opening the Paleo-Tethys behind them. These terranes accreted to Asia during Late Devonian and Permian. Second, in the Late Carboniferous to Early Permian, Cimmerian terranes opened Meso-Tethys Ocean; Sibumasu and Qiangtang were added to south-east Asia during Late Permian and Early Jurassic. Third, in the Late Triassic to Late Jurassic, Lhasa , Burma , Woyla terranes opened the Neo-Tethys Ocean; Lhasa collided with Asia during the Early Cretaceous, and Burma and Woyla during the Late Cretaceous. [ 22 ]
Gondwana's long, northern margin remained a mostly passive margin throughout the Paleozoic. The Early Permian opening of the Neo-Tethys Ocean along this margin produced a long series of terranes, many of which were and still are being deformed in the Himalayan orogeny . These terranes are, from Turkey to north-eastern India: the Taurides in southern Turkey; the Lesser Caucasus Terrane in Georgia; the Sanand, Alborz, and Lut terranes in Iran; the Mangysglak Terrane in the Caspian Sea; the Afghan Terrane; the Karakorum Terrane in northern Pakistan; and the Lhasa and Qiangtang terranes in Tibet. The Permian–Triassic widening of the Neo-Tethys pushed all these terranes across the Equator and over to Eurasia. [ 23 ]
During the Neoproterozoic to Paleozoic phase of the Terra Australis Orogen , a series of terranes were rafted from the proto-Andean margin when the Iapetus Ocean opened, to be added back to Gondwana during the closure of that ocean. [ 24 ] During the Paleozoic, some blocks which helped to form parts of the Southern Cone of South America, include a piece transferred from Laurentia when the west edge of Gondwana scraped against southeast Laurentia in the Ordovician . [ 25 ] This is the Cuyania or Precordillera terrane of the Famatinian orogeny in northwest Argentina which may have continued the line of the Appalachians southwards. [ 26 ] Chilenia terrane accreted later against Cuyania. [ 27 ] The collision of the Patagonian terrane with the southwestern Gondwanan occurred in the late Paleozoic. Subduction-related igneous rocks from beneath the North Patagonian Massif have been dated at 320–330 million years old, indicating that the subduction process initiated in the early Carboniferous. [ 28 ] This was relatively short-lived (lasting about 20 million years), and initial contact of the two landmasses occurred in the mid-Carboniferous, [ 28 ] [ 29 ] with broader collision during the early Permian. [ 29 ] In the Devonian, an island arc named Chaitenia accreted to Patagonia in what is now south-central Chile. [ 30 ]
Gondwana and Laurasia formed the Pangaea supercontinent during the Carboniferous. Pangaea began to break up in the Mid-Jurassic when the Central Atlantic opened . [ 32 ]
In the western end of Pangaea, the collision between Gondwana and Laurasia closed the Rheic and Paleo-Tethys oceans. The obliquity of this closure resulted in the docking of some northern terranes in the Marathon , Ouachita , Alleghanian , and Variscan orogenies, respectively. Southern terranes, such as Chortis and Oaxaca , on the other hand, remained largely unaffected by the collision along the southern shores of Laurentia. Some Peri-Gondwanan terranes, such as Yucatán and Florida , were buffered from collisions by major promontories. Other terranes, such as Carolina and Meguma , were directly involved in the collision. The final collision resulted in the Variscan- Appalachian Mountains , stretching from present-day Mexico to southern Europe. Meanwhile, Baltica collided with Siberia and Kazakhstania which resulted in the Uralian orogeny and Laurasia . Pangaea was finally amalgamated in the Late Carboniferous-Early Permian, but the oblique forces continued until Pangaea began to rift in the Triassic. [ 33 ]
In the eastern end, collisions occurred slightly later. The North China , South China , and Indochina blocks rifted from Gondwana during the middle Paleozoic and opened the Proto-Tethys Ocean . North China docked with Mongolia and Siberia during the Carboniferous–Permian, followed by South China. The Cimmerian blocks then rifted from Gondwana to form the Paleo-Tethys and Neo-Tethys oceans in the Late Carboniferous, and docked with Asia during the Triassic and Jurassic. Western Pangaea began to rift while the eastern end was still being assembled. [ 34 ]
The formation of Pangaea and its mountains had a tremendous impact on global climate and sea levels, which resulted in glaciations and continent-wide sedimentation. In North America, the base of the Absaroka sequence coincides with the Alleghanian and Ouachita orogenies and are indicative of a large-scale change in the mode of deposition far away from the Pangaean orogenies. Ultimately, these changes contributed to the Permian–Triassic extinction event and left large deposits of hydrocarbons, coal, evaporite, and metals. [ 35 ]
The breakup of Pangaea began with the Central Atlantic magmatic province (CAMP) between South America, Africa, North America, and Europe. CAMP covered more than seven million square kilometres over a few million years, reached its peak at c. 200 Ma , and coincided with the Triassic–Jurassic extinction event . [ 36 ] The reformed Gondwanan continent was not precisely the same as that which had existed before Pangaea formed; for example, most of Florida and southern Georgia and Alabama is underlain by rocks that were originally part of Gondwana, but this region stayed attached to North America when the Central Atlantic opened . [ 37 ]
Antarctica, the centre of the supercontinent, shared boundaries with all other Gondwana continents and the fragmentation of Gondwana propagated clockwise around it. The break-up was the result of the eruption of the Karoo-Ferrar igneous province , one of the Earth's most extensive large igneous provinces (LIP) c. 200 to 170 Ma , but the oldest magnetic anomalies between South America, Africa, and Antarctica are found in what is now the southern Weddell Sea where initial break-up occurred during the Jurassic c. 180 to 160 Ma . [ 38 ]
Gondwana began to break up in the early Jurassic following the extensive and fast emplacement of the Karoo-Ferrar flood basalts c. 184 Ma . Before the Karoo plume initiated rifting between Africa and Antarctica , it separated a series of smaller continental blocks from Gondwana's southern, Proto-Pacific margin (along what is now the Transantarctic Mountains ): the Antarctic Peninsula , Marie Byrd Land , Zealandia , and Thurston Island ; the Falkland Islands and Ellsworth–Whitmore Mountains (in Antarctica) were rotated 90° in opposite directions; and South America south of the Gastre Fault (often referred to as Patagonia ) was pushed westward. [ 39 ] The history of the Africa-Antarctica break-up can be studied in great detail in the fracture zones and magnetic anomalies flanking the Southwest Indian Ridge . [ 40 ]
The Madagascar block and the Mascarene Plateau , stretching from the Seychelles to Réunion , were broken off India, causing Madagascar and Insular India to be separate landmasses : elements of this break-up nearly coincide with the Cretaceous–Paleogene extinction event . The India–Madagascar–Seychelles separations appear to coincide with the eruption of the Deccan basalts , whose eruption site may survive as the Réunion hotspot . The Seychelles and the Maldives are now separated by the Central Indian Ridge .
During the initial break-up in the Early Jurassic, a marine transgression swept over the Horn of Africa covering Triassic planation surfaces with sandstone , limestone , shale , marls and evaporites . [ 41 ] [ 42 ]
East Gondwana, comprising Antarctica, Madagascar, India, and Australia, began to separate from Africa. East Gondwana then began to break up c. 132.5 to 96 Ma when India moved northwest from Australia-Antarctica. [ 43 ] The Indian plate and the Australian plate are now separated by the Capricorn plate and its diffuse boundaries. [ 44 ] During the opening of the Indian Ocean, the Kerguelen hotspot first formed the Kerguelen Plateau on the Antarctic plate c. 118 to 95 Ma and then the Ninety East Ridge on the Indian plate at c. 100 Ma . [ 45 ] The Kerguelen Plateau and the Broken Ridge , the southern end of the Ninety East Ridge, are now separated by the Southeast Indian Ridge .
Separation between Australia and East Antarctica began c. 132 Ma with seafloor spreading occurring c. 96 Ma . A shallow seaway developed over the South Tasman Rise during the Early Cenozoic and as oceanic crust started to separate the continents during the Eocene c. 35.5 Ma global ocean temperature dropped significantly. [ 46 ] A dramatic shift from arc- to rift magmatism c. 100 Ma separated Zealandia , including New Zealand , the Campbell Plateau , Chatham Rise , Lord Howe Rise , Norfolk Ridge , and New Caledonia , from West Antarctica c. 84 Ma . [ 47 ]
The opening of the South Atlantic Ocean divided West Gondwana (South America and Africa), but there is considerable debate over the exact timing of this break-up. Rifting propagated from south to north along Triassic–Early Jurassic lineaments, but intra-continental rifts also began to develop within both continents in Jurassic–Cretaceous sedimentary basins, subdividing each continent into three sub-plates. Rifting began c. 190 Ma at Falkland latitudes, forcing Patagonia to move relative to the still static remainder of South America and Africa, and this westward movement lasted until the Early Cretaceous 126.7 Ma . From there rifting propagated northward during the Late Jurassic c. 150 Ma or Early Cretaceous c. 140 Ma most likely forcing dextral movements between sub-plates on either side. South of the Walvis Ridge and Rio Grande Rise the Paraná and Etendeka magmatics resulted in further ocean-floor spreading c. 130 to 135 Ma and the development of rifts systems on both continents, including the Central African Rift System and the Central African Shear Zone which lasted until c. 85 Ma . At Brazilian latitudes spreading is more difficult to assess because of the lack of palaeo-magnetic data, but rifting occurred in Nigeria at the Benue Trough c. 118 Ma . North of the Equator the rifting began after 120.4 Ma and continued until c. 100 to 96 Ma . [ 48 ] Dinosaur footprints representing identical species assemblages are known from opposite sides of the South Atlantic (Brazil and Cameroon ) dating to around 120 million years ago , suggesting that some form of land connection still existed between Africa and South America as recently as the early Aptian . [ 49 ]
The first phases of Andean orogeny in the Jurassic and Early Cretaceous were characterised by extensional tectonics , rifting , the development of back-arc basins and the emplacement of large batholiths . [ 50 ] [ 51 ] This development is presumed to have been linked to the subduction of cold oceanic lithosphere . [ 51 ] During the mid to Late Cretaceous ( c. 90 million years ago ), the Andean orogeny changed significantly in character. [ 50 ] [ 51 ] Warmer and younger oceanic lithosphere is believed to have started to be subducted beneath South America around this time. Such kind of subduction is held responsible not only for the intense contractional deformation that different lithologies were subject to, but also the uplift and erosion known to have occurred from the Late Cretaceous onward. [ 51 ] Plate tectonic reorganisation since the mid-Cretaceous might also have been linked to the opening of the South Atlantic Ocean . [ 50 ] Another change related to mid-Cretaceous plate tectonic rearrangement was the change of subduction direction of the oceanic lithosphere that went from having south-east motion to having a north-east motion about 90 million years ago. [ 52 ] While subduction direction changed, it remained oblique (and not perpendicular) to the coast of South America, and the direction change affected several subduction zone -parallel faults including Atacama , Domeyko and Liquiñe-Ofqui . [ 51 ] [ 52 ]
Insular India began to collide with Asia circa 70 Ma , forming the Indian subcontinent , since which more than 1,400 km (870 mi) of crust has been absorbed by the Himalayan - Tibetan orogen. During the Cenozoic, the orogen resulted in the construction of the Tibetan Plateau between the Tethyan Himalayas in the south and the Kunlun and Qilian mountains in the north. [ 53 ]
Later, South America was connected to North America via the Isthmus of Panama , cutting off a circulation of warm water and thereby making the Arctic colder, [ 54 ] as well as allowing the Great American Interchange .
The break-up of Gondwana can be said to continue in eastern Africa at the Afar triple junction , which separates the Arabian , African , and Somali plates, resulting in rifting in the Red Sea and East African Rift . [ 55 ]
In the Early Cenozoic , Australia was still connected to Antarctica c. 35–40° south of its current location and both continents were largely unglaciated. [ 56 ] This was one end of the Antarctic land bridge , the other connecting Antarctica to South America. [ 57 ] A rift between the two developed but remained an embayment until the Eocene-Oligocene boundary when the Circumpolar Current developed and the glaciation of Antarctica began. [ 56 ]
Australia was warm and wet during the Paleocene and dominated by rainforests. The opening of the Tasman Gateway at the Eocene-Oligocene boundary ( 33 Ma ) resulted in abrupt cooling but the Oligocene became a period of high rainfall with swamps in southeastern Australia. During the Miocene, a warm and humid climate developed with pockets of rainforests in central Australia, but before the end of the period, colder and drier climate severely reduced this rainforest. A brief period of increased rainfall in the Pliocene was followed by drier climate which favoured grassland. Since then, the fluctuation between wet interglacial periods and dry glacial periods has developed into the present arid regime. Australia has thus experienced various climate changes over a 15-million-year period with a gradual decrease in precipitation. [ 58 ]
The Tasman Gateway between Australia and Antarctica began to open c. 40 to 30 Ma . Palaeontological evidence indicates the Antarctic Circumpolar Current (ACC) was established in the Late Oligocene c. 23 Ma with the full opening of the Drake Passage and the deepening of the Tasman Gateway. The oldest oceanic crust in the Drake Passage, however, is 34 to 29 Ma -old which indicates that the spreading between the Antarctic and South American plates began near the Eocene-Oligocene boundary. [ 59 ] Deep sea environments in Tierra del Fuego and the North Scotia Ridge during the Eocene and Oligocene indicate a "Proto-ACC" opened during this period. Later, 26 to 14 Ma , a series of events severally restricted the Proto-ACC: change to shallow marine conditions along the North Scotia Ridge; closure of the Fuegan Seaway, the deep sea that existed in Tierra del Fuego; and uplift of the Patagonian Cordillera. This, together with the reactivated Iceland plume , contributed to global warming. During the Miocene, the Drake Passage began to widen, and as water flow between South America and the Antarctic Peninsula increased, the renewed ACC resulted in cooler global climate. [ 60 ]
Since the Eocene, the northward movement of the Australian Plate has resulted in an arc-continent collision with the Philippine and Caroline plates and the uplift of the New Guinea Highlands . [ 61 ] From the Oligocene to the late Miocene, the climate in Australia, dominated by warm and humid rainforests before this collision, began to alternate between open forest and rainforest before the continent became the arid or semiarid landscape it is today. [ 62 ]
The adjective "Gondwanan" is in common use in biogeography when referring to patterns of distribution of living organisms, typically when the organisms are restricted to two or more of the now-discontinuous regions that were once part of Gondwana, including the Antarctic flora . [ 8 ] For example, the plant family Proteaceae , known from all continents in the Southern Hemisphere, has a "Gondwanan distribution" and is often described as an archaic, or relict , lineage. The distributions in the Proteaceae is, nevertheless, the result of both Gondwanan rafting and later oceanic dispersal. [ 63 ]
During the Silurian, Gondwana extended from the Equator (Australia) to the South Pole (North Africa and South America) whilst Laurasia was located on the Equator opposite to Australia. A short-lived Late Ordovician glaciation was followed by a Silurian Hot House period. [ 64 ] The End-Ordovician extinction , which resulted in 27% of marine invertebrate families and 57% of genera going extinct, occurred during this shift from Ice House to Hot House. [ 65 ]
By the end of the Ordovician, Cooksonia , a slender, ground-covering plant, became the first known vascular plant to establish itself on land. This first colonisation occurred exclusively around the Equator on landmasses then limited to Laurasia and, in Gondwana, to Australia. In the late Silurian, two distinctive lineages, zosterophylls and rhyniophytes , had colonised the tropics. The former evolved into the lycopods that were to dominate the Gondwanan vegetation over a long period, whilst the latter evolved into horsetails and gymnosperms . Most of Gondwana was located far from the Equator during this period and remained a lifeless and barren landscape. [ 66 ]
West Gondwana drifted north during the Devonian , bringing Gondwana and Laurasia close together. Global cooling contributed to the Late Devonian extinction (19% of marine families and 50% of genera went extinct) and glaciation occurred in South America. Before Pangaea had formed, terrestrial plants, such as pteridophytes , began to diversify rapidly resulting in the colonisation of Gondwana. The Baragwanathia Flora, found only in the Yea Beds of Victoria, Australia, occurs in two strata separated by 1,700 m (5,600 ft) or 30 Ma; the upper assemblage is more diverse and includes Baragwanathia, the first primitive herbaceous lycopod to evolve from the zosterophylls. During the Devonian, giant club mosses replaced the Baragwanathia Flora, introducing the first trees, and by the Late Devonian this first forest was accompanied by the progymnosperms , including the first large trees Archaeopteris . [ 67 ] The Late Devonian extinction probably also resulted in osteolepiform fishes evolving into the amphibian tetrapods , the earliest land vertebrates, in Greenland and Russia. The only traces of this evolution in Gondwana are amphibian footprints and a single jaw from Australia. [ 68 ]
The closure of the Rheic Ocean and the formation of Pangaea in the Carboniferous resulted in the rerouting of ocean currents that initiated an Ice House period. As Gondwana began to rotate clockwise, Australia shifted south to more temperate latitudes. An ice cap initially covered most of southern Africa and South America but spread to eventually cover most of the supercontinent, except northernmost Africa-South America. Giant lycopod and horsetail forests continued to evolve in tropical Laurasia together with a diversified assemblage of true insects. In Gondwana, in contrast, ice and, in Australia, volcanism decimated the Devonian flora to a low-diversity seed fern flora – the pteridophytes were increasingly replaced by the gymnosperms which were to dominate until the Mid-Cretaceous. Australia, however, was still located near the Equator during the Early Carboniferous, and during this period, temnospondyl and lepospondyl amphibians and the first amniote reptilians evolved, all closely related to the Laurasian fauna, but spreading ice eventually drove these animals away from Gondwana entirely. [ 69 ]
The Gondwana ice sheet melted, and sea levels dropped during the Permian and Triassic global warming. During this period, the extinct glossopterids colonised Gondwana and reached peak diversity in the Late Permian when coal-forming forests covered all of Gondwana. The period also saw the evolution of Voltziales , one of the few plant orders to survive the Permian–Triassic extinction (57% of marine families and 83% of genera went extinct) and which came to dominate in the Late Permian and from whom true conifers evolved. Tall lycopods and horsetails dominated the wetlands of Gondwana in the Early Permian. Insects co-evolved with glossopterids across Gondwana and diversified with more than 200 species in 21 orders by the Late Permian, many known from South Africa and Australia. Beetles and cockroaches remained minor elements in this fauna. Tetrapod fossils from the Early Permian have only been found in Laurasia but they became common in Gondwana later during the Permian. The arrival of the therapsids resulted in the first plant-vertebrate-insect ecosystem. [ 70 ]
During the Mid- to Late Triassic, hot-house conditions coincided with a peak in biodiversity – the end-Permian extinction was enormous and so was the radiation that followed. Two families of conifers, Podocarpaceae and Araucariaceae , dominated Gondwana in the Early Triassic, but Dicroidium , an extinct genus of fork-leaved seed ferns, dominated woodlands and forests of Gondwana during most of the Triassic. Conifers evolved and radiated during the period, with six of eight extant families already present before the end of it. Bennettitales and Pentoxylales , two now extinct orders of gymnospermous plants, evolved in the Late Triassic and became important in the Jurassic and Cretaceous. It is possible that gymnosperm biodiversity surpassed later angiosperm biodiversity and that the evolution of angiosperms began during the Triassic but, if so, in Laurasia rather than in Gondwana. Two Gondwanan classes, lycophytes and sphenophytes , saw a gradual decline during the Triassic while ferns, though never dominant, managed to diversify. [ 71 ]
The brief period of icehouse conditions during the Triassic–Jurassic extinction event had a dramatic impact on dinosaurs but left plants largely unaffected. The Jurassic was mostly one of hot-house conditions and, while vertebrates managed to diversify in this environment, plants have left little evidence of such development, apart from Cheiroleidiacean conifers and Caytoniales and other groups of seed ferns. In terms of biomass, the Jurassic flora was dominated by conifer families and other gymnosperms that had evolved during the Triassic. The Pteridophytes that had dominated during the Paleozoic were now marginalised, except for ferns. In contrast to Laurentia, very few insect fossils have been found in Gondwana, to a considerable extent because of widespread deserts and volcanism. While plants had a cosmopolitan distribution, dinosaurs evolved and diversified in a pattern that reflects the Jurassic break-up of Pangaea. [ 72 ]
The Cretaceous saw the arrival of the angiosperms , or flowering plants, a group that probably evolved in western Gondwana (South America–Africa). From there the angiosperms diversified in two stages: the monocots and magnoliids evolved in the Early Cretaceous, followed by the hammamelid dicots . By the Mid-Cretaceous, angiosperms constituted half of the flora in northeastern Australia. There is, however, no obvious connection between this spectacular angiosperm radiation and any known extinction event nor with vertebrate/insect evolution. Insect orders associated with pollination, such as beetles , flies , butterflies and moths , wasps, bees, ants , radiated continuously from the Permian-Triassic, long before the arrival of the angiosperms. Well-preserved insect fossils have been found in the lake deposits of the Santana Formation in Brazil, the Koonwarra Lake fauna in Australia, and the Orapa diamond mine in Botswana. [ 73 ]
Dinosaurs continued to prosper but, as the angiosperm diversified, conifers, bennettitaleans and pentoxylaleans disappeared from Gondwana c. 115 Ma together with the specialised herbivorous ornithischians , whilst generalist browsers, such as several families of sauropodomorph Saurischia , prevailed. The Cretaceous–Paleogene extinction event killed off all dinosaurs except birds, but plant evolution in Gondwana was hardly affected. [ 73 ] Gondwanatheria is an extinct group of non- therian mammals with a Gondwanan distribution (South America, Africa, Madagascar, India, Zealandia and Antarctica) during the Late Cretaceous and Palaeogene. [ 74 ] Xenarthra and Afrotheria , two placental clades, are of Gondwanan origin and probably began to evolve separately c. 105 Ma when Africa and South America separated. [ 75 ]
The laurel forests of Australia, New Caledonia, and New Zealand have a number of species related to those of the laurissilva of Valdivia, through the connection of the Antarctic flora . These include gymnosperms and the deciduous species of Nothofagus , as well as the New Zealand laurel, Corynocarpus laevigatus , and Laurelia novae-zelandiae . New Caledonia and New Zealand became separated from Australia by continental drift 85 million years ago. The islands still retain plants that originated in Gondwana and spread to the Southern Hemisphere continents later.
Africa
Antarctica
Asia
Australia
Europe
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Afro-Eurasia
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Oceania | https://en.wikipedia.org/wiki/Gondwana |
Gongronella is a genus of fungi belonging to the family Cunninghamellaceae . [ 1 ]
The genus has cosmopolitan distribution . [ 1 ]
Species: [ 1 ] | https://en.wikipedia.org/wiki/Gongronella |
A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία ( gōnía ) ' angle ' and μέτρον ( métron ) ' measure '. [ 1 ] The protractor is a commonly used type in the fields of mechanics, engineering, and geometry.
The first known description of a goniometer, based on the astrolabe , was by Gemma Frisius in 1538.
A protractor is a measuring instrument , typically made of transparent plastic, for measuring angles . Some protractors are simple half-discs or full circles. More advanced protractors, such as the bevel protractor , have one or two swinging arms, which can be used to help measure the angle.
Most protractors measure angles in degrees (°). Radian-scale protractors measure angles in radians . Most protractors are divided into 180 equal parts. Some precision protractors further divide degrees into arcminutes . A protractor divided in centiturns is normally called a " percentage protractor ".
A bevel protractor is a graduated circular protractor with one pivoted arm; used for measuring or marking off angles. Sometimes Vernier scales are attached to give more precise readings. It has wide application in architectural and mechanical drawing, although its use is decreasing with the availability of modern drawing software or CAD .
Universal bevel protractors are also used by toolmakers; as they measure angles by mechanical contact they are classed as mechanical protractors. [ 2 ] [ 3 ]
The bevel protractor is used to establish and test angles to very close tolerances. It reads to 5 arcminutes (5′ or 1 / 12 °) and can measure angles from 0° to 450°.
The bevel protractor consists of a beam, a graduated dial, and a blade which is connected to a swivel plate (with Vernier scale) by a thumb nut and clamp. When the edges of the beam and blade are parallel, a small mark on the swivel plate coincides with the zero line on the graduated dial. To measure an angle between the beam and the blade of 90° or less, the reading may be obtained directly from the graduation number on the dial indicated by the mark on the swivel plate. To measure an angle of over 90°, subtract the number of degrees as indicated on the dial from 180°, as the dial is graduated from opposite zero marks to 90° each way.
Since the spaces, both on the main scale and the Vernier scale, are numbered both to the right and the left from zero, any angle can be measured. The readings can be taken either to the right or to the left, according to the direction in which the zero on the main scale is moved.
Prior to the invention of the theodolite , the goniometer was used in surveying . The application of triangulation to geodesy was described in the second (1533) edition of Cosmograficus liber by Petri Appiani as a 16-page appendix by Frisius entitled Libellus de locorum describendorum ratione . [ 4 ]
The Bellini–Tosi direction finder was a type of radio direction finder that was widely used from World War I to World War II . It used the signals from two crossed antennas, or four individual antennas simulating two crossed ones, to re-create the radio signal in a small area between two loops of wire. The operator could then measure the angle to the target radio source by performing direction finding within this small area. The advantage to the Bellini–Tosi system is that the antennas do not move, allowing them to be built at any required size.
The basic technique remains in use, although the equipment has changed dramatically. Goniometers are widely used for military and civil purposes, [ 5 ] e.g. interception of satellite and naval communications on the French warship Dupuy de Lôme uses multiple goniometers.
In crystallography , goniometers are used for measuring angles between crystal faces. They are also used in X-ray diffraction to rotate the samples. The groundbreaking investigations of physicist Max von Laue and colleagues into the atomic structure of crystals in 1912 involved a goniometer.
Goniophotometers measure the spatial distribution of light visible to the human eye (often luminous intensity ) at specific angular positions, usually covering all spherical angles.
A goniometer is used to document initial and subsequent range of motion, at the visits for occupational injuries, and by disability evaluators to determine a permanent disability. This is to evaluate progress, and also for medico-legal purposes. It is a tool to evaluate Waddell's signs (findings that may indicate symptom magnification.)
In physical therapy, occupational therapy, Orthotics and prosthetics and athletic training, a goniometer measures range of motion of limbs and joints of the body. These measurements help accurately track progress in a rehabilitation program. When a patient has decreased range of motion, a therapist assesses the joint before performing an intervention, and continues to use the tool to monitor progress. The therapist can take these range of motion measurements at any joint. They typically require knowledge about the anatomy of the body, particularly bony landmarks. For example, when measuring the knee joint, the therapist places the axis (point of rotation) on the lateral epicondyle of the femur, and lines up the stationary arm with the greater trochanter of the femur . Finally, the therapist lines up the moveable arm of the goniometer with the lateral malleolus of the fibula , and records a measurement using the degree scale on the circular portion of the tool. Reading accuracy is sometimes a problem with goniometers. Issues with the intra-measure (between measures) and inter-tester (between clinicians) reliability may increase as the experience of the examiner decreases. Some studies suggest that these errors can be anywhere between 5 and 10 degrees. [ citation needed ]
These goniometers come in different forms that some argue increase reliability. [ 6 ] [ 7 ] The universal standard goniometer is a plastic or metal tool with 1 degree increments. The arms are usually not longer than 12-inches, so it can be hard to accurately pinpoint the exact landmark for measurement. The telescopic-armed goniometer is more reliable—with a plastic circular axis like a classic goniometer, but with arms that extend to as long as two feet in either direction.
More recently in the twenty-first century, smartphone application developers have created mobile applications that provide the functions of a goniometer. These applications (such as Knee Goniometer and Goniometer Pro) use the accelerometers in phones to calculate joint angles. Recent research supports these applications and their devices as reliable and valid tools with as much accuracy as a universal goniometer. [ 8 ] [ 9 ] [ 10 ]
Modern rehabilitative therapy motion capture systems perform goniometry at the very least measuring active range of motion. [ 11 ] While in some cases accuracy may be inferior to a goniometer, measuring angles with a motion capture system is superior at measuring during dynamic, as opposed to static situations. Furthermore, using a traditional goniometer takes valuable time. In the clinical context, performing manual measurements takes valuable time and may not be practical.
In surface science , an instrument called a contact angle goniometer or tensiometer measures the static contact angle , advancing and receding contact angles, and sometimes surface tension. The first contact angle goniometer was designed by William Zisman of the United States Naval Research Laboratory in Washington, D.C. and manufactured by ramé-hart (now ramé-hart instrument company), New Jersey, USA. The original manual contact angle goniometer used an eyepiece with a microscope. Today's contact angle goniometer uses a camera and software to capture and analyze the drop shape, and is better suited for dynamic and advanced studies.
Contact angle goniometers can also determine the surface tension for any liquid in gas or the interfacial tension between any two liquids. If the difference in densities between the two fluids is known, the surface tension or interfacial tension can be calculated by the pendant drop method. An advanced instrument often called a goniometer / tensiometer includes software tools that measure surface tension and interfacial tension using the pendant drop, inverted pendant drop, and sessile drop methods, in addition to contact angle . A centrifugal adhesion balance relates the contact angles to the adhesion of the drop to the surface. A gonioreflectometer measures the reflectivity of a surface at a number of angles.
A positioning goniometer or goniometric stage is a device that rotates an object precisely about a fixed axis in space. It is similar to a linear stage —however, rather than move linearly relative to its base, the stage platform rotates partially about a fixed axis above the mounting surface of the platform. Positioning goniometers typically use a worm drive with a partial worm wheel fixed to the underside of the stage platform meshing with a worm in the base. The worm gear may be rotated manually, or by a motor in automated positioning systems.
The included cutting angles of all kinds of sharp edge blades are measured using a laser reflecting goniometer. Developed by the Cutlery and Allied Trades Research Association (CATRA) in the UK, a range of devices can accurately determine the cutting edge profile, including a rounding of the tip to ½°. The included angle of a blade is important in controlling its cutting ability and edge strength—i.e., a low angle makes a thin sharp edge optimized for cutting softer materials, while a large angle makes a thick edge that is less sharp but stronger, which may be better for cutting harder materials.
Used doctor blades , from gravure and other printing and coating processes, can be inspected with a goniometer, typically with a built-in light source, to examine the blade edge for wear and correct angles. A difference in angle from that set on the machine may indicate excessive pressure, and a range of angles ("rounding") probably indicates a lack of stiffness, or wear, in the blade holder assembly. | https://en.wikipedia.org/wiki/Goniometer |
A goniometer is often included in analog audio equipment to display a Lissajous figure which shows the amount of stereo (that is, phase differences ) in a dual-channel signal. [ 1 ] It allows the sound technician to adjust for optimal stereo and determine the makeup of errors such as an inverted signal. Many goniometers also provide a VU or PPM as a secondary function. [ 2 ] A goniometer adapted for surround metering is called a 'jellyfish display'. This term was coined by DK-Technologies A/S from Denmark.
Its function is to plot a signal on a two-dimensional area so that the correlation between the two axes (audio channels, or phases ) becomes apparent.
The channels are plotted on diagonal axes; a left-channel-only signal would form a diagonal line running top-left to bottom-right and a right-channel-only signal would form the opposite diagonal running top-right to bottom-left. A signal on both channels would provide components on both axes and thus expand the plot into two dimensions: a mono signal would produce a straight line angled according to balance; whereas a stereo signal, being asymmetrical, would produce a highly volatile image with a concentration of lines towards the center of the goniometer (see image).
An audio technician would typically begin a session by adjusting the equipment (usually with a 1 kHz mono tone ) so that the output produces a vertical plot line. If one channel were phase-inverted, it would result in the plot line being a horizontal instead of vertical, a sure sign of problems. As for mono signals, a half-inverted signal would be reduced to (near) silence.
The persistence of a CRT display is a desired effect on goniometers because the signal display is very dynamic, and the overall shape or envelope of the signal is the object of interest. In fact, good digital and software goniometers provide artificial and even user-adjustable persistence.
The goniometer proves useful because it provides very dense information in an analog and surprisingly intuitive form. From the display, one can get a good feel for the audio levels for each channel and the amount of stereo and its compatibility as a mono signal. | https://en.wikipedia.org/wiki/Goniometer_(audio) |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Goniometric_function |
A gonioreflectometer is a device for measuring a bidirectional reflectance distribution function (BRDF).
The device consists of a light source illuminating the material to be measured and a sensor that captures light reflected from that material. The light source should be able to illuminate and the sensor should be able to capture data from a hemisphere around the target. The hemispherical rotation dimensions of the sensor and light source are the four dimensions of the BRDF. The 'gonio' part of the word refers to the device's ability to measure at different angles.
Several similar devices have been built and used to capture data for similar functions. Most of these devices use a camera instead of the light intensity-measuring sensor to capture a two-dimensional sample of the target. Examples include: | https://en.wikipedia.org/wiki/Gonioreflectometer |
In biology , gonochorism is a sexual system where there are two sexes and each individual organism is either male or female . [ 1 ] The term gonochorism is usually applied in animal species, the vast majority of which are gonochoric. [ 2 ] : 212–222
Gonochorism contrasts with simultaneous hermaphroditism but it may be hard to tell if a species is gonochoric or sequentially hermaphroditic e.g. parrotfish , Patella ferruginea . [ 3 ] However, in gonochoric species individuals remain either male or female throughout their lives. [ 4 ] Species that reproduce by thelytokous parthenogenesis and do not have males can still be classified as gonochoric. [ 5 ] [ clarification needed ]
The term is derived from Greek gone 'generation' + chorizein 'to separate'. [ 6 ] The term gonochorism originally came from German Gonochorismus . [ 7 ]
Gonochorism is also referred to as unisexualism or gonochory.
Gonochorism has evolved independently multiple times. [ 8 ] It is very evolutionarily stable in animals. [ 9 ] Its stability and advantages have received little attention. [ 10 ] : 46 Gonochorism owes its origin to the evolution of anisogamy , [ 11 ] but it is unclear if the evolution of anisogamy first led to hermaphroditism or gonochorism. [ 2 ] : 213
Gonochorism is thought to be the ancestral state in polychaetes , [ 9 ] : 126 Hexacorallia , [ 12 ] : 74 nematodes , [ 13 ] : 62 and hermaphroditic fishes . Gonochorism is thought to be ancestral in hermaphroditic fishes because it is widespread in basal clades of fish and other vertebrate lineages. [ 14 ]
Two papers from 2008 have suggested that transitions between hermaphroditism and gonochorism or vice versa have occurred in animals between 10 and 20 times. [ 15 ] In a 2017 study involving 165 taxon groups, more evolutionary transitions from gonochorism to hermaphroditism were found than the reverse. [ 16 ]
The term gonochorism is most often used for animal species, an estimated 95% of which are gonochoric. [ 17 ] It is very common in vertebrate species, 99% of which are gonochoric. [ 18 ] [ 19 ] Ninety-eight percent of fishes are gonochoric. [ 20 ] Mammals (including humans [ 21 ] [ 22 ] ) and birds are solely gonochoric. [ 23 ] Tardigrades are almost always gonochoric. [ 24 ] Seventy-five percent of snails are gonochoric. [ 25 ] Most arthropods including a majority of crustaceans are gonochoric. [ 26 ] [ 27 ]
In animals, sex is most often genetically determined, but may be determined by other mechanisms. For example, alligators use temperature-dependent sex determination during egg incubation.
The term gonochorism is not usually applied to plants . Vascular plants which have single-sex individuals are called dioecious , [ 28 ] while bryophytes with single-sex individuals are dioicous . [ 29 ] In flowering plants , individual flowers may be hermaphroditic (i.e., with both stamens and ovaries) or dioecious (unisexual), having either no stamens (i.e., no male parts) or no ovaries (i.e., no female parts). Among flowering plants with unisexual flowers, some also produce hermaphrodite flowers, and the three types may occur in different arrangements on the same or separate plants. Plant species can thus be hermaphrodite, monoecious , dioecious , trioecious , polygamomonoecious , polygamodioecious , andromonoecious , or gynomonoecious .
Examples of species with gonochoric or dioecious pollination include hollies and kiwifruit . In these plants the male plant that supplies the pollen is referred to as the pollenizer .
Gonochorism stands in contrast to other reproductive strategies such as asexual reproduction and hermaphroditism . Closely related taxa can have differing sexual strategies – for example, the genus Ophryotrocha contains species that are gonochoric and species that are hermaphrodites. [ 30 ]
The sex of an individual may also change during its lifetime – this sequential hermaphroditism can, for example, be found in parrotfish [ 31 ] [ 32 ] and cockles . [ citation needed ] | https://en.wikipedia.org/wiki/Gonochorism |
Gonocytes are the precursors of spermatogonia that differentiate in the testis from primordial germ cells around week 7 of embryonic development and exist up until the postnatal period, when they become spermatogonia. [ 1 ] Despite some uses of the term to refer to the precursors of oogonia , it was generally restricted to male germ cells. [ 1 ] [ 2 ] Germ cells operate as vehicles of inheritance by transferring genetic and epigenetic information from one generation to the next. Male fertility is centered around continual spermatogonia which is dependent upon a high stem cell population. Thus, the function and quality of a differentiated sperm cell is dependent upon the capacity of its originating spermatogonial stem cell (SSC). [ 3 ]
Gonocytes represent the germ cells undergoing the successive, short-term and migratory stages of development. This occurs between the time they inhabit the forming gonads on the genital ridge to the time they migrate to the basement membrane of the seminiferous cords . Gonocyte development consists of several phases of cell proliferation , differentiation , migration and apoptosis . [ 4 ] [ 5 ] The abnormal development of gonocytes leads to fertility-related diseases. [ 6 ]
They are also identified as prespermatogonia, prospermatogonia and primitive germ cells, although gonocyte is most common. [ 7 ]
Gonocytes are described as large and spherical, with a prominent nucleus and two nucleoli . [ 1 ] The term, gonocyte, was created in 1957 by Canadian scientists Yves Clermont and Bernard Perey. [ 2 ] They considered it essential to study the origin of spermatogonia and carried out a study on rats to investigate this. [ 8 ] In 1987, Clermont referred to gonocytes as the cells that differentiate into type A spermatogonia , which differentiate into type B spermatogonia and spermatocytes . [ 2 ]
Very few studies used gonocytes to also refer to the female germ cells in the ovarium primordium. [ 9 ] The specification of gonocytes to be confined to male germ cells occurred after foundational differences between the mechanisms of male and female fetal germ cells were uncovered. Some scientists prefer the terms “prospermatogonia” and “prespermatogonia” for their functional clarity. [ 7 ] [ 9 ]
Later studies found that the process from primordial germ cell to spermatogonial development is gradual, without clear gene expression markers to distinguish the precursor cells. [ 2 ] A 2006 study found that some gonocytes differentiate straight into committed spermatogonia (type B) rather than spermatogonial stem cells (type A). [ 1 ]
Gonocytes are long-lived precursor germ cells responsible for the production of spermatogonial stem cells (SSCs). Gonocytes relate to both fetal and neonatal germ cells from the point at which they enter the testis primordial until they reach the base membrane at the seminiferous cords and differentiate. At the time of gastrulation, certain cells are set aside for later gamete development. These cells are called post migratory germ cells (PGCs). The gonocyte population develops from the post migratory germ cells (PGCs) around embryonic day (ED) 15. [ 10 ] At this point of development, PGCs become dormant and remain inactivated until birth. Shortly after birth, the cell cycle continues and the production of postnatal spermatogonia commences. [ 11 ] Gonocytes migrate to the basement membrane to proliferate. Gonocytes that do not migrate undergo apoptosis and are cleared from the seminiferous epithelium. [ 12 ] Spermatogonia are formed in infancy and differentiate throughout adult life. [ 13 ]
There are currently two proposed models for the formation of the spermatogonial lineage during neonatal development. Both models theorize that the gonocyte population develops from a subset of post migratory germ cells (PGCs) but, differ in the proposed subsets of derived gonocytes. One of the models proposes that the PGCs give rise to a single subset of pluripotent gonocytes that either become SSCs from which progenitors then arise or differentiate into type A spermatogonia directly. The other model proposes that the PGCs give rise to multiple predetermined subsets of gonocytes that produce the foundational SSC pool, initial progenitor spermatogonial population, and initial differentiating type A spermatogonia. [ 3 ]
The development of germ cells can be divided into two phases. The first phases involves the fetal and neonatal phases of germ cell development that lead to the formation of the SSCs. The second phase is spermatogenesis , which is a cycle of regulated mitosis , meiosis and differentiation (via spermiogenesis ) leading to the production of mature spermatozoa , also known as sperm cells. [ 4 ] [ 14 ] [ 15 ]
Gonocytes are functionally present during the first phase of germ cell maturation and development. [ 5 ] [ 14 ] This period consists of the primordial germ cells (PGC), the initial cells that commence germ cell development in the embryo , [ 16 ] and the gonocytes, which after being differentiated from PGCs, undergo regulated proliferation, differentiation, migration and apoptosis to produce the SSCs. [ 4 ] [ 5 ] Gonocytes therefore correspond to the developmental stages between the PGCs and SSCs.
Gonocytes are formed from the differentiation of PGCs. [ 5 ] Embryonic cells initiate germ cell development in the proximal epiblast located near the extra-embryonic ectoderm by the release of bone morphogenetic protein 4 (BMP4) and BMP8b . These proteins specify embryonic cells into PGCs expressing the genes PRDM1 and PRDM14 at embryonic day (E) 6.25. The PGCs which are positively stained by alkaline phosphatase and expressing Stella at E7.25 are also specified. [ 17 ] [ 18 ] In between E7.5 and E12.5, these PGCs migrate towards the genital ridge , where they form the testicular cords, via the cytokine interactions of the CXCR4 and c-Kit membrane receptors and their ligands SDF1 and SCF respectively. [ 19 ] [ 20 ] [ 21 ] During this migratory period, PGCs undergo epigenetic reprogramming through genome -wide DNA demethylation . [ 20 ] Once resident in the genital ridge, these germ cells and surrounding supporting cells undergo sex determination driven by the expression of the SRY gene. [ 22 ] It is only after these developmental steps that the germ cells present in the developed testicular cords are identified as gonocytes. [ 5 ]
In order to provide the long-term production of sperm, gonocytes undergo proliferation to produce a populate pool of SSCs. [ 5 ] [ 14 ] [ 6 ] Once enclosed by Sertoli cells to form the testicular cords, gonocytes undergo a succession of differing fetal and neonatal periods of mitosis, with a phase of quiescence in between. [ 4 ] The mitotic activity that occurs in the neonatal period is necessary for the migration of gonocytes to the basement membrane of the seminiferous cords in order to differentiate into the SSCs. [ 4 ] [ 5 ] As many populations of gonocytes are in different stages of development, mitotic and quiescent gonocytes coexist in neonatal developing testes . [ 9 ]
Proliferation in fetal and neonatal gonocytes is differently regulated. Retinoic acid (RA), the bioactive metabolite of retinal , is a morphogen shown to modulate fetal gonocyte proliferation. Investigation of fetal gonocyte activity in organ cultures recorded RA to slightly stimulate proliferation. [ 23 ] [ 24 ] Moreover, RA inhibited differentiation by stopping the fetal gonocytes from entering mitotic arrest while simultaneously triggering apoptosis. RA, by decreasing the overall fetal gonocyte population via apoptosis, is speculated to allow the elimination of mutated and dysfunctional germ cells. [ 24 ] The activation of protein kinase C by phorbol ester PMA also decreased fetal gonocyte mitotic activity. [ 25 ]
There are a number of factors that influence neonatal gonocyte proliferation, including 17β‐estradiol (E2), Leukemia inhibitory factor (LIF), platelet-derived growth factor (PDGF) -BB, and RA. The production of PDGF-BB and E2 by surrounding Sertoli cells activate their respective receptors on neonatal gonocytes, triggering proliferation via an interactive, crosstalk mechanism. [ 26 ] [ 27 ] The regulation of LIF is speculated to allow gonocytes to become sensitive to Sertoli cell factors that trigger proliferation, such as PDGF-BB and E2. [ 28 ] Compared to fetal gonocytes, RA exerts a similar functional role in neonatal gonocytes; It simultaneously stimulates proliferation and apoptosis for regulation of gonocyte and future SSCs population. [ 5 ] [ 23 ] [ 24 ]
The migration of gonocytes to the basement membrane of the seminiferous cords is necessary for their differentiation into SSCs. [ 4 ] [ 14 ] [ 6 ] This process is regulated by different factors.
Various studies provide comprehensive comparison of the expression of c-Kit on the membrane of cells and migratory-related behavior, for example PGCs. [ 29 ] Although c-Kit expression is evident in a small fraction of neonatal gonocytes, [ 30 ] they also express of PDGF receptor beta (PDGFRβ) on their membrane to aid in their migration. [ 31 ] Inhibition of PDGF receptors and c-Kit by in vivo treatment of imatinib , an inhibitory drug , interrupted migration, leading to a number of gonocytes centrally located in the seminiferous cords. [ 26 ]
The ADAM-Integrin - Tetraspanin complexes, which is a family of proteins , also mediate gonocyte migration. These complexes consist of various proteins that bind to integrins found on the basement membrane of the seminiferous cords and at locations where spermatogonia normally reside, allowing the gonocyte to migrate and bind to the basement membrane. [ 32 ]
The differentiation of gonocytes to SSC only occur once the cells have established close contact with the basement membrane in the seminiferous cords. [ 5 ] [ 14 ] RA is the best characterised activator of gonocyte differentiation. [ 5 ] De novo synthesis of RA involves retinol , the precursor to RA, being transported to the membrane receptor STRA6 by the retinol-binding protein released by Sertoli cells. Binding of retinol to STRA6 endocytoses retinol into the cell, whereby it undergoes oxidation reactions to form RA. RA is also directly transported from the surrounding Sertoli cells or the vasculature . RA internalization triggers a variety of pathways that modulate the differentiation, such as PDGF receptor pathways and Janus kinase 2 (JAK2) signaling pathway. [ 24 ]
Anti-Müllerian hormone (AMH), a glycoprotein gonadal hormone produced by Sertoli cells in early development, is the only hormone to significantly increase the number of successfully differentiated gonocytes. [ 33 ]
The timing of differentiation is regulated by NOTCH signaling . [ 34 ] The functional components of the NOTCH signaling pathway are expressed and released by both developing and adult Sertoli cells. [ 35 ] Activation of the signaling pathway is crucial for gonocyte development as it triggers gonocytes to depart from quiescence and enter into differentiation. Over activation of the pathway allows effective inhibition of quiescence and gonocyte differentiation. [ 34 ] [ 35 ]
Gonocytes are large cells with a spherical euchromatic nucleus , two nucleoli and a surrounding, ring-like cytosol . [ 9 ] [ 36 ] Throughout the majority of their developmental period, gonocytes are structurally supported by the cytoplasmic extensions of surrounding Sertoli cells and are suspended by Sertoli cell nuclei from the basement membrane. [ 9 ] [ 37 ] [ 5 ] Gonocytes are attached to Sertoli cells by gap junctions , [ 37 ] desmosome junctions [ 5 ] and a number of different cell adhesion molecules such as connexin 43 , PB-cadherin and NCAM [ 9 ] for regulation of cell-to-cell communication . [ 37 ] [ 5 ] Gonocytes dissociate from these junctions and migrate so that the basal side of the cell is in close proximity with the basement membrane, where they undergo phenotypic changes and take the appearance of spermatogonia. [ 9 ] [ 5 ]
Dysfunctional development in germ cells plays a significant role in fertility-related diseases . [ 5 ] [ 6 ] The development of PGCs to gonocytes, and gonocyte differentiation to SSCs is critical for adult fertility and the defective growth often leads to infertility . [ 5 ]
Testicular germ cell tumors , that occur primarily in young adults, are the consequent of preinvasive cells called carcinoma in situ (CIS). [ 38 ] The development of CIS is due to fetal germ cells, such as gonocytes, arrested in quiescence and unable to properly differentiate. [ 38 ] [ 39 ] This leads to malignant transformation of the germ cells until it becomes an overt germ cell cancer after puberty . [ 39 ]
Cryptorchidism , also known as undescended testis, is a common birth defect affecting male genital formation. [ 40 ] Individuals diagnosed with cryptorchidism are often at risk of testicular cancer and infertility due to dysfunction in the development of the neonatal germ cells, in particular, the disruption of the differentiation of gonocytes into adult dark-spermatogonia. [ 6 ] It is proposed that this dysfunction is a product of heat stress caused by the undescended testes remaining in the abdomen and unable to regulate its temperature which is often accomplished by the scrotum . [ 41 ] | https://en.wikipedia.org/wiki/Gonocyte |
The Good Clinical Practice Directive (Directive 2005/28/EC of 8 April 2005 of the European Parliament and of the Council) lays down principles and detailed guidelines for good clinical practice as regards conducting clinical trials of medicinal products for human use, as well as the requirements for authorisation of the manufacturing or importation of such products.
The directive deals with the following items: | https://en.wikipedia.org/wiki/Good_Clinical_Practice_Directive |
Good engineering practice ( GEP ) is engineering and technical activities that ensure that a company manufactures products of the required quality as expected (e.g., by the relevant regulatory authorities). Good engineering practices are to ensure that the development and/or manufacturing effort consistently generates deliverables that support the requirements for qualification or validation. Good engineering practices are applied to all industries that require engineering. [ 1 ]
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Good_engineering_practice |
Current good manufacturing practices ( cGMP ) are those conforming to the guidelines recommended by relevant agencies. Those agencies control the authorization and licensing of the manufacture and sale of food and beverages , [ 1 ] cosmetics , [ 2 ] pharmaceutical products , [ 3 ] dietary supplements , [ 4 ] and medical devices . [ 5 ] These guidelines provide minimum requirements that a manufacturer must meet to assure that their products are consistently high in quality, from batch to batch, for their intended use.
The rules that govern each industry may differ significantly; however, the main purpose of GMP is always to prevent harm from occurring to the end user. [ 2 ] Additional tenets include ensuring the end product is free from contamination, that it is consistent in its manufacture, that its manufacture has been well documented, that personnel are well trained, and that the product has been checked for quality more than just at the end phase. [ 2 ] GMP is typically ensured through the effective use of a quality management system (QMS). [ 1 ] : "The Basis for GMP", [ 2 ]
Good manufacturing practice, along with good agricultural practice , good laboratory practice and good clinical practice , are overseen by regulatory agencies in the United Kingdom, United States, Canada, various European countries, China, India and other countries.
Good manufacturing practice guidelines provide guidance for manufacturing, testing, and quality assurance in order to ensure that a manufactured product is safe for human consumption or use. Many countries have legislated that manufacturers follow GMP procedures and create their own GMP guidelines that correspond with their legislation.
All guidelines follow a few basic principles: [ 2 ] [ 6 ]
Good manufacturing practice is recommended with the goal of safeguarding the health of consumers and patients as well as producing quality products. In the United States, a food or drug may be deemed "adulterated" if it has passed all of the specifications tests but is found to be manufactured in a facility or condition which violates or does not comply with current good manufacturing guideline.
GMP standards are not prescriptive instructions on how to manufacture products. They are a series of performance based requirements that must be met during manufacturing. [ 7 ] When a company is setting up its quality program and manufacturing process, there may be many ways it can fulfill GMP requirements. It is the company's responsibility to determine the most effective and efficient quality process that both meets business and regulatory needs. [ 1 ] : "Decision Makers' Summary", [ 2 ]
Regulatory agencies have recently begun to look at more fundamental quality metrics of manufacturers than just compliance with basic GMP regulations. US-FDA has found that manufacturers who have implemented quality metrics programs [ 8 ] gain a deeper insight into employee behaviors that impact product quality.
In its Guidance for Industry "Data Integrity and Compliance With Drug CGMP" US-FDA states “it is the role of management with executive responsibility to create a quality culture where employees understand that data integrity is an organizational core value and employees are encouraged to identify and promptly report data integrity issues.” [ 9 ] Australia's Therapeutic Goods Administration has said that recent data integrity failures have raised questions about the role of quality culture in driving behaviors. [ 10 ] In addition, non-governmental organizations such as the International Society for Pharmaceutical Engineering (ISPE) and the Parenteral Drug Association (PDA) have developed information and resources to help pharmaceutical companies better understand why quality culture is important and how to assess the current situation within a site or organization. [ 11 ]
GMP is enforced in the United States by the U.S. Food and Drug Administration (FDA), under Title 21 CFR . The regulations use the phrase "current good manufacturing practices" (CGMP) to describe these guidelines. [ 12 ] [ 13 ] [ 14 ] [ 15 ] Courts may theoretically hold that a product is adulterated even if there is no specific regulatory requirement that was violated as long as the process was not performed according to industry standards. [ 16 ] However, since June 2007, a different set of CGMP requirements have applied to all manufacturers of dietary supplements , with additional supporting guidance issued in 2010. [ 4 ] Additionally, in the U.S., medical device manufacturers must follow what are called "quality system regulations" which are deliberately harmonized with ISO requirements, not necessarily CGMPs. [ 14 ]
The World Health Organization (WHO) version of GMP is used by pharmaceutical regulators and the pharmaceutical industry in over 100 countries worldwide, primarily in the developing world. [ 3 ] The European Union 's GMP (EU GMP) enforces similar requirements to WHO GMP, as does the FDA's version in the US. Similar GMPs are used in other countries, with Australia, Canada, Japan, Saudi Arabia, Singapore, Philippines], Vietnam and others having highly developed/sophisticated GMP requirements. [ 17 ] In the United Kingdom, the Medicines Act (1968) covers most aspects of GMP in what is commonly referred to as "The Orange Guide," which is named so because of the color of its cover; it is officially known as Rules and Guidance for Pharmaceutical Manufacturers and Distributors . [ 18 ]
Since the 1999 publication of Good Manufacturing Practice for Active Pharmaceutical Ingredients , by the International Conference on Harmonization (ICH), GMPs now apply in those countries and trade groupings that are signatories to ICH (the EU, Japan and the U.S.), and applies in other countries (e.g., Australia, Canada, Singapore) which adopt ICH guidelines for the manufacture and testing of active raw materials. [ 17 ]
Within the European Union GMP inspections are performed by National Regulatory Agencies. GMP inspections are performed in Canada by the Health Products and Food Branch Inspectorate; [ 19 ] in the United Kingdom by the Medicines and Healthcare products Regulatory Agency (MHRA); [ 20 ] in the Republic of Korea (South Korea) by the Ministry of Food and Drug Safety (MFDS); [ 21 ] in Australia by the Therapeutic Goods Administration (TGA); [ 22 ] in Bangladesh by the Directorate General of Drug Administration (DGDA); [ 23 ] in South Africa by the Medicines Control Council (MCC); [ 24 ] in Brazil by the National Health Surveillance Agency (ANVISA); [ 25 ] in India by state Food and Drugs Administrations (FDA), reporting to the Central Drugs Standard Control Organization ; [ 26 ] in Pakistan by the Drug Regulatory Authority of Pakistan ; [ 27 ] in Nigeria by NAFDAC ; [ 28 ] and by similar national organizations worldwide. Each of the inspectorates carries out routine GMP inspections to ensure that drug products are produced safely and correctly. Additionally, many countries perform pre-approval inspections (PAI) for GMP compliance prior to the approval of a new drug for marketing.
Regulatory agencies (including the FDA in the U.S. and regulatory agencies in many European nations) are authorized to conduct unannounced inspections, though some are scheduled. [ 12 ] [ 18 ] [ 21 ] [ 22 ] [ 23 ] [ 24 ] [ 26 ] [ 27 ] [ 28 ] FDA routine domestic inspections are usually unannounced, but must be conducted according to 704(a) of the Food, Drug and Cosmetic Act (21 USCS § 374), which requires that they are performed at a "reasonable time". Courts have held that any time the firm is open for business is a reasonable time for an inspection. [ 29 ]
Other good-practice systems, along the same lines as GMP, exist:
Collectively, these and other good-practice requirements are referred to as " GxP " requirements, all of which follow similar philosophies. Other examples include good guidance practice and good tissue practice. | https://en.wikipedia.org/wiki/Good_manufacturing_practice |
Within the branch of materials science known as material failure theory , the Goodman relation (also called a Goodman diagram , a Goodman-Haigh diagram , a Haigh diagram or a Haigh-Soderberg diagram ) is an equation used to quantify the interaction of mean and alternating stresses on the fatigue life of a material . [ 1 ] The equation is typically presented as a linear curve of mean stress vs. alternating stress that provides the maximum number of alternating stress cycles a material will withstand before failing from fatigue. [ 2 ] [ 3 ]
A scatterplot of experimental data shown on an amplitude versus mean stress plot can often be approximated by a parabola known as the Gerber line , which can in turn be (conservatively) approximated by a straight line called the Goodman line . [ 1 ] [ 4 ]
The relations can be represented mathematically as:
( n σ m σ b ) 2 + n σ a σ w = 1 {\displaystyle ({\frac {n\sigma _{\text{m}}}{\sigma _{\text{b}}}})^{2}+{\frac {n\sigma _{\text{a}}}{\sigma _{\text{w}}}}=1} , Gerber Line (parabola)
where σ a {\displaystyle \sigma _{\text{a}}} is the stress amplitude, σ m {\displaystyle \sigma _{\text{m}}} is the mean stress, σ w {\displaystyle \sigma _{\text{w}}} is the fatigue limit for completely reversed loading, σ b {\displaystyle \sigma _{\text{b}}} is the ultimate tensile strength of the material and n {\displaystyle n} is the factor of safety .
The Gerber parabola is indication of the region just beneath the failure points during experiment.
The Goodman line connects σ b {\displaystyle \sigma _{\text{b}}} on the abscissa and σ w {\displaystyle \sigma _{\text{w}}} on the ordinate. The Goodman line is much safer consideration than the Gerber parabola because it is completely inside the Gerber parabola and excludes some of area which is nearby to failure region.
The Soderberg Line connects σ y {\displaystyle \sigma _{\text{y}}} on the abscissa and σ w {\displaystyle \sigma _{\text{w}}} on the ordinate, which is more conservative consideration and much safer. σ y {\displaystyle \sigma _{\text{y}}} is the yield strength of the material. [ 5 ] [ 6 ]
The general trend given by the Goodman relation is one of decreasing fatigue life with increasing mean stress for a given level of alternating stress. The relation can be plotted to determine the safe cyclic loading of a part; if the coordinate given by the mean stress and the alternating stress lies under the curve given by the relation, then the part will survive. If the coordinate is above the curve, then the part will fail for the given stress parameters. [ 7 ] | https://en.wikipedia.org/wiki/Goodman_relation |
In computer networks, goodput (a portmanteau of good and throughput ) is the application-level throughput of a communication; i.e. the number of useful information bits delivered by the network to a certain destination per unit of time. The amount of data considered excludes protocol overhead bits as well as retransmitted data packets. This is related to the amount of time from the first bit of the first packet sent (or delivered) until the last bit of the last packet is delivered.
For example, if a file is transferred, the goodput that the user experiences corresponds to the file size in bits divided by the file transfer time. The goodput is always lower than the throughput (the gross bit rate that is transferred physically), which generally is lower than network access connection speed (the channel capacity or bandwidth ).
Examples of factors that cause lower goodput than throughput are:
Over Ethernet, files are broken down into individual chunks for transmission. These chunks are no larger than the maximum transmission unit of IP over Ethernet, or 1500 bytes . Each packet requires 20 bytes of IPv4 header information and 20 bytes of TCP header information, leaving 1460 bytes per packet for file data ( Linux and macOS [ 1 ] are further limited to 1448 bytes as they also carry a 12-byte time stamp). The data is transmitted over Ethernet in a frame, which imposes a 26 byte overhead per packet. Given these overheads, the maximum goodput is 1460/1526 × 100 Mbit/s which is 95.67 megabits per second or 11.959 megabytes per second .
Note that this example doesn't consider additional Ethernet overhead, such as the interframe gap (a minimum of 96 bit times), or collisions (which have a variable impact, depending on the network load). TCP adds the overhead of acknowledgements (which along with the round-trip delay time and the TCP window size in effect rate-limit each individual TCP connection, see bandwidth-delay product ). This example does not consider the overhead of the HTTP protocol itself, which becomes relevant when transferring small files.
The goodput is a ratio between delivered amount of information, and the total delivery time. This delivery time includes: | https://en.wikipedia.org/wiki/Goodput |
In mathematical logic , Goodstein's theorem is a statement about the natural numbers , proved by Reuben Goodstein in 1944, which states that every Goodstein sequence (as defined below) eventually terminates at 0. Laurence Kirby and Jeff Paris [ 1 ] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory ). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen 's 1943 direct proof of the unprovability of ε 0 -induction in Peano arithmetic. The Paris–Harrington theorem gave another example.
Kirby and Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna ) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for Goodstein sequences, Kirby and Paris showed that it cannot be proven in Peano arithmetic alone. [ 1 ]
Goodstein sequences are defined in terms of a concept called "hereditary base- n notation". This notation is very similar to usual base- n positional notation , but the usual notation does not suffice for the purposes of Goodstein's theorem.
To achieve the ordinary base- n notation, where n is a natural number greater than 1, an arbitrary natural number m is written as a sum of multiples of powers of n :
where each coefficient a i satisfies 0 ≤ a i < n , and a k ≠ 0 . For example, to achieve the base 2 notation , one writes
Thus the base-2 representation of 35 is 100011, which means 2 5 + 2 + 1 . Similarly, 100 represented in base-3 is 10201:
Note that the exponents themselves are not written in base- n notation. For example, the expressions above include 2 5 and 3 4 , and 5 > 2, 4 > 3.
To convert a base-n notation (which is a step in achieving base- n representation) to a hereditary base- n notation, first rewrite all of the exponents as a sum of powers of n (with the limitation on the coefficients 0 ≤ a i < n ). Then rewrite any exponent inside the exponents in base- n notation (with the same limitation on the coefficients), and continue in this way until every number appearing in the expression (except the bases themselves) is written in base- n notation.
For example, while 35 in ordinary base-2 notation is 2 5 + 2 + 1 , it is written in hereditary base-2 notation as
using the fact that 5 = 2 2 1 + 1. Similarly, 100 in hereditary base-3 notation is
The Goodstein sequence G m {\displaystyle G_{m}} of a number m is a sequence of natural numbers. The first element in the sequence G m {\displaystyle G_{m}} is m itself. To get the second, G m ( 2 ) {\displaystyle G_{m}(2)} , write m in hereditary base-2 notation, change all the 2s to 3s, and then subtract 1 from the result. In general, the 1 + n th term, G m ( n + 1 ) {\displaystyle G_{m}(n+1)} , of the Goodstein sequence of m is as follows:
Early Goodstein sequences terminate quickly. For example, G 3 {\displaystyle G_{3}} terminates at the 6th step:
Later Goodstein sequences increase for a very large number of steps. For example, G 4 {\displaystyle G_{4}} OEIS : A056193 starts as follows:
Elements of G 4 {\displaystyle G_{4}} continue to increase for a while, but at base 3 ⋅ 2 402 653 209 {\displaystyle 3\cdot 2^{402\,653\,209}} ,
they reach the maximum of 3 ⋅ 2 402 653 210 − 1 {\displaystyle 3\cdot 2^{402\,653\,210}-1} , stay there for the next 3 ⋅ 2 402 653 209 {\displaystyle 3\cdot 2^{402\,653\,209}} steps, and then begin their descent.
However, even G 4 {\displaystyle G_{4}} doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase. G 19 {\displaystyle G_{19}} increases much more rapidly and starts as follows:
8 8 8 − 1 = 7 ⋅ 8 7 ⋅ 8 7 + 7 ⋅ 8 6 + 7 ⋅ 8 5 + 7 ⋅ 8 4 + 7 ⋅ 8 3 + 7 ⋅ 8 2 + 7 ⋅ 8 + 7 {\displaystyle 8^{8^{8}}-1=7\cdot 8^{7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}+7\cdot 8^{3}+7\cdot 8^{2}+7\cdot 8+7}} + 7 ⋅ 8 7 ⋅ 8 7 + 7 ⋅ 8 6 + 7 ⋅ 8 5 + 7 ⋅ 8 4 + 7 ⋅ 8 3 + 7 ⋅ 8 2 + 7 ⋅ 8 + 6 + ⋯ {\displaystyle {}+7\cdot 8^{7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}+7\cdot 8^{3}+7\cdot 8^{2}+7\cdot 8+6}+\cdots } + 7 ⋅ 8 8 + 2 + 7 ⋅ 8 8 + 1 + 7 ⋅ 8 8 {\displaystyle {}+7\cdot 8^{8+2}+7\cdot 8^{8+1}+7\cdot 8^{8}} + 7 ⋅ 8 7 + 7 ⋅ 8 6 + 7 ⋅ 8 5 + 7 ⋅ 8 4 {\displaystyle {}+7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}} + 7 ⋅ 8 3 + 7 ⋅ 8 2 + 7 ⋅ 8 + 7 {\displaystyle {}+7\cdot 8^{3}+7\cdot 8^{2}+7\cdot 8+7}
7 ⋅ 9 7 ⋅ 9 7 + 7 ⋅ 9 6 + 7 ⋅ 9 5 + 7 ⋅ 9 4 + 7 ⋅ 9 3 + 7 ⋅ 9 2 + 7 ⋅ 9 + 7 {\displaystyle 7\cdot 9^{7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}+7\cdot 9^{3}+7\cdot 9^{2}+7\cdot 9+7}} + 7 ⋅ 9 7 ⋅ 9 7 + 7 ⋅ 9 6 + 7 ⋅ 9 5 + 7 ⋅ 9 4 + 7 ⋅ 9 3 + 7 ⋅ 9 2 + 7 ⋅ 9 + 6 + ⋯ {\displaystyle {}+7\cdot 9^{7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}+7\cdot 9^{3}+7\cdot 9^{2}+7\cdot 9+6}+\cdots } + 7 ⋅ 9 9 + 2 + 7 ⋅ 9 9 + 1 + 7 ⋅ 9 9 {\displaystyle {}+7\cdot 9^{9+2}+7\cdot 9^{9+1}+7\cdot 9^{9}} + 7 ⋅ 9 7 + 7 ⋅ 9 6 + 7 ⋅ 9 5 + 7 ⋅ 9 4 {\displaystyle {}+7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}} + 7 ⋅ 9 3 + 7 ⋅ 9 2 + 7 ⋅ 9 + 6 {\displaystyle {}+7\cdot 9^{3}+7\cdot 9^{2}+7\cdot 9+6}
In spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.
Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G m {\displaystyle G_{m}} , we construct a parallel sequence P m {\displaystyle P_{m}} of ordinal numbers in Cantor normal form which is strictly decreasing and terminates. A common misunderstanding of this proof is to believe that G m {\displaystyle G_{m}} goes to 0 {\displaystyle 0} because it is dominated by P m {\displaystyle P_{m}} . Actually, the fact that P m {\displaystyle P_{m}} dominates G m {\displaystyle G_{m}} plays no role at all. The important point is: G m ( k ) {\displaystyle G_{m}(k)} exists if and only if P m ( k ) {\displaystyle P_{m}(k)} exists (parallelism), and comparison between two members of G m {\displaystyle G_{m}} is preserved when comparing corresponding entries of P m {\displaystyle P_{m}} . [ 2 ] Then if P m {\displaystyle P_{m}} terminates, so does G m {\displaystyle G_{m}} . By infinite regress , G m {\displaystyle G_{m}} must reach 0 {\displaystyle 0} , which guarantees termination.
We define a function f = f ( u , k ) {\displaystyle f=f(u,k)} which computes the hereditary base k {\displaystyle k} representation of u {\displaystyle u} and then replaces each occurrence of the base k {\displaystyle k} with the first infinite ordinal number ω {\displaystyle \omega } . For example, f ( 100 , 3 ) = f ( 3 3 1 + 1 + 2 ⋅ 3 2 + 1 , 3 ) = ω ω 1 + 1 + ω 2 ⋅ 2 + 1 = ω ω + 1 + ω 2 ⋅ 2 + 1 {\displaystyle f(100,3)=f(3^{3^{1}+1}+2\cdot 3^{2}+1,3)=\omega ^{\omega ^{1}+1}+\omega ^{2}\cdot 2+1=\omega ^{\omega +1}+\omega ^{2}\cdot 2+1} .
Each term P m ( n ) {\displaystyle P_{m}(n)} of the sequence P m {\displaystyle P_{m}} is then defined as f ( G m ( n ) , n + 1 ) {\displaystyle f(G_{m}(n),n+1)} . For example, G 3 ( 1 ) = 3 = 2 1 + 2 0 {\displaystyle G_{3}(1)=3=2^{1}+2^{0}} and P 3 ( 1 ) = f ( 2 1 + 2 0 , 2 ) = ω 1 + ω 0 = ω + 1 {\displaystyle P_{3}(1)=f(2^{1}+2^{0},2)=\omega ^{1}+\omega ^{0}=\omega +1} . Addition, multiplication and exponentiation of ordinal numbers are well defined.
We claim that f ( G m ( n ) , n + 1 ) > f ( G m ( n + 1 ) , n + 2 ) {\displaystyle f(G_{m}(n),n+1)>f(G_{m}(n+1),n+2)} :
Let G m ′ ( n ) {\displaystyle G'_{m}(n)} be G m ( n ) {\displaystyle G_{m}(n)} after applying the first, base-changing operation in generating the next element of the Goodstein sequence,
but before the second minus 1 operation in this generation.
Observe that G m ( n + 1 ) = G m ′ ( n ) − 1 {\displaystyle G_{m}(n+1)=G'_{m}(n)-1} .
Then f ( G m ( n ) , n + 1 ) = f ( G m ′ ( n ) , n + 2 ) {\displaystyle f(G_{m}(n),n+1)=f(G'_{m}(n),n+2)} . Now we apply the minus 1 operation, and f ( G m ′ ( n ) , n + 2 ) > f ( G m ( n + 1 ) , n + 2 ) {\displaystyle f(G'_{m}(n),n+2)>f(G_{m}(n+1),n+2)} , as G m ′ ( n ) = G m ( n + 1 ) + 1 {\displaystyle G'_{m}(n)=G_{m}(n+1)+1} .
For example, G 4 ( 1 ) = 2 2 {\displaystyle G_{4}(1)=2^{2}} and G 4 ( 2 ) = 2 ⋅ 3 2 + 2 ⋅ 3 + 2 {\displaystyle G_{4}(2)=2\cdot 3^{2}+2\cdot 3+2} , so f ( 2 2 , 2 ) = ω ω {\displaystyle f(2^{2},2)=\omega ^{\omega }} and f ( 2 ⋅ 3 2 + 2 ⋅ 3 + 2 , 3 ) = ω 2 ⋅ 2 + ω ⋅ 2 + 2 {\displaystyle f(2\cdot 3^{2}+2\cdot 3+2,3)=\omega ^{2}\cdot 2+\omega \cdot 2+2} , which is strictly smaller. Note that in order to calculate f ( G m ( n ) , n + 1 ) {\displaystyle f(G_{m}(n),n+1)} , we first need to write G m ( n ) {\displaystyle G_{m}(n)} in hereditary base n + 1 {\displaystyle n+1} notation, as for instance the expression ω ω − 1 {\displaystyle \omega ^{\omega }-1} is not an ordinal.
Thus the sequence P m {\displaystyle P_{m}} is strictly decreasing. As the standard order < on ordinals is well-founded , an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals terminates (and cannot be infinite). But P m ( n ) {\displaystyle P_{m}(n)} is calculated directly from G m ( n ) {\displaystyle G_{m}(n)} . Hence the sequence G m {\displaystyle G_{m}} must terminate as well, meaning that it must reach 0 {\displaystyle 0} .
While this proof of Goodstein's theorem is fairly easy, the Kirby–Paris theorem , [ 1 ] which shows that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic.
The above proof still works if the definition of the Goodstein sequence is changed so that the base-changing operation replaces each occurrence of the base b {\displaystyle b} with b + 2 {\displaystyle b+2} instead of b + 1 {\displaystyle b+1} .
More generally, let b 1 {\displaystyle b_{1}} , b 2 {\displaystyle b_{2}} , b 3 , … {\displaystyle b_{3},\ldots } be any non-decreasing sequence of integers with b 1 ≥ 2 {\displaystyle b_{1}\geq 2} .
Then let the ( n + 1 ) {\displaystyle (n+1)} st
term G m ( n + 1 ) {\displaystyle G_{m}(n+1)} of the extended Goodstein sequence of m {\displaystyle m} be as
follows:
A simple modification of the above proof shows that this sequence still terminates. For example, if b n = 4 {\displaystyle b_{n}=4} and if b n + 1 = 9 {\displaystyle b_{n+1}=9} ,
then f ( 3 ⋅ 4 4 4 + 4 , 4 ) = 3 ω ω ω + ω = f ( 3 ⋅ 9 9 9 + 9 , 9 ) {\displaystyle f(3\cdot 4^{4^{4}}+4,4)=3\omega ^{\omega ^{\omega }}+\omega =f(3\cdot 9^{9^{9}}+9,9)} , hence the ordinal f ( 3 ⋅ 4 4 4 + 4 , 4 ) {\displaystyle f(3\cdot 4^{4^{4}}+4,4)} is strictly greater than the ordinal f ( ( 3 ⋅ 9 9 9 + 9 ) − 1 , 9 ) . {\displaystyle f{\big (}(3\cdot 9^{9^{9}}+9)-1,9{\big )}.}
The extended version is in fact the one considered in Goodstein's original paper, [ 3 ] where Goodstein proved that it is equivalent to the restricted ordinal theorem (i.e. the claim that transfinite induction below ε 0 is valid), and gave a finitist proof for the case where m ≤ b 1 b 1 b 1 {\displaystyle m\leq b_{1}^{b_{1}^{b_{1}}}} (equivalent to transfinite induction up to ω ω ω {\displaystyle \omega ^{\omega ^{\omega }}} ).
The extended Goodstein's theorem without any restriction on the sequence b n is not formalizable in Peano arithmetic (PA), since such an arbitrary infinite sequence cannot be represented in PA. This seems to be what kept Goodstein from claiming back in 1944 that the extended Goodstein's theorem is unprovable in PA due to Gödel's second incompleteness theorem and Gentzen's proof of the consistency of PA using ε 0 -induction. [ 4 ] However, inspection of Gentzen's proof shows that it only needs the fact that there is no primitive recursive strictly decreasing infinite sequence of ordinals, so limiting b n to primitive recursive sequences would have allowed Goodstein to prove an unprovability result. [ 4 ] Furthermore, with the relatively elementary technique of the Grzegorczyk hierarchy , it can be shown that every primitive recursive strictly decreasing infinite sequence of ordinals
can be "slowed down" so that it can be transformed to a Goodstein sequence where b n = n + 1 {\displaystyle b_{n}=n+1} , thus giving an alternative proof to the same result Kirby and Paris proved. [ 4 ]
The Goodstein function , G : N → N {\displaystyle {\mathcal {G}}:\mathbb {N} \to \mathbb {N} } , is defined such that G ( n ) {\displaystyle {\mathcal {G}}(n)} is the length of the Goodstein sequence that starts with n . (This is a total function since every Goodstein sequence terminates.) The extremely high growth rate of G {\displaystyle {\mathcal {G}}} can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions H α {\displaystyle H_{\alpha }} in the Hardy hierarchy , and the functions f α {\displaystyle f_{\alpha }} in the fast-growing hierarchy of Löb and Wainer:
Some examples:
(For Ackermann function and Graham's number bounds see fast-growing hierarchy#Functions in fast-growing hierarchies .)
Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine ; thus the function which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequence reaches 0 , returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function. | https://en.wikipedia.org/wiki/Goodstein's_theorem |
In biology, the Goodwin model describes negative feedback oscillators in cellular systems, [ 1 ] for example, circadian rhythms or enzymatic regulation (such as lactose in bacteria). The Goodwin model, though, shows no stable limit cycles .
limit cycles can exist, see references. [ 2 ] [ 3 ] But not in the original Goodwin model which only has two variables.
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Goodwin_model_(biology) |
Google Docs is an online word processor and part of the free, web-based Google Docs Editors suite offered by Google . Google Docs is accessible via a web browser as a web-based application and is also available as a mobile app on Android and iOS and as a desktop application on Google's ChromeOS .
Google Docs allows users to create and edit documents online while collaborating with users in real-time. Edits are tracked by the user making the edit, with a revision history presenting changes. [ 4 ] An editor's position is highlighted with an editor-specific color and cursor, and a permissions system regulates what users can do. Updates have introduced features using machine learning , including "Explore", offering search results based on the contents of a document, and "Action items", allowing users to assign tasks to other users. [ 5 ]
Google Docs supports opening and saving documents in the standard OpenDocument format as well as in Rich text format , plain Unicode text, zipped HTML , and Microsoft Word . Exporting to PDF and EPUB formats is implemented.
Google Docs originated from Writely, a web-based word processor created by the software company Upstartle and launched in August 2005. [ 6 ] [ 7 ] It began as an experiment by programmers Sam Schillace, Steve Newman, and Claudia Carpenter, trying out the then-new Ajax technology and the "contentEditable" HTML feature. [ 7 ] On March 9, 2006, Google announced that it had acquired Upstartle. [ 8 ] [ 9 ] Google would release a new product, based on Writely, called Google documents on October 10, 2006. [ 10 ] In July 2009, Google dropped the beta testing status from Google Docs. [ 11 ] In March 2010, Google acquired DocVerse, an online document collaboration company. DocVerse allowed multiple users to collaborate online on Microsoft Word documents, like other Microsoft Office formats, such as Excel and PowerPoint . [ 12 ] Improvements based on DocVerse were announced and deployed in April 2010. [ 13 ] In June 2012, Google acquired Quickoffice , a freeware proprietary productivity suite for mobile devices. [ 14 ] In October 2012, Google renamed the Google Drive products, and Google Documents became Google Docs. At the same time, Google Chrome App versions of Google Docs, Google Sheets , and Google Slides were released, which provided shortcuts to the service on Chrome's new tab page. [ 15 ] In February 2019, Google announced grammar suggestions in Docs, expanding their spell check using machine translation techniques to help catch tricky grammatical errors. [ 16 ] In March of 2023, Google Docs, with Slides and Sheets, introduced a new UI theme. [ 17 ] On the 19th March 2024, Google announced that Google Docs, would be folded into the existing G-Suite system, which will incorporate all of Google’s services under one-roof.
Google Docs is available as a web application supported on Google Chrome , Firefox , Microsoft Edge , and Safari web browsers. [ 18 ] Users can access all Docs, as well as other files, collectively through the Google Drive website. In June 2014, Google rolled out a dedicated website homepage for Docs that contains only files created with the service. [ 19 ] In 2014, Google launched a dedicated mobile app for Docs on the Android and iOS mobile operating systems. [ 20 ] [ 21 ] [ 22 ] The mobile website for Docs was updated in 2015 with a "simpler, more uniform" interface, and while users can read files through the mobile websites, users trying to edit will be redirected towards the dedicated mobile app, thus preventing editing on the mobile web. [ 23 ]
Google Docs and the other apps in the Google Drive suite serve as a tool for collaborative editing of documents in real time. Documents can be shared, opened, and edited by multiple users simultaneously, and users can see character-by-character changes as other collaborators make edits. Changes are automatically saved to Google 's servers, and a revision history is automatically kept so past edits may be viewed and reverted. [ 24 ] To resolve concurrent edits from different users, Google Docs uses an operational transformation method based on the Jupiter algorithm, where the document is stored as a list of changes . [ 25 ] [ 26 ]
An editor's current position is represented with an editor-specific color/cursor, so if another editor happens to be viewing that part of the document, they can see edits as they occur. A sidebar chat functionality allows collaborators to discuss edits. The revision history allows users to see the additions made to a document, with each author distinguished by color. Only adjacent revisions can be compared, and users cannot control how frequently revisions are saved. Files can be exported to a user's local computer in a variety of formats ( ODF , HTML , PDF , RTF , Text , Office Open XML ).
In March 2014, Google introduced add-ons, new tools from third-party developers that add more features to Google Docs. [ 27 ] To view and edit documents offline on a computer, users need to use the Google Chrome web browser. A Chrome extension , Google Docs Offline, allows users to enable offline support for Docs files on the Google Drive website. [ 28 ] The Android and iOS apps natively support offline editing. [ 29 ] [ 30 ]
In June 2014, Google introduced "Suggested edits" in Google Docs; as part of the "commenting access" permission, participants can come up with suggestions for edits that the author can accept or reject, in contrast to full editing ability. [ 21 ] In October 2016, Google announced "Action items" for Docs. If a user writes phrases such as "Ryan to follow up on the keynote script", the service will intelligently assign that action to "Ryan". Google states this will make it easier for other collaborators to see which person is responsible for what task. When a user visits Google Drive, Docs, Sheets, or Slides, any files with tasks assigned to them will be highlighted with a badge. [ 31 ]
A basic research tool was introduced in 2012. [ 32 ] [ 33 ] [ 34 ] This was expanded into "Explore" in September 2016, which has additional functionality through machine learning . [ 35 ] [ 36 ] [ 37 ] In Google Docs, Explore shows relevant Google search results based on information in the document, simplifying information gathering. Users can also mark specific document text, press Explore, and see search results based on the marked text only.
In December 2016, Google introduced a quick citations feature to Google Docs. The quick citation tool allows users to "insert citations as footnotes with the click of a button" on the web through the Explore feature introduced in September. The citation feature also marked the launch of the Explore functionalities in G Suite for Education accounts. [ 38 ] [ 39 ] [ 40 ]
Limits to insertable file sizes, overall document length, and size are listed below: [ 42 ] [ 43 ]
Google Docs and the Google Docs Editors suite are free of charge for use by individuals but are also available as part of Google's business-centered Google Workspace , enabling additional business-focused functionality on payment of a monthly subscription. [ 44 ]
A simple find-and-replace tool is available. Google offers an extension for the Google Chrome web browser called Office editing for Docs, Sheets and Slides that enables users to view and edit Microsoft Word documents on Google Chrome via the Docs app. The extension can be used for opening Office files stored on the computer using Chrome, as well as for opening Office files encountered on the web (in the form of email attachments, web search results, etc.) without having to download them. The extension is installed on ChromeOS by default. [ 45 ] Google Cloud Connect was a plug-in for Microsoft Office 2003, 2007, and 2010 that could automatically store and synchronize any Word document to Google Docs (before the introduction of Drive) in Google Docs or Microsoft Office formats. The online copy was automatically updated each time the Microsoft Word document was saved. Microsoft Word documents could be edited offline and synchronized later when online. Google Cloud Connect maintained previous Microsoft Word document versions and allowed multiple users to collaborate by working on the same document at the same time. [ 46 ] [ 47 ] Google Cloud Connect was discontinued in April 2013 as, according to Google, Google Drive achieves all of the above tasks, "with better results". [ 48 ]
In January 2022, Google announced the text watermark feature to the word processor, allowing users to create or import watermarks to a document. In addition to text watermarks, image watermarks can also be added to the document. [ 49 ] [ 50 ]
In July 2024, Google announced that Google Docs would begin fully supporting Markdown syntax. [ 51 ] This built on Google's announcement in March 2022 that it had added an opt-in feature to automatically detect Markdown within Google Docs. [ 52 ]
In a December 2016 review of Google Docs and the Drive software suite, Edward Mendelsohn of PC Magazine wrote that the suite was "visually elegant" with "effortless collaboration", but that Docs, as paired with Sheets and Slides , was "less powerful than desktop-based suites". Comparing Google's Office suite with Microsoft Office and Apple's iWork , he stated that "Docs exists only in your Web browser", meaning that users have a "more limited feature set" than "the spacious, high-powered setting of a desktop app". He wrote that offline support required a plug-in, describing it as "less convenient than a desktop app, and you have to remember to install it before you need it". Mendelsohn praised the user interface , describing it as "elegant, highly usable" with "fast performance", and that the revision history "alerts you to recent changes, and stores fine-grained records of revisions". Regarding the Explore functionality, he credited it for being the "niftiest new feature" in the suite and that it surpassed comparable features in Microsoft Office. He described the quality of imports of Word files as "impressive fidelity". He summarized by praising Docs and the Drive suite for having "the best balance of speed and power, and the best collaboration features, too", while noting that "it lacks a few features offered by Microsoft Office 365, but it was also faster to load and save in our testing". [ 53 ]
In May 2017, a phishing attack impersonated a Google Docs sharing email spread on the Internet. The attack sent emails pretending to be someone the target knew, requesting to share a document with them. Once the link in the email was pressed, users were directed to a real Google account permissions page where the phishing software, a third-party app named "Google Docs", requested access to the user's Google account. Once granted, the software received access to the user's Gmail messages and address book and sent new fraudulent document invitations to their contacts. [ 54 ] The phishing attack was described by media outlets as "massive" [ 55 ] and "widespread", [ 56 ] and The Next Web ' s Napier Lopez wrote that it's "very easy to fall for". [ 55 ] One of the reasons the attack was so effective was that its email messages passed through spam and security software, and used a real Google address. [ 57 ] Within hours, the attack was stopped and fixed by Google, with a spokesperson stating "We have taken action to protect users against an email impersonating Google Docs and have disabled offending accounts. We've removed the fake pages, pushed updates through Safe Browsing, and our abuse team is working to prevent this kind of spoofing from happening again". [ 58 ] [ 59 ]
On the same day, Google updated Gmail on Android to feature protection from phishing attacks. [ 60 ] [ 61 ] [ 62 ] Media outlets noticed that, while the added protection was announced on the same day as the attack, it "may not have prevented this week's attack, however, as that attack involved a malicious and fake "Google Docs" app that was hosted on Google's own domain". [ 62 ] In early May 2017, Ars Technica reported that "at least three security researchers" had raised issues about the threat, one of them in October 2011, and that the attacker or attackers behind the actual incident "may have copied the technique from a proof of concept posted by one security researcher to GitHub in February". Furthermore, the report noted that Google had been repeatedly warned by researchers about the potential threat, with security researcher Greg Carson telling Ars Technica that "I don't think Google fully understood how severely this could be abused, but certainly, hackers did". [ 63 ]
In October 2017, Google released a server-side update to its codebase, which started incorrectly flagging random documents as unspecified violations of its "Terms of Service" policies. A fix was released shortly afterward, [ 64 ] [ 65 ] though the issue became noteworthy for the extent of Google's control over users' content, including its analysis of the contents of documents, as well as for its ability to shut users out at any time, including during critical moments of work. [ 66 ] [ 67 ] | https://en.wikipedia.org/wiki/Google_Docs |
[ 1 ] Mac OS 2.20.2700.1 (November 10, 2016 ; 8 years ago ( 2016-11-10 ) ) [±] [ 2 ] Android 2.24.3535.3.231113858 (January 27, 2019 ; 6 years ago ( 2019-01-27 ) ) [±]
Mac OS 1.10.1389.101 (March 26, 2013 ; 12 years ago ( 2013-03-26 ) ) [±]
Google Japanese Input ( Google 日本語入力 , Gūguru Nihongo Nyūryoku ) is an input method published by Google for the entry of Japanese text on a computer. Since its dictionaries are generated automatically from the Internet , it supports typing of personal names , Internet slang, neologisms and related terms. Google Japanese Input can be used on Windows, macOS, and ChromeOS.
Google also releases an open-source version under the name mozc . It can be used on Linux , Windows , macOS , Android , and ChromeOS . It does not use Google's closed-source algorithms for generating dictionary data from online sources. [ 4 ]
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Google_Japanese_Input |
Google Messages [ 4 ] (formerly known as Messenger , Android Messages , and Messages by Google ) is a text messaging software application developed by Google for its Android and Wear OS mobile operating systems. It is also available as a web app .
Google's official universal messaging platform for the Android ecosystem, Messages employs SMS , MMS , and Rich Communication Services (RCS) . Starting in 2023, Google has RCS activated by default on participating Android devices, [ 5 ] similar to the implementation of iMessage on Apple devices. [ 6 ]
The original code for Android SMS messaging was released in 2009 integrated into the Operating System. [ 7 ] It was released as a standalone application independent of Android with the release of Android 5.0 Lollipop in 2014, replacing Google Hangouts as the default SMS app on Google's Nexus line of phones. [ 8 ]
In 2018, Messages adopted RCS messages and evolved to send larger data files, sync with other apps, and even create mass messages. [ 9 ] This was in preparation for when Google launched messages for web. [ 10 ]
In December 2019, Google began to introduce support for Rich Communication Services (RCS) messaging via an RCS service hosted by Google, referred to in the user interface as "chat features". [ 11 ] This was followed by a wider global rollout throughout 2020. [ 12 ]
The app surpassed 1 billion installs in April 2020, [ 13 ] doubling its number of installs in less than a year. [ 14 ]
Initially, RCS did not support end-to-end encryption . [ 15 ] In June 2021, Google introduced end-to-end encryption in Messages by default using the Signal Protocol , for all one-to-one RCS-based conversations, [ 16 ] [ 17 ] [ 18 ] [ 19 ] for all RCS group chats in December 2022 for beta users, [ 20 ] [ 21 ] and for all RCS users by August 2023, as well as enabling RCS for all users by default to encourage encryption. [ 22 ] In July 2023, Google announced it would build the Message Layer Security (MLS) end-to-end encryption protocol into Google Messages. [ 23 ]
Beginning with the Samsung Galaxy S21 , Messages replaces Samsung's in-house Messages app as the default text messaging app for One UI for some regions and carriers. [ 24 ] In April 2021, the app began to receive UI modifications on Samsung devices to follow aspects of One UI, including pushing the top of the message list towards the middle of the screen to improve ergonomics. [ 25 ] [ 26 ]
In February 2023, Google began to replace references to "chat features" in the Messages user interface with "RCS". [ 27 ] In August 2023, Google announced that Messages will use RCS by default for all users unless they opt out, to allow them to benefit from secure messaging. [ 22 ] In December 2023, with the arrival of several new features, the app was renamed "Google Messages". [ 4 ]
In July 2024, Samsung announced it would no longer pre-install Samsung Messages on its Galaxy devices in some regions, starting with the Galaxy Z Fold 6 and Flip, favoring Google Messages instead. [ 28 ]
Some of the most important features in Google Messages are: [ 29 ] | https://en.wikipedia.org/wiki/Google_Messages |
Google Wave , later known as Apache Wave , is a discontinued software framework for real-time collaborative online editing . Originally developed by Google and announced on May 28, 2009, [ 1 ] [ 2 ] [ 3 ] it was renamed to Apache Wave when the project was adopted by the Apache Software Foundation as an incubator project in 2010.
Wave was a web-based computing platform and communications protocol designed to merge key features of communications media , such as email, instant messaging , wikis , and social networking . [ 4 ] Communications using the system can be synchronous or asynchronous . Software extensions provide contextual spelling and grammar checking , automated language translation [ 2 ] and other features. [ 5 ]
Initially released only to developers, a preview release of Google Wave was extended to 100,000 users in September 2009, each allowed to invite additional users. Google accepted most requests submitted starting November 29, 2009, soon after the September extended release of the technical preview. On May 19, 2010, it was released to the general public. [ 6 ]
On August 4, 2010, Google announced the suspension of stand-alone Wave development and the intent of maintaining the web site at least for the remainder of the year; [ 7 ] on November 22, 2011, they announced that existing Waves would become read-only in January 2012, and all Waves would be deleted in April 2012. [ 8 ] Development was handed over to the Apache Software Foundation which started to develop a server-based product called Wave in a Box . [ 9 ] [ 10 ] [ 11 ] Apache Wave never reached a full release and was discontinued on January 15, 2018. [ 12 ]
The science fiction television series Firefly provided the inspiration for the project's name. [ 13 ] In the series, a wave is an electronic communication, often consisting of a video call or video message. [ 13 ] During the developer preview, a number of references were made to the series, such as Lars Rasmussen replying to a message with "shiny", a word used in the series to mean cool or good , and the crash message of Wave being a popular quotation from the series: "Curse your sudden but inevitable betrayal!" [ 2 ] [ 14 ] Another common error message, "Everything's shiny, Cap'n. Not to fret!" is a quote from Kaylee Frye in the 2005 motion-picture Firefly reworking, Serenity , and it is matched with a sign declaring that "This wave is experiencing some turbulence and might explode. If you don't want to explode..." which is another reference to the opening of the film.
During an event in Amsterdam , Netherlands , [ 15 ] it became apparent that the 60-strong team that was then working on Wave in Sydney used Joss Whedon -related references to describe, among others, the sandbox version of Wave called Dollhouse after the TV series by Firefly producer Joss Whedon, which was aired on Fox in the US. The development of external extensions was codenamed "Serenity", after the spaceship used in Firefly and Serenity .
Google released most of the source code as free software , [ 2 ] allowing the public to develop its features through extensions. [ 2 ] Google allowed third parties to build their own Wave services (be it private or commercial) because it wanted the Wave protocol to replace the e-mail protocol. [ 2 ] [ 16 ] [ 17 ] Initially, Google was the only Wave service provider, but it was hoped that other service providers would launch their own Wave services, possibly designing their own unique web-based clients as is common with many email service providers. The possibility also existed for native Wave clients to be made, as demonstrated with their CLI -based console client. [ 18 ]
Google released initial free software components of Wave: [ 19 ]
In addition, Google provided some detail about later phases of the free software release: [ 18 ]
Google Wave was a new Internet communications platform. It was written in Java using OpenJDK and its web interface used the Google Web Toolkit . Google Wave worked like previous messaging systems such as email and Usenet , but instead of sending a message along with its entire thread of previous messages, or requiring all responses to be stored in each user's inbox for context, message documents (referred to as waves ) that contain complete threads of multimedia messages (blips) were perpetually stored on a central server. Waves were shared with collaborators who could be added or removed from the wave at any point during a wave's existence.
Waves, described by Google as " equal parts conversation and document ", were hosted XML documents that allowed seamless and low latency concurrent modifications. [ 20 ] Any participant of a wave could reply anywhere within the message, edit any part of the wave, and add participants at any point in the process. Each edit/reply was a blip and users can reply to individual blips within waves. Recipients were notified of changes/replies in all waves in which they were active and, upon opening a wave, could review those changes in chronological order. In addition, waves were live. All replies/edits were visible in real-time, letter-by-letter, as they were typed by the other collaborators. Multiple participants could edit a single wave simultaneously in Google Wave. Thus, waves could function not only as e-mails and threaded conversations but also as an instant messaging service when many participants were online at the same time. A wave could repeatedly shift roles between e-mail and instant messaging depending on the number of users editing it concurrently. The ability to show messages as they are typed could be disabled, similar to conventional instant messaging. [ 4 ]
The ability to modify a wave at any location let users create collaborative documents, edited in a manner akin to wikis . Waves could easily link to other waves. In many respects, it was a more advanced forum. [ 21 ] It could be read and known to exist by only one person, or by two or more and could also be public, available for reading and writing to everyone on the Wave. [ 22 ] [ citation needed ]
The history of each wave was stored within it. Collaborators could use a playback feature to observe the order in which it was edited, blips that were added, and who was responsible for what in the wave. [ 4 ] [ 5 ] The history could also be searched by a user to view and/or modify specific changes, such as specific kinds of changes or messages from a single user. [ 2 ]
During the initial launch of Google Wave, invitations were widely sought by users and were sold on auction sites. [ 23 ] Those who received invitations and decided to test Google Wave could not communicate with their contacts on their regular email accounts. The initial spread of Wave was very restricted.
Google Wave initially received positive press coverage for its design [ 24 ] and potential uses. [ 25 ] [ 26 ] After its demise, it was criticized for trying to merge "all forms of communication in a single, crowded space". [ 27 ]
On August 4, 2010, Google announced Wave would no longer be developed as a stand-alone product due to a lack of interest. [ 28 ] Google's statement surprised many in the industry and user community.
Google later clarified the Wave service would be available until April 2012, giving users the opportunity to use the export functionality to keep a local copy of their waves in PDF format. [ 29 ]
Response to the news of the end of development came from Wave users in the form of a website. [ 30 ] After their announcement in early August 2010, the website recorded over 49,000 supporter registrations urging Google Wave's continuation. [ 31 ]
In retrospect, the lack of success of Google Wave was attributed among other things to its complicated user interface resulting from a product that merged features of email, instant messengers and wikis but ultimately failed to do anything significantly better than the existing solutions. [ 32 ]
Chris Dawson of online technology magazine Zdnet discussed inconsistencies in the reasoning of Google in deciding to end support for Wave, [ 28 ] mentioning its "deep involvement" in developing social media networks, to which many of Wave's capabilities are ideally suited.
Google Wave was accepted by the Apache Software Foundation 's Incubator program under the project name Apache Wave. The Google Wave Developer blog was updated with news of the change on December 6, 2010. [ 33 ] A Wave Proposal page with details on the project's goals was created on the Apache Foundation's Incubator Wiki. [ 34 ]
Wave in a Box is the current server implementation of Apache Wave. Currently, there are no demo servers available. [ 35 ]
In 2016, several discussions took place within the Apache Wave community, aiming to tackle the stagnation and crisis state of the project. The Apache Software Foundation mentor of Apache Wave, Upayavira, [ 36 ] was concerned by the project's stagnation, but framed SwellRT (a fork which re-engineered Wave into a backend-as-a-service for building apps) as Wave's potential savior. [ 37 ] Eventually, Wave was approved to continue within Apache incubator program, and a copy of SwellRT codebase was placed in the Apache Wave repository in order to grant the Wave community access to it. [ 38 ] In this regard, Intellectual Property of SwellRT was transferred to the Apache Foundation in 2017. [ 39 ] Still this was not sufficient to resurrect Wave's developer community, and SwellRT continued as independent project.
The Wave project retired on January 15, 2018, having never left incubator status. [ 40 ]
Google Wave is extensible through an application programming interface (API). It provides extensions in the form of Gadgets and Robots , and is embeddable by dropping interactive windows into a given wave on external sites, such as blog sites. [ 2 ] [ 41 ]
The last version of robots API is 2.0. [ 42 ]
Google Wave also supports extension installers, which bundle back-end elements (robots and gadgets) and front-end user interface elements into an integrated package. Users may install extensions directly within the Wave client using an extension installer.
Google Wave extensions are add-ins that may be installed on Google Wave to enhance its functionality. They may be Internet bots (robots) to automate common tasks, or gadgets to extend or change user interaction features, e.g., posting blips on microblog feeds or providing RSVP recording mechanisms. [ 2 ] [ 4 ] [ 41 ]
Over 150 Google Wave extensions have been developed either in the form of Gadgets or Robots. [ 43 ]
A robot is an automated participant on a wave. It can read the contents of a wave in which it participates, modify its contents, add or remove participants, and create new blips or new waves. Robots perform actions in response to events. For example, a robot might publish the contents of a wave to a public blog site and update the wave with user comments.
Robots may be added as participants to the Wave itself. In theory, a robot can be added anywhere a human participant can be involved.
Gadget extensions are applications that run within the wave, and to which all participants have access. Robots and Gadgets can be used together, but they generally serve different purposes. A gadget is an application users could participate with, many of which are built on Google's OpenSocial platform. A good comparison would be iGoogle gadgets or Facebook applications.
The gadget is triggered based on the user action. They can be best described as applications installed on a mobile phone. For example, a wave might include a sudoku gadget that lets the wave participants compete to see who can solve the puzzle first.
Gadgets may be added to individual waves and all the participants share and interact with the gadget.
Google Wave provides federation using an extension of Extensible Messaging and Presence Protocol (XMPP), the free Wave Federation Protocol . Being an open protocol, anyone can use it to build a custom Wave system and become a wave provider. [ 44 ] The use of an open protocol is intended to parallel the openness and ease of adoption of the e-mail protocol and, like e-mail, allow communication regardless of provider. Google hoped that waves would replace e-mail as the dominant form of Internet communication. [ 2 ] [ 16 ] [ 17 ] In this way, Google intended to be only one of many wave providers [ 2 ] [ 16 ] [ 17 ] and to also be used as a supplement to e-mail, instant messaging , FTP , etc.
A key feature of the protocol is that waves are stored on the service provider's servers instead of being sent between users. Waves are federated; copies of waves and wavelets are distributed by the wave provider of the originating user to the providers of all other participants in a particular wave or wavelet so all participants have immediate access to up-to-date content. The originating wave server is responsible for hosting, processing, and concurrency control of waves. [ 16 ] [ 17 ] The protocol allows private reply wavelets within parent waves, where other participants have no access or knowledge of them. [ 16 ] [ 17 ]
Security for the communications is provided via Transport Layer Security authentication, and encrypted connections and waves/wavelets are identified uniquely by a service provider's domain name and ID strings. User-data is not federated, that is, not shared with other wave providers.
Besides Apache Wave itself, there were other open-source variants of servers and clients with different percentage of Wave Federation and Wave Protocol support. Wave was re-engineered into a backend-as-a-service solution by the SwellRT project. Wave was adopted in different forms for corporate applications by Novell for Novell Pulse , [ 45 ] or by SAP for Cloudave, [ 46 ] and community projects such as PyOfWave or Kune .
The following servers were compatible with the Google Wave protocol: | https://en.wikipedia.org/wiki/Google_Wave |
In mathematics , the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh . The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation
satisfying x > y > 1 {\displaystyle x>y>1} and n , m > 2 {\displaystyle n,m>2} are
and
Davenport, Lewis & Schinzel (1961) showed that, for each pair of fixed exponents m {\displaystyle m} and n {\displaystyle n} , this equation has only finitely many solutions. But this proof depends on Siegel's finiteness theorem , which is ineffective. Nesterenko & Shorey (1998) showed that, if m − 1 = d r {\displaystyle m-1=dr} and n − 1 = d s {\displaystyle n-1=ds} with d ≥ 2 {\displaystyle d\geq 2} , r ≥ 1 {\displaystyle r\geq 1} , and s ≥ 1 {\displaystyle s\geq 1} , then max ( x , y , m , n ) {\displaystyle \max(x,y,m,n)} is bounded by an effectively computable constant depending only on r {\displaystyle r} and s {\displaystyle s} . Yuan (2005) showed that for m = 3 {\displaystyle m=3} and odd n {\displaystyle n} , this equation has no solution ( x , y , n ) {\displaystyle (x,y,n)} other than the two solutions given above.
Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions ( x , y , m , n ) {\displaystyle (x,y,m,n)} to the equations with prime divisors of x {\displaystyle x} and y {\displaystyle y} lying in a given finite set and that they may be effectively computed. He & Togbé (2008) showed that, for each fixed x {\displaystyle x} and y {\displaystyle y} , this equation has at most one solution.
For fixed x (or y ), equation has at most 15 solutions, and at most two unless x is either odd prime power times a power of two , or in the finite set {15, 21, 30, 33, 35, 39, 45, 51, 65, 85, 143, 154, 713}, in which case there are at most three solutions. Furthermore, there is at most one solution if the odd part of x is squareful unless x has at most two distinct odd prime factors or x is in a finite set {315, 495, 525, 585, 630, 693, 735, 765, 855, 945, 1035, 1050, 1170, 1260, 1386, 1530, 1890, 1925, 1950, 1953, 2115, 2175, 2223, 2325, 2535, 2565, 2898, 2907, 3105, 3150, 3325, 3465, 3663, 3675, 4235, 5525, 5661, 6273, 8109, 17575, 39151}. If x is a power of two , there is at most one solution except for x = 2, in which case there are two known solutions. In fact, max ( m , n ) < 4 x {\displaystyle \max(m,n)<4^{x}} and y < 2 2 x {\displaystyle y<2^{2^{x}}} .
The Goormaghtigh conjecture may be expressed as saying that 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2) are the only two numbers that are repunits with at least 3 digits in two different bases. | https://en.wikipedia.org/wiki/Goormaghtigh_conjecture |
A gooseneck (or goose neck ) is a 180° pipe fitting at the top of a vertical pipe that prevents entry of water. Common implementations of goosenecks are ventilator piping or ducting for bathroom and kitchen exhaust fans, ship holds, landfill methane vent pipes, or any other piping implementation exposed to the weather where water ingress would be undesired. It is so named because the word comes from the similarity of pipe fitting to the bend in a goose 's neck. [ 1 ]
Gooseneck may also refer to a style of kitchen or bathroom faucet with a long vertical pipe terminating in a 180° bend.
To avoid hydrocarbon accumulation, a thermosiphon should be installed at the low point of the gooseneck. [ citation needed ]
Leaded goosenecks are short sections of lead pipe (1’ to 2’ long) used during the early 1900s up to World War Two in supplying water to a customer. These lead tubes could be easily bent, and allowed for a flexible connection between rigid service piping. The bent segments of pipe often took the shape of a goose's neck, and are referred to as “ lead goosenecks .” Lead is no longer permitted in new water systems or new building construction.
Goosenecks (also referred to as pigtails ) [ 2 ] are in-line components of a water service (i.e. piping, valves, fittings, tubing, and accessories) running from the distribution system water main to a meter or building inlet. The valve used to connect a small-diameter service line to a water main is called a corporation stop (also called a tap, or corp stop). One gooseneck joins the corporation stop to the water service pipe work. A second gooseneck links the supply pipeline to a water meter located outside the building.
This industry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gooseneck_(piping) |
Gordan's lemma is a lemma in convex geometry and algebraic geometry . It can be stated in several ways.
The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma".
There are topological and algebraic proofs.
Let σ {\displaystyle \sigma } be the dual cone of the given rational polyhedral cone. Let u 1 , … , u r {\displaystyle u_{1},\dots ,u_{r}} be integral vectors so that σ = { x ∣ ⟨ u i , x ⟩ ≥ 0 , 1 ≤ i ≤ r } . {\displaystyle \sigma =\{x\mid \langle u_{i},x\rangle \geq 0,1\leq i\leq r\}.} Then the u i {\displaystyle u_{i}} 's generate the dual cone σ ∨ {\displaystyle \sigma ^{\vee }} ; indeed, writing C for the cone generated by u i {\displaystyle u_{i}} 's, we have: σ ⊂ C ∨ {\displaystyle \sigma \subset C^{\vee }} , which must be the equality. Now, if x is in the semigroup
then it can be written as
where n i {\displaystyle n_{i}} are nonnegative integers and 0 ≤ r i ≤ 1 {\displaystyle 0\leq r_{i}\leq 1} . But since x and the first sum on the right-hand side are integral, the second sum is a lattice point in a bounded region, and so there are only finitely many possibilities for the second sum (the topological reason). Hence, S σ {\displaystyle S_{\sigma }} is finitely generated.
The proof [ 3 ] is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra C [ S ] {\displaystyle \mathbb {C} [S]} is a finitely generated algebra over C {\displaystyle \mathbb {C} } . To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup S of Z d {\displaystyle \mathbb {Z} ^{d}} ,
Put A = C [ S ] {\displaystyle A=\mathbb {C} [S]} , which has a basis χ a , a ∈ S {\displaystyle \chi ^{a},\,a\in S} . It has Z {\displaystyle \mathbb {Z} } -grading given by
By assumption, A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that C [ S + ] = ⊕ 0 ∞ A n {\displaystyle \mathbb {C} [S^{+}]=\oplus _{0}^{\infty }A_{n}} is a finitely generated algebra over A 0 {\displaystyle A_{0}} . Now, the semigroup S 0 = S ∩ { x ∣ ⟨ x , v ⟩ = 0 } {\displaystyle S_{0}=S\cap \{x\mid \langle x,v\rangle =0\}} is the image of S under a linear projection, thus finitely generated and so A 0 = C [ S 0 ] {\displaystyle A_{0}=\mathbb {C} [S_{0}]} is finitely generated. Hence, S + {\displaystyle S^{+}} is finitely generated then.
Lemma : Let A be a Z {\displaystyle \mathbb {Z} } -graded ring. If A is a Noetherian ring, then A + = ⊕ 0 ∞ A n {\displaystyle A^{+}=\oplus _{0}^{\infty }A_{n}} is a finitely generated A 0 {\displaystyle A_{0}} -algebra.
Proof: Let I be the ideal of A generated by all homogeneous elements of A of positive degree. Since A is Noetherian, I is actually generated by finitely many f i ′ s {\displaystyle f_{i}'s} , homogeneous of positive degree. If f is homogeneous of positive degree, then we can write f = ∑ i g i f i {\textstyle f=\sum _{i}g_{i}f_{i}} with g i {\displaystyle g_{i}} homogeneous. If f has sufficiently large degree, then each g i {\displaystyle g_{i}} has degree positive and strictly less than that of f . Also, each degree piece A n {\displaystyle A_{n}} is a finitely generated A 0 {\displaystyle A_{0}} -module. (Proof: Let N i {\displaystyle N_{i}} be an increasing chain of finitely generated submodules of A n {\displaystyle A_{n}} with union A n {\displaystyle A_{n}} . Then the chain of the ideals N i A {\displaystyle N_{i}A} stabilizes in finite steps; so does the chain N i = N i A ∩ A n . {\displaystyle N_{i}=N_{i}A\cap A_{n}.} ) Thus, by induction on degree, we see A + {\displaystyle A^{+}} is a finitely generated A 0 {\displaystyle A_{0}} -algebra.
A multi- hypergraph over a certain set V {\displaystyle V} is a multiset of subsets of V {\displaystyle V} (it is called "multi-hypergraph" since each hyperedge may appear more than once). A multi-hypergraph is called regular if all vertices have the same degree . It is called decomposable if it has a proper nonempty subset that is regular too. For any integer n , let D ( n ) {\displaystyle D(n)} be the maximum degree of an indecomposable multi-hypergraph on n vertices. Gordan's lemma implies that D ( n ) {\displaystyle D(n)} is finite. [ 1 ] Proof : for each subset S of vertices, define a variable x S (a non-negative integer). Define another variable d (a non-negative integer). Consider the following set of n equations (one equation per vertex): ∑ S ∋ v x S − d = 0 for all v ∈ V {\displaystyle \sum _{S\ni v}x_{S}-d=0{\text{ for all }}v\in V} Every solution ( x , d ) denotes a regular multi-hypergraphs on V {\displaystyle V} , where x defines the hyperedges and d is the degree. By Gordan's lemma, the set of solutions is generated by a finite set of solutions, i.e., there is a finite set M {\displaystyle M} of multi-hypergraphs, such that each regular multi-hypergraph is a linear combination of some elements of M {\displaystyle M} . Every non-decomposable multi-hypergraph must be in M {\displaystyle M} (since by definition, it cannot be generated by other multi-hypergraph). Hence, the set of non-decomposable multi-hypergraphs is finite. | https://en.wikipedia.org/wiki/Gordan's_lemma |
Gordana Todorov (born July 24, 1949) [ 1 ] is a mathematician working in noncommutative algebra , representation theory , Artin algebras , and cluster algebras . She is a professor of mathematics at Northeastern University . [ 2 ]
Todorov earned her Ph.D. in 1978, at Brandeis University . Her dissertation, Almost Split Sequences in the Representation Theory of Certain Classes of Artin Algebras , was supervised by Maurice Auslander . [ 3 ]
Todorov is married to mathematician Kiyoshi Igusa . [ 4 ] The Igusa–Todorov functions [ 5 ] and Igusa–Todorov endomorphism algebras [ 6 ] are named for their joint work. Todorov is also the namesake of Todorov's theorem on preprojective partitions, [ 7 ] and the Gentle–Todorov theorem on abelian categories . [ 8 ] | https://en.wikipedia.org/wiki/Gordana_Todorov |
The Gordon A. McKay Award is an annual prize given by the Meteoritical Society to the student who gives the best oral presentation at its annual meeting. [ 1 ] This award honors the memory of Gordon A. McKay (1945–2008), a NASA planetary scientist specializing in lunar and Martian geochemistry . It was established in 2008 and comes with a prize of $1,000 and a certificate.
Source: Previous Winners, Meteoritical Society Accessed 4/20/2025 | https://en.wikipedia.org/wiki/Gordon_A._McKay_Award |
The Gordon E. Moore Medal for Outstanding Achievement in Solid State Science and Technology (formerly the Solid-State Science and Technology Award) was established by The Electrochemical Society in 1971 to recognize individuals distinguished for outstanding contributions to solid-state science and technology. The award is presented every two years, and recipients receive a silver medal, wall plaque, cash prize, Society Life membership, and a complimentary meeting registration. [ 1 ]
Despite the fact that the solid-state community represented a major force in The Electrochemical Society, there was no form of recognition at the Society level of achievements in the field prior to the establishment of this award. [ 2 ] Known as the Solid-State Science and Technology Award until 2005, the award was then renamed after Intel co-founder and author of Moore's Law , Gordon E. Moore , who is a long-time member of The Electrochemical Society. [ 3 ]
As listed by ECS: [ 4 ]
This article about materials science is a stub . You can help Wikipedia by expanding it .
This science awards article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gordon_E._Moore_Medal_for_Outstanding_Achievement_in_Solid_State_Science_and_Technology |
Gordon Woods (July 14, 1952 – August 20, 2009) was an American veterinary scientist who co-created Idaho Gem , the world's first cloned mule . Idaho Gem was the first clone born in the horse family. [ 1 ]
Woods was raised in northern Idaho. He obtained his bachelor's degree from the University of Idaho . Woods received a doctorate of veterinary medicine from Colorado State University . He later obtained a second doctorate in reproductive biology from the University of Wisconsin–Madison . [ 1 ]
Woods first taught veterinary medicine at Cornell University . [ 1 ]
Woods founded the Northwest Equine Reproduction Laboratory in Idaho in 1986. He moved to Moscow, Idaho, and he taught at Washington State University in Pullman, WA until he joined the faculty of the University of Idaho in 1988 as an Animal and Veterinary Science Department professor. [ 1 ]
Woods, along with colleagues Dr. Dirk Vanderwall of University of Idaho and Dr. Ken White of Utah State University , led a team of scientists in 2003 that cloned Idaho Gem , the world's first cloned mule . The cloning of Idaho Gem was a part of a larger scientific study intended to understand human diseases. Horses , mules , and other equines have lower rates of cancer than humans. Woods, Vanderwall, White and their team hoped that the cloning of mules and other equines would provide an important scientific insight into the different cancer rates between humans and equines. Woods was particularly interested in the role that calcium played in the development of cancer. Horses and mules have less calcium in their cell walls than humans. [ 1 ]
Woods' colleague, Dirk Vanderwall, later explained Woods' goals during the Idaho Gem cloning, "That certainly was another primary focus of Gordon's...to use the horse as a model to try to understand age-onset diseases in people. Gordon's hypothesis was that excessive intracellular calcium in human cells could be an underlying factor in age-onset diseases." [ 1 ]
Woods departed the University of Idaho in 2007 and joined the faculty of Colorado State University . [ 1 ] He became a professor in the school's College of Veterinary Medicine and Biomedical Sciences.
Woods died unexpectedly at the Medical Center of the Rockies in Loveland, Colorado , at the age of 57. He was survived by his wife, Shauna, to whom he had been married for 37 years, four children, and six grandchildren. [ 1 ] | https://en.wikipedia.org/wiki/Gordon_Woods |
In mathematical physics , the Gordon decomposition [ 1 ] (named after Walter Gordon ) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.
For any solution ψ {\displaystyle \psi } of the massive Dirac equation,
the Lorentz covariant number-current j μ = ψ ¯ γ μ ψ {\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi } may be expressed as
where
is the spinor generator of Lorentz transformations ,
and
is the Dirac adjoint .
The corresponding momentum-space version for plane wave solutions u ( p ) {\displaystyle u(p)} and u ¯ ( p ′ ) {\displaystyle {\bar {u}}(p')} obeying
is
where
One sees that from Dirac's equation that
and, from the adjoint of Dirac's equation,
Adding these two equations yields
From Dirac algebra , one may show that Dirac matrices satisfy
Using this relation,
which amounts to just the Gordon decomposition, after some algebra.
The second, spin-dependent, part of the current coupled to the photon field, − A μ j μ {\displaystyle -A_{\mu }j^{\mu }} yields, up to an ignorable total divergence,
that is, an effective Pauli moment term , − ( e ℏ / 2 m c ) B → ⋅ ψ † σ → ψ {\displaystyle -(e\hbar /2mc){\vec {B}}\cdot \psi ^{\dagger }{\vec {\sigma }}\psi } .
This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires m ≠ 0 {\displaystyle m\neq 0} .
If one assumed that the given solution has energy E = | k | 2 + m 2 {\textstyle E={\sqrt {|{\mathbf {k} }|^{2}+m^{2}}}} so that ψ ( r , t ) = ψ ( r ) e − i E t {\textstyle \psi (\mathbf {r} ,t)=\psi ({\mathbf {r} })e^{-iEt}} , one might obtain a decomposition that is valid for both massive and massless cases. [ 2 ]
Using the Dirac equation again, one finds that
Here γ = ( γ 1 , γ 2 , γ 3 ) {\displaystyle {\boldsymbol {\gamma }}=(\gamma ^{1},\gamma ^{2},\gamma ^{3})} , and S = ψ † S ^ ψ {\displaystyle {\mathbf {S} }=\psi ^{\dagger }{\hat {\mathbf {S} }}\psi } with ( S ^ x , S ^ y , S ^ z ) = ( Σ 23 , Σ 31 , Σ 12 ) , {\displaystyle ({\hat {S}}_{x},{\hat {S}}_{y},{\hat {S}}_{z})=(\Sigma ^{23},\Sigma ^{31},\Sigma ^{12}),} so that
where σ = ( σ x , σ y , σ z ) {\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})} is the vector of Pauli matrices .
With the particle-number density identified with ρ = ψ † ψ {\displaystyle \rho =\psi ^{\dagger }\psi } , and for a near plane-wave
solution of finite extent, one may interpret the first term in the decomposition as the current j f r e e = e ρ k / E = e ρ v {\displaystyle {\mathbf {j} }_{\rm {free}}=e\rho {\mathbf {k} }/E=e\rho {\mathbf {v} }} , due to particles moving at speed v = k / E {\displaystyle {\mathbf {v} }={\mathbf {k} }/E} .
The second term, j b o u n d = ( e / E ) ∇ × S {\displaystyle {\mathbf {j} }_{\rm {bound}}=(e/E)\nabla \times {\mathbf {S} }} is the current due to the gradients in the intrinsic magnetic moment density. The magnetic moment itself is found by integrating by parts to show that
For a single massive particle in its rest frame , where E = m {\displaystyle E=m} , the magnetic moment reduces to
where | S | = ℏ / 2 {\displaystyle |{\mathbf {S} }|=\hbar /2} and g = 2 {\displaystyle g=2} is the Dirac value of the gyromagnetic ratio .
For a single massless particle obeying the right-handed Weyl equation, the spin-1/2 is locked to the direction k ^ {\displaystyle {\hat {\mathbf {k} }}} of its kinetic momentum and the magnetic moment becomes [ 3 ]
For both the massive and massless cases, one also has an expression for the momentum density as part of the symmetric Belinfante–Rosenfeld stress–energy tensor
Using the Dirac equation one may evaluate T B R 0 μ = ( E , P ) {\displaystyle T_{\rm {BR}}^{0\mu }=({\mathcal {E}},{\mathbf {P} })} to find the energy density to be E = E ψ † ψ {\displaystyle {\mathcal {E}}=E\psi ^{\dagger }\psi } , and the momentum density,
If one used the non-symmetric canonical energy-momentum tensor
one would not find the bound spin-momentum contribution.
By an integration by parts one finds that the spin contribution to the total angular momentum is
This is what is expected, so the division by 2 in the spin contribution to the momentum density is necessary. The absence of a division by 2 in the formula for the current reflects the g = 2 {\displaystyle g=2} gyromagnetic ratio of the electron. In other words, a spin-density gradient is twice as effective at making an electric current as it is at contributing to the linear momentum.
Motivated by the Riemann–Silberstein vector form of Maxwell's equations , Michael Berry [ 4 ] uses the Gordon strategy to obtain gauge-invariant expressions for the intrinsic spin angular-momentum density for solutions to Maxwell's equations .
He assumes that the solutions are monochromatic and uses the phasor expressions E = E ( r ) e − i ω t {\displaystyle \mathbf {E} =\mathbf {E} (\mathbf {r} )e^{-i\omega t}} , H = H ( r ) e − i ω t {\displaystyle \mathbf {H} =\mathbf {H} (\mathbf {r} )e^{-i\omega t}} . The time average of the Poynting vector momentum density is then given by ⟨ P ⟩ = 1 4 c 2 [ E ∗ × H + E × H ∗ ] = ϵ 0 4 i ω [ E ∗ ⋅ ( ∇ E ) − ( ∇ E ∗ ) ⋅ E + ∇ × ( E ∗ × E ) ] = μ 0 4 i ω [ H ∗ ⋅ ( ∇ H ) − ( ∇ H ∗ ) ⋅ H + ∇ × ( H ∗ × H ) ] . {\displaystyle {\begin{aligned}\langle \mathbf {P} \rangle &={\frac {1}{4c^{2}}}[{\mathbf {E} }^{*}\times {\mathbf {H} }+{\mathbf {E} }\times {\mathbf {H} }^{*}]\\&={\frac {\epsilon _{0}}{4i\omega }}[{\mathbf {E} }^{*}\cdot (\nabla {\mathbf {E} })-(\nabla {\mathbf {E} }^{*})\cdot {\mathbf {E} }+\nabla \times ({\mathbf {E} }^{*}\times {\mathbf {E} })]\\&={\frac {\mu _{0}}{4i\omega }}[{\mathbf {H} }^{*}\cdot (\nabla {\mathbf {H} })-(\nabla {\mathbf {H} }^{*})\cdot {\mathbf {H} }+\nabla \times ({\mathbf {H} }^{*}\times {\mathbf {H} })].\end{aligned}}} We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as H ∗ ⋅ ( ∇ H ) {\displaystyle {\mathbf {H} }^{*}\cdot (\nabla {\mathbf {H} })} the scalar product is between the fields so that the vector character is determined by the ∇ {\displaystyle \nabla } .
As P tot = P free + P bound , {\displaystyle \mathbf {P} _{\text{tot}}=\mathbf {P} _{\text{free}}+{\mathbf {P} }_{\text{bound}},} and for a fluid with intrinsic angular momentum density S {\displaystyle \mathbf {S} } we have P bound = 1 2 ∇ × S , {\displaystyle \mathbf {P} _{\text{bound}}={\frac {1}{2}}\nabla \times \mathbf {S} ,} these identities suggest that the spin density can be identified as either S = μ 0 2 i ω H ∗ × H {\displaystyle \mathbf {S} ={\frac {\mu _{0}}{2i\omega }}{\mathbf {H} }^{*}\times {\mathbf {H} }} or S = ϵ 0 2 i ω E ∗ × E . {\displaystyle \mathbf {S} ={\frac {\epsilon _{0}}{2i\omega }}{\mathbf {E} }^{*}\times {\mathbf {E} }.} The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state – i.e. when E = i σ c B {\displaystyle {\mathbf {E} }=i\sigma c{\mathbf {B} }} where the helicity σ {\displaystyle \sigma } takes the values ± 1 {\displaystyle \pm 1} for light that is right or left circularly polarized respectively. In other cases they may differ. | https://en.wikipedia.org/wiki/Gordon_decomposition |
See text.
Gordonia is a genus of gram-positive to gram-variable, aerobic, catalase-positive bacterium in the Actinomycetota , [ 1 ] [ 2 ] closely related to the Rhodococcus , Mycobacterium , Skermania , and Nocardia genera. [ 2 ] Gordonia bacteria are non-motile, and non-sporulating. [ 2 ] Gordonia is from the same lineage that includes Mycobacterium tuberculosis . [ 2 ] The genus was discovered by Tsukamura in 1971 and named after American bacteriologist Ruth Gordon . [ 2 ] Many species are often found in the soil, [ 1 ] while other species have been isolated from aquatic environments. [ 2 ] Some species have been associated with problems like sludge bulking and foaming in wastewater treatment plants. [ 3 ] [ 4 ] Gordonia species are rarely known to cause infections in humans. [ 5 ]
Some pathogenic instances of Gordonia have been reported to cause skin and soft tissue infections, including bacteremia and cutaneous infections. Though infections are generally treated with antibiotics, surgical procedures are sometimes used to contain infections. [ 6 ] Some investigations have found that 28 °C is the ideal temperature for the growth of Gordonia bacteria. [ 1 ] Gordonia species often have high G-C base pair contents in DNA, ranging from 63% to 69%. [ 6 ]
Some species of Gordonia , such as Gordonia rubripertincta , produce colonies that have a bright orange or orange-red color. [ 1 ]
Some strains of Gordonia have recently garnered interest in the biotechnology industry due to their ability to degrade environmental pollutants. [ 7 ]
Gordonia bronchialis has occasionally shown pathogenicity, infecting sternal wounds from surgery. [ 8 ] However, since G. bronchialis infections can present with minimal and mild symptoms, few reports of G. bronchialis infections have been documented. [ 9 ]
Gordonia can infect immunocompetent and immunocompromised individuals. [ 9 ]
Gordonia species are able to degrade various environmental pollutants toxins and other natural compounds that cannot regularly be biodegraded. Two common materials, natural and synthetic isoprene rubber ( cis-1,4-polyisoprene ), can be biodegraded and used as a carbon and energy source by Gordonia. [ 8 ] Gordonia are commonly detected in activated sludge wastewater treatment plants, where they along with other mycolic acid containing actinomycetes are well known contributors to sludge foaming issues that impede biomass settling and process performance. [ 10 ] [ 11 ]
Gordonia species are also being studied as hosts to bacteriophages , or bacteria-parasitizing viruses. Because of their relatedness to Mycobacterium , Gordonia were used as hosts in the SEA-PHAGES project, [ 12 ] greatly contributing to the number of isolated Gordonia phages. According to the Actinobacteriophage Database PhagesDb.org , more than 2,806 Gordonia -infecting types of bacteriophages have been identified as of April 26, 2023. [ 13 ] Research with bacteriophages parasitizing Gordonia and other genera can be used to develop bacteriophage therapies for drug-resistant human, animal, and plant bacterial infections; contamination prevention in food processing facilities; targeted gene delivery ; and more. [ 14 ]
Gordonia comprises the following species: [ 15 ] | https://en.wikipedia.org/wiki/Gordonia_(bacterium) |
Gordonia sp. nov. Q8 is a bacterium in the phylum of Actinomycetota . [ 1 ] It was discovered in 2017 as one of eighteen new species isolated from the Jiangsu Wei5 oilfield in East China with the potential for bioremediation . [ 2 ] [ 3 ] Strain Q8 is rod-shaped and gram-positive with dimensions 1.0–4.0 μm × 0.5–1.2 μm and an optimal growth temperature of 40 °C. [ 2 ] Phylogenetically, it is most closely related to Gordonia paraffinivorans and Gordonia alkaliphila , both of which are known bioremediators. Q8 was assigned as a novel species based on a <70% ratio of DNA homology with other Gordonia bacteria. [ 2 ]
Bioremediation is the process by which polluted soil, water, and other natural materials are treated to encourage growth of microorganisms which can degrade contaminants. It is generally considered more cost-effective and sustainable as compared to other methods of ecosystem restoration . Q8 was chosen for study as a bioremediator due to its ability to grow on media which includes the polycyclic aromatic hydrocarbons (PAHs) naphthalene and pyrene. PAHs are the products of incomplete combustion of fossil fuels and are considered toxic and carcinogenic , in particular to aquatic organisms. [ 4 ] Sixteen PAHs are listed as priority pollutants by the U.S. Environmental Protection Agency because of their association with cancer in aquatic animals and increased mutagenicity of sediments. [ 2 ]
PAH-degrading microorganisms are commonly found in polluted areas such as oil wells, where they utilize PAHs as their sole carbon and energy source. [ 5 ] [ 2 ] The study with Q8 demonstrated that the bacterium could degrade nearly all naphthalene and pyrene with 1–2 weeks, indicating that Q8 can grow in the presence of and rapidly degrade PAHs. [ 2 ] Q8 can also significantly reduce the viscosity of oil, making it more soluble in water and easier to utilize by other bacteria in a process known as petroleum bioremediation. [ 6 ] [ 2 ] Other Gordonia have been used to remove boat lubricants from water using a similar mechanism. [ 7 ] The process of PAH degradation by Q8 results in products including benzene , hydroxyl and methyl groups, and oxidized oil. Proportions of saturated and aromatic hydrocarbons decrease while resins and asphaltenes increase. [ 2 ] Compared to its closest relatives, Q8 is more efficient at removal of PAHs, suggesting future favorability as a bioremediator. | https://en.wikipedia.org/wiki/Gordonia_sp._nov._Q8 |
In mathematics , the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian.
The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are isotopic . The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic.
These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere .
The theorem was proved by Cameron Gordon and John Luecke . Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem , combinatorial techniques in the style of Litherland, thin position , and Scharlemann cycles .
For link complements, it is not in fact true that links are determined by their complements. For example, JHC Whitehead proved that there are infinitely many links whose complements are all homeomorphic to the Whitehead link . His construction is to twist along a disc spanning an unknotted component (as is the case for either component of the Whitehead link). Another method is to twist along an annulus spanning two components. Gordon proved that for the class of links where these two constructions are not possible there are finitely many links in this class with a given complement. | https://en.wikipedia.org/wiki/Gordon–Luecke_theorem |
In queueing theory , a discipline within the mathematical theory of probability , the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers where customers cannot leave the network. [ 1 ] Jackson's theorem cannot be applied to closed networks because the queue length at a node in the closed network is limited by the population of the network. The Gordon–Newell theorem calculates the open network solution and then eliminates the infeasible states by renormalizing the probabilities. Calculation of the normalizing constant makes the treatment more awkward as the whole state space must be enumerated. Buzen's algorithm or mean value analysis can be used to calculate the normalizing constant more efficiently. [ 2 ]
A network of m interconnected queues is known as a Gordon–Newell network [ 3 ] or closed Jackson network [ 4 ] if it meets the following conditions:
In a closed Gordon–Newell network of m queues, with a total population of K individuals, write ( k 1 , k 2 , … , k m ) {\displaystyle \scriptstyle {(k_{1},k_{2},\ldots ,k_{m})}} (where k i is the length of queue i ) for the state of the network and S ( K , m ) for the state space
Then the equilibrium state probability distribution exists and is given by
where service times at queue i are exponentially distributed with parameter μ i . The normalizing constant G ( K ) is given by
and e i is the visit ratio, calculated by solving the simultaneous equations | https://en.wikipedia.org/wiki/Gordon–Newell_theorem |
In mathematics , specifically finite group theory , the Gorenstein–Harada theorem , proved by Daniel Gorenstein and Koichiro Harada , classifies the finite simple groups of sectional 2-rank at most 4. [ 1 ] [ 2 ] It is part of the classification of finite simple groups . [ 3 ]
Finite simple groups of section 2 with rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3, which implies that they have to be of either component type or of characteristic 2 type . Therefore, the Gorenstein–Harada theorem splits the problem of classifying finite simple groups into these two sub-cases.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gorenstein–Harada_theorem |
Gorilla Glass , developed and manufactured by Corning , is a brand of chemically strengthened glass now in its ninth generation. Designed to be thin, light, and damage-resistant, its surface strength and crack-resistance are achieved through immersion in a hot potassium - salt ion-exchange bath. [ 1 ]
The alkali - aluminosilicate sheet glass is primarily used as cover glass for portable electronic devices, including smartphones , smartwatches , portable media players , portable computer displays, and television screens. [ 2 ] It is manufactured in Harrodsburg , Kentucky ; Asan , South Korea ; [ 3 ] and Taiwan . As of October 2017, Gorilla Glass was used in approximately five billion devices worldwide. [ 4 ] Despite its market dominance, Gorilla Glass faces competition from similar products, including AGC Inc. 's Dragontrail , Schott AG 's Xensation, and synthetic sapphire . [ 5 ] [ 6 ] [ 7 ] [ 4 ]
Corning experimented with chemically strengthened glass in 1960 as part of a "Project Muscle" initiative. Replacement of smaller sodium ions with larger potassium ones by a chemical treatment in order to improve the compressive strength of the surface layer of a glass was first developed by Steven Kistler in 1962. [ 8 ] [ 9 ] Soon Corning researchers found that addition of aluminium and zirconium oxides improved the qualities even further. [ 10 ] [ 9 ]
Within a few years they had developed a "muscled glass" [ 11 ] marketed as Chemcor . The product was used until the early 1990s in commercial and industrial applications, including automotive, aviation and pharmaceutical uses, [ 11 ] notably in approximately 100 Dodge Dart and Plymouth Barracuda racing cars in 1968, where minimizing the vehicle's weight was essential. [ 12 ]
Experimentation was revived in 2005, investigating whether the glass could be made thin enough for use in consumer electronics. [ 13 ] Although not called Gorilla Glass at the time, it was brought into commercial use with the launch of the iPhone in June 2007. [ 14 ] [ 15 ] The iPhone that Steve Jobs revealed in January 2007 still featured a plastic display. The day after he held up the plastic iPhone on stage, Jobs complained about scratches that had developed on the phone's display after carrying it around in his pocket. Apple then contacted Corning and asked for a thin, toughened glass to be used in its new phone. [ 16 ] The scratch-resistant glass that shipped [ 17 ] on the first-generation iPhone would eventually come to be known as Gorilla Glass, officially introduced in February 2008. [ 18 ] Corning further developed the material for a variety of smartphones and other consumer electronics devices for a range of companies. [ 19 ] [ 20 ] [ 21 ]
Corning markets the material's primary properties as its high scratch-resistance (protective coating) and its hardness (with a Vickers hardness test rating of 622 to 701), [ 22 ] which allows the glass to be thin without being fragile. The glass can be recycled. [ 19 ]
In December 2015, Ford announced that it would use the material for the windshield and rear window of its second-generation Ford GT sports car going on sale in 2016. [ 23 ] It later spread to mainstream models such as the Ford F-150 and Jeep Wrangler . [ 24 ]
During its manufacture, the glass is toughened by ion exchange . The material is immersed in a molten alkaline potassium salt at a temperature of approximately 400 °C (750 °F), [ 25 ] wherein smaller sodium ions in the glass are replaced by larger potassium ions from the salt bath. The larger ions occupy more volume and thereby create a surface layer of high residual compressive stress , giving the glass surface increased strength, the ability to contain flaws, [ 26 ] and overall crack-resistance, [ 27 ] making it resistant to damage from everyday use. [ 25 ]
Gorilla Glass, initially featured on the original iPhone upon its release in June 2007, [ 14 ] was formally unveiled in February 2008. [ 18 ] By 2010, the glass had been used in approximately 20% of mobile handsets worldwide, or about 200 million units. [ 28 ]
Gorilla Glass 2 was introduced in January 2012. [ 29 ] The second generation is 20% thinner than the original Gorilla Glass. [ 30 ] It was first used on the Samsung Galaxy S III . [ 31 ] In October 2012, Corning announced that over one billion mobile devices used Gorilla Glass. [ 32 ]
Gorilla Glass 3 was introduced at CES 2013 on January 7. [ 33 ] According to Corning, the material is up to three times more scratch-resistant than the previous version, with enhanced ability to resist deep scratches that typically weaken glass. [ 34 ] [ 35 ] Promotional material for Gorilla Glass 3 claims that it is 40% more scratch-resistant in addition to being more flexible. [ 36 ] The design of Gorilla Glass 3 was Corning's first use of atomic-scale modeling before the material was melted in laboratories, with the prediction of the optimal composition obtained through the application of rigidity theory . [ 37 ] The first phone to use Gorilla Glass 3 was the Samsung Galaxy S4 . [ 38 ]
When Gorilla Glass 3 was announced, Corning indicated that areas for future improvements included reducing reflectivity and susceptibility to fingerprint smudges, changing surface treatments, and the way the glass is finished. [ 30 ] Antimicrobial Gorilla Glass, with antibacterial ionic silver incorporated into its surface, was introduced and demonstrated at CES 2014 in January. [ 39 ]
Gorilla Glass 4 was introduced in November 2014. [ 40 ] It has better damage resistance and capability to be made thinner with the same performance as its predecessor. [ 41 ] It was first used on the Samsung Galaxy Alpha . [ 42 ]
Gorilla Glass 5 was introduced in July 2016. [ 43 ] It offers better resistance to cracking from drops and was first used on the Samsung Galaxy Note 7 . [ 44 ]
Gorilla Glass SR+ was introduced in August 2016. [ 45 ] It was designed for wearable mobile devices, focusing on toughness, scratch resistance, and optical clarity. [ 46 ] It was first used on the Samsung Gear S3 smartwatch. [ 47 ]
Gorilla Glass 6 was introduced in July 2018. [ 48 ] It has the scratch resistance of Gorilla Glass 5, but is designed to withstand multiple drops from even greater heights. It was first used on the Samsung Galaxy S10 . [ 49 ]
Gorilla Glass DX and DX+ were introduced in July 2018, following the launch of Gorilla Glass 6. [ 50 ] An extension of Gorilla Glass SR+, Gorilla Glass DX features enhanced antireflective optics with the same scratch resistance of Gorilla Glass, while Gorilla Glass DX+ provides enhanced antireflective optics with superior scratch resistance. While primarily intended for wearable mobile devices, developments were underway to adapt these new glass composites to larger form-factor devices. [ 50 ] Gorilla Glass DX+ was first used on the Samsung Galaxy Watch . [ 51 ]
Gorilla Glass 3+ was introduced in August 2019. [ 52 ] It was designed for intermediate and value-segment mobile devices, and its drop performance bridges the gap between Gorilla Glass 3 and Gorilla Glass 5 in this market segment. [ 52 ]
Gorilla Glass Victus was introduced in July 2020. [ 53 ] It improves both drop and scratch performance and is claimed by Corning to be twice as scratch-resistant as Gorilla Glass 6. [ 53 ] It was first used on the Samsung Galaxy Note 20 Ultra . [ 54 ]
In July 2021, Corning announced that it will bring its Gorilla Glass DX and DX+ glass composites to cover smartphone camera lenses and said that Samsung would be the first customer to adopt them. [ 55 ] The first Samsung phones to use Gorilla Glass DX protection for the cameras were the Galaxy Z Fold 3 and the Galaxy Z Flip 3 . [ 56 ]
Gorilla Glass Victus 2 was introduced in November 2022. [ 57 ] The second-generation Victus has a new composition for improved drop performance on rough surfaces like concrete. In terms of scratch resistance, it is said to be similar to its predecessor. [ 58 ] However, according to one user report, the glass is susceptible to micro-scratches, and it has been suggested that this could be due to the use of softer materials for improved resistance to shattering. [ 59 ] [ 60 ] Gorilla Glass Victus 2 was first used on the Samsung Galaxy S23 series. [ 61 ]
Gorilla Glass's ninth generation, branded as Gorilla Armor, was introduced in January 2024. [ 62 ] Although Corning hasn't directly compared it to the previous generation, the company claims that the glass performs up to three times better in drop tests onto concrete and is four times more scratch-resistant than competitive aluminosilicate cover glasses. Additionally, it enhances visual clarity by reducing reflectance by up to 75% compared to a typical glass surface, thereby improving display readability. It was first used on the Samsung Galaxy S24 Ultra . [ 62 ]
Gorilla Glass 7i was introduced in June 2024, specifically targeting intermediate and value-segment smartphones. It offers enhanced drop protection and scratch resistance, outperforming competitive lithium aluminosilicate glasses. According to Corning, Gorilla Glass 7i can withstand drops from up to one meter (3.3 ft) onto rough surfaces, an improvement over similar materials that typically fail at half that height, while also being up to two times as scratch-resistant. [ 63 ]
Gorilla Armor 2 was introduced in January 2025, marking the introduction of glass-ceramic materials to the Gorilla Glass lineup. [ 64 ] Building upon Gorilla Armor's focus on drop performance and anti-reflective properties, Armor 2 further enhances drop performance through its glass-ceramic composition. In Corning's lab tests, Armor 2 survives drops of up to 2.2 meters (7.2 ft) onto a surface replicating concrete, while maintaining its scratch resistance. It was first used on the Samsung Galaxy S25 Ultra . [ 64 ]
Superfest is a chemically hardened glass also known as Ceverit and CV-Glas. It was developed in East Germany in the 1970s.
On October 26, 2011, Corning announced the commercial launch of Lotus Glass , designed for OLED and next-generation LC displays. [ 75 ] The intrinsic thermal consistency of Lotus Glass allows it to retain its shape and quality during high-temperature processing. Decreased compaction and variation during the crystallization and activation step further reduce stress and distortions to the substrate. This enables tighter design rules in advanced backplanes for higher resolution and faster response time. [ 76 ] According to Corning, Gorilla Glass is specifically a cover glass for the exterior of display devices while Lotus Glass is designed as a glass substrate to be used within liquid crystal display panels. In other words, a single product could incorporate both Gorilla Glass and Lotus Glass. [ 77 ]
On February 2, 2012, Corning Incorporated and Samsung Mobile Display Co., Ltd. signed an agreement to establish a new equity venture for the manufacture of specialty glass substrates for the OLED device market in Korea. The joint venture is based on Lotus Glass. [ 78 ] Lotus XT Glass became available in 2013. [ 79 ]
In 2012, Corning introduced Willow Glass , [ 80 ] a flexible glass based on borosilicate glass , [ 81 ] launched for use as a display substrate.
Ceramic Shield , a ceramic-hardened front glass, was co-developed with Apple and is used on all iPhones from iPhone 12 onwards (except the third-generation iPhone SE ). [ 82 ] | https://en.wikipedia.org/wiki/Gorilla_Glass |
Gorō Azumaya ( 東屋 五郎 , Azumaya Gorō , February 26, 1920– July 8, 2010) was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. [ 1 ] [ 2 ] His advisor was Shokichi Iyanaga . At the time of his death he was an emeritus professor at Indiana University .
This article about a Japanese scientist is a stub . You can help Wikipedia by expanding it .
This article about an Asian mathematician is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Goro_Azumaya |
In mathematics , Gosper's algorithm , due to Bill Gosper , is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a (1) + ... + a ( n ) = S ( n ) − S (0), where S ( n ) is a hypergeometric term (i.e., S ( n + 1)/ S ( n ) is a rational function of n ); then necessarily a ( n ) is itself a hypergeometric term, and given the formula for a ( n ) Gosper's algorithm finds that for S ( n ).
Step 1: Find a polynomial p such that, writing b ( n ) = a ( n )/ p ( n ), the ratio b ( n )/ b ( n − 1) has the form q ( n )/ r ( n ) where q and r are polynomials and no q ( n ) has a nontrivial factor with r ( n + j ) for j = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in closed form.)
Step 2: Find a polynomial ƒ such that S ( n ) = q ( n + 1)/ p ( n ) ƒ ( n ) a ( n ). If the series is summable in closed form then clearly a rational function ƒ with this property exists; in fact it must always be a polynomial, and an upper bound on its degree can be found. Determining ƒ (or finding that there is no such ƒ ) is then a matter of solving a system of linear equations. [ 1 ]
Gosper's algorithm can be used to discover Wilf–Zeilberger pairs , where they exist. Suppose that F ( n + 1, k ) − F ( n , k ) = G ( n , k + 1) − G ( n , k ) where F is known but G is not. Then feed a ( k ) := F ( n + 1, k ) − F ( n , k ) into Gosper's algorithm. (Treat this as a function of k whose coefficients happen to be functions of n rather than numbers; everything in the algorithm works in this setting.) If it successfully finds S ( k ) with S ( k ) − S ( k − 1) = a ( k ), then we are done: this is the required G . If not, there is no such G .
Gosper's algorithm finds (where possible) a hypergeometric closed form for the indefinite sum of hypergeometric terms. It can happen that there is no such closed form, but that the sum over all n , or some particular set of values of n, has a closed form. This question is only meaningful when the coefficients are themselves functions of some other variable. So, suppose a(n,k) is a hypergeometric term in both n and k : that is, a ( n , k )/ a ( n − 1, k ) and a ( n , k )/ a ( n , k − 1) are rational functions of n and k . Then Zeilberger's algorithm and Petkovšek's algorithm may be used to find closed forms for the sum over k of a ( n , k ).
Bill Gosper discovered this algorithm in the 1970s while working on the Macsyma computer algebra system at SAIL and MIT . | https://en.wikipedia.org/wiki/Gosper's_algorithm |
In the field of mathematics, the Goss zeta function, named after David Goss , is an analogue of the Riemann zeta function for function fields . Sheats (1998) proved that it satisfies an analogue of the Riemann hypothesis . Kapranov (1995) proved results for a higher-dimensional generalization of the Goss zeta function.
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Gothenburg International Bioscience Business School ( GIBBS ) is in Gothenburg , Sweden .
The education is a collaboration between Sahlgrenska Academy at University of Gothenburg and Chalmers University of Technology and is a part of the Center for Intellectual Property Studies , CIP. Most of the collaboration is during the first year with a shared curriculum studying with peers from various backgrounds such as law, engineering, life sciences, and management. The second year students work in groups with the innovation project with the aim to commercialize an innovation.
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Gottfried Maria Hugo Köthe (25 December 1905 – 30 April 1989) was an Austrian mathematician working in abstract algebra and functional analysis .
In 1923 Köthe enrolled in the University of Graz . He started studying chemistry, but switched to mathematics a year later after meeting the philosopher Alfred Kastil . In 1927 he submitted his thesis Beiträge zu Finslers Grundlegung der Mengenlehre ("Contributions to Finsler's foundations of set theory") and was awarded a doctorate. After spending a year in Zürich working with Paul Finsler , Köthe received a fellowship to visit the University of Göttingen , where he attended the lectures of Emmy Noether and Bartel van der Waerden on the emerging subject of abstract algebra. He began working in ring theory and in 1930 published the Köthe conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal. By a recommendation of Emmy Noether, he was appointed an assistant of Otto Toeplitz in Bonn University in 1929–1930. During this time he began transition to functional analysis. He continued scientific collaboration with Toeplitz for several years afterward.
Köthe's Habilitationsschrift , Schiefkörper unendlichen Ranges über dem Zentrum ("Skew fields of infinite rank over the center"), was accepted in 1931. He became Privatdozent at University of Münster under Heinrich Behnke . During World War II he was involved in coding work. In 1946 he was appointed the director of the Mathematics Institute at the University of Mainz and he served as a dean (1948–1950) and a rector of the university (1954–1956). In 1957 he became the founding director of the Institute for Applied Mathematics at the University of Heidelberg and served as a rector of the university (1960–1961).
Köthe's best known work has been in the theory of topological vector spaces . In 1960, volume 1 of his seminal monograph Topologische lineare Räume was published (the second edition was translated into English in 1969). It was not until 1979 that volume 2 appeared, this time written in English. He also made contributions to the theory of lattices . | https://en.wikipedia.org/wiki/Gottfried_Köthe |
Gottfried Wilhelm Leibniz (or Leibnitz ; [ a ] 1 July 1646 [ O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician , philosopher , scientist and diplomat who is credited, alongside Sir Isaac Newton , with the creation of calculus in addition to many other branches of mathematics , such as binary arithmetic and statistics . Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. [ 16 ] He is a prominent figure in both the history of philosophy and the history of mathematics . He wrote works on philosophy , theology , ethics , politics , law , history , philology , games , music , and other studies. Leibniz also made major contributions to physics and technology , and anticipated notions that surfaced much later in probability theory , biology , medicine , geology , psychology , linguistics and computer science .
Leibniz contributed to the field of library science , developing a cataloguing system (at the Herzog August Library in Wolfenbüttel , Germany) that came to serve as a model for many of Europe's largest libraries. [ 17 ] [ 18 ] His contributions to a wide range of subjects were scattered in various learned journals , in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German. [ 19 ] [ b ]
As a philosopher, he was a leading representative of 17th-century rationalism and idealism . As a mathematician, his major achievement was the development of differential and integral calculus, independently of Newton's contemporaneous developments. [ 21 ] Leibniz's notation has been favored as the conventional and more exact expression of calculus. [ 22 ] [ 23 ] [ 24 ] In addition to his work on calculus, he is credited with devising the modern binary number system, which is the basis of modern communications and digital computing; [ 25 ] however, Thomas Harriot had devised the same system decades before. [ 26 ] He envisioned the field of combinatorial topology as early as 1679, [ 27 ] and helped initiate the field of fractional calculus . [ 28 ] [ 29 ]
In the 20th century, Leibniz's notions of the law of continuity and the transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis . He was also a pioneer in the field of mechanical calculators . While working on adding automatic multiplication and division to Pascal's calculator , he was the first to describe a pinwheel calculator in 1685 [ 30 ] and invented the Leibniz wheel , later used in the arithmometer , the first mass-produced mechanical calculator.
In philosophy and theology , Leibniz is most noted for his optimism , i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created , a view sometimes lampooned by other thinkers, such as Voltaire in his satirical novella Candide . Leibniz, along with René Descartes and Baruch Spinoza , was one of the three influential early modern rationalists . His philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions. The work of Leibniz anticipated modern logic and still influences contemporary analytic philosophy , such as its adopted use of the term " possible world " to define modal notions.
Gottfried Leibniz was born on 1 July [ OS : 21 June] 1646, in Leipzig , Saxony, to Friedrich Leibniz (1597–1652) and Catharina Schmuck (1621–1664). [ 31 ] He was baptized two days later at St. Nicholas Church, Leipzig ; his godfather was the Lutheran theologian Martin Geier [ de ] . [ 32 ] His father died when he was six years old, and Leibniz was raised by his mother. [ 33 ]
Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig , where he also served as dean of philosophy. The boy inherited his father's personal library. He was given free access to it from the age of seven, shortly after his father's death. While Leibniz's schoolwork was largely confined to the study of a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years. [ 34 ] Access to his father's library, largely written in Latin , also led to his proficiency in the Latin language, which he achieved by the age of 12. At the age of 13 he composed 300 hexameters of Latin verse in a single morning for a special event at school. [ 35 ]
In April 1661 he enrolled in his father's former university at age 14. [ 36 ] [ 1 ] [ 37 ] There he was guided, among others, by Jakob Thomasius , previously a student of Friedrich. Leibniz completed his bachelor's degree in Philosophy in December 1662. He defended his Disputatio Metaphysica de Principio Individui ( Metaphysical Disputation on the Principle of Individuation ), [ 38 ] which addressed the principle of individuation , on 9 June 1663 [ O.S. 30 May], presenting an early version of monadic substance theory. Leibniz earned his master's degree in Philosophy on 7 February 1664. In December 1664 he published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum ( An Essay of Collected Philosophical Problems of Right ), [ 38 ] arguing for both a theoretical and a pedagogical relationship between philosophy and law. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665. [ 39 ] His dissertation was titled De conditionibus ( On Conditions ). [ 38 ]
In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria ( On the Combinatorial Art ), the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. [ 38 ] [ 40 ] De Arte Combinatoria was inspired by Ramon Llull 's Ars Magna and contained a proof of the existence of God , cast in geometrical form, and based on the argument from motion . [ citation needed ]
His next goal was to earn his license and Doctorate in Law, which normally required three years of study. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. [ 41 ] [ 42 ] Leibniz subsequently left Leipzig. [ 43 ]
Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, which he had probably been working on earlier in Leipzig. [ 44 ] The title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure ( Inaugural Disputation on Ambiguous Legal Cases ). [ 38 ] Leibniz earned his license to practice law and his Doctorate in Law in November 1666. He next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction". [ 45 ]
As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Many posthumously published editions of his writings presented his name on the title page as " Freiherr G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility . [ 46 ]
Leibniz's first position was as a salaried secretary to an alchemical society in Nuremberg . [ 47 ] He knew fairly little about the subject at that time but presented himself as deeply learned. He soon met Johann Christian von Boyneburg (1622–1672), the dismissed chief minister of the Elector of Mainz , Johann Philipp von Schönborn . [ 48 ] Von Boyneburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for the Electorate. [ 49 ] In 1669, Leibniz was appointed assessor in the Court of Appeal. Although von Boyneburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674. [ 50 ]
Von Boyneburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. After Leibniz's service to the Elector there soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main force in European geopolitics during Leibniz's adult life was the ambition of Louis XIV of France , backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows: France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies . In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion, [ 51 ] but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting, late implementation of Leibniz's plan, after the Eastern hemisphere colonial supremacy in Europe had already passed from the Dutch to the British.
Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus . He met Nicolas Malebranche and Antoine Arnauld , the leading French philosophers of the day, and studied the writings of Descartes and Pascal , unpublished as well as published. [ 52 ] He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus ; they corresponded for the rest of their lives. [ citation needed ]
When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673. [ 53 ] There Leibniz came into acquaintance of Henry Oldenburg and John Collins . He met with the Royal Society where he demonstrated a calculating machine that he had designed and had been building since 1670. The machine was able to execute all four basic operations (adding, subtracting, multiplying, and dividing), and the society quickly made him an external member. [ citation needed ]
The mission ended abruptly when news of the Elector's death (12 February 1673) reached them. Leibniz promptly returned to Paris and not, as had been planned, to Mainz. [ 54 ] The sudden deaths of his two patrons in the same winter meant that Leibniz had to find a new basis for his career. [ citation needed ]
In this regard, a 1669 invitation from Duke John Frederick of Brunswick to visit Hanover proved to have been fateful. Leibniz had declined the invitation, but had begun corresponding with the duke in 1671. In 1673, the duke offered Leibniz the post of counsellor. Leibniz very reluctantly accepted the position two years later, only after it became clear that no employment was forthcoming in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court. [ 55 ]
In 1675 he tried to get admitted to the French Academy of Sciences as a foreign honorary member, but it was considered that there were already enough foreigners there and so no invitation came. He left Paris in October 1676.
Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where Newton accused him of having seen his unpublished work on calculus in advance. [ 56 ] This was alleged to be evidence supporting the accusation, made decades later, that he had stolen calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met van Leeuwenhoek , the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza , who had just completed, but had not published, his masterwork, the Ethics . [ 57 ] Spinoza died very shortly after Leibniz's visit.
In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.
Leibniz began promoting a project to use windmills to improve the mining operations in the Harz Mountains. This project did little to improve mining operations and was shut down by Duke Ernst August in 1685. [ 55 ]
Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach , the consort of her grandson, the future George II . To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain . [ 58 ]
The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire . The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of England, once both King William III and his sister-in-law and successor, Queen Anne , were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament .
The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum . That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.
The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.
Leibniz was appointed Librarian of the Herzog August Library in Wolfenbüttel , Lower Saxony , in 1691.
In 1708, John Keill , writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarised Newton's calculus. [ 59 ] Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of calculus.
In 1712, Leibniz began a two-year residence in Vienna , where he was appointed Imperial Court Councillor to the Habsburgs . On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain , under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714. In 1716, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Bad Pyrmont and met Leibniz, who took interest in Russian matters since 1708 and was appointed advisor in 1711. [ 60 ]
Leibniz died in Hanover in 1716. At the time, he was so out of favor that neither George I (who happened to be near Hanover at that time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences , neither organization saw fit to honor his death. His grave went unmarked for more than 50 years. He was, however, eulogized by Fontenelle , before the French Academy of Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans , a niece of the Electress Sophia.
Leibniz never married. He proposed to an unknown woman at age 50, but changed his mind when she took too long to decide. [ 61 ] He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had paid him fairly well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy . [ 62 ]
He was charming, well-mannered, and not without humor and imagination. [ 63 ] He had many friends and admirers all over Europe. He was identified as a Protestant and a philosophical theist . [ 64 ] [ 65 ] [ 66 ] [ 67 ] Leibniz remained committed to Trinitarian Christianity throughout his life. [ 68 ]
Leibniz's philosophical thinking appears fragmented because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and letters to correspondents. He wrote two book-length philosophical treatises, of which only the Théodicée of 1710 was published in his lifetime.
Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics , which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche and Antoine Arnauld . This led to an extensive correspondence with Arnauld; [ 69 ] it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances". [ 70 ] Between 1695 and 1705, he composed his New Essays on Human Understanding , a lengthy commentary on John Locke 's 1690 An Essay Concerning Human Understanding , but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie , composed in 1714 and published posthumously, consists of 90 aphorisms.
Leibniz also wrote a short paper, "Primae veritates" ("First Truths"), first published by Louis Couturat in 1903 (pp. 518–523) [ 71 ] summarizing his views on metaphysics . The paper is undated; that he wrote it while in Vienna in 1689 was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–1690. [ 72 ] Couturat's reading of this paper influenced much 20th-century thinking about Leibniz, especially among analytic philosophers . After a meticulous study (informed by the 1999 additions to the critical edition) of all of Leibniz's philosophical writings up to 1688, Mercer (2001) disagreed with Couturat's reading. [ clarification needed ]
Leibniz met Baruch Spinoza in 1676, read some of his unpublished writings, and had since been influenced by some of Spinoza's ideas. [ citation needed ] While Leibniz befriended him and admired Spinoza's powerful intellect, he was also dismayed by Spinoza's conclusions, [ 73 ] especially when these were inconsistent with Christian orthodoxy.
Unlike Descartes and Spinoza, Leibniz had a university education in philosophy. He was influenced by his Leipzig professor Jakob Thomasius , who also supervised his BA thesis in philosophy. [ 9 ] Leibniz also read Francisco Suárez , a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle , but the established philosophical ideas in which he was educated influenced his view of their work.
Leibniz variously invoked one or another of seven fundamental philosophical Principles: [ 74 ]
Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted. [ 80 ]
Leibniz's best known contribution to metaphysics is his theory of monads , as exposited in Monadologie . He proposes his theory that the universe is made of an infinite number of simple substances known as monads. [ 81 ] Monads can also be compared to the corpuscles of the mechanical philosophy of René Descartes and others. These simple substances or monads are the "ultimate units of existence in nature". Monads have no parts but still exist by the qualities that they have. These qualities are continuously changing over time, and each monad is unique. They are also not affected by time and are subject to only creation and annihilation. [ 82 ] Monads are centers of force ; substance is force, while space , matter , and motion are merely phenomenal. He argued, against Newton, that space , time , and motion are completely relative: [ 83 ] "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions." [ 84 ] Einstein, who called himself a "Leibnizian", wrote in the introduction to Max Jammer 's book Concepts of Space that Leibnizianism was superior to Newtonianism, and his ideas would have dominated over Newton's had it not been for the poor technological tools of the time; Joseph Agassi argues that Leibniz paved the way for Einstein's theory of relativity . [ 85 ]
Leibniz's proof of God can be summarized in the Théodicée . [ 86 ] Reason is governed by the principle of contradiction and the principle of sufficient reason . Using the principle of reasoning, Leibniz concluded that the first reason of all things is God. [ 86 ] All that we see and experience is subject to change, and the fact that this world is contingent can be explained by the possibility of the world being arranged differently in space and time. The contingent world must have some necessary reason for its existence. Leibniz uses a geometry book as an example to explain his reasoning. If this book was copied from an infinite chain of copies, there must be some reason for the content of the book. [ 87 ] Leibniz concluded that there must be the " monas monadum " or God.
The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of pre-established harmony , each monad follows a pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic.
Monads are purported to have gotten rid of the problematic:
The Theodicy [ 88 ] tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds . It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws. [ 89 ]
Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity. [ 90 ] This project was motivated in part by Leibniz's belief, shared by many philosophers and theologians during the Enlightenment , in the rational and enlightened nature of the Christian religion. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science).
In the view of Leibniz, because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism: [ 91 ] if God is all good , all wise , and all powerful , then how did evil come into the world ? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions, and ineffective actions in the exercise of their free will . God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin) and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good. [ 92 ]
Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination.
For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome (VI). Along these lines, he declares that every type of perfection "pertains to him (God) in the highest degree" (I). Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly (III). Because God cannot act imperfectly, the decisions he makes pertaining to the world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured. However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition (IV). In accordance with this, many act as rebels, but Leibniz says that the only way we can truly love God is by being content "with all that comes to us according to his will" (IV).
Because God is "an absolutely perfect being" (I), Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of (III). His syllogism then ends with the statement that God has made the world perfectly in all ways. This also affects how we should view God and his will. Leibniz states that, in lieu of God's will, we have to understand that God "is the best of all masters" and he will know when his good succeeds, so we, therefore, must act in conformity to his good will—or as much of it as we understand (IV). In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so. Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition (II). Leibniz then asserts that different principles and geometry cannot simply be from the will of God, but must follow from his understanding. [ 93 ]
Leibniz wrote: " Why is there something rather than nothing? The sufficient reason ... is found in a substance which ... is a necessary being bearing the reason for its existence within itself." [ 94 ] Martin Heidegger called this question "the fundamental question of metaphysics". [ 95 ] [ 96 ]
Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:
The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right. [ 97 ] [ 98 ] [ 99 ]
Leibniz's calculus ratiocinator , which resembles symbolic logic , can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda [ 100 ] that can now be read as groping attempts to get symbolic logic—and thus his calculus —off the ground. These writings remained unpublished until the appearance of a selection edited by Carl Immanuel Gerhardt (1859). Louis Couturat published a selection in 1901; by this time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege .
Leibniz thought symbols were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this. His notation for calculus is an example of his skill in this regard. Leibniz's passion for symbols and notation, as well as his belief that these are essential to a well-running logic and mathematics, made him a precursor of semiotics . [ 101 ]
But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including Egyptian hieroglyphics , Chinese characters , and the symbols of astronomy and chemistry , he deemed not real. [ 102 ] Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character:
It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus. [ 103 ]
Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering . Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.
Because Leibniz was a mathematical novice when he first wrote about the characteristic , at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established. [ 104 ] Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as Turing completeness , where computation was used to define equivalent universal languages (see Turing degree ).
Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege . [ 105 ] Leibniz enunciated the principal properties of what we now call conjunction , disjunction , negation , identity , set inclusion , and the empty set . The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:
The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse .
Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his History of Western Philosophy , Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.
Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims (e.g., that each of the fundamental monads mirrors the whole universe) follow logically from Leibniz's conscious choice to reject relations between things as unreal. He regarded such relations as (real) qualities of things (Leibniz admitted unary predicates only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of De Morgan , Peirce , Schröder and Russell himself, now standard in predicate logic . Notably, Leibniz also declared space and time to be inherently relational. [ 106 ]
Leibniz's 1690 discovery of his algebra of concepts [ 107 ] [ 108 ] (deductively equivalent to the Boolean algebra ) [ 109 ] and the associated metaphysics, are of interest in present-day computational metaphysics . [ 110 ]
Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa , ordinate , tangent , chord , and the perpendicular (see History of the function concept ). [ 111 ] In the 18th century, "function" lost these geometrical associations. Leibniz was also one of the pioneers in actuarial science , calculating the purchase price of life annuities and the liquidation of a state's debt. [ 112 ]
Leibniz's research into formal logic, also relevant to mathematics, is discussed in the preceding section . The best overview of Leibniz's writings on calculus may be found in Bos (1974). [ 113 ]
Leibniz, who invented one of the earliest mechanical calculators, said of calculation : "For it is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used." [ 114 ]
Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix , in order to find a solution to the system if it existed. [ 115 ] This method was later called Gaussian elimination . Leibniz laid down the foundations and theory of determinants , although the Japanese mathematician Seki Takakazu also discovered determinants independently of Leibniz. [ 116 ] [ 117 ] His works show calculating the determinants using cofactors. [ 118 ] Calculating the determinant using cofactors is named the Leibniz formula . Finding the determinant of a matrix using this method proves impractical with large n , requiring to calculate n! products and the number of n-permutations. [ 119 ] He also solved systems of linear equations using determinants, which is now called Cramer's rule . This method for solving systems of linear equations based on determinants was found in 1684 by Leibniz ( Gabriel Cramer published his findings in 1750). [ 117 ] Although Gaussian elimination requires O ( n 3 ) {\displaystyle O(n^{3})} arithmetic operations, linear algebra textbooks still teach cofactor expansion before LU factorization . [ 120 ] [ 121 ]
The Leibniz formula for π states that
Leibniz wrote that circles "can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted". [ 122 ] However this formula is only accurate with a large number of terms, using 10,000,000 terms to obtain the correct value of π / 4 to 8 decimal places. [ 123 ] Leibniz attempted to create a definition for a straight line while attempting to prove the parallel postulate . [ 124 ] While most mathematicians defined a straight line as the shortest line between two points, Leibniz believed that this was merely a property of a straight line rather than the definition. [ 125 ]
Leibniz is credited, along with Isaac Newton , with the invention of calculus (differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = f ( x ) . [ 126 ] He introduced several notations used to this day, for instance the integral sign ∫ ( ∫ f ( x ) d x {\displaystyle \displaystyle \int f(x)\,dx} ), representing an elongated S, from the Latin word summa , and the d used for differentials ( d y d x {\displaystyle {\frac {dy}{dx}}} ), from the Latin word differentia . Leibniz did not publish anything about his calculus until 1684. [ 127 ] Leibniz expressed the inverse relation of integration and differentiation, later called the fundamental theorem of calculus , by means of a figure [ 128 ] in his 1693 paper Supplementum geometriae dimensoriae... . [ 129 ] However, James Gregory is credited for the theorem's discovery in geometric form, Isaac Barrow proved a more generalized geometric version, and Newton developed supporting theory. The concept became more transparent as developed through Leibniz's formalism and new notation. [ 130 ] The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule .
Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley , in a tract called The Analyst and also in De Motu , criticized these. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms. [ 131 ]
Leibniz introduced fractional calculus in a letter written to Guillaume de l'Hôpital in 1695. [ 29 ] At the same time, Leibniz wrote to Johann Bernoulli about derivatives of "general order". [ 28 ] In the correspondence between Leibniz and John Wallis in 1697, Wallis's infinite product for 1 2 {\displaystyle {\frac {1}{2}}} π is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation d 1 / 2 y {\displaystyle {d}^{1/2}{y}} to denote the derivative of order 1 2 {\displaystyle {\frac {1}{2}}} . [ 28 ]
From 1711 until his death, Leibniz was engaged in a dispute with John Keill , Newton and others, over whether Leibniz had invented calculus independently of Newton .
The use of infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass , [ 132 ] but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential . Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory , in the context of a field of hyperreal numbers . The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity , while the standard part function implements the Leibnizian transcendental law of homogeneity .
Leibniz was the first to use the term analysis situs , [ 133 ] later used in the 19th century to refer to what is now known as topology . There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal , argues:
Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler , in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ... [It] is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics. [ 134 ]
But Hideaki Hirano argues differently, quoting Mandelbrot : [ 135 ]
To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in "packing", ... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms, he states ...: "I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets." This claim can be proved today. [ 136 ]
Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: Natura non facit saltus . [ 77 ] We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.
He envisioned the field of combinatorial topology as early as 1679, in his work titled Characteristica Geometrica, as he "tried to formulate basic geometric properties of figures, to use special symbols to represent them, and to combine these properties under operations so as to produce new ones." [ 27 ]
Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings .
Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton . He devised a new theory of motion ( dynamics ) based on kinetic energy and potential energy , which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695. [ 137 ]
Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space , time and motion are relative, not absolute: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions." [ 84 ]
Leibniz held a relational notion of space and time, against Newton's substantivalist views. [ 138 ] [ 139 ] [ 140 ] According to Newton's substantivalism, space and time are entities in their own right, existing independently of things. Leibniz's relationalism, in contrast, describes space and time as systems of relations that exist between objects. The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light.
One of Leibniz's projects was to recast Newton's theory as a vortex theory . [ 141 ] However, his project went beyond vortex theory, since at its heart there was an attempt to explain one of the most difficult problems in physics, that of the origin of the cohesion of matter . [ 141 ]
The principle of sufficient reason has been invoked in recent cosmology , and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. In addition to his theories about the nature of reality, Leibniz's contributions to the development of calculus have also had a major impact on physics.
Leibniz's vis viva (Latin for "living force") is m v 2 , twice the modern kinetic energy . He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. [ 142 ] Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes and Voltaire in France; hence academics in those countries tended to neglect Leibniz's idea. Leibniz knew of the validity of conservation of momentum. In reality, both energy and momentum are conserved (in closed systems ), so both approaches are valid.
By proposing that the Earth has a molten core, he anticipated modern geology. In embryology , he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology , he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea , unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory . [ 143 ] In medicine, he exhorted the physicians of his time—with some results—to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.
Psychology had been a central interest of Leibniz. [ 144 ] [ 145 ] He appears to be an "underappreciated pioneer of psychology" [ 146 ] He wrote on topics which are now regarded as fields of psychology: attention and consciousness , memory , learning ( association ), motivation (the act of "striving"), emergent individuality , the general dynamics of development ( evolutionary psychology ). His discussions in the New Essays and Monadology often rely on everyday observations such as the behaviour of a dog or the noise of the sea, and he develops intuitive analogies (the synchronous running of clocks or the balance spring of a clock). He also devised postulates and principles that apply to psychology: the continuum of the unnoticed petites perceptions to the distinct, self-aware apperception , and psychophysical parallelism from the point of view of causality and of purpose: "Souls act according to the laws of final causes, through aspirations, ends and means. Bodies act according to the laws of efficient causes, i.e. the laws of motion. And these two realms, that of efficient causes and that of final causes, harmonize with one another." [ 147 ] This idea refers to the mind-body problem, stating that the mind and brain do not act upon each other, but act alongside each other separately but in harmony. [ 148 ] Leibniz, however, did not use the term psychologia . [ 149 ] Leibniz's epistemological position—against John Locke and English empiricism ( sensualism )—was made clear: "Nihil est in intellectu quod non fuerit in sensu, nisi intellectu ipse." – "Nothing is in the intellect that was not first in the senses, except the intellect itself." [ 150 ] Principles that are not present in sensory impressions can be recognised in human perception and consciousness: logical inferences, categories of thought, the principle of causality and the principle of purpose ( teleology ).
Leibniz found his most important interpreter in Wilhelm Wundt , founder of psychology as a discipline. Wundt used the "… nisi intellectu ipse" quotation 1862 on the title page of his Beiträge zur Theorie der Sinneswahrnehmung (Contributions on the Theory of Sensory Perception) and published a detailed and aspiring monograph on Leibniz. [ 151 ] Wundt shaped the term apperception , introduced by Leibniz, into an experimental psychologically based apperception psychology that included neuropsychological modelling – an excellent example of how a concept created by a great philosopher could stimulate a psychological research program. One principle in the thinking of Leibniz played a fundamental role: "the principle of equality of separate but corresponding viewpoints." Wundt characterized this style of thought ( perspectivism ) in a way that also applied for him—viewpoints that "supplement one another, while also being able to appear as opposites that only resolve themselves when considered more deeply." [ 152 ] [ 153 ] Much of Leibniz's work went on to have a great impact on the field of psychology. [ 154 ] Leibniz thought that there are many petites perceptions, or small perceptions of which we perceive but of which we are unaware. He believed that by the principle that phenomena found in nature were continuous by default, it was likely that the transition between conscious and unconscious states had intermediary steps. [ 155 ] For this to be true, there must also be a portion of the mind of which we are unaware at any given time. His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep . In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious . Leibniz was a direct influence on Ernst Platner , who is credited with originally coining the term Unbewußtseyn (unconscious). [ 156 ] Additionally, the idea of subliminal stimuli can be traced back to his theory of small perceptions. [ 157 ] Leibniz's ideas regarding music and tonal perception went on to influence the laboratory studies of Wilhelm Wundt. [ 158 ]
In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine . He worked to set up a coherent medical training program, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade . He even proposed something akin to what much later emerged as game theory . In sociology he laid the ground for communication theory .
In 1906, Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxi , he urged that theory be combined with practical application, and thus has been claimed as the father of applied science . He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin , he created a steam engine . He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains , but did not succeed. [ 159 ]
Leibniz may have been the first computer scientist and information theorist. [ 160 ] Early in life, he documented the binary numeral system ( base 2), then revisited that system throughout his career. [ 161 ] While Leibniz was examining other cultures to compare his metaphysical views, he encountered an ancient Chinese book I Ching . Leibniz interpreted a diagram which showed yin and yang and corresponded it to a zero and one. [ 162 ] More information can be found in the Sinophology section. Leibniz had similarities with Juan Caramuel y Lobkowitz and Thomas Harriot , who independently developed the binary system, as he was familiar with their works on the binary system. [ 163 ] Juan Caramuel y Lobkowitz worked extensively on logarithms including logarithms with base 2. [ 164 ] Thomas Harriot's manuscripts contained a table of binary numbers and their notation, which demonstrated that any number could be written on a base 2 system. [ 165 ] Regardless, Leibniz simplified the binary system and articulated logical properties such as conjunction, disjunction, negation, identity, inclusion, and the empty set. [ 166 ] He anticipated Lagrangian interpolation and algorithmic information theory . His calculus ratiocinator anticipated aspects of the universal Turing machine . In 1961, Norbert Wiener suggested that Leibniz should be considered the patron saint of cybernetics . [ 167 ] Wiener is quoted with "Indeed, the general idea of a computing machine is nothing but a mechanization of Leibniz's Calculus Ratiocinator." [ 168 ]
In 1671, Leibniz began to invent a machine that could execute all four arithmetic operations, gradually improving it over a number of years. This " stepped reckoner " attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover by a craftsman working under his supervision. They were not an unambiguous success because they did not fully mechanize the carry operation . Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations. [ 169 ] Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher in 2010. [ 170 ] In 1693, Leibniz described a design of a machine which could, in theory, integrate differential equations, which he called "integraph". [ 171 ]
Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace . In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards. [ 172 ] [ 173 ] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.
Later in Leibniz's career (after the death of von Boyneburg), Leibniz moved to Paris and accepted a position as a librarian in the Hanoverian court of Johann Friedrich, Duke of Brunswick-Luneburg. [ 174 ] Leibniz's predecessor, Tobias Fleischer, had already created a cataloging system for the Duke's library but it was a clumsy attempt. At this library, Leibniz focused more on advancing the library than on the cataloging. For instance, within a month of taking the new position, he developed a comprehensive plan to expand the library. He was one of the first to consider developing a core collection for a library and felt "that a library for display and ostentation is a luxury and indeed superfluous, but a well-stocked and organized library is important and useful for all areas of human endeavor and is to be regarded on the same level as schools and churches". [ 175 ] Leibniz lacked the funds to develop the library in this manner. After working at this library, by the end of 1690 Leibniz was appointed as privy-councilor and librarian of the Bibliotheca Augusta at Wolfenbüttel. It was an extensive library with at least 25,946 printed volumes. [ 175 ] At this library, Leibniz sought to improve the catalog. He was not allowed to make complete changes to the existing closed catalog, but was allowed to improve upon it so he started on that task immediately. He created an alphabetical author catalog and had also created other cataloging methods that were not implemented. While serving as librarian of the ducal libraries in Hanover and Wolfenbüttel , Leibniz effectively became one of the founders of library science . Seemingly, Leibniz paid a good deal of attention to the classification of subject matter, favoring a well-balanced library covering a host of numerous subjects and interests. [ 176 ] Leibniz, for example, proposed the following classification system in the Otivm Hanoveranvm Sive Miscellanea (1737): [ 176 ] [ 177 ]
He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University . He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg . Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library . [ citation needed ]
He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis , calculus ratiocinator , and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages (e.g., Esperanto and its rivals), symbolic logic , even the World Wide Web .
Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Académie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg , Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works. [ 178 ]
Leibniz's writings on law, ethics, and politics [ 179 ] were long overlooked by English-speaking scholars, but this has changed of late. [ 180 ]
While Leibniz was no apologist for absolute monarchy like Hobbes , or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke , views invoked in support of liberalism, in 18th-century America and later elsewhere. The following excerpt from a 1695 letter to Baron J. C. Boyneburg's son Philipp is very revealing of Leibniz's political sentiments:
As for ... the great question of the power of sovereigns and the obedience their peoples owe them, I usually say that it would be good for princes to be persuaded that their people have the right to resist them, and for the people, on the other hand, to be persuaded to obey them passively. I am, however, quite of the opinion of Grotius , that one ought to obey as a rule, the evil of revolution being greater beyond comparison than the evils causing it. Yet I recognize that a prince can go to such excess, and place the well-being of the state in such danger, that the obligation to endure ceases. This is most rare, however, and the theologian who authorizes violence under this pretext should take care against excess; excess being infinitely more dangerous than deficiency. [ 181 ]
In 1677, Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and would be free to vote their consciences; [ 182 ] this is sometimes considered an anticipation of the European Union . He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.
But at the same time, he arrived to propose an interreligious and multicultural project to create a universal system of justice, which required from him a broad interdisciplinary perspective. In order to propose it, he combined linguistics (especially sinology), moral and legal philosophy, management, economics, and politics. [ 183 ]
Leibniz trained as a legal academic, but under the tutelage of Cartesian-sympathiser Erhard Weigel we already see an attempt to solve legal problems by rationalist mathematical methods (Weigel's influence being most explicit in the Specimen Quaestionum Philosophicarum ex Jure collectarum ( An Essay of Collected Philosophical Problems of Right )). For example, the Disputatio Inauguralis de Casibus Perplexis in Jure ( Inaugural Disputation on Ambiguous Legal Cases ) [ 184 ] uses early combinatorics to solve some legal disputes, while the 1666 De Arte Combinatoria ( On the Art of Combination ) [ 185 ] includes simple legal problems by way of illustration.
The use of combinatorial methods to solve legal and moral problems seems, via Athanasius Kircher and Daniel Schwenter to be of Llullist inspiration: Ramón Llull attempted to solve ecumenical disputes through recourse to a combinatorial mode of reasoning he regarded as universal (a mathesis universalis). [ 186 ]
In the late 1660s the enlightened Prince-Bishop of Mainz Johann Philipp von Schönborn announced a review of the legal system and made available a position to support his current law commissioner. Leibniz left Franconia and made for Mainz before even winning the role. On reaching Frankfurt am Main Leibniz penned The New Method of Teaching and Learning the Law, by way of application. [ 187 ] The text proposed a reform of legal education and is characteristically syncretic, integrating aspects of Thomism, Hobbesianism, Cartesianism and traditional jurisprudence. Leibniz's argument that the function of legal teaching was not to impress rules as one might train a dog, but to aid the student in discovering their own public reason, evidently impressed von Schönborn as he secured the job.
Leibniz's next major attempt to find a universal rational core to law and so found a legal "science of right", [ 188 ] came when Leibniz worked in Mainz from 1667–72. Starting initially from Hobbes' mechanistic doctrine of power, Leibniz reverted to logico-combinatorial methods in an attempt to define justice. [ 189 ] As Leibniz's so-called Elementa Juris Naturalis advanced, he built in modal notions of right (possibility) and obligation (necessity) in which we see perhaps the earliest elaboration of his possible worlds doctrine within a deontic frame. [ 190 ] While ultimately the Elementa remained unpublished, Leibniz continued to work on his drafts and promote their ideas to correspondents up until his death.
Leibniz devoted considerable intellectual and diplomatic effort to what would now be called an ecumenical endeavor, seeking to reconcile the Roman Catholic and Lutheran churches. In this respect, he followed the example of his early patrons, Baron von Boyneburg and the Duke John Frederick —both cradle Lutherans who converted to Catholicism as adults—who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others. (The House of Brunswick remained Lutheran, because the Duke's children did not follow their father.) These efforts included corresponding with French bishop Jacques-Bénigne Bossuet , and involved Leibniz in some theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation .
Leibniz the philologist was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way. In 1710, he applied ideas of gradualism and uniformitarianism to linguistics in a short essay. [ 191 ] He refuted the belief, widely held by Christian scholars of the time, that Hebrew was the primeval language of the human race. At the same time, he rejected the idea of unrelated language groups and considered them all to have a common source. [ 192 ] He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto- Swedish was the ancestor of the Germanic languages . He puzzled over the origins of the Slavic languages and was fascinated by classical Chinese . Leibniz was also an expert in the Sanskrit language. [ 193 ]
He published the princeps editio (first modern edition) of the late medieval Chronicon Holtzatiae , a Latin chronicle of the County of Holstein .
Leibniz was perhaps the first major European intellectual to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China. He apparently read Confucius Sinarum Philosophus in the first year of its publication. [ 195 ] He came to the conclusion that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic . He noted how the I Ching hexagrams correspond to the binary numbers from 000000 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired. [ 196 ] Leibniz communicated his ideas of the binary system representing Christianity to the Emperor of China, hoping it would convert him. [ 193 ] Leibniz was one of the western philosophers of the time who attempted to accommodate Confucian ideas to prevailing European beliefs. [ 197 ]
Leibniz's attraction to Chinese philosophy originates from his perception that Chinese philosophy was similar to his own. [ 195 ] The historian E.R. Hughes suggests that Leibniz's ideas of "simple substance" and " pre-established harmony " were directly influenced by Confucianism, pointing to the fact that they were conceived during the period when he was reading Confucius Sinarum Philosophus . [ 195 ]
While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna between May 1688 and February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains . His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican , and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics .
When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée , [ 198 ] whose supposed central argument Voltaire lampooned in his popular book Candide , which concludes with the character Candide saying, " Non liquet " (it is not clear), a term that was applied during the Roman Republic to a legal verdict of "not proven". Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description. Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff , whose dogmatic and facile outlook did Leibniz's reputation much harm. Leibniz also influenced David Hume , who read his Théodicée and used some of his ideas. [ 199 ] In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent proponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.
Leibniz's reputation began to recover with the 1765 publication of the Nouveaux Essais . In 1768, Louis Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by a number of editions, including those edited by Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp, and Mollat. Publication of Leibniz's correspondence with notables such as Antoine Arnauld , Samuel Clarke , Sophia of Hanover , and her daughter Sophia Charlotte of Hanover , began.
In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics . [ 200 ] Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. They made Leibniz somewhat respectable among 20th-century analytical and linguistic philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann ). For example, Leibniz's phrase salva veritate , meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine 's writings. Nevertheless, the secondary literature on Leibniz did not really blossom until after World War II. This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1900–1985) through his translations and his interpretive essays in LeClerc (1973). Leibniz's philosophy was also highly regarded by Gilles Deleuze, [ 201 ] who in 1988 published The Fold: Leibniz and the Baroque .
Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive. [ 202 ] Analytic and contemporary philosophy continue to invoke his notions of identity , individuation , and possible worlds . Work in the history of 17th- and 18th-century ideas has revealed more clearly the 17th-century "Intellectual Revolution" that preceded the better-known Industrial and commercial revolutions of the 18th and 19th centuries.
In Germany, various important institutions were named after Leibniz. In Hanover in particular, he is the namesake for some of the most important institutions in the town:
Outside of Hanover:
Awards:
In 1985, the German government created the Leibniz Prize , offering an annual award of 1.55 million euros for experimental results and 770,000 euros for theoretical ones. It was the world's largest prize for scientific achievement prior to the Fundamental Physics Prize .
The collection of manuscript papers of Leibniz at the Gottfried Wilhelm Leibniz Bibliothek – Niedersächische Landesbibliothek was inscribed on UNESCO 's Memory of the World Register in 2007. [ 203 ]
Leibniz still receives popular attention. The Google Doodle for 1 July 2018 celebrated Leibniz's 372nd birthday. [ 204 ] [ 205 ] [ 206 ] Using a quill , his hand is shown writing "Google" in binary ASCII code.
One of the earliest popular but indirect expositions of Leibniz was Voltaire 's satire Candide , published in 1759. Leibniz was lampooned as Professor Pangloss, described as "the greatest philosopher of the Holy Roman Empire ".
Leibniz also appears as one of the main historical figures in Neal Stephenson 's series of novels The Baroque Cycle . Stephenson credits readings and discussions concerning Leibniz for inspiring him to write the series. [ 207 ]
Leibniz also stars in Adam Ehrlich Sachs's novel The Organs of Sense .
The German biscuit Choco Leibniz is named after Leibniz, a famous resident of Hanover where the manufacturer Bahlsen is based.
Leibniz mainly wrote in three languages: scholastic Latin , French and German. During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée . (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg , most notably the "De jure suprematum" a major consideration of the nature of sovereignty .) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain , which Leibniz had withheld from publication after the death of John Locke . Only in 1895, when Bodemann completed his catalogue of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1700, remains unpublished, and much of what is published has appeared only in recent decades. The more than 67,000 records of the Leibniz Edition's Catalogue cover almost all of his known writings and the letters from him and to him. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described in a letter as follows:
I cannot tell you how extraordinarily distracted and spread out I am. I am trying to find various things in the archives; I look at old papers and hunt up unpublished documents. From these I hope to shed some light on the history of the [House of] Brunswick. I receive and answer a huge number of letters. At the same time, I have so many mathematical results, philosophical thoughts, and other literary innovations that should not be allowed to vanish that I often do not know where to begin. [ 208 ]
The extant parts of the critical edition [ 209 ] of Leibniz's writings are organized as follows:
The systematic cataloguing of all of Leibniz's Nachlass began in 1901. It was hampered by two world wars and then by decades of German division into two states, separating scholars and scattering portions of his literary estates. The ambitious project has had to deal with writings in seven languages, contained in some 200,000 written and printed pages. In 1985 it was reorganized and included in a joint program of German federal and state ( Länder ) academies. Since then the branches in Potsdam , Münster , Hanover and Berlin have jointly published 57 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance works.
The year given is usually that in which the work was completed, not of its eventual publication.
Six important collections of English translations are Wiener (1951), Parkinson (1966), Loemker (1969), Ariew and Garber (1989), Woolhouse and Francks (1998), and Strickland (2006). The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe . [ 209 ]
An updated bibliography of more than 25.000 titles is available at Leibniz Bibliographie . | https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz |
Friedrich Ludwig Gottlob Frege ( / ˈ f r eɪ ɡ ə / ; [ 7 ] German: [ˈɡɔtloːp ˈfreːɡə] ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena , and is understood by many to be the father of analytic philosophy , concentrating on the philosophy of language , logic , and mathematics . Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle , and one of the most profound philosophers of mathematics ever. [ 8 ]
His contributions include the development of modern logic in the Begriffsschrift and work in the foundations of mathematics . His book the Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn . His philosophical papers " On Sense and Reference " and " The Thought " are also widely cited. The former argues for two different types of meaning and descriptivism . In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism , concerning numbers and propositions respectively.
Frege was born in 1848 in Wismar , Mecklenburg-Schwerin (today part of Mecklenburg-Vorpommern in northern Germany). His father, Carl (Karl) Alexander Frege (1809–1866), was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege (née Bialloblotzky, 12 January 1815 – 14 October 1898); her mother was Auguste Amalia Maria Ballhorn, a descendant of Philipp Melanchthon [ 9 ] and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century. [ 10 ] Frege was a Lutheran. [ 11 ]
In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote a textbook on the German language for children aged 9–13, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren (2nd ed., Wismar 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862) (Help book for teaching German to children from 9 to 13 years old), the first section of which dealt with the structure and logic of language .
Frege studied at Große Stadtschule Wismar [ de ] and graduated in 1869. [ 12 ] Teacher of mathematics and natural science Gustav Adolf Leo Sachse (1843–1909), who was also a poet, played an important role in determining Frege's future scientific career, encouraging him to continue his studies at his own alma mater the University of Jena . [ 13 ]
Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation . In the four semesters of his studies, he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe (1840–1905; physicist, mathematician, and inventor). Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturer Carl Zeiss AG, he was in a position to advance Frege's career. After Frege's graduation, they came into closer correspondence. [ citation needed ]
His other notable university teachers were Christian Philipp Karl Snell (1806–86; subjects: use of infinitesimal analysis in geometry, analytic geometry of planes , analytical mechanics, optics, physical foundations of mechanics); Hermann Karl Julius Traugott Schaeffer (1824–1900; analytic geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines ); and the philosopher Kuno Fischer (1824–1907; Kantian and critical philosophy ). [ citation needed ]
Starting in 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures of Rudolf Friedrich Alfred Clebsch (1833–72; analytic geometry), Ernst Christian Julius Schering (1824–97; function theory), Wilhelm Eduard Weber (1804–91; physical studies, applied physics), Eduard Riecke (1845–1915; theory of electricity), and Hermann Lotze (1817–81; philosophy of religion). Many of the philosophical doctrines of the mature Frege have parallels in Lotze; it has been the subject of scholarly debate whether or not there was a direct influence on Frege's views arising from his attending Lotze's lectures. [ citation needed ]
In 1873, Frege attained his doctorate under Schering.
Frege married Margarete Katharina Sophia Anna Lieseberg (15 February 1856 – 25 June 1904) on 14 March 1887. [ 12 ] The couple had at least two children, who unfortunately died when young. Years later, they adopted a son, Alfred. Little else is known about Frege's family life, however. [ 14 ]
Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens [ Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic ], Halle a/S: Verlag von Louis Nebert, 1879 marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables . Frege's goal was to show that mathematics grows out of logic , and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic. [ 15 ]
In effect, Frege invented axiomatic predicate logic , in large part thanks to his invention of quantified variables , which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality . Previous logic had dealt with the logical constants and , or , if... then... , not , and some and all , but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a sentence like "every boy loves some girl" and "some girl is loved by every boy" could be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".
A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem , a fundamental statement of number theory that there are an infinite number of prime numbers . Frege's "conceptual notation", however, can represent such inferences. [ 16 ] The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica (3 vols., 1910–13, by Bertrand Russell , 1872–1970, and Alfred North Whitehead , 1861–1947), to Russell's theory of descriptions , to Kurt Gödel 's (1906–78) incompleteness theorems , and to Alfred Tarski 's (1901–83) theory of truth, is ultimately due to Frege.
One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism : unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example, a generalized form of law of trichotomy , were derived within what Frege understood to be pure logic.
This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic ( Die Grundlagen der Arithmetik , 1884). Later, in his Basic Laws of Arithmetic ( Grundgesetze der Arithmetik , vol. 1, 1893; vol. 2, 1903; vol. 2 was published at his own expense), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift , though not without some significant changes. The one truly new principle was one he called the Basic Law V : the "value-range" of the function f ( x ) is the same as the "value-range" of the function g ( x ) if and only if ∀ x [ f ( x ) = g ( x )].
The crucial case of the law may be formulated in modern notation as follows. Let { x | Fx } denote the extension of the predicate Fx , that is, the set of all Fs, and similarly for Gx . Then Basic Law V says that the predicates Fx and Gx have the same extension if and only if ∀x[ Fx ↔ Gx ]. The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.)
In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x ". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Frege opened the Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This letter and Frege's reply are translated in Jean van Heijenoort 1967.)
Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse , and hence is worthless (indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummett 1973), but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways:
Frege's work in logic had little international attention until 1903, when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege. The diagrammatic notation that Frege used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's Principia Mathematica (3 vols.) appeared in 1910–13, the dominant approach to mathematical logic was still that of George Boole (1815–64) and his intellectual descendants, especially Ernst Schröder (1841–1902). Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891–1970) and other admirers, particularly Bertrand Russell [ 19 ] : 2 and Ludwig Wittgenstein (1889–1951). [ 20 ] : 357
Frege is one of the founders of analytic philosophy , whose work on logic and language gave rise to the linguistic turn in philosophy. His contributions to the philosophy of language include:
As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as "What is a number?" or "What objects do number-words ('one', 'two', etc.) refer to?" But in pursuing these matters, he eventually found himself analysing and explaining what meaning is, and thus came to several conclusions that proved highly consequential for the subsequent course of analytic philosophy and the philosophy of language.
Frege's 1892 paper, " On Sense and Reference " ("Über Sinn und Bedeutung"), introduced his influential distinction between sense ("Sinn") and reference ("Bedeutung", which has also been translated as "meaning", or "denotation"). While conventional accounts of meaning took expressions to have just one feature (reference), Frege introduced the view that expressions have two different aspects of significance: their sense and their reference.
Reference (or "Bedeutung") applied to proper names , where a given expression (say the expression "Tom") simply refers to the entity bearing the name (the person named Tom). Frege also held that propositions had a referential relationship with their truth-value (in other words, a statement "refers" to the truth-value it takes). By contrast, the sense (or "Sinn") associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the "mode of presentation" of the item referred to, and there can be multiple modes of representation for the same referent.
The distinction can be illustrated thus: In their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor", which for logical purposes is an unanalysable whole, and the functional expression "the King of the United Kingdom", which contains the significant parts "the King of ξ" and "United Kingdom", have the same referent , namely, the person best known as King Charles III . But the sense of the word " United Kingdom " is a part of the sense of the latter expression, but no part of the sense of the "full name" of King Charles.
These distinctions were disputed by Bertrand Russell, especially in his paper " On Denoting "; the controversy has continued into the present, fueled especially by Saul Kripke 's famous lectures " Naming and Necessity ".
Frege's original papers having been destroyed in the Second World War , in 1954 Dummett studied what transcriptions had survived of his Nachlass , including fragments of a 1924 diary. [ 21 ] [ 22 ] Dummett, an anti-racism activist as well as a Frege scholar, later recounted how he had been deeply shocked to discover from this that the man he had "revered" as "an absolutely rational man" was, at the end of his life, a 'virulent anti-Semite ' of "extreme right-wing opinions". [ 23 ] [ 24 ]
The diary fragments were finally published in 1994. [ 25 ] with an English translation following in 1996. [ 26 ] Written in the last year of his life, at the age of 76, it contains opposition to the parliamentary system, universal suffrage, democrats, socialism and liberals, and hostility toward Catholics and the French as well as the Jews. [ 27 ] Frege thought Jews ought at least be deprived of certain political rights. [ 28 ] And, although he had held friendly relations with Jews in real life (among his students was Gershom Scholem who greatly valued his teaching), Frege wrote that it would be best if Jews would "get lost, or better would like to disappear from Germany." [ 29 ]
Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck ", but then sympathized with General Ludendorff . In an entry dated 5 May 1924 Frege expressed some agreement with an article published in Houston Stewart Chamberlain's Deutschlands Erneuerung which praised Adolf Hitler . [ 29 ] Some interpretations have been written about that time. [ 30 ]
Frege was described by his students as a highly introverted person, seldom entering into dialogues with others and mostly facing the blackboard while lecturing. He was, however, known to occasionally show wit and even bitter sarcasm during his classes. [ 31 ]
Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879), Halle an der Saale: Verlag von Louis Nebert ( online version ).
Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884), Breslau: Verlag von Wilhelm Koebner ( online version ).
Grundgesetze der Arithmetik , Band I (1893); Band II (1903), Jena: Verlag Hermann Pohle ( online version) .
" Function and Concept " (1891)
" On Sense and Reference " (1892)
" Concept and Object " (1892)
"What is a Function?" (1904)
Logical Investigations (1918–1923). Frege intended that the following three papers be published together in a book titled Logische Untersuchungen ( Logical Investigations ). Though the German book never appeared, the papers were published together in Logische Untersuchungen , ed. G. Patzig, Vandenhoeck & Ruprecht, 1966, and English translations appeared together in Logical Investigations , ed. Peter Geach, Blackwell, 1975.
Philosophy
Logic and mathematics
Historical context | https://en.wikipedia.org/wiki/Gottlob_Frege |
The Gough–Joule effect (a.k.a. Gow–Joule effect ) is originally the tendency of elastomers to contract when heated if they are under tension . Elastomers that are not under tension do not see this effect. The term is also used more generally to refer to the dependence of the temperature of any solid on the mechanical deformation. [ 1 ] This effect can be observed in nylon strings of classical guitars, whereby the string contracts as a result of heating. [ 2 ] The effect is due to the decrease of entropy when long chain molecules are stretched.
If an elastic band is first stretched and then subjected to heating, it will shrink rather than expand. This effect was first observed by John Gough in 1802, and was investigated further by James Joule in the 1850s, when it then became known as the Gough–Joule effect. [ 3 ] [ 4 ] Examples in Literature:
The effect is important in O-ring seal design, where the seals can be mounted in a peripherally compressed state in hot applications to prolong life. [ 7 ] The effect is also relevant to rotary seals which can bind if the seal shrinks due to overheating.
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gough–Joule_effect |
In mathematics the Gould polynomials G n ( x ; a , b ) are polynomials introduced by H. W. Gould and named by Roman in 1984. [ 1 ] They are given by [ 2 ]
where
This polynomial -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gould_polynomials |
The Gould–Jacobs reaction is an organic synthesis for the preparation of quinolines and 4‐hydroxyquinoline derivatives. The Gould–Jacobs reaction is a series of reactions. The series of reactions begins with the condensation/substitution of an aniline with alkoxy methylenemalonic ester or acyl malonic ester, producing anilidomethylenemalonic ester. Then through a 6 electron cyclization process, 4-hydroxy-3-carboalkoxyquinoline is formed, which exist mostly in the 4-oxo form. Saponification results in the formation of an acid. This step is followed by decarboxylation to give 4-hydroxyquinoline. [ 1 ] The Gould–Jacobs reaction is effective for anilines with electron‐donating groups at the meta ‐position. [ 2 ]
Specifically, 4-quinolinol can be synthesized. [ 3 ] In this reaction aniline or an aniline derivative first reacts with malonic acid derivative ethyl ethoxymethylenemalonate with substitution of the ethoxy group by nitrogen. A benzannulation takes place by application of heat to a quinoline. The ester group is hydrolysed by sodium hydroxide to the carboxylic acid and decarboxylation again by application of heat to 4-hydroxyquinoline .
Extension of the Gould-Jacobs approach can prepare unsubstituted parent heterocycles with fused pyridine ring of Skraup type (see Skraup reaction ). [ 1 ]
Further reading: [ 4 ] [ 5 ] [ 2 ]
The mechanism for the Gould–Jacobs reaction begins with a nucleophilic attack from the amine nitrogen follows by the loss of ethanol to form the condensation product. A 6 electron cyclization reaction with the loss of another ethanol molecule forms a quinoline (ethyl 4-oxo-4,4a-dihydroquinoline-3-carboxylate). The enol form can be represented from the keto form through keto-enol tautomerism. Protonation of the nitrogen forms ethyl 4-oxo-1,4-dihydroquinoline-3-carboxylate.
An example is the synthesis of 4,7-dichloroquinoline . [ 6 ]
Another example is in the synthesis of antimalarials as aminoalkylamino derivatives of 2,3-dihydrofuroquinolines [ 8 ]
The Gould reaction is also used to convert 5-aminoindole to quinolines for the purpose of synthesizing pyrazolo[4,3- c ]pyrrolo[3,2- f ]quinolin-3-one derivatives as modified pyrazoloquinolinone analogs. These compounds have the potential to act as antagonists at central benzodiazepine receptors (BZRs) in Xenopus laevis oocytes. [ 9 ]
The Gould‐Jacobs reaction has also been used both conventionally with condensation steps and acyclic intermediated and with single step microwave irradiation to synthesize ethyl 4‐oxo‐8,10‐substituted‐4,8‐dihydropyrimido[1,2‐c]pyrrolo[3,2‐e]pyrimidine‐3‐carboxylates. [ 10 ] | https://en.wikipedia.org/wiki/Gould–Jacobs_reaction |
Goursat's lemma , named after the French mathematician Édouard Goursat , is an algebraic theorem about subgroups of the direct product of two groups .
It can be stated more generally in a Goursat variety (and consequently it also holds in any Maltsev variety ), from which one recovers a more general version of Zassenhaus' butterfly lemma . In this form, Goursat's lemma also implies the snake lemma .
Goursat's lemma for groups can be stated as follows.
An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa.
Notice that if H {\displaystyle H} is any subgroup of G × G ′ {\displaystyle G\times G'} (the projections p 1 : H → G {\displaystyle p_{1}:H\to G} and p 2 : H → G ′ {\displaystyle p_{2}:H\to G'} need not be surjective), then the projections from H {\displaystyle H} onto p 1 ( H ) {\displaystyle p_{1}(H)} and p 2 ( H ) {\displaystyle p_{2}(H)} are surjective. Then one can apply Goursat's lemma to H ≤ p 1 ( H ) × p 2 ( H ) {\displaystyle H\leq p_{1}(H)\times p_{2}(H)} .
To motivate the proof, consider the slice S = { g } × G ′ {\displaystyle S=\{g\}\times G'} in G × G ′ {\displaystyle G\times G'} , for any arbitrary g ∈ G {\displaystyle g\in G} . By the surjectivity of the projection map to G {\displaystyle G} , this has a non trivial intersection with H {\displaystyle H} . Then essentially, this intersection represents exactly one particular coset of N ′ {\displaystyle N'} . Indeed, if we have elements ( g , a ) , ( g , b ) ∈ S ∩ H {\displaystyle (g,a),(g,b)\in S\cap H} with a ∈ p N ′ ⊂ G ′ {\displaystyle a\in pN'\subset G'} and b ∈ q N ′ ⊂ G ′ {\displaystyle b\in qN'\subset G'} , then H {\displaystyle H} being a group, we get that ( e , a b − 1 ) ∈ H {\displaystyle (e,ab^{-1})\in H} , and hence, ( e , a b − 1 ) ∈ N ′ {\displaystyle (e,ab^{-1})\in N'} . It follows that ( g , a ) {\displaystyle (g,a)} and ( g , b ) {\displaystyle (g,b)} lie in the same coset of N ′ {\displaystyle N'} . Thus the intersection of H {\displaystyle H} with every "horizontal" slice isomorphic to G ′ ∈ G × G ′ {\displaystyle G'\in G\times G'} is exactly one particular coset of N ′ {\displaystyle N'} in G ′ {\displaystyle G'} .
By an identical argument, the intersection of H {\displaystyle H} with every "vertical" slice isomorphic to G ∈ G × G ′ {\displaystyle G\in G\times G'} is exactly one particular coset of N {\displaystyle N} in G {\displaystyle G} .
All the cosets of N , N ′ {\displaystyle N,N'} are present in the group H {\displaystyle H} , and by the above argument, there is an exact 1:1 correspondence between them. The proof below further shows that the map is an isomorphism.
Before proceeding with the proof , N {\displaystyle N} and N ′ {\displaystyle N'} are shown to be normal in G × { e ′ } {\displaystyle G\times \{e'\}} and { e } × G ′ {\displaystyle \{e\}\times G'} , respectively. It is in this sense that N {\displaystyle N} and N ′ {\displaystyle N'} can be identified as normal in G and G' , respectively.
Since p 2 {\displaystyle p_{2}} is a homomorphism , its kernel N is normal in H . Moreover, given g ∈ G {\displaystyle g\in G} , there exists h = ( g , g ′ ) ∈ H {\displaystyle h=(g,g')\in H} , since p 1 {\displaystyle p_{1}} is surjective. Therefore, p 1 ( N ) {\displaystyle p_{1}(N)} is normal in G , viz:
It follows that N {\displaystyle N} is normal in G × { e ′ } {\displaystyle G\times \{e'\}} since
The proof that N ′ {\displaystyle N'} is normal in { e } × G ′ {\displaystyle \{e\}\times G'} proceeds in a similar manner.
Given the identification of G {\displaystyle G} with G × { e ′ } {\displaystyle G\times \{e'\}} , we can write G / N {\displaystyle G/N} and g N {\displaystyle gN} instead of ( G × { e ′ } ) / N {\displaystyle (G\times \{e'\})/N} and ( g , e ′ ) N {\displaystyle (g,e')N} , g ∈ G {\displaystyle g\in G} . Similarly, we can write G ′ / N ′ {\displaystyle G'/N'} and g ′ N ′ {\displaystyle g'N'} , g ′ ∈ G ′ {\displaystyle g'\in G'} .
On to the proof. Consider the map H → G / N × G ′ / N ′ {\displaystyle H\to G/N\times G'/N'} defined by ( g , g ′ ) ↦ ( g N , g ′ N ′ ) {\displaystyle (g,g')\mapsto (gN,g'N')} . The image of H {\displaystyle H} under this map is { ( g N , g ′ N ′ ) ∣ ( g , g ′ ) ∈ H } {\displaystyle \{(gN,g'N')\mid (g,g')\in H\}} . Since H → G / N {\displaystyle H\to G/N} is surjective, this relation is the graph of a well-defined function G / N → G ′ / N ′ {\displaystyle G/N\to G'/N'} provided g 1 N = g 2 N ⟹ g 1 ′ N ′ = g 2 ′ N ′ {\displaystyle g_{1}N=g_{2}N\implies g_{1}'N'=g_{2}'N'} for every ( g 1 , g 1 ′ ) , ( g 2 , g 2 ′ ) ∈ H {\displaystyle (g_{1},g_{1}'),(g_{2},g_{2}')\in H} , essentially an application of the vertical line test .
Since g 1 N = g 2 N {\displaystyle g_{1}N=g_{2}N} (more properly, ( g 1 , e ′ ) N = ( g 2 , e ′ ) N {\displaystyle (g_{1},e')N=(g_{2},e')N} ), we have ( g 2 − 1 g 1 , e ′ ) ∈ N ⊂ H {\displaystyle (g_{2}^{-1}g_{1},e')\in N\subset H} . Thus ( e , g 2 ′ − 1 g 1 ′ ) = ( g 2 , g 2 ′ ) − 1 ( g 1 , g 1 ′ ) ( g 2 − 1 g 1 , e ′ ) − 1 ∈ H {\displaystyle (e,g_{2}'^{-1}g_{1}')=(g_{2},g_{2}')^{-1}(g_{1},g_{1}')(g_{2}^{-1}g_{1},e')^{-1}\in H} , whence ( e , g 2 ′ − 1 g 1 ′ ) ∈ N ′ {\displaystyle (e,g_{2}'^{-1}g_{1}')\in N'} , that is, g 1 ′ N ′ = g 2 ′ N ′ {\displaystyle g_{1}'N'=g_{2}'N'} .
Furthermore, for every ( g 1 , g 1 ′ ) , ( g 2 , g 2 ′ ) ∈ H {\displaystyle (g_{1},g_{1}'),(g_{2},g_{2}')\in H} we have ( g 1 g 2 , g 1 ′ g 2 ′ ) ∈ H {\displaystyle (g_{1}g_{2},g_{1}'g_{2}')\in H} . It follows that this function is a group homomorphism .
By symmetry, { ( g ′ N ′ , g N ) ∣ ( g , g ′ ) ∈ H } {\displaystyle \{(g'N',gN)\mid (g,g')\in H\}} is the graph of a well-defined homomorphism G ′ / N ′ → G / N {\displaystyle G'/N'\to G/N} . These two homomorphisms are clearly inverse to each other and thus are indeed isomorphisms .
As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder – Schreier theorem in Goursat varieties. | https://en.wikipedia.org/wiki/Goursat's_lemma |
The Gouy balance , invented by the French physicist Louis Georges Gouy , is a device for measuring the magnetic susceptibility of a sample. The Gouy balance operates on magnetic torque , by placing the sample on a horizontal arm or beam suspended by a thin fiber, and placing either a permanent magnet or electromagnet on the other end of the arm, there is a magnetic field applied to the system, causing the coil to experience a torque causing the arm to twist or rotate. The angle of rotation can then be calculated.
Amongst a wide range of interest in optics, Brownian motion, and experimental physics, Gouy also had a strong intrigue for the phenomena of magnetism. Gouy derived a mathematical expression showing that force is proportional to volume susceptibility for the interaction of material in a uniform magnetic field in 1889. From this derivation, Gouy proposed that balance measurements taken for tubes of material suspended in a magnetic field could evaluate his expression for volume susceptibility. Though Gouy never tested the scientific suggestion himself, this simple and inexpensive method became the foundation for measuring magnetic susceptibility and the blueprint for the Gouy balance. [ 1 ]
Quincke made note in 1888 that liquid meniscuses within capillaries moved under the influences of magnetic fields, demonstrating that pressure changes may be related to its magnetic propensity. Gouy became interested by this hypothesis and subsequently formulated an interaction expression of materials within cylinder designations in uniform magnetic fields, displaying how the force would be proportional to volume susceptibility. He established his own hypothesis that measurements be made by tube materials being weighed in magnetic fields from a balance. For unknown reasons, he never introduced this concept himself, although it was eventually replicated by others over its simplicity, thus emerging as a regular means to measure magnetic susceptibility.
The Gouy balance measures the apparent change in the mass of the sample as it is repelled or attracted by the region of high magnetic field between the poles. [ 2 ] Some commercially available balances have a port at their base for this application.
In use, a long, cylindrical sample to be tested is suspended from a balance , partially entering between the poles of a magnet . The sample can be in solid or liquid form, and is often placed in a cylindrical container such as a test tube. Solid compounds are generally ground into a fine powder to allow for uniformity within the sample. [ 3 ] The sample is suspended between the magnetic poles through an attached thread or string. [ 2 ] The experimental procedure requires two separate reading to be performed. An initial balance reading is performed on the sample of interest without a magnetic field ( m a ). A subsequent balance reading is taken with an applied magnetic field ( m b ). The difference between these two readings relates to the magnetic force on the sample ( m b – m a ). [ 2 ] [ 4 ]
The apparent change in mass from the two balance readings is a result of magnetic force on the sample. The magnetic force is applied across the gradient of a strong and weak magnetic field. A sample with a paramagnetic compound will be pulled down towards the magnetic, and provide a positive difference in apparent mass m b – m a . Diamagnetic compounds can either exhibit no apparent change in weight or a negative change as the sample is slightly repelled by the applied magnetic field. [ 5 ] With a paramagnetic sample, the magnetic induction is stronger than the applied field and magnetic susceptibility is positive. A diamagnetic sample has a magnetic induction much weaker than the applied field, and a respective negative magnetic susceptibility. [ 6 ] The following mathematical equation relates the apparent change in mass to the volume susceptibility of the sample:
F = ( m b − m a ) g = 1 2 ( K 2 − K 1 ) A H 2 {\displaystyle {\text{F}}=(m_{b}-m_{a})g={\frac {1}{2}}(K_{2}-K_{1})AH^{2}} [ 3 ]
In a practical device, the whole assembly of balance and magnet is enclosed in a glass box to ensure that the weight measurement is not affected by air currents. The sample can also be enclosed in a thermostat in order to make measurements at different temperatures. [ 7 ] Since it requires a large and powerful electromagnet, the Gouy balance is a stationary instrument permanently set up on a bench. [ 2 ] The apparatus is often placed on a marble balance table in a non-ventilated room to minimize the vibrations and disruption from the environment. [ 6 ] The stationary magnetic of a Gouy balance is often an electromagnet connected to a power source, since balance recordings with and without the applied magnetic field are required of the procedure. | https://en.wikipedia.org/wiki/Gouy_balance |
In thermodynamics and thermal physics , the Gouy-Stodola theorem is an important theorem for the quantification of irreversibilities in an open system , and aids in the exergy analysis of thermodynamic processes . It asserts that the rate at which work is lost during a process, or at which exergy is destroyed, is proportional to the rate at which entropy is generated, and that the proportionality coefficient is the temperature of the ambient heat reservoir. [ 1 ] In the literature, the theorem often appears in a slightly modified form, changing the proportionality coefficient. [ 2 ]
The theorem is named jointly after the French physicist Georges Gouy and Slovak physicist Aurel Stodola , who demonstrated the theorem in 1889 and 1905 respectively. [ 3 ] [ 4 ] Gouy used it while working on exergy and utilisable energy, and Stodola while working on steam and gas engines. [ 5 ] [ 6 ] [ 7 ] [ 8 ]
The Gouy-Stodola theorem is often applied upon an open thermodynamic system, which can exchange heat with some thermal reservoirs . It holds both for systems which cannot exchange mass, and systems which mass can enter and leave. [ 2 ] [ 9 ]
Observe such a system, as sketched in the image shown, as it is going through some process . It is in contact with multiple reservoirs, of which one, that at temperature T 0 {\displaystyle T_{0}} , is the environment reservoir. During the process, the system produces work and generates entropy. Under these conditions, the theorem has two general forms.
The reversible work is the maximal useful work which can be obtained, W r e v = W m a x {\displaystyle W_{rev}=W_{max}} , and can only be fully utilized in an ideal reversible process . An irreversible process produces some work W a c t u a l {\displaystyle W_{actual}} , which is less than W r e v {\displaystyle W_{rev}} . The lost work is then W l o s t = W r e v − W a c t u a l {\displaystyle W_{lost}=W_{rev}-W_{actual}} ; in other words, W l o s t {\displaystyle W_{lost}} is the work which was lost or not exploited during the process due to irreversibilities. [ 2 ] [ 10 ] In terms of lost work, the theorem generally states W ˙ l o s t = T 0 S g ˙ {\displaystyle {\dot {W}}_{lost}=T_{0}{\dot {S_{g}}}} where W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} is the rate at which work is lost, and S g ˙ {\displaystyle {\dot {S_{g}}}} is the rate at which entropy is generated. Time derivatives are denoted by dots. The theorem, as stated above, holds only for the entire thermodynamic universe - the system along with its surroundings, together: W ˙ l o s t , t o t = T 0 S ˙ g , t o t {\displaystyle {\dot {W}}_{lost,tot}=T_{0}{\dot {S}}_{g,tot}} where the index "tot" denotes the total quantities produced within or by the entire universe.
Note that W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} is a relative quantity, in that it is measured in relation to a specific thermal reservoir. In the above equations, W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} is defined in reference to the environment reservoir, at T 0 {\displaystyle T_{0}} . When comparing the actual process to an ideal, reversible process between the same endpoints (in order to evaluate W r e v {\displaystyle W_{rev}} , so as to find the value of W l o s t {\displaystyle W_{lost}} ), only the heat interaction with the reference reservoir T 0 {\displaystyle T_{0}} is allowed to vary. The heat interactions between the system and other reservoirs are kept the same. So, if a different reference reservoir T r e f {\displaystyle T_{ref}} is chosen, the theorem would read W ˙ l o s t , t o t = T r e f S ˙ g , t o t {\displaystyle {\dot {W}}_{lost,tot}=T_{ref}{\dot {S}}_{g,tot}} , where this time W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} is in relation to T r e f {\displaystyle T_{ref}} , and in the corresponding reversible process, only the heat interaction with T r e f {\displaystyle T_{ref}} is different. [ 2 ]
By integrating over the lifetime of the process, the theorem can also be expressed in terms of final quantities, rather than rates: W l o s t , t o t = T 0 S g , t o t {\displaystyle {W}_{lost,tot}=T_{0}{S}_{g,tot}} . [ 10 ]
The theorem also holds for adiabatic processes . That is, for closed systems, which are not in thermal contact with any heat reservoirs.
Similarly to the non-adiabatic case, the lost work is measured relative to some reference reservoir T 0 {\displaystyle T_{0}} . Even though the process itself is adiabatic, the corresponding reversible process may not be, and might require heat exchange with the reference reservoir. Thus, this can be thought of as a special case of the above statement of the theorem - an adiabatic process is one for which the heat interactions with all reservoirs are zero, and in the reversible process, only the heat interaction with the reference thermal reservoir may be different. [ 2 ] [ 9 ]
The adiabatic case of the theorem holds also for the other formulation of the theorem, presented below.
The exergy of the system is the maximal amount of useful work that the system can generate, during a process which brings it to equilibrium with its environment, or the amount of energy available. During an irreversible process , such as heat exchanges with reservoirs, exergy is destroyed. Generally, the theorem states that ψ ˙ d = T 0 S g ˙ {\displaystyle {\dot {\psi }}_{d}=T_{0}{\dot {S_{g}}}} where ψ ˙ d {\displaystyle {\dot {\psi }}_{d}} is the rate at which exergy is destroyed, and S g ˙ {\displaystyle {\dot {S_{g}}}} is the rate at which entropy is generated. [ 2 ] [ 9 ] As above, time derivatives are denoted by dots.
Unlike the lost work formulation, this version of the theorem holds for both the system (the control volume) and for its surroundings (the environment and the thermal reservoirs) separately: ψ ˙ d , s y s = T 0 S ˙ g , s y s {\displaystyle {\dot {\psi }}_{d,sys}=T_{0}{\dot {S}}_{g,sys}} and ψ ˙ d , s u r r = T 0 S ˙ g , s u r r {\displaystyle {\dot {\psi }}_{d,surr}=T_{0}{\dot {S}}_{g,surr}} where the index "sys" denotes quantities produced within or by the system itself, and "surr" within or by the surroundings. Therefore, summing these two forms, the theorem also holds for the thermodynamic universe as a whole: ψ ˙ d , t o t = ψ ˙ d , s y s + ψ ˙ d , s u r r = T 0 S ˙ g , s y s + T 0 S ˙ g , s u r r = T 0 S ˙ g , t o t {\displaystyle {\dot {\psi }}_{d,tot}={\dot {\psi }}_{d,sys}+{\dot {\psi }}_{d,surr}=T_{0}{\dot {S}}_{g,sys}+T_{0}{\dot {S}}_{g,surr}=T_{0}{\dot {S}}_{g,tot}} where the index "tot" denotes the total quantities of the entire universe.
Thus, the exergy formulation of the theorem is less limited, as it can be applied on different regions separately. Nevertheless, the work form is used more often.
The proof of the theorem, in both forms, uses the first law of thermodynamics , writing out the terms W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} , ψ ˙ d {\displaystyle {\dot {\psi }}_{d}} , and S g ˙ {\displaystyle {\dot {S_{g}}}} in the relevant regions, and comparing them.
In many cases, it is preferable to use a slightly modified version of the Gouy-Stodola theorem in work form, where T 0 {\displaystyle T_{0}} is replaced by some effective temperature. When this is done, it often enlarges the scope of the theorem, and adapts it to be applicable to more systems or situations. For example, the corrections elaborated below are only necessary when the system exchanges heat with more than one reservoir - if it exchanges heat only at the environmental temperature T 0 {\displaystyle T_{0}} , the simple form above holds true. [ 11 ] Additionally, modifications may change the reversible process to which the real process is compared in calculating W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} .
The modified theorem then reads W ˙ l o s t = T e f f S g ˙ {\displaystyle {\dot {W}}_{lost}=T_{eff}{\dot {S_{g}}}} where T e f f {\displaystyle T_{eff}} is the effective temperature.
For a flow process, let s 1 {\displaystyle s_{1}} denote the specific entropy (entropy per unit mass) at the inlet, where mass flows in, and s 2 {\displaystyle s_{2}} the specific entropy at the outlet, where mass flows out. Similarly, denote the specific enthalpies by h 1 {\displaystyle h_{1}} and h 2 {\displaystyle h_{2}} . The inlet and outlet, in this case, function as initial and final states a process: mass enters the system at an initial state (the inlet, indexed "1"), undergoes some process, and then leaves at a final state (the outlet, indexed "2").
This process is then compared to a reversible process, with the same initial state, but with a (possibly) different final state. The theoretical specific entropy and enthalpy after this ideal, isentropic process are given by s 2 , r e v {\displaystyle s_{2,rev}} and h 2 , r e v {\displaystyle h_{2,rev}} , respectively. When the actual process is compared to this theoretical reversible process and W ˙ l o s t {\displaystyle {\dot {W}}_{lost}} is evaluated, the proper effective temperature is given by T e f f = h 2 − h 2 , r e v s 2 − s 2 , r e v {\displaystyle T_{eff}={\frac {h_{2}-h_{2,rev}}{s_{2}-s_{2,rev}}}} In general, T e f f {\displaystyle T_{eff}} lies somewhere in between the final temperature in the actual process T 2 {\displaystyle T_{2}} and the final temperature in the theoretical reversible process T 2 , r e v {\displaystyle T_{2,rev}} . [ 1 ] [ 2 ] [ 11 ]
This equation above can sometimes be simplified. If both the pressure and the specific heat capacity remain constant, then the changes in enthalpy and entropy can be written in terms of the temperatures, and [ 2 ] [ 11 ] [ 12 ] [ 13 ] T e f f = T 2 − T 2 , r e v ln T 2 − ln T 2 , r e v = T 2 − T 2 , r e v ln T 2 T 2 , r e v {\displaystyle T_{eff}={\frac {T_{2}-T_{2,rev}}{\ln T_{2}-\ln T_{2,rev}}}={\frac {T_{2}-T_{2,rev}}{\ln {\frac {T_{2}}{T_{2,rev}}}}}} However, it is important to note that this version of the theorem doesn't relate the exact values which the original theorem does. Specifically, in comparing the actual process to a reversible one, the modified version allows the final state to be different between the two. This is in contrast to the original version, wherein reversible process is constructed to match so that the final states are the same. [ 2 ] [ 11 ]
In general, the Gouy-Stodola theorem is used to quantify irreversibilities in a system and to perform exergy analysis. That is, it allows one to take a thermodynamic system and better understand how inefficient it is (energy-wise), how much work is lost, how much room there is for improvement and where. The second law of thermodynamics states, in essence, that the entropy of a system only increases. Over time, thermodynamic systems tend to gain entropy and lose energy (in approaching equilibrium): thus, the entropy is "somehow" related to how much exergy or potential for useful work a system has. The Gouy-Stodola theorem provides a concrete link. For the most part, this is how the theorem is used - to find and quantify inefficiencies in a system.
A flow process is a type of thermodynamic process , where matter flows in and out of an open system called the control volume . Such a process may be steady , meaning that the matter and energy flowing into and out of the system are constant through time. It can also be unsteady, or transient, meaning that the flows may change and differ at different times.
Many proofs of the theorem demonstrate it specifically for flow systems. Thus, the theorem is particularly useful in performing exergy analysis on such systems. [ 2 ] [ 14 ] [ 15 ]
The Gouy-Stodola theorem is often applied to refrigeration cycles . These are thermodynamic cycles or mechanical systems where external work can be used to move heat from low temperature sources to high temperature sinks, or vice versa. Specifically, the theorem is useful in analyzing vapor compression and vapor absorption refrigeration cycles.
The theorem can help identify which components of a system have major irreversibilities, and how much exergy they destroy. It can be used to find at which temperatures the performance is optimal, or what size system should be constructed. Overall, that is, the Gouy-Stodola theorem is a tool to find and quantify inefficiencies in a system, and can point to how to minimize them - this is the goal of exergy analysis. When the theorem is used for these purposes, it is usually applied in its modified form. [ 11 ] [ 12 ] [ 13 ] [ 16 ] [ 17 ]
Macroscopically, the theorem may be useful environmentally, in ecophysics. An ecosystem is a complex system, where many factors and components interact, some biotic and some abiotic. The Gouy-Stodola theorem can find how much entropy is generated by each part of the system, or how much work is lost. Where there is human interference in an ecosystem, whether the ecosystem continues to exist or is lost may depend on how many irreversibilities it can support. The amount of entropy which is generated or the amount of work the system can perform may vary. Hence, two different states (for example, a healthy forest versus one which has undergone significant deforestation) of the same ecosystem may be compared in terms of entropy generation, and this may be used to evaluate the sustainability of the ecosystem under human interference. [ 18 ] [ 19 ]
The theorem is also useful on a more microscopic scale, in biology . Living systems, such as cells , can be analyzed thermodynamically. They are rather complex systems, where many energy transformations occur, and they often waste heat. Hence, the Gouy-Stodola theorem may be useful, in certain situations, to perform exergy analysis on such systems. In particular, it may help to highlight differences between healthy and diseased cells.
Generally, the theorem may find applications in fields of biomedicine , or where biology and physics cross over, such as biochemical engineering thermodynamics . [ 3 ] [ 20 ]
A variational principle in physics, such as the principle of least action or Fermat's principle in optics, allows one to describe the system in a global manner and to solve it using the calculus of variations . In thermodynamics, such a principle would allow a Lagrangian formulation. The Gouy-Stodola theorem can be used as the basis for such a variational principle, in thermodynamics. It has been proven to satisfy the necessary conditions. [ 3 ] [ 4 ] [ 10 ]
This is fundamentally different from most of the theorem's other uses - here, it isn't being applied in order to locate components with irreversibilities or loss of exergy, but rather helps give some more general information about the system. | https://en.wikipedia.org/wiki/Gouy–Stodola_theorem |
govdex is an Australian government initiative designed to facilitate business process collaboration across policy portfolios, administrative jurisdictions and agencies. The service is designed to promote effective and efficient information sharing, providing governance, tools, methods and re-usable technical components across Australian government. [ 1 ]
govdex is managed by the Australian Government Information Management Office (AGIMO), in the Department of Finance and Deregulation (Australia) .
A major component of govdex is a wiki -based framework and other collaboration tools. These tools enable Australian Government entities to establish online communities of practice and manage collaborative initiatives across government and non-government stakeholders.
The primary govdex principles/tenets are: [ 2 ]
The key components of govdex include: [ 2 ]
govdex does not include an integrated email solution, such as that found in Yahoo Groups. It does not provide an automatic update to members following a post to a specific group, or entry on a specific discussion board. | https://en.wikipedia.org/wiki/Govdex |
The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables ) change when one or more of the known (i.e. independent ) variables change.
Physical systems can be modeled phenomenologically at various levels of sophistication, with each level capturing a different degree of detail about the system. A governing equation represents the most detailed and fundamental phenomenological model currently available for a given system.
For example, at the coarsest level, a beam is just a 1D curve whose torque is a function of local curvature. At a more refined level , the beam is a 2D body whose stress-tensor is a function of local strain-tensor, and strain-tensor is a function of its deformation. The equations are then a PDE system. Note that both levels of sophistication are phenomenological, but one is deeper than the other. As another example, in fluid dynamics, the Navier-Stokes equations are more refined than Euler equations .
As the field progresses and our understanding of the underlying mechanisms deepens, governing equations may be replaced or refined by new, more accurate models that better represent the system's behavior. These new governing equations can then be considered the deepest level of phenomenological model at that point in time.
A mass balance , also called a material balance , is an application of conservation of mass to the analysis of physical systems. It is the simplest governing equation, and it is simply a budget (balance calculation) over the quantity in question:
The governing equations [ 1 ] [ 2 ] in classical physics that are
lectured [ 3 ] [ 4 ] [ 5 ] [ 6 ] at universities are listed below.
The basic equations in classical continuum mechanics are all balance equations , and as such each of them contains a time-derivative term which calculates how much the dependent variable change with time. For an isolated, frictionless / inviscid system the first four equations are the familiar conservation equations in classical mechanics.
Darcy's law of groundwater flow has the form of a volumetric flux caused by a pressure gradient. A flux in classical mechanics is normally not a governing equation, but usually a defining equation for transport properties . Darcy's law was originally established as an empirical equation, but is later shown to be derivable as an approximation of Navier-Stokes equation combined with an empirical composite friction force term. This explains the duality in Darcy's law as a governing equation and a defining equation for absolute permeability.
The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equations in classical mechanics exposed to establishing of simpler approximations.
Some examples of governing differential equations in classical continuum mechanics are
A famous example of governing differential equations within biology is
A governing equation may also be a state equation , an equation describing the state of the system, and thus actually be a constitutive equation that has "stepped up the ranks" because the model in question was not meant to include a time-dependent term in the equation. This is the case for a model of an oil production plant which on the average operates in a steady state mode. Results from one thermodynamic equilibrium calculation are input data to the next equilibrium calculation together with some new state parameters, and so on. In this case the algorithm and sequence of input data form a chain of actions, or calculations, that describes change of states from the first state (based solely on input data) to the last state that finally comes out of the calculation sequence. | https://en.wikipedia.org/wiki/Governing_equation |
The United States Government Accountability Office ( GAO ) is an independent, nonpartisan government agency within the legislative branch that provides auditing , evaluative , and investigative services for the United States Congress . [ 2 ] It is the supreme audit institution of the federal government of the United States . It identifies its core "mission values" as: accountability, integrity, and reliability. [ 3 ] It is also known as the "congressional watchdog". [ 4 ] The agency is headed by the Comptroller General of the United States . The comptroller general is appointed by the president with the advice and consent of the Senate. When a vacancy occurs in the office of the comptroller general, Congress establishes a commission to recommend individuals to the president. [ 5 ] The commission consists of the following:
The commission must recommend at least three individuals to the president, and the president may request that the commission recommend additional individuals. The president then selects an individual from those recommended to nominate as the new comptroller general. The president's nomination must be confirmed by the Senate's Committee on Homeland Security & Governmental Affairs before being voted on by the full Senate. [ 6 ]
The current comptroller general is Gene Dodaro , who has served in the position since March 13, 2008. [ 5 ]
The work of the GAO is done at the request of congressional committees or subcommittees or is mandated by public laws or committee reports. It also undertakes research under the authority of the comptroller general. It supports congressional oversight by:
As a result of its work, GAO produces:
The GAO also produces special publications on specific issues of general interest to many Americans, such as its report on the fiscal future of the United States, GAO's role in the federal bid protest process, and critical issues for congressional consideration related to improving the nation's image abroad.
The GAO is headquartered in Washington, D.C. and maintains an additional 11 field offices around the country. Each field office contains several mission teams, but not every mission team is represented at each field office.
The GAO is composed of 15 mission teams that work on reports in a given subject area. [ 7 ] Missions teams are headed by a Managing Director which fall under the Senior Executive Service . The current slate of mission teams is:
In addition to its mission teams, the GAO also has 16 operations and staff components that support their work and carryout other agency functions, including its bid decisions.
The GAO was established as the General Accounting Office by the Budget and Accounting Act of 1921. The act required the head of the GAO to:
investigate, at the seat of government or elsewhere, all matters relating to the receipt, disbursement, and application of public funds, and shall make to the President ... and to Congress ... reports [and] recommendations looking to greater economy or efficiency in public expenditures. [ 8 ]
According to the GAO's current mission statement, the agency exists to support the Congress in meeting its constitutional responsibilities and to help improve the performance and ensure the accountability of the federal government for the benefit of the American people.
The name was changed in 2004 to the Government Accountability Office by the GAO Human Capital Reform Act to better reflect the mission of the office. [ 9 ] [ 10 ] [ 11 ] The GAO's auditors conduct not only financial audits, but also engage in a wide assortment of performance audits.
Over the years, the GAO has been referred to as "The Congressional Watchdog" and "The Taxpayers' Best Friend" for its frequent audits and investigative reports that have uncovered waste and inefficiency in government. News media often draw attention to the GAO's work by publishing stories on the findings, conclusions, and recommendations of its reports. Members of Congress also frequently cite the GAO's work in statements to the press, congressional hearings, and floor debates on proposed legislation. In 2007 the Partnership for Public Service ranked the GAO second on its list of the best places to work in the federal government and Washingtonian magazine included the GAO on its 2007 list of great places to work in Washington, a list that encompasses the public, private, and non-profit sectors.
The GAO is headed by the comptroller general of the U.S. , a professional and non-partisan position in the U.S. government. The comptroller general is appointed by the president , by and with the advice and consent of the Senate , for a fifteen-year, non-renewable term. The president selects a nominee from a list of at least three individuals recommended by an eight-member bipartisan, bicameral commission of congressional leaders. During such a term, the comptroller general has standing to pursue litigation to compel access to federal agency information. The comptroller general may not be removed by the president, but only by Congress through impeachment or joint resolution for specific reasons. [ 12 ] Since 1921, there have been only eight comptrollers general, and no formal attempt has ever been made to remove a comptroller general.
Labor-management relations became fractious during the nine-year tenure of the seventh comptroller general, David M. Walker . On September 19, 2007, GAO analysts voted by a margin of two to one (897–445), in a 75% turnout, to establish the first union in the GAO's 86-year history. The analysts voted to affiliate with the International Federation of Professional and Technical Engineers (IFPTE), a member union of the AFL–CIO . There are more than 1,800 analysts in the GAO analysts bargaining unit; the local voted to name itself IFPTE Local 1921, in honor of the date of the GAO's establishment. On February 14, 2008, the GAO analysts' union approved its first-ever negotiated pay contract with management; of just over 1,200 votes, 98% were in favor of the contract. [ 13 ]
The GAO also establishes standards for audits of government organizations, programs, activities, and functions, and of government assistance received by contractors, nonprofit organizations, and other nongovernmental organizations. These standards, often referred to as Generally Accepted Government Auditing Standards (GAGAS), are to be followed by auditors and audit organizations when required by law, regulation, agreement, contract, or policy. These standards pertain to auditors' professional qualifications, the quality of audit effort, and the characteristics of professional and meaningful audit reports.
In 1992, the GAO hosted the XIV INCOSAI , the fourteenth triennial convention of the International Organization of Supreme Audit Institutions (INTOSAI). [ 14 ]
The GAO is a United States government electronic data provider, as all of its reports are available on its website, [ 15 ] except for certain reports whose distribution is limited to official use in order to protect national security. [ citation needed ] The variety of their reports' topics range from Federal Budget and Fiscal Issues to Financial Management, Education, Retirement Issues, Defense, Homeland Security, Administration of Justice, Health Care, Information Management and Technology, Natural Resources, Environment, International Affairs, Trade, Financial Markets, Housing, Government Management and Human Capital, and Science and Technology Assessments and Analytics. The GAO often produces highlights of its reports that serve as a statement for the record for various subcommittees of the United States Congress.
Most GAO studies and reports are initiated by requests from members of Congress, including requests mandated in statute, and so reflect concerns of current political import, for example to study the impact of a government-wide hiring freeze. [ 16 ] Many reports are issued periodically and take a long view of US agencies' operations. [ citation needed ] The GAO also produces annual reports on key issues [ 17 ] including duplication and cost savings [ 18 ] and High-Risk Update. [ 19 ]
The GAO prepares some 900 reports annually. [ 20 ] The GAO publishes reports and information relating to, inter alia :
Each year the GAO issues an audit report on the financial statements of the United States Government. The 2010 Financial Report of the United States Government was released on December 21, 2010. [ 21 ] The accompanying press release states that the GAO 'cannot render an opinion on the 2010 consolidated financial statements of the federal government , because of widespread material internal control weaknesses, significant uncertainties, and other limitations'. [ 21 ]
As part of its initiative to advocate sustainability , the GAO publishes a Federal Fiscal Outlook Report, [ 22 ] as well as data relating to the deficit . [ 23 ] The U.S. deficit is presented on a cash rather than accruals basis, although the GAO notes that the accrual deficit "provides more information on the longer-term implications of the government's annual operations". [ 23 ] In FY 2010, the US federal government had a net operating cost of $2,080 billion, although since this includes accounting provisions (estimates of future liabilities), the cash deficit is $1,294 billion. [ 24 ]
The most recent GAO strategic plan, for 2018–2023, sets out four goals, namely: [ 25 ]
The Forensic Audits and Investigative Service (FAIS) team provides Congress with high-quality forensic audits and investigations of fraud, waste, and abuse; other special investigations; and security and vulnerability assessments. Its work cuts across a diverse array of government programs administered by the IRS, the Centers for Medicare and Medicaid Services, the Department of Veterans Affairs, and the Department of Homeland Security, among others. In April 2024, the GAO published a report entitled "Fraud Risk Management," which concluded that between the years 2018 to 2024, the U.S. federal government lost an estimated $233 billion to $521 billion annually due to fraud. [ 26 ]
Unsuccessful bidders for government contracts may submit protests if they have reason to challenge an agency's decision, and the GAO may then release a report on the decision, redacted if necessary. Various GAO decisions have confirmed that:
In reviewing protests of an agency’s evaluation, [GAO] does not reevaluate proposals, rather, we review the evaluation to determine if it was reasonable, consistent with the solicitation’s evaluation scheme and procurement statutes and regulations, and adequately documented. [ 27 ]
There is a facility within the Bid Protest Regulations for the GAO to recommend reimbursement of a bidder's protest costs if the procuring agency takes corrective action in response to a protest. The circumstances justifying bid protest cost reimbursement must involve "undue delay" by the agency in "taking corrective action in the face of a clearly meritorious protest". [ 28 ]
The GAO confirmed in 2014 that its jurisdiction includes investigation of protests raising allegations of agency violation of the Procurement Integrity Act . [ 29 ]
After the closing of the Office of Technology Assessment (OTA) in 1995, Congress directed the GAO to conduct a technology assessment (TA) pilot program. Between 2002 and 2005, three reports were completed: use of biometrics for border security, [ 30 ] cyber security for critical infrastructure protection , [ 31 ] and technologies for protecting structures in wildland fires. [ 32 ] The GAO reports and technology assessments, which are made available to the public, have become essential vehicles for understanding science and technology (S&T) implications of policies considered by the Congress.
Since 2008, Congress has established a permanent TA function within the GAO. This new operational role augments GAO's performance audits related to S&T issues, including effectiveness and efficiency of U.S. federal programs. In 2010, the GAO joined the European Parliamentary Technology Assessment (EPTA) as an associate member. [ citation needed ] In 2019, the GAO established a new mission team, the Science, Technology Assessment, and Analytics team, which has primary responsibility for technology assessments. [ 33 ]
The GAO has published a TA Design Handbook to help technology assessment teams analyze the impact of technology and make complex issues more easily understood and useful to policymakers. [ 34 ] The GAO defines TA as the "thorough and balanced analysis of significant primary, secondary, indirect, and delayed interactions of a technological innovation with society, the environment, and the economy and the present and foreseen consequences and impacts of those interactions." [ 35 ] Recognizing that the effects of those interactions can have implications, the GAO has in some of its products included policy options. The Technology Assessment section of its website lists GAO's public TA reports. [ 36 ] | https://en.wikipedia.org/wiki/Government_Accountability_Office |
Government Institute of Ceramic Technology is statewide institution in Andhra Pradesh. It is located in Gudur in Tirupati Dist. It is established in 1952. Government Institute of Ceramic Technology is an autonomous institute offering Diploma in Ceramic Technology that cater to the changing needs of industry, business and community at large using need based curricular delivered in a dynamic learning environment. AICTE approved full-time programs are offered to candidates selected as per POLYCET conducted by the government of Andhra Pradesh. The polytechnic also maintains relations with accreditations bodies like All India Council for Technical Education (AICTE) and State Board of Technical Education (SBTETAP).
It is located in Malavya Nagar in Gudur, Tirupati Dist. It has a 10 acres of land with hostel facilities to students. It is offering only Diploma in Ceramic Technology. It is a sandwich course . One year implant training will be provided. It is a three and half years diploma course. The intake of the course is 60. The students are admitted through AP POLYCET entrance exam and seats are allotted through AP POLYCET counselling
Cultural activities are conducted in the second semester of the year for annual college day function. Various sports and games are also held within the campus ground and the institute actively takes part in Inter Polytechnic Sports and Games Meet (IPSGM) every year. | https://en.wikipedia.org/wiki/Government_Institute_of_Ceramic_Technology |
Early research and development:
Merging the networks and creating the Internet:
Commercialization, privatization, broader access leads to the modern Internet:
Examples of Internet services:
The Government Open Systems Interconnection Profile ( GOSIP ) was a specification that profiled open networking products for procurement by governments in the late 1980s and early 1990s.
In practice, from 1995 interest in OSI implementations declined, and worldwide the deployment of standards-based networking services since have been predominantly based on the Internet protocol suite . [ 7 ] However, the Defense Messaging System continued to be based on the OSI protocols X.400 and X.500 , due to their integrated security capabilities. | https://en.wikipedia.org/wiki/Government_Open_Systems_Interconnection_Profile |
In statistics , Gower's distance between two mixed-type objects is a similarity measure that can handle different types of data within the same dataset and is particularly useful in cluster analysis or other multivariate statistical techniques. Data can be binary, ordinal, or continuous variables. It works by normalizing the differences between each pair of variables and then computing a weighted average of these differences. The distance was defined in 1971 by Gower [ 1 ] and it takes values between 0 and 1 with smaller values indicating higher similarity.
For two objects i {\displaystyle i} and j {\displaystyle j} having p {\displaystyle p} descriptors, the similarity S {\displaystyle S} is defined as: S i j = ∑ k = 1 p w i j k s i j k ∑ k = 1 p w i j k , {\displaystyle S_{ij}={\frac {\sum _{k=1}^{p}w_{ijk}s_{ijk}}{\sum _{k=1}^{p}w_{ijk}}},}
where the w i j k {\displaystyle w_{ijk}} are non-negative weights usually set to 1 {\displaystyle 1} [ 2 ] and s i j k {\displaystyle s_{ijk}} is the similarity between the two objects regarding their k {\displaystyle k} -th variable. If the variable is binary or ordinal, the values of s i j k {\displaystyle s_{ijk}} are 0 or 1, with 1 denoting equality. If the variable is continuous, s i j k = 1 − | x i − x j | R k {\displaystyle s_{ijk}=1-{\frac {|x_{i}-x_{j}|}{R_{k}}}} with R k {\displaystyle R_{k}} being the range of k {\displaystyle k} -th variable and thus ensuring 0 ≤ s i j k ≤ 1 {\displaystyle 0\leq s_{ijk}\leq 1} . As a result, the overall similarity S i j {\displaystyle S_{ij}} between two objects is the weighted average of the
similarities calculated for all their descriptors. [ 3 ]
In its original exposition, the distance does not treat ordinal variables in a special manner. In the 1990s, first Kaufman and Rousseeuw [ 4 ] and later Podani [ 5 ] suggested extensions where the ordering of an ordinal feature is used. For example, Podani obtains relative rank differences as s i j k = 1 − | r i − r j | max { r } − min { r } {\displaystyle s_{ijk}=1-{\frac {|r_{i}-r_{j}|}{\max {\{r\}}-\min {\{r\}}}}} with r {\displaystyle r} being the ranks corresponding to the ordered categories of the k {\displaystyle k} -th variable.
Many programming languages and statistical packages, such as R , Python , etc., include implementations of Gower's distance. The implementations may follow Kaufmann and Rousseeuw's extensions, which change the similarity for continuous variables to s i j k = | x i − x j | R k {\displaystyle s_{ijk}={\frac {|x_{i}-x_{j}|}{R_{k}}}} [ 6 ] | https://en.wikipedia.org/wiki/Gower's_distance |
gpsOne is the brand name for a cellphone chipset manufactured by Qualcomm for mobile phone tracking . It uses A-GPS or Assisted-GPS to locate the phone more quickly, accurately and reliably than by GPS alone, especially in places with poor GPS reception.
Some vendors are also looking at GPS phone technology as a method of implementing location-based solutions, such as:
gpsOne can operate in four modes:
Since introduction in 2000, the gpsOne chipset has been adopted by 40+ vendors, and is used in more than 250 cellphone models worldwide.
More than 300 million gpsOne enabled handsets are currently on the market, making it one of the most widely deployed solutions.
Product website Archived 2011-05-27 at the Wayback Machine
The gpsOne XTRA MSB assistance data format: | https://en.wikipedia.org/wiki/GpsOne |
gpsd is a computer software program that collects data from a Global Positioning System (GPS) receiver and provides the data via an Internet Protocol (IP) network to potentially multiple client applications in a server-client application architecture. Gpsd may be run as a daemon to operate transparently as a background task of the server. The network interface provides a standardized data format for multiple concurrent client applications, such as Kismet or GPS navigation software .
Gpsd is commonly used on Unix-like operating systems. [ 2 ] [ 3 ] [ 4 ] It is distributed as free software under the 2-clause BSD license .
gpsd provides a TCP/IP service by binding to port 2947 by default. [ 5 ] It communicates via that socket by accepting commands, and returning results. These commands use a JSON -based syntax and provide JSON responses. [ 5 ] Multiple clients can access the service concurrently.
The application supports many types of GPS receivers with connections via serial ports , USB , and Bluetooth . Starting in 2009, gpsd also supports AIS receivers. [ 6 ]
gpsd supports interfacing with the Network Time Protocol (NTP) server ntpd via shared memory to enable setting the host platform's time via the GPS clock.
gpsd was originally written by Remco Treffkorn with Derrick Brashear, then maintained by Russell Nelson . [ 7 ] It is now maintained by Eric S. Raymond . [ 8 ] [ 9 ] | https://en.wikipedia.org/wiki/Gpsd |
Grabyo is a browser-based live video production suite integrated with other social media platforms such as Facebook, YouTube, Instagram, Snapchat, Twitter, and Periscope. Sports federations and media companies use cloud-based technology to produce professional-quality live streams and video clips for digital audiences.
Founded in 2013, the company produces and distributes live shows (such as sports or music events) and video clips (such as pre-match warm-ups, behind-the-scene activities, and instant highlights). It is used to build digital fan bases, drive TV audiences and generate revenue from third-party sponsors and pay-TV subscriptions. Its customers include major sports rights owners and media companies such as La Liga , [ 1 ] NHL , [ 2 ] Eurosport , [ 3 ] Sky Sports , [ 4 ] FIFA World Cup , [ 5 ] FIA Formula E Championship , The Championships, Wimbledon , [ 6 ] the Premier League and Real Madrid C.F. [ 7 ]
Grabyo ranked 77th in the Financial Times' FT 1000 Europe's Fastest Growing Companies 2018. [ 8 ]
The company's investors include Oliver Slipper , Nicole Junkermann , Cesc Fàbregas , Thierry Henry , Robin van Persie and Tony Parker . [ 9 ] | https://en.wikipedia.org/wiki/Grabyo |
Various definitions, see text
Gracilicutes ( Latin : gracilis , slender, and cutis , skin, referring to the cell wall) is a clade in bacterial phylogeny. [ 2 ]
Traditionally gram staining results were most commonly used as a classification tool, consequently until the advent of molecular phylogeny , the Kingdom Monera (as the domains Bacteria and Archaea were known then) was divided into four phyla , [ 1 ] [ 3 ]
This classification system was abandoned in favour of the three-domain system based on molecular phylogeny started by C. Woese. [ 5 ] [ 6 ]
Using hand-drawn schematics rather than standard molecular phylogenetic analysis, Gracilicutes was revived in 2006 by Cavalier-Smith as an infrakindgom containing the phyla Spirochaetota , Sphingobacteria (FCB), Planctobacteria (PVC), and Proteobacteria . [ 7 ] It is a gram-negative clade that branched off from other bacteria just before the evolutionary loss of the outer membrane or capsule , and just after the evolution of flagella . [ 7 ] Most notably, this author assumed an unconventional tree of life placing Chloroflexota near the origin of life and Archaea as a close relative of Actinomycetota . This taxon is not generally accepted and the three-domain system is followed. [ 8 ]
A taxon called Hydrobacteria was defined in 2009 from a molecular phylogenetic analysis of core genes. It is in contrast to the other major group of eubacteria called Terrabacteria . [ 9 ] Some researchers have used the name Gracilicutes in place of Hydrobacteria , but this does not agree with the original description of Gracilicutes by Gibbons and Murray, noted above, which included cyanobacteria and did not follow the three-domain system . Also as noted above, the use of Gracilicutes by Cavalier-Smith can be rejected because it was a major alteration of an earlier taxonomic name, was not based on a statistical analysis, and did not follow the three-domain system . The most recent genomic analyses have supported the division of Bacteria into two major superphyla, corresponding to Terrabacteria and Hydrobacteria . [ 10 ] [ 11 ]
The phylogenetic tree according to the phylogenetic analyzes of Battistuzzi and Hedges (2009) is the following and with a molecular clock calibration. [ 9 ]
Recent phylogenetic analyzes have found that proteobacteria are a paraphyletic phylum that could encompass several recently discovered candidate phyla and other phyla such as Acidobacteriota , Chrysiogenota , Deferribacterota , and possibly Aquificota . This suggests that Gracilicutes or Hydrobacteria as a clade may comprise several candidates more closely related to Proteobacteria, Spirochaetes, PVC group, and FCB group than to bacteria from the clade Terrabacteria . Some of these phyla were classified as part of the proteobacteria. For example, Cavalier-Smith in his proposal of the 6 kingdoms included Acidobacteriota , Aquificota , Chrysiogenota , and Deferribacterota as part of the proteobacteria. [ 7 ]
Phylogenetic analyzes have found roughly the following phylogeny between the major and some more closely related phyla. [ 12 ] [ 13 ] [ 14 ] [ 15 ]
Fusobacteriota (sometimes included in Terrabacteria .)
Spirochaetota
Poribacteria
FCB /" Sphingobacteria "
PVC /" Planctobacteria "
Elusimicrobiota
Aerophobetes
Rokubacteria
Acidobacteriota
Aminicenantes
Dadabacteria
Thermodesulfobacteriota
Myxococcota
Modulibacteria
Tectomicrobia
Nitrospinota
Nitrospirota
Chrysiogenota
Deferribacterota
Dependentiae
Campylobacterota
Aquificota (sometimes included in Terrabacteria.)
Alphaproteobacteria
Zetaproteobacteria
Acidithiobacillia
Betaproteobacteria
Gammaproteobacteria
According to the phylogenetic analysis of Hug (2016), the relationships could be the following. [ 16 ]
The following graph shows Cavalier-Smith 's version of the tree of life, indicating the status of Gracilicutes. However, this tree is not supported by any molecular analysis so it should not be considered phylogenetic.
Cavalier-Smith's Tree of Life, 2006 [ cstol 1 ]
Chlorobacteria
Hadobacteria
Cyanobacteria
Spirochaetae
Sphingobacteria (FCB)
Planctobacteria (PVC)
Proteobacteria s.l.
Eurybacteria
Endobacteria (Bacillota)
Actinobacteria
Archaea
Eukarya
Legend: [A] Gram-negative with a peptidoglycan cell wall like Chlorosome . [B] Oxygenic Photosynthesis , Omp85 and four new catalases . [C] Glycobacterial revolution: outer membrane with insertion of lipopolysaccharides , hopanoids , diaminopimelic acid , ToIC and TonB . [D] Phycobilin chromophores . [E] Flagella . [F] Four sections: an amino acid in HSP60 and FtsZ and a domain in RNA polymerases β and σ. [G] Endospores . [H] Gram-positive Bacteria: hypertrophy of the wall peptidoglycan , sortase enzyme and a loss of the outer membrane. [I] Glycerol 1-P dehydrogenase . [J] Proteasome and phosphatidylinositol . [K] Neomura revolution: Replacement of peptidoglycan by glycoproteins and lipoproteins . [L] Reverse DNA gyrase and ether lipid isoprenoids . [M] Phagocytosis . | https://en.wikipedia.org/wiki/Gracilicutes |
Gracility is slenderness, the condition of being gracile , which means slender. It derives from the Latin adjective gracilis ( masculine or feminine ), or gracile ( neuter ), [ 1 ] which in either form means slender, and when transferred for example to discourse takes the sense of "without ornament", "simple" or various similar connotations . [ 2 ]
In Glossary of Botanic Terms , B. D. Jackson speaks dismissively [ 3 ] of an entry in earlier dictionary of A. A. Crozier [ 4 ] as follows: "Gracilis (Lat.), slender. Crozier has the needless word 'gracile'". However, his objection would be hard to sustain in current usage; apart from the fact that gracile is a natural and convenient term, it is hardly a neologism . The Shorter Oxford English Dictionary [ 5 ] gives the source date for that usage as 1623 and indicates the word is misused (through association with grace ) for "gracefully slender". [ 5 ] This misuse is unfortunate at least, because the terms gracile and grace are unrelated: the etymological root of grace is the Latin word gratia from gratus , meaning 'pleasing', [ 5 ] and has nothing to do with slenderness or thinness. [ citation needed ]
In biology, the term is in common use, whether as English or Latin:
In biological taxonomy , gracile is the specific name or specific epithet for various species. Where the gender is appropriate, the form is gracilis . Examples include:
The same root appears in the names of some genera and higher taxa : | https://en.wikipedia.org/wiki/Gracility |
A grade beam or grade beam footing is a component of a building's foundation. It consists of a reinforced concrete beam that transmits the load from a bearing wall into spaced foundations such as pile caps or caissons. [ 1 ] It is used in conditions where the surface soil's load-bearing capacity is less than the anticipated design loads .
A grade beam differs from a wall footing because a grade beam is designed for bending and typically spans between pile caps or caissons, while a wall footing bears on soil and transmits the weight of the wall directly into the ground. It also differs from a strap beam because a grade beam is reinforced to distribute the weight of a wall to separate foundations, [ 2 ] while a strap beam is designed to redistribute the weight of a column between footings.
Grade beams may also be used in conjunction with spread footings , in a case with large moments from lateral loads, in order to reduce the size of each spread footing. [ citation needed ] | https://en.wikipedia.org/wiki/Grade_beam |
In telecommunications engineering , and in particular teletraffic engineering , the quality of voice service is specified by two measures: the grade of service ( GoS ) and the quality of service ( QoS ).
Grade of service is the probability of a call in a circuit group being blocked or delayed for more than a specified interval, expressed as a vulgar fraction or decimal fraction . This is always with reference to the busy hour when the traffic intensity is the greatest. Grade of service may be viewed independently from the perspective of incoming versus outgoing calls, and is not necessarily equal in each direction or between different source-destination pairs. "Grade of Service" sometimes means a measure of inbound call center traffic to verify adherence to conditions to measure the success of customers served.
On the other hand, the quality of service which a single circuit is designed or conditioned to provide, e.g. voice grade or program grade is called the quality of service. Quality criteria for such circuits may include equalization for amplitude over a specified band of frequencies, or in the case of digital data transported via analogue circuits, may include equalization for phase . Criteria for mobile quality of service in cellular telephone circuits include the probability of abnormal termination of the call.
When a user attempts to make a telephone call, the routing equipment handling the call has to determine whether to accept the call, reroute the call to alternative equipment, or reject the call entirely. Rejected calls occur as a result of heavy traffic loads (congestion) on the system and can result in the call either being delayed or lost. If a call is delayed, the user simply has to wait for the traffic to decrease, however if a call is lost then it is removed from the system. [ 1 ]
The Grade of Service is one aspect of the quality a customer can expect to experience when making a telephone call. [ 2 ] In a Loss System, the Grade of Service is described as that proportion of calls that are lost due to congestion in the busy hour. [ 3 ] For a Lost Call system, the Grade of Service can be measured using Equation 1 . [ 4 ]
For a delayed call system, the Grade of Service is measured using three separate terms: [ 1 ]
The Grade of Service can be measured using different sections of a network. When a call is routed from one end to another, it will pass through several exchanges. If the Grade of Service is calculated based on the number of calls rejected by the final circuit group, then the Grade of Service is determined by the final circuit group blocking criteria. If the Grade of Service is calculated based on the number of rejected calls between exchanges, then the Grade of Service is determined by the exchange-to-exchange blocking criteria. [ 1 ]
The Grade of Service should be calculated using both the access networks and the core networks as it is these networks that allow a user to complete an end-to-end connection. [ 4 ] Furthermore, the Grade of Service should be calculated from the average of the busy hour traffic intensities of the 30 busiest traffic days of the year. This will cater for most scenarios as the traffic intensity will seldom exceed the reference level.
The grade of service is a measure of the ability of a user to access a trunk system during the busiest hour. The busy is based upon customer demand at the busiest hour during a week month or year.
Different telecommunications applications require different Qualities of Service. For example, if a telecommunications service provider decides to offer different qualities of voice connection, then a premium voice connection will require a better connection quality compared to an ordinary voice connection. Thus different Qualities of Service are appropriate, depending on the intended use. To help telecommunications service providers to market their different services, each service is placed into a specific class. Each Class of Service determines the level of service required. [ 4 ]
To identify the Class of Service for a specific service, the network's switches and routers examine the call based on several factors. Such factors can include: [ 2 ]
In broadband networks, the Quality of Service is measured using two criteria. The first criterion is the probability of packet losses or delays in already accepted calls. The second criterion refers to the probability that a new incoming call will be rejected or blocked. To avoid the former, broadband networks limit the number of active calls so that packets from established calls will not be lost due to new calls arriving. As in circuit-switched networks, the Grade of Service can be calculated for individual switches or for the whole network. [ 5 ]
The telecommunications provider is usually aware of the required Grade of Service for a particular product. To achieve and maintain a given Grade of Service, the operator must ensure that sufficient telecommunications circuits or routes are available to meet a specific level of demand. It should also be kept in mind that too many circuits will create a situation where the operator is providing excess capacity which may never be used, or at the very least may be severely underutilized. This adds costs which must be borne by other parts of the network. To determine the correct number of circuits that are required, telecommunications service providers make use of Traffic Tables. [ 4 ] An example of a Traffic Table can be viewed in Figure 1 . [ 4 ] It follows that in order for a telecommunications network to continue to offer a given Grade of Service, the number of circuits provided in a circuit group must increase (non-linearly) if the traffic intensity increases. [ 4 ]
To calculate the Grade of Service of a specified group of circuits or routes, Agner Krarup Erlang used a set of assumptions that relied on the network losing calls when all circuits in a group were busy. These assumptions are: [ 4 ]
From these assumptions Erlang developed the Erlang-B formula which describes the probability of congestion in a circuit group. The probability of congestion gives the Grade of Service experienced. [ 4 ]
To determine the Grade of Service of a network when the traffic load and number of circuits are known, telecommunications network operators make use of Equation 2 , which is the Erlang-B equation. [ 4 ]
A = Expected traffic intensity in Erlangs, N = Number of circuits in group.
This equation allows operators to determine whether each of their circuit groups meet the required Grade of Service, simply by monitoring the reference traffic intensity.
(For delay networks, the Erlang-C formula allows network operators to determine the probability of delay depending on peak traffic and the number of circuits. [ 4 ] ) | https://en.wikipedia.org/wiki/Grade_of_service |
In algebra , given a commutative ring R , the graded-symmetric algebra of a graded R -module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form:
for homogeneous elements x , y in M of degree | x |, | y |. By construction, a graded-symmetric algebra is graded-commutative ; i.e., x y = ( − 1 ) | x | | y | y x {\displaystyle xy=(-1)^{|x||y|}yx} and is universal for this.
In spite of the name, the notion is a common generalization of a symmetric algebra and an exterior algebra : indeed, if V is a (non-graded) R - module , then the graded-symmetric algebra of V with trivial grading is the usual symmetric algebra of V . Similarly, the graded-symmetric algebra of the graded module with V in degree one and zero elsewhere is the exterior algebra of V .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Graded-symmetric_algebra |
In mathematics , in the branch of combinatorics , a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers . ρ must satisfy the following two properties:
The value of the rank function for an element of the poset is called its rank . Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset . A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. [ 1 ] [ 2 ]
Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram .
Some examples of graded posets (with the rank function in parentheses) are:
A bounded poset [ 4 ] admits a grading if and only if all maximal chains in P have the same length: [ 5 ] setting the rank of the least element to 0 then determines the rank function completely. This covers many finite cases of interest; see picture for a negative example. However, unbounded posets can be more complicated.
A candidate rank function, compatible with the ordering, makes a poset into graded poset if and only if, whenever one has x < z with z of rank n + 1, an element y of rank n can be found with x ≤ y < z . This condition is sufficient because if z is taken to be a cover of x , the only possible choice is y = x showing that the ranks of x and z differ by 1, and it is necessary because in a graded poset one can take for y any element of maximal rank with x ≤ y < z , which always exists and is covered by z .
Often a poset comes with a natural candidate for a rank function; for instance if its elements are finite subsets of some base set B , one can take the number of elements of those subsets. Then the criterion just given can be more practical than the definition because it avoids mention of covers. For instance if B is itself a poset, and P consists of its finite lower sets (subsets for which with every one of its elements, all smaller elements are also in the subset), then the criterion is automatically satisfied, since for lower sets x ⊊ z there is always a maximal element of z that is absent from x , and it can be removed from z to form y .
In some common posets such as the face lattice of a convex polytope there is a natural grading by dimension , which if used as rank function would give the minimal element, the empty face, rank −1. In such cases it might be convenient to bend the definition stated above by adjoining the value −1 to the set of values allowed for the rank function. Allowing arbitrary integers as rank would however give a fundamentally different notion; for instance the existence of a minimal element would no longer be assured.
A graded poset (with positive integer ranks) cannot have any elements x for which arbitrarily long chains with greatest element x exist, as otherwise it would have to have elements of arbitrarily small (and eventually negative) rank. For instance, the integers (with the usual order) cannot be a graded poset, nor can any interval (with more than one element) of rational or real numbers . (In particular, graded posets are well-founded , meaning that they satisfy the descending chain condition (DCC): they do not contain any infinite descending chains . [ 6 ] ) Henceforth we shall therefore only consider posets in which this does not happen. This implies that whenever x < y we can get from x to y by repeatedly choosing a cover, finitely many times. It also means that (for positive integer rank functions) compatibility of ρ with the ordering follows from the requirement about covers. As a variant of the definition of a graded poset, Birkhoff [ 7 ] allows rank functions to have arbitrary (rather than only nonnegative) integer values. In this variant, the integers can be graded (by the identity function) in his setting, and the compatibility of ranks with the ordering is not redundant. As a third variant, Brightwell and West [ 8 ] define a rank function to be integer-valued, but don't require its compatibility with the ordering; hence this variant can grade even e.g. the real numbers by any function, as the requirement about covers is vacuous for this example.
Note that graded posets need not satisfy the ascending chain condition (ACC): for instance, the natural numbers contain the infinite ascending chain 0 < 1 < 2 < … {\displaystyle 0<1<2<\dots } .
A poset is graded if and only if every connected component of its comparability graph is graded, so further characterizations will suppose this comparability graph to be connected. On each connected component the rank function is only unique up to a uniform shift (so the rank function can always be chosen so that the elements of minimal rank in their connected component have rank 0).
If P has a least element Ô then being graded is equivalent to the condition that for any element x all maximal chains in the interval [Ô, x ] have the same length. This condition is necessary since every step in a maximal chain is a covering relation, which should change the rank by 1. The condition is also sufficient, since when it holds, one can use the mentioned length to define the rank of x (the length of a finite chain is its number of "steps", so one less than its number of elements), and whenever x covers y , adjoining x to a maximal chain in [Ô, y ] gives a maximal chain in [Ô, x ].
If P also has a greatest element Î (so that it is a bounded poset ), then the previous condition can be simplified to the requirement that all maximal chains in P have the same (finite) length. This suffices, since any pair of maximal chains in [Ô, x ] can be extended by a maximal chain in [ x , Î] to give a pair of maximal chains in P .
Many authors in combinatorics define graded posets in such a way that all minimal elements of P must have rank 0, and moreover that there is a maximal rank r that is the rank of any maximal element. Then being graded means that all maximal chains have length r , as is indicated above. In this case one says that P has rank r .
Furthermore, in this case, to the rank levels are associated the rank numbers or Whitney numbers W 0 , W 1 , W 2 , . . . {\displaystyle W_{0},W_{1},W_{2},...} . These numbers are defined by W i {\displaystyle W_{i}} = number of elements of P having rank i .
The Whitney numbers are connected with a lot of important combinatorial theorems . The classic example is Sperner's theorem , which can be formulated as follows:
For the power set P ( S ) {\displaystyle {\mathcal {P}}(S)} of every finite set S {\displaystyle S} the maximum cardinality of a Sperner family equals the maximum Whitney number .
This means:
Every finite power set has the Sperner property | https://en.wikipedia.org/wiki/Graded_poset |
In mathematics , the term " graded " has a number of meanings, mostly related:
In abstract algebra , it refers to a family of concepts:
In other areas of mathematics: | https://en.wikipedia.org/wiki/Graded_structure |
A grader , also commonly referred to as a road grader , motor grader , or simply blade , is a form of heavy equipment with a long blade used to create a flat surface during grading . Although the earliest models were towed behind horses, and later tractors , most modern graders are self-propelled and thus technically "motor graders".
Typical graders have three axles , with the steering wheels in front, followed by the grading blade or mouldboard, then a cab and engine atop tandem rear axles. Some graders also have front-wheel drives for improved performance. Some graders have optional rear attachments, such as a ripper , scarifier , or compactor . A blade forward of the front axle may also be added. For snowplowing and some dirt grading operations, a main blade extension can also be mounted.
Capacities range from a blade width of 2.50 to 7.30 m (8 to 24 ft) and engines from 93–373 kW (125–500 hp ). Certain graders can operate multiple attachments, or be designed for specialized tasks like underground mining.
In civil engineering "rough grading" is performed by heavy equipment such as wheel tractor-scrapers and bulldozers . Graders are used to "finish grade", with the angle, tilt (or pitch), and height of their blade capable of being adjusted to a high level of precision.
Graders are commonly used in the construction and maintenance of dirt and gravel roads. In constructing paved roads , they prepare a wide flat base course for the final road surface . Graders are also used to set native soil or gravel foundation pads to finish grade before the construction of large buildings. Graders can produce canted surfaces for drainage or safety. They may be used to produce drainage ditches with shallow V-shaped cross-sections on either side of highways.
Steering is performed via a steering wheel , or a joystick capable of controlling both the angle and cant of the front wheels. Many models also allow frame articulation between the front and rear axles, which allows a smaller turning radius in addition to allowing the operator to adjust the articulation angle to aid in the efficiency of moving material. Other implement functions are typically hydraulically powered and can be directly controlled by levers , or by joystick inputs or electronic switches controlling electrohydraulic servo valves .
Graders are also outfitted with modern digital grade control technologies, such as those manufactured by Topcon Positioning Systems, Inc. , Trimble Navigation , Leica Geosystems , or Mikrofyn. [ 1 ] These may combine both laser and GPS guidance to establish precise grade control and (potentially) "stateless" construction. Manufacturers such as John Deere have also begun to integrate these technologies during construction. [ 2 ]
Early graders were drawn by humans and draft animals . The Fresno Scraper is a machine pulled by horses used for constructing canals and ditches in sandy soil. The design of the Fresno Scraper forms the basis of most modern earthmoving scrapers, having the ability to scrape and move a quantity of soil, and also to discharge it at a controlled depth, thus quadrupling the volume which could be handled manually. The Fresno scraper was invented in 1883 by James Porteous. Working with farmers in Fresno, California, he had recognised the dependence of the Central San Joaquin Valley on irrigation, and the need for a more efficient means of constructing canals and ditches in the sandy soil. In perfecting the design of his machine, Porteous made several revisions on his own and also traded ideas with William Deidrick, Frank Dusy, and Abijah McCall, who invented and held patents on similar scrapers. [ citation needed ]
The era of motorization by traction engines , steam tractors , motor trucks , and tractors saw such towed graders grow in size and productivity. The first self-propelled grader was made in 1920 by the Russell Grader Manufacturing Company, which called it the Russell Motor Hi-Way Patrol. These early graders were created by adding the grader blade as an attachment to a generalist tractor unit. After purchasing the company in 1928, Caterpillar went on to truly integrate the tractor and grader into one design—at the same time replacing crawler tracks with wheels to yield the first rubber-tire self-propelled grader, the Caterpillar Auto Patrol, released in 1931. [ 3 ]
In addition to their use in road construction, graders may also be used to perform roughly equivalent work.
In some locales such as Northern Europe , Canada , and places in the United States , graders are often used in municipal and residential snow removal . In scrubland and grassland areas of Australia and Africa , graders are often an essential piece of equipment on ranches , large farms , and plantations to make dirt tracks where the absence of rocks and trees means bulldozers are not required. | https://en.wikipedia.org/wiki/Grader |
Gradient-index ( GRIN ) optics is the branch of optics covering optical effects produced by a gradient of the refractive index of a material. Such gradual variation can be used to produce lenses with flat surfaces, or lenses that do not have the aberrations typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that is spherical, axial, or radial.
The lens of the eye is the most obvious example of gradient-index optics in nature. In the human eye , the refractive index of the lens varies from approximately 1.406 in the central layers down to 1.386 in less dense layers of the lens. [ 1 ] This allows the eye to image with good resolution and low aberration at both short and long distances. [ 2 ]
Another example of gradient index optics in nature is the common mirage of a pool of water appearing on a road on a hot day. The pool is actually an image of the sky, apparently located on the road since light rays are being refracted (bent) from their normal straight path. This is due to the variation of refractive index between the hot, less dense air at the surface of the road, and the denser cool air above it. The variation in temperature (and thus density) of the air causes a gradient in its refractive index, causing it to increase with height. [ 3 ] This index gradient causes refraction of light rays (at a shallow angle to the road) from the sky, bending them into the eye of the viewer, with their apparent location being the road's surface.
The Earth's atmosphere acts as a GRIN lens, allowing observers to see the sun for a few minutes after it is actually below the horizon, and observers can also view stars that are below the horizon. [ 3 ] This effect also allows for observation of electromagnetic signals from satellites after they have descended below the horizon, as in radio occultation measurements.
The ability of GRIN lenses to have flat surfaces simplifies the mounting of the lens, which makes them useful where many very small lenses need to be mounted together, such as in photocopiers and scanners . [ 4 ] The flat surface also allows a GRIN lens to be easily optically aligned to a fiber , to produce collimated output, making it applicable for endoscopy as well as for in vivo calcium imaging and optogenetic stimulation in brain. [ 5 ]
In imaging applications, GRIN lenses are mainly used to reduce aberrations. The design of such lenses involves detailed calculations of aberrations as well as efficient manufacture of the lenses. A number of different materials have been used for GRIN lenses including optical glasses, plastics, germanium , zinc selenide , and sodium chloride . [ 4 ]
Certain optical fibres ( graded-index fibres ) are made with a radially-varying refractive index profile; this design strongly reduces the modal dispersion of a multi-mode optical fiber . The radial variation in refractive index allows for a sinusoidal height distribution of rays within the fibre, preventing the rays from leaving the core . This differs from traditional optical fibres, which rely on total internal reflection , in that all modes of the GRIN fibres propagate at the same speed, allowing for a higher temporal bandwidth for the fibre. [ 6 ]
Antireflection coatings are typically effective for narrow ranges of frequency or angle of incidence. Graded-index materials are less constrained. [ 7 ]
An axial gradient lens has been used to concentrate sunlight onto solar cells, capturing as much as 90% of incident light when the sun is not at an optimal angle. [ 8 ]
GRIN lenses are made by several techniques:
In 1854, J C Maxwell suggested a lens whose refractive index distribution would allow for every region of space to be sharply imaged. Known as the Maxwell fisheye lens , it involves a spherical index function and would be expected to be spherical in shape as well. [ 15 ] This lens, however, is impractical to make and has little usefulness since only points on the surface and within the lens are sharply imaged and extended objects suffer from extreme aberrations. In 1905, R. W. Wood used a dipping technique creating a gelatin cylinder with a refractive index gradient that varied symmetrically with the radial distance from the axis. Disk-shaped slices of the cylinder were later shown to have plane faces with radial index distribution. He showed that even though the faces of the lens were flat, they acted like converging and diverging lens depending on whether the index was a decreasing or increasing relative to the radial distance. [ 16 ] In 1964, a posthumous book of R. K. Luneburg was published in which he described a lens that focuses incident parallel rays of light onto a point on the opposite surface of the lens. [ 17 ] This also limited the applications of the lens because it was difficult to use it to focus visible light; however, it had some usefulness in microwave applications. Some years later several new techniques have been developed to fabricate lenses of the Wood type. Since then at least the thinner GRIN lenses can possess surprisingly good imaging properties considering their very simple mechanical construction, while thicker GRIN lenses found application e.g. in Selfoc rods . [ 18 ]
An inhomogeneous gradient-index lens possesses a refractive index whose change follows the function n = f ( x , y , z ) {\displaystyle n=f(x,y,z)} of the coordinates of the region of interest in the medium. According to Fermat's principle , the light path integral ( L ), taken along a ray of light joining any two points of a medium , is stationary relative to its value for any nearby curve joining the two points. The light path integral is given by the equation
where prime corresponds to d/d s. [ 19 ] The light path integral is able to characterize the path of light through the lens in a qualitative manner, such that the lens may be easily reproduced in the future.
The refractive index gradient of GRIN lenses can be mathematically modelled according to the method of production used. For example, GRIN lenses made from a radial gradient index material, such as SELFOC Microlens , [ 20 ] have a refractive index that varies according to: | https://en.wikipedia.org/wiki/Gradient-index_optics |
In differential topology , a mathematical discipline, and more specifically in Morse theory , a gradient-like vector field is a generalization of gradient vector field .
The primary motivation is as a technical tool in the construction of Morse functions , to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.
Given a Morse function f on a manifold M, a gradient-like vector field X for the function f is, informally:
Formally: [ 1 ]
and on which X equals the gradient of f.
The associated dynamical system of a gradient-like vector field, a gradient-like dynamical system , is a special case of a Morse–Smale system .
This differential geometry -related article is a stub . You can help Wikipedia by expanding it .
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gradient-like_dynamical_systems |
In mathematics , the gradient conjecture , due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka ( University of Savoie , France), Tadeusz Mostowski ( Warsaw University , Poland) and Adam Parusiński ( University of Angers , France).
The conjecture states that given a real-valued analytic function f defined on R n and a trajectory x ( t ) of the gradient vector field of f having a limit point x 0 ∈ R n , where f has an isolated critical point at x 0 , there exists a limit (in the projective space PR n −1 ) for the secant lines from x ( t ) to x 0 , as t tends to zero.
The proof depends on a theorem due to Stanisław Łojasiewicz .
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gradient_conjecture |
Gradient echo is a magnetic resonance imaging (MRI) sequence that has wide variety of applications, from magnetic resonance angiography to perfusion MRI and diffusion MRI . Rapid imaging acquisition allows it to be applied to 2D and 3D MRI imaging. Gradient echo uses magnetic gradients to generate a signal, instead of using 180 degrees radiofrequency pulse like spin echo ; thus leading to faster image acquisition time. [ 1 ]
Unlike spin-echo sequence, a gradient echo sequence does not use a 180 degrees RF pulse to make the spins of particles coherent. Instead, the gradient echo uses magnetic gradients to manipulate the spins, allowing the spins to dephase and rephase when required. After an excitation pulse (usually less than 90 degrees), the spins are dephased after a period of time (due to free induction decay ) and also by applying a reversed magnetic gradient to decay the spins. [ 2 ] No signal is produced because the spins are not coherent. When the spins are rephased via a magnetic gradient, they become coherent, and thus signal (or "echo") is generated to form images. Unlike spin echo, gradient echo does not need to wait for transverse magnetisation to decay completely before initiating another sequence, thus it requires very short repetition times (TR), and therefore to acquire images in a short time. [ 2 ]
After echo is formed, some transverse magnetisations remains because of short TR. [ 2 ] Manipulating gradients during this time will produce images with different contrast. There are three main methods of manipulating contrast at this stage, namely steady-state free-precession (SSFP) that does not spoil the remaining transverse magnetisation, but attempts to recover them in subsequent RF pulses (thus producing T2-weighted images); the sequence with spoiler gradient that averages the transverse magnetisations in subsequent RF pulses by rotating residual transverse magnetisation into longitudinal plane and longitudinal magnetisation into transverse planes (thus producing mixed T1 and T2-weighted images), and RF spoiler that vary the phases of RF pulse to eliminates the transverse magnetisation, thus producing pure T1-weighted images. [ 1 ] [ 2 ]
Gradient echo uses a flip angle smaller than 90 degrees, thus longitudinal magnetisation is not eliminated while flipping the spins. The larger the flip angle, the higher the T1 weighing of the tissue because more longitudinal magnetisation most recover to produce a difference in signals between the tissues. [ 2 ]
Steady-state free precession imaging (SSFP) or balanced SSFP is an MRI technique which uses short repetition times (TR) and low flip angles (about 10 degrees) to achieve steady state of longitudinal magnetizations as the magnetizations does not decay completely nor achieving full T1 relaxation. [ 1 ] While spoiled gradient-echo sequences refer to a steady state of the longitudinal magnetization only, SSFP gradient-echo sequences include transverse coherences (magnetizations) from overlapping multi-order spin echoes and stimulated echoes. This is usually accomplished by refocusing the phase-encoding gradient in each repetition interval in order to keep the phase integral (or gradient moment) constant. Fully balanced SSFP MRI sequences achieve a phase of zero by refocusing all imaging gradients.
MP-RAGE (magnetization-prepared rapid acquisition with gradient echo) [ 3 ] improves images of multiple sclerosis cortical lesions. [ 4 ]
At the end of the reading, the residual transverse magnetization can be terminated (through the application of suitable gradients and the excitation through pulses with a variable phase radiofrequency) or maintained.
In the first case there is a spoiled sequence, such as the fast low-angle shot MRI (FLASH MRI) sequence, while in the second case there are steady-state free precession imaging (SSFP) sequences.
In-phase (IP) and out-of-phase (OOP) sequences correspond to paired gradient echo sequences using the same repetition time (TR) but with two different echo times (TE). [ 5 ] This can detect even microscopic amounts of fat, which has a drop in signal on OOP compared to IP. Among renal tumors that do not show macroscopic fat, such a signal drop is seen in 80% of the clear cell type of renal cell carcinoma as well as in minimal fat angiomyolipoma . [ 6 ]
T 2 *-weighted imaging can be created as a postexcitation refocused gradient echo sequence with small flip angle. The sequence of a GRE T 2 *WI requires high uniformity of the magnetic field. [ 7 ]
VIBE (volumetric interpolated breath-hold examination) is an MRI sequence that produces T1-weighted gradient echo images in three-dimensions (3D). Apart from lower fluid signal intensity than a typical T1-weighted image, other appearances of VIBE images is similar to a typical T1-weighted image. Since its acquisition is only 30 seconds, suitable for breath-holding, it is used in breast and abdominal imaging to obtain high-resolution images minimising respiratory movement artifacts. VIBE images have low contrast in soft tissues and cartilage but have high contrast between the bony cortex and bone marrow. Bony lesions such as callus and fibrous tissue can also be readily distinguished from surrounding cortical bone because high contrast between the bone lesions and the bony cortex. [ 8 ] | https://en.wikipedia.org/wiki/Gradient_echo |
Gradient enhanced NMR is a method for obtaining high resolution nuclear magnetic resonance spectra without the need for phase cycling . Gradient methodology is used extensively for two purposes, either rephasing (selection) or dephasing (elimination) of a particular magnetization transfer pathway. It includes the application of magnetic field gradient pulses to select specific coherences . By using actively shielded gradients, a gradient pulse is applied during the evolution period of the selected coherence to dephase the transverse magnetization and another gradient pulse refocuses the desired coherences remaining during the acquisition period.
*Ralph E. Hurd, Gradient-Enhanced Spectroscopy, Journal of magnetic resonance. 87, 422-428 (1990)
This nuclear magnetic resonance –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gradient_enhanced_NMR_spectroscopy |
The gradient theorem , also known as the fundamental theorem of calculus for line integrals , says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n -dimensional) rather than just the real line.
If φ : U ⊆ R n → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q , then
∫ γ ∇ φ ( r ) ⋅ d r = φ ( q ) − φ ( p ) {\displaystyle \int _{\gamma }\nabla \varphi (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} =\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)}
where ∇ φ denotes the gradient vector field of φ .
The gradient theorem implies that line integrals through gradient fields are path-independent . In physics this theorem is one of the ways of defining a conservative force . By placing φ as potential, ∇ φ is a conservative field . Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field . Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.
If φ is a differentiable function from some open subset U ⊆ R n to R and r is a differentiable function from some closed interval [ a , b ] to U (Note that r is differentiable at the interval endpoints a and b . To do this, r is defined on an interval that is larger than and includes [ a , b ] .), then by the multivariate chain rule , the composite function φ ∘ r is differentiable on [ a , b ] :
d d t ( φ ∘ r ) ( t ) = ∇ φ ( r ( t ) ) ⋅ r ′ ( t ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(\varphi \circ \mathbf {r} )(t)=\nabla \varphi (\mathbf {r} (t))\cdot \mathbf {r} '(t)}
for all t in [ a , b ] . Here the ⋅ denotes the usual inner product .
Now suppose the domain U of φ contains the differentiable curve γ with endpoints p and q . (This is oriented in the direction from p to q ). If r parametrizes γ for t in [ a , b ] (i.e., r represents γ as a function of t ), then
∫ γ ∇ φ ( r ) ⋅ d r = ∫ a b ∇ φ ( r ( t ) ) ⋅ r ′ ( t ) d t = ∫ a b d d t φ ( r ( t ) ) d t = φ ( r ( b ) ) − φ ( r ( a ) ) = φ ( q ) − φ ( p ) , {\displaystyle {\begin{aligned}\int _{\gamma }\nabla \varphi (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} &=\int _{a}^{b}\nabla \varphi (\mathbf {r} (t))\cdot \mathbf {r} '(t)\mathrm {d} t\\&=\int _{a}^{b}{\frac {d}{dt}}\varphi (\mathbf {r} (t))\mathrm {d} t=\varphi (\mathbf {r} (b))-\varphi (\mathbf {r} (a))=\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right),\end{aligned}}}
where the definition of a line integral is used in the first equality, the above equation is used in the second equality, and the second fundamental theorem of calculus is used in the third equality. [ 1 ]
Even if the gradient theorem (also called fundamental theorem of calculus for line integrals ) has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per differentiable curve component. [ 2 ]
Suppose γ ⊂ R 2 is the circular arc oriented counterclockwise from (5, 0) to (−4, 3) . Using the definition of a line integral ,
∫ γ y d x + x d y = ∫ 0 π − tan − 1 ( 3 4 ) ( ( 5 sin t ) ( − 5 sin t ) + ( 5 cos t ) ( 5 cos t ) ) d t = ∫ 0 π − tan − 1 ( 3 4 ) 25 ( − sin 2 t + cos 2 t ) d t = ∫ 0 π − tan − 1 ( 3 4 ) 25 cos ( 2 t ) d t = 25 2 sin ( 2 t ) | 0 π − tan − 1 ( 3 4 ) = 25 2 sin ( 2 π − 2 tan − 1 ( 3 4 ) ) = − 25 2 sin ( 2 tan − 1 ( 3 4 ) ) = − 25 ( 3 / 4 ) ( 3 / 4 ) 2 + 1 = − 12. {\displaystyle {\begin{aligned}\int _{\gamma }y\,\mathrm {d} x+x\,\mathrm {d} y&=\int _{0}^{\pi -\tan ^{-1}\!\left({\frac {3}{4}}\right)}((5\sin t)(-5\sin t)+(5\cos t)(5\cos t))\,\mathrm {d} t\\&=\int _{0}^{\pi -\tan ^{-1}\!\left({\frac {3}{4}}\right)}25\left(-\sin ^{2}t+\cos ^{2}t\right)\mathrm {d} t\\&=\int _{0}^{\pi -\tan ^{-1}\!\left({\frac {3}{4}}\right)}25\cos(2t)\mathrm {d} t\ =\ \left.{\tfrac {25}{2}}\sin(2t)\right|_{0}^{\pi -\tan ^{-1}\!\left({\tfrac {3}{4}}\right)}\\[.5em]&={\tfrac {25}{2}}\sin \left(2\pi -2\tan ^{-1}\!\!\left({\tfrac {3}{4}}\right)\right)\\[.5em]&=-{\tfrac {25}{2}}\sin \left(2\tan ^{-1}\!\!\left({\tfrac {3}{4}}\right)\right)\ =\ -{\frac {25(3/4)}{(3/4)^{2}+1}}=-12.\end{aligned}}}
This result can be obtained much more simply by noticing that the function f ( x , y ) = x y {\displaystyle f(x,y)=xy} has gradient ∇ f ( x , y ) = ( y , x ) {\displaystyle \nabla f(x,y)=(y,x)} , so by the Gradient Theorem:
∫ γ y d x + x d y = ∫ γ ∇ ( x y ) ⋅ ( d x , d y ) = x y | ( 5 , 0 ) ( − 4 , 3 ) = − 4 ⋅ 3 − 5 ⋅ 0 = − 12. {\displaystyle \int _{\gamma }y\,\mathrm {d} x+x\,\mathrm {d} y=\int _{\gamma }\nabla (xy)\cdot (\mathrm {d} x,\mathrm {d} y)\ =\ xy\,|_{(5,0)}^{(-4,3)}=-4\cdot 3-5\cdot 0=-12.}
For a more abstract example, suppose γ ⊂ R n has endpoints p , q , with orientation from p to q . For u in R n , let | u | denote the Euclidean norm of u . If α ≥ 1 is a real number, then
∫ γ | x | α − 1 x ⋅ d x = 1 α + 1 ∫ γ ( α + 1 ) | x | ( α + 1 ) − 2 x ⋅ d x = 1 α + 1 ∫ γ ∇ | x | α + 1 ⋅ d x = | q | α + 1 − | p | α + 1 α + 1 {\displaystyle {\begin{aligned}\int _{\gamma }|\mathbf {x} |^{\alpha -1}\mathbf {x} \cdot \mathrm {d} \mathbf {x} &={\frac {1}{\alpha +1}}\int _{\gamma }(\alpha +1)|\mathbf {x} |^{(\alpha +1)-2}\mathbf {x} \cdot \mathrm {d} \mathbf {x} \\&={\frac {1}{\alpha +1}}\int _{\gamma }\nabla |\mathbf {x} |^{\alpha +1}\cdot \mathrm {d} \mathbf {x} ={\frac {|\mathbf {q} |^{\alpha +1}-|\mathbf {p} |^{\alpha +1}}{\alpha +1}}\end{aligned}}}
Here the final equality follows by the gradient theorem, since the function f ( x ) = | x | α +1 is differentiable on R n if α ≥ 1 .
If α < 1 then this equality will still hold in most cases, but caution must be taken if γ passes through or encloses the origin, because the integrand vector field | x | α − 1 x will fail to be defined there. However, the case α = −1 is somewhat different; in this case, the integrand becomes | x | −2 x = ∇(log | x |) , so that the final equality becomes log | q | − log | p | .
Note that if n = 1 , then this example is simply a slight variant of the familiar power rule from single-variable calculus.
Suppose there are n point charges arranged in three-dimensional space, and the i -th point charge has charge Q i and is located at position p i in R 3 . We would like to calculate the work done on a particle of charge q as it travels from a point a to a point b in R 3 . Using Coulomb's law , we can easily determine that the force on the particle at position r will be
F ( r ) = k q ∑ i = 1 n Q i ( r − p i ) | r − p i | 3 {\displaystyle \mathbf {F} (\mathbf {r} )=kq\sum _{i=1}^{n}{\frac {Q_{i}(\mathbf {r} -\mathbf {p} _{i})}{\left|\mathbf {r} -\mathbf {p} _{i}\right|^{3}}}}
Here | u | denotes the Euclidean norm of the vector u in R 3 , and k = 1/(4 πε 0 ) , where ε 0 is the vacuum permittivity .
Let γ ⊂ R 3 − { p 1 , ..., p n } be an arbitrary differentiable curve from a to b . Then the work done on the particle is
W = ∫ γ F ( r ) ⋅ d r = ∫ γ ( k q ∑ i = 1 n Q i ( r − p i ) | r − p i | 3 ) ⋅ d r = k q ∑ i = 1 n ( Q i ∫ γ r − p i | r − p i | 3 ⋅ d r ) {\displaystyle W=\int _{\gamma }\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} =\int _{\gamma }\left(kq\sum _{i=1}^{n}{\frac {Q_{i}(\mathbf {r} -\mathbf {p} _{i})}{\left|\mathbf {r} -\mathbf {p} _{i}\right|^{3}}}\right)\cdot \mathrm {d} \mathbf {r} =kq\sum _{i=1}^{n}\left(Q_{i}\int _{\gamma }{\frac {\mathbf {r} -\mathbf {p} _{i}}{\left|\mathbf {r} -\mathbf {p} _{i}\right|^{3}}}\cdot \mathrm {d} \mathbf {r} \right)}
Now for each i , direct computation shows that
r − p i | r − p i | 3 = − ∇ 1 | r − p i | . {\displaystyle {\frac {\mathbf {r} -\mathbf {p} _{i}}{\left|\mathbf {r} -\mathbf {p} _{i}\right|^{3}}}=-\nabla {\frac {1}{\left|\mathbf {r} -\mathbf {p} _{i}\right|}}.}
Thus, continuing from above and using the gradient theorem,
W = − k q ∑ i = 1 n ( Q i ∫ γ ∇ 1 | r − p i | ⋅ d r ) = k q ∑ i = 1 n Q i ( 1 | a − p i | − 1 | b − p i | ) {\displaystyle W=-kq\sum _{i=1}^{n}\left(Q_{i}\int _{\gamma }\nabla {\frac {1}{\left|\mathbf {r} -\mathbf {p} _{i}\right|}}\cdot \mathrm {d} \mathbf {r} \right)=kq\sum _{i=1}^{n}Q_{i}\left({\frac {1}{\left|\mathbf {a} -\mathbf {p} _{i}\right|}}-{\frac {1}{\left|\mathbf {b} -\mathbf {p} _{i}\right|}}\right)}
We are finished. Of course, we could have easily completed this calculation using the powerful language of electrostatic potential or electrostatic potential energy (with the familiar formulas W = −Δ U = − q Δ V ). However, we have not yet defined potential or potential energy, because the converse of the gradient theorem is required to prove that these are well-defined, differentiable functions and that these formulas hold ( see below ). Thus, we have solved this problem using only Coulomb's Law, the definition of work, and the gradient theorem.
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative ), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
Theorem — If F is a path-independent vector field, then F is the gradient of some scalar-valued function. [ 3 ]
It is straightforward to show that a vector field is path-independent if and only if the integral of the vector field over every closed loop in its domain is zero. Thus the converse can alternatively be stated as follows: If the integral of F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.
Suppose U is an open , path-connected subset of R n , and F : U → R n is a continuous and path-independent vector field. Fix some element a of U , and define f : U → R by f ( x ) := ∫ γ [ a , x ] F ( u ) ⋅ d u {\displaystyle f(\mathbf {x} ):=\int _{\gamma [\mathbf {a} ,\mathbf {x} ]}\mathbf {F} (\mathbf {u} )\cdot \mathrm {d} \mathbf {u} } Here γ [ a , x ] is any (differentiable) curve in U originating at a and terminating at x . We know that f is well-defined because F is path-independent.
Let v be any nonzero vector in R n . By the definition of the directional derivative , ∂ f ( x ) ∂ v = lim t → 0 f ( x + t v ) − f ( x ) t = lim t → 0 ∫ γ [ a , x + t v ] F ( u ) ⋅ d u − ∫ γ [ a , x ] F ( u ) ⋅ d u t = lim t → 0 1 t ∫ γ [ x , x + t v ] F ( u ) ⋅ d u {\displaystyle {\begin{aligned}{\frac {\partial f(\mathbf {x} )}{\partial \mathbf {v} }}&=\lim _{t\to 0}{\frac {f(\mathbf {x} +t\mathbf {v} )-f(\mathbf {x} )}{t}}\\&=\lim _{t\to 0}{\frac {\int _{\gamma [\mathbf {a} ,\mathbf {x} +t\mathbf {v} ]}\mathbf {F} (\mathbf {u} )\cdot \mathrm {d} \mathbf {u} -\int _{\gamma [\mathbf {a} ,\mathbf {x} ]}\mathbf {F} (\mathbf {u} )\cdot d\mathbf {u} }{t}}\\&=\lim _{t\to 0}{\frac {1}{t}}\int _{\gamma [\mathbf {x} ,\mathbf {x} +t\mathbf {v} ]}\mathbf {F} (\mathbf {u} )\cdot \mathrm {d} \mathbf {u} \end{aligned}}} To calculate the integral within the final limit, we must parametrize γ [ x , x + t v ] . Since F is path-independent, U is open, and t is approaching zero, we may assume that this path is a straight line, and parametrize it as u ( s ) = x + s v for 0 < s < t . Now, since u' ( s ) = v , the limit becomes lim t → 0 1 t ∫ 0 t F ( u ( s ) ) ⋅ u ′ ( s ) d s = d d t ∫ 0 t F ( x + s v ) ⋅ v d s | t = 0 = F ( x ) ⋅ v {\displaystyle \lim _{t\to 0}{\frac {1}{t}}\int _{0}^{t}\mathbf {F} (\mathbf {u} (s))\cdot \mathbf {u} '(s)\,\mathrm {d} s={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{0}^{t}\mathbf {F} (\mathbf {x} +s\mathbf {v} )\cdot \mathbf {v} \,\mathrm {d} s{\bigg |}_{t=0}=\mathbf {F} (\mathbf {x} )\cdot \mathbf {v} } where the first equality is from the definition of the derivative with a fact that the integral is equal to 0 at t = 0, and the second equality is from the first fundamental theorem of calculus . Thus we have a formula for ∂ v f , (one of ways to represent the directional derivative ) where v is arbitrary; for f ( x ) := ∫ γ [ a , x ] F ( u ) ⋅ d u {\displaystyle f(\mathbf {x} ):=\int _{\gamma [\mathbf {a} ,\mathbf {x} ]}\mathbf {F} (\mathbf {u} )\cdot \mathrm {d} \mathbf {u} } (see its full definition above), its directional derivative with respect to v is ∂ f ( x ) ∂ v = ∂ v f ( x ) = D v f ( x ) = F ( x ) ⋅ v {\displaystyle {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {v} }}=\partial _{\mathbf {v} }f(\mathbf {x} )=D_{\mathbf {v} }f(\mathbf {x} )=\mathbf {F} (\mathbf {x} )\cdot \mathbf {v} } where the first two equalities just show different representations of the directional derivative. According to the definition of the gradient of a scalar function f , ∇ f ( x ) = F ( x ) {\displaystyle \nabla f(\mathbf {x} )=\mathbf {F} (\mathbf {x} )} , thus we have found a scalar-valued function f whose gradient is the path-independent vector field F (i.e., F is a conservative vector field.), as desired. [ 3 ]
To illustrate the power of this converse principle, we cite an example that has significant physical consequences. In classical electromagnetism , the electric force is a path-independent force; i.e. the work done on a particle that has returned to its original position within an electric field is zero (assuming that no changing magnetic fields are present).
Therefore, the above theorem implies that the electric force field F e : S → R 3 is conservative (here S is some open , path-connected subset of R 3 that contains a charge distribution). Following the ideas of the above proof, we can set some reference point a in S , and define a function U e : S → R by
U e ( r ) := − ∫ γ [ a , r ] F e ( u ) ⋅ d u {\displaystyle U_{e}(\mathbf {r} ):=-\int _{\gamma [\mathbf {a} ,\mathbf {r} ]}\mathbf {F} _{e}(\mathbf {u} )\cdot \mathrm {d} \mathbf {u} }
Using the above proof, we know U e is well-defined and differentiable, and F e = −∇ U e (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: W = −Δ U ). This function U e is often referred to as the electrostatic potential energy of the system of charges in S (with reference to the zero-of-potential a ). In many cases, the domain S is assumed to be unbounded and the reference point a is taken to be "infinity", which can be made rigorous using limiting techniques. This function U e is an indispensable tool used in the analysis of many physical systems.
Many of the critical theorems of vector calculus generalize elegantly to statements about the integration of differential forms on manifolds . In the language of differential forms and exterior derivatives , the gradient theorem states that
∫ ∂ γ ϕ = ∫ γ d ϕ {\displaystyle \int _{\partial \gamma }\phi =\int _{\gamma }\mathrm {d} \phi }
for any 0-form , ϕ , defined on some differentiable curve γ ⊂ R n (here the integral of ϕ over the boundary of the γ is understood to be the evaluation of ϕ at the endpoints of γ ).
Notice the striking similarity between this statement and the generalized Stokes’ theorem , which says that the integral of any compactly supported differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative d ω over the whole of Ω , i.e.,
∫ ∂ Ω ω = ∫ Ω d ω {\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }\mathrm {d} \omega }
This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension.
The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds. In particular, suppose ω is a form defined on a contractible domain , and the integral of ω over any closed manifold is zero. Then there exists a form ψ such that ω = d ψ . Thus, on a contractible domain, every closed form is exact . This result is summarized by the Poincaré lemma . | https://en.wikipedia.org/wiki/Gradient_theorem |
Grading in civil engineering and landscape architectural construction is the work of ensuring a level base, or one with a specified slope , [ 1 ] for a construction work such as a foundation , the base course for a road or a railway , or landscape and garden improvements, or surface drainage. The earthworks created for such a purpose are often called the sub-grade or finished contouring (see diagram).
Regrading is the process of grading for raising and/or lowering the levels of land. Such a project can also be referred to as a regrade .
Regrading may be done on a small scale (as in preparation of a house site) [ 3 ] or on quite a large scale (as in major reconfiguration of the terrain of a city, such as the Denny Regrade in Seattle ). [ 2 ]
Regrading is typically performed to make land more level (flatter), in which case it is sometimes called levelling . [ 4 ] ) Levelling can have the consequence of making other nearby slopes steeper, and potentially unstable or prone to erosion.
In the case of gravel roads and earthworks for certain purposes, grading forms not just the base but the cover and surface of the finished construction, and is often called finished grade . [ 5 ]
After the existing conditions of the limit of work has been surveyed, surveyors will set stakes in places that are to be regraded. These stakes have marks on them that either give a finished grade to the design of the project, or have CUT/FILL marks which specify how much dirt is to be added or subtracted. All grade marks are relative to site benchmarks that have been established. [ 6 ] The regrading work is then often done using heavy machinery like bulldozers and excavators to roughly prepare an area, then a grader is used for a finer finish.
In the environmental design professions, grading and regrading are a specifications and construction component in landscape design , landscape architecture , and architecture projects. It is used for buildings or outdoor amenities regarding foundations and footings , slope terracing and stabilizing, aesthetic contouring, and directing surface runoff drainage of stormwater and domestic/irrigation runoff flows.
Reasons for regrading include:
Potential problems and consequences from regrading include: | https://en.wikipedia.org/wiki/Grading_(earthworks) |
In pathology , grading is a measure of the cell appearance in tumors and other neoplasms . Some pathology grading systems apply only to malignant neoplasms ( cancer ); others apply also to benign neoplasms. The neoplastic grading is a measure of cell anaplasia (reversion of differentiation ) in the sampled tumor and is based on the resemblance of the tumor to the tissue of origin. [ 1 ] Grading in cancer is distinguished from staging , which is a measure of the extent to which the cancer has spread .
Pathology grading systems classify the microscopic cell appearance abnormality and deviations in their rate of growth with the goal of predicting developments at tissue level (see also the 4 major histological changes in dysplasia ).
Cancer is a disorder of cell life cycle alteration that leads (non-trivially) to excessive cell proliferation rates, typically longer cell lifespans and poor differentiation. The grade score (numerical: G1 up to G4) increases with the lack of cellular differentiation - it reflects how much the tumor cells differ from the cells of the normal tissue they have originated from (see 'Categories' below). Tumors may be graded on four-tier, three-tier, or two-tier scales, depending on the institution and the tumor type.
The histologic tumor grade score along with the metastatic (whole-body-level cancer-spread) staging are used to evaluate each specific cancer patient, develop their individual treatment strategy and to predict their prognosis. A cancer that is very poorly differentiated is called anaplastic .
Grading systems are also different for many common types of cancer, though following a similar pattern with grades being increasingly malignant over a range of 1 to 4. If no specific system is used, the following general grades are most commonly used, and recommended by the American Joint Commission on Cancer and other bodies: [ 2 ]
Of the many cancer-specific schemes, the Gleason system , [ 3 ] named after Donald Floyd Gleason , used to grade the adenocarcinoma cells in prostate cancer is the most famous. This system uses a grading score ranging from 2 to 10. Lower Gleason scores describe well-differentiated less aggressive tumors.
Other systems include the Bloom-Richardson grading system for breast cancer and the Fuhrman system for kidney cancer . Invasive-front grading is useful as well in oral squamous cell carcinoma. [ 4 ]
For soft-tissue sarcoma two histological grading systems are used : the National Cancer Institute (NCI) system and the French Federation of Cancer Centers Sarcoma Group (FNCLCC) system. [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Grading_(tumors) |
A gradiometer measures the gradient (numerical rate of change) of a physical quantity , such as a magnetic field or gravity. [ 1 ]
There are at least two types of gradiometer to measure magnetic fields :
Each sensor type responds differently to certain spatial signals.
Axial gradiometers are good for measuring depth, while planar gradiometers can measure weak signals even under a lot of noise. [ 2 ]
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gradiometer |
A gradsect or gradient-directed transect is a low-input, high-return sampling method where the aim is to maximise information about the distribution of biota in any area of study. Most living things are rarely distributed at random , their placement being largely determined by a hierarchy of environmental factors. For this reason, standard statistical designs based on purely random sampling or systematic (e.g. grid-based) systems tend to be less efficient in recovering information about the distribution of taxa than sample designs that are purposively directed instead along deterministic environmental gradients.
Ecologists have long been aware of the significance of environmental gradient based approaches to better understand community dynamics and this is reflected especially in the work of Robert Whittaker (1967) [ 1 ] and others. Although in practice, life-scientists intuitively sample gradients, until the early 1980s there was little formal theoretical or empirical support for such an approach, sample design being driven largely by traditional statistical methods based on probability theory incorporating random sampling .
Intensively sampled landscape-based surveys in Australia provided a reference platform for developing and testing a less logistically demanding and yet statistically acceptable gradient-based survey design that avoided the need for random or purely grid-based sampling. These initial studies [ 2 ] and subsequently developed statistical support for purposive, gradient-based survey [ 3 ] provided a formalized, practical alternative to more logistically demanding traditional designs. It was here the term gradsect was coined that coupled purposive, transect sampling with a hierarchical framework of environmental gradients considered to be key determinants of species distribution.
In constructing a gradsect, existing information is initially reviewed in which a hierarchy of environmental gradients is first identified either by visual means (maps, aerial photographs etc..) or through numerical analysis or spatial analysis of institutional or other data sources. A typical regional gradsect for example may be constructed according to a primary climate gradient (temperature, moisture, seasonality) then a secondary gradient ( geomorphology , lithology , major and minor drainage systems), a tertiary gradient possibly represented by a local soil catena or local land use farming system or finer scale gradient levels representing local vegetational sequences. Through an inspection of spatial overlays of all gradients, a minimum number of sample locations is then purposively located to reflect, as far as possible, total environmental variation. For logistic and other purposes (such as improving the capacity to locate rare species) the steepest gradients are usually selected. In this way an ideal gradsect is constructed that may then be modified to accommodate logistic tradeoffs. The selection discipline requires that the fullest possible range of each hierarchical level is sampled. This commonly results in a set of progressively nested clusters of sample sites contained within the overarching primary gradient that may not reflect a linear distribution. At relatively local landscape scale, a primary gradients may be represented by salinity levels or water depth as in tidal wetlands or micro-topographic relief as in forest margins or a riparian zone . For most practical purposes, transects are commonly laid out along contours perpendicular to the main direction of the gradient. Iterative spatial analysis of environmental layers over a digital elevation model can then be used to identify areas requiring additional sampling thereby improving environmental representativeness. [ 4 ]
Initial studies in gradsect development revealed considerable logistic and other advantages over more traditional non-gradient-based survey designs concerned primarily with random sampling. This finding is now widely supported especially in biodiversity and other areas of environmental surveying and conservation design (see Applications next). Apart from improved logistic efficiency, the gradsect method seeks to maximise environmental representativeness which has the dual advantage of potentially improving location of rarities and enhancing spatial modelling of species distribution. Because the underlying statistical model is not based on probability theory , gradsect sampling cannot be used to estimate numbers of species or other biological attributes per unit area. For that purpose some measure of random sampling needs to be built into the sample design.
Since the publication of gradsect theory in 1984, subsequent vegetational and landscape studies in regional Australia (Austin and Heyligers 1989); [ 5 ] Ludwig and Tongway (1995 [ 6 ] ) were followed by a successful evaluation of the method in faunal surveys in South Africa (Wessels et al. [ 7 ] ). Since then applications involving gradsects have ranged from habitat suitability studies of fungi (Shearer and Crane 2011 [ 8 ] ), termites (Gillison et al. 2003 [ 9 ] ) other macro invertebrates (Lawes et al. 2005 [ 10 ] ); birds (Damalas 2005 [ 11 ] ) small and large mammals (Laurance 1994; [ 12 ] Ramono et al. 2009 [ 13 ] ). Vegetation studies using gradsects have been widely applied in many countries ranging from tidal wetlands (Parker et al. 2011 [ 14 ] ) and agricultural cropping systems and forested landscape mosaics (Gillison et al. 2004 [ 15 ] ) to infectious diseases (Boone et al. 2000 [ 16 ] ). At broader geographic and national scales (Grossman et al., 1998, [ 17 ] 2007; [ 18 ] USA/NPS 2012 [ 19 ] ) gradsects have been applied to guide field sampling and forest mapping in mountainous terrain (Sandman and Lertzmann 2003 [ 20 ] ) as well as wide-ranging remote sensing applications (Mallinis et al. 2008; [ 21 ] Rocchini et al. 2011 [ 22 ] ). | https://en.wikipedia.org/wiki/Gradsect |
Gradshteyn and Ryzhik ( GR ) is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is Table of Integrals, Series, and Products .
Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors.
At some stage a German and English dual-language translation became available, followed by Polish, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain , but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well.
One of the valuable characteristics of Gradshteyn and Ryzhik compared to similar compilations is that most listed integrals are referenced. The literature list contains 92 main entries and 140 additional entries (in the eighth English edition). The integrals are classified by numbers, which haven't changed from the fourth Russian up to the seventh English edition (the numbering in older editions as well as in the eighth English edition is not fully compatible).
The book does not only contain the integrals, but also lists additional properties and related special functions .
It also includes tables for integral transforms .
Another advantage of Gradshteyn and Ryzhik compared to computer algebra systems is the fact that all special functions and constants used in the evaluation of the integrals are listed in a registry as well, thereby allowing reverse lookup of integrals based on special functions or constants.
On the downsides, Gradshteyn and Ryzhik has become known to contain a relatively high number of typographical errors even in newer editions, which has repeatedly led to the publication of extensive errata lists. Earlier English editions were also criticized for their poor translation of mathematical terms [ 1 ] [ 2 ] [ 3 ] and mediocre print quality. [ 1 ] [ 2 ] [ 4 ] [ 5 ]
The work was originally compiled by the Russian mathematicians Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик , German: Jossif Moissejewitsch Ryschik ) [ 6 ] [ nb 1 ] and Izrail Solomonovich Gradshteyn (Russian: Израиль Соломонович Градштейн , German: Israil Solomonowitsch Gradstein ). [ 6 ] [ nb 2 ] While some contents were original, significant portions were collected from other previously existing integral tables like David Bierens de Haan 's Nouvelles tables d'intégrales définies (1867), [ 6 ] [ 7 ] Václav Jan Láska 's Sammlung von Formeln der reinen und angewandten Mathematik (1888–1894) [ 6 ] [ 8 ] or Edwin Plimpton Adams ' and Richard Lionel Hippisley 's Smithsonian Mathematical Formulae and Tables of Elliptic Functions (1922). [ 6 ] [ 9 ]
The first edition, which contained about 5 000 formulas, [ 10 ] [ 11 ] [ nb 3 ] was authored by Ryzhik, [ nb 1 ] who had already published a book on special functions in 1936 [ 6 ] [ 12 ] and died during World War II around 1941. [ 6 ] Not announcing this fact, his compilation was published posthumously [ 6 ] [ nb 1 ] in 1943, followed by a second corrected edition in his name in 1948. [ nb 4 ]
The third edition (1951) was worked on by Gradshteyn, [ 13 ] who also introduced the chapter numbering system in decimal notation . Gradshteyn planned considerable expansion for the fourth edition, a work he could not finish due to his own death. [ 6 ] [ nb 2 ] Therefore, the fourth (1962/1963) and fifth (1971) editions were continued by Yuri Veniaminovich Geronimus (Russian: Юрий Вениаминович Геронимус , German: Juri Weniaminowitsch Geronimus ) [ 6 ] [ nb 5 ] and Michail Yulyevich Tseytlin (Russian: Михаил Ю́льевич Цейтлин , German: Michael Juljewitsch Zeitlin ). [ nb 6 ] The fourth edition contained about 12 000 formulas already. [ 14 ] [ nb 3 ]
Based on the third Russian edition, the first German-English edition with 5 400 formulas [ 15 ] [ nb 3 ] was published in 1957 by the East-German Deutscher Verlag der Wissenschaften (DVW) with German translations by Christa [ nb 7 ] and Lothar Berg [ nb 8 ] and the English texts by Martin Strauss. [ nb 9 ] In Zentralblatt für Mathematik Karl Prachar wrote: [ 16 ]
" Die sehr reichhaltigen Tafeln wurden nur aus dem Russischen ins Deutsche und Englische übersetzt. " (Translation: The very comprehensive tables were only translated into German and English language.)
In 1963, it was followed by the second edition, a reprint edition with a four-page inlet listing corrections compiled by Eldon Robert Hansen .
Derived from the 1963 edition, but considerably expanded and split into two volumes, the third German-English edition by Ludwig Boll [ nb 10 ] was finally published by MIR Moscow in 1981 (with permission of DVW and distributed through Verlag Harri Deutsch in the Western world); it incorporated the material of the fifth Russian edition (1971) as well. [ nb 11 ]
Pending this third German-English edition an English-only edition by Alan Jeffrey [ nb 12 ] was published in 1965. Lacking a clear designation by itself it was variously known as first, third or fourth English edition, as it was based on the then-current fourth Russian edition. The formulas were photographically reproduced and the text translated. This still held true for the expanded fourth English edition in 1980, which added chapters 10 to 17. [ 17 ]
Both of these editions saw multiple print runs each incorporating newly found corrections. Starting with the third printing, updated table entries were marked by adding a small superscript number to the entry number indicating the corresponding print run ("3" etc.), a convention carried over into later editions by continuing to increase the superscript number as kind of a revision number (no longer directly corresponding with actual print runs).
The fifth edition (1994), which contained close to 20 000 formulas, [ 18 ] [ nb 3 ] was electronically reset [ 3 ] in preparation for a CD-ROM issue of the fifth edition (1996) and in anticipation of further editions. Since the sixth edition (2000), now corresponding with superscript number "10", Daniel Zwillinger [ nb 13 ] started contributing as well. The last edition being edited by Jeffrey before his death [ nb 12 ] was the seventh English edition published in 2007 (with superscript number "11"). [ 19 ] This edition has been retranslated back into Russian as " seventh Russian edition" in 2011. [ 20 ] [ nb 11 ]
For the eighth edition (2014/2015, with superscript number "12") Zwillinger took over the role of the editor. He was assisted by Victor Hugo Moll . [ 21 ] [ nb 14 ] In order to make room for additional information without increasing the size of the book significantly, the former chapters 11 (on algebraic inequalities ), chapters 13 to 16 (on matrices and related results, determinants , norms , ordinary differential equations ) and chapter 18 (on z-transforms ) worth about 50 pages in total were removed and some chapters renumbered (12 to 11, 17 to 12). This edition contains more than 10 000 entries. [ 21 ] [ nb 3 ]
In 1995, Alan Jeffrey published his Handbook of Mathematical Formulas and Integrals . [ 22 ] It was partially based on the fifth English edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and meant as an companion, but written to be more accessible for students and practitioners. [ 22 ] It went through four editions up to 2008. [ 22 ] [ 23 ] [ 24 ] [ 25 ] The fourth edition also took advantage of changes incorporated into the seventh English edition of Gradshteyn and Ryzhik. [ 25 ]
Inspired by a 1988 paper in which Ilan Vardi [ de ] proved several integrals in Gradshteyn and Ryzhik , [ 26 ] Victor Hugo Moll with George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references. [ 27 ] In the foreword of the book Irresistible Integrals (2004), they wrote: [ 28 ]
It took a short time to realize that this task was monumental.
Nevertheless, the efforts have meanwhile resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles [ 29 ] [ 30 ] [ 31 ] of which papers 1 to 28 [ a ] have been published in issues 14 to 26 of Scientia , Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015 [ 60 ] and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015) already. [ 61 ] [ 62 ] | https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik |
Graduate School of Engineering and Faculty of Engineering (京都大学大学院工学研究科・工学部) is one of schools at the Kyoto University . The Faculty ( Undergraduate ) and the Graduate School operate as one.
According to the QS World University Rankings by Subject 2020 in the field of Engineering & Technology, KU is ranked third in Japan after University of Tokyo and TokyoTech . [ 1 ]
In 1897, College of Science and Engineering (理工科大学) was established with the establishment of Imperial University of Kyoto . [ 2 ] It was divided into College of Engineering (工科大学) and College of Science in 1914. [ 2 ]
College of Engineering was reorganized into Faculty of Engineering (工学部) in 1919. [ 2 ]
In 1953, Graduate School of Engineering (工学研究科) was established. [ 2 ] [ 3 ]
In the 1990s, A four-year plan to emphasize graduate school education was implemented and some departments were converted into independent graduate schools; Graduate School of Energy Science was established in 1996 and Graduate School of Informatics in 1998. [ 2 ] [ 3 ]
It consists of the undergraduate schools, departments and centers. [ 4 ]
The Faculty of Engineering has 6 Undergraduate Schools, some of which have more specialized courses. [ 4 ]
Graduate School of Engineering has 17 departments for research and graduate education. [ 4 ] | https://en.wikipedia.org/wiki/Graduate_School_of_Engineering_and_Faculty_of_Engineering,_Kyoto_University |
Graduate Women in Science ( GWIS ), formerly known as Sigma Delta Epsilon ( ΣΔΕ ), is an international professional organization for women in science . It was established as a scientific women's fraternity in 1921 at Cornell University , United States. It played an important role for women scientists for some fifty years when they were not allowed membership in most mainstream scientific organizations. GWIS is a non-profit 501(c)(3) organization with over 1,000 active members and more than 30 active chapters.
Sigma Delta Epsilon was established at Cornell University in Ithaca, New York by Adele Lewis Grant on May 24, 1921. It was founded as a fraternity for women pursuing graduate degrees in the sciences. [ 1 ] [ 2 ] [ 3 ] Its stated purpose was "to further interest in science, recognize women involved in science, and unite them through friendship". [ 3 ]
Initially, Sigma Delta Epsilon had 25 student members and eight honorary members, who were professional women who had achieved recognition in science. [ 2 ] Its first officers were Adele Lewis Grant, president; Katherine Van Winkle , vice president; Josephine Overton Sonders, secretary; and Hazel Elizabeth Branch , treasurer. [ 2 ] [ 4 ] Sigma Delta Epsilon had a fraternity house where its members could live. [ 2 ]
In 1922, a similar local group for women at the University of Wisconsin–Madison agreed to merge with Sigma Delta Epsilon, establishing a national fraternity. [ 1 ] Its purpose was "to further interest in science, to provide a fraternity for the recognition of women in science, and to bring them together in a fraternal relationship". [ 1 ]
Sigma Delta Epsilon was incorporated in the state of New York in April 1922. [ 1 ] It held its first national convention on April 20, 1922. [ 4 ] It joined the American Association for the Advancement of Science (AAAS) as an associate member in 1936 and as an affiliated member in 1939. [ 1 ] [ 5 ] In this era when mainstream scientific organizations did not give women full membership, Sigma Delta Epsilon "filled an important niche", according to Margaret Rossiter. [ 6 ] Hazel Fox was the only woman on the AAAS Council at the time, as a representative of Sigma Delta Epsilon. [ 7 ]
One of the organization's early activities was collecting money to distribute to other members needing research funds. [ 8 ] In 1931, Sigma Delta Epsilon established a formal Fellowships Fund. Its first research fellowship was awarded in 1941. [ 1 ] In 1970, Eloise Gerry established a fellowship, the first within the organization to be funded by a single individual. [ 4 ]
By the early 1970s, the fraternity was struggling from an increasing anti-fraternity sentiment on college campuses and competition from previously male-only organizations. [ 6 ] Hoping to counter this, the fraternity changed its name to Sigma Delta Epsilon Graduate Women in Science in December 1971. [ 6 ] [ 4 ] This was shortened to Graduate Women in Science (GWIS) on April 21, 2016. [ 4 ] An international chapter was established in 2013. [ 9 ]
The motto of Graduate Women in Science is "United in Friendship through Science". [ 10 ] Its guiding principles or pillars are Connect, Lead, and Empower. [ 11 ] [ 9 ]
Its badge is a Nile key with the Greek letters ΣΔΕ in black enamel on its crossbar. Attached to the key are a benzene ring, a thunderbolt, and the nabla. [ 1 ] Its colors are those of the spectrum . [ 1 ]
Graduate Women in Science is a non-profit 501(c)(3) organization that works to connect, lead, and empower women in science . Its mission is "building a global community to inspire, support, recognize, and empower women in science." [ 9 ] [ 4 ] It has over 1,000 members and dozens of chapters spread across the United States, as well as an international chapter that was established in 2013. [ 9 ] Its national office is in Mullica Hill, New Jersey . [ 9 ]
Graduate Women in Science offers grants, awards , and fellowships . [ 9 ] [ 4 ] It serves an international network of women scientists and promotes the participation and representation of women in science-related events. [ 9 ] The GWIS National Meeting is held annually in June. [ 12 ] It also sponsors mentoring, webinars , and seminars featuring its member's research. [ 13 ] [ 14 ] [ 15 ] The society publishes a monthly newsletter, GWIS Connect, and GWIS Lead , a periodical that features women leaders in science. [ 16 ]
Membership in the Graduate Women in Science is open to anyone who has at least a bachelor's degree in a scientific discipline and engineering, or equivalent professional experience. [ 8 ]
Graduate Women in Science has chartered more than 50 chapters and has more than 30 active chapters. [ 17 ] [ 18 ] [ 1 ] | https://en.wikipedia.org/wiki/Graduate_Women_in_Science |
The Graduate of Pharmacy ( Ph.G. ) is an obsolete academic pharmacy degree. It was superseded by the Bachelor of Pharmacy degree (B.Pharm.) in the early part of the 20th century. [ 1 ]
This pharmacy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Graduate_of_Pharmacy |
[ 1 ] Graduated optimization is a global optimization technique that attempts to solve a difficult optimization problem by initially solving a greatly simplified problem, and progressively transforming that problem (while optimizing) until it is equivalent to the difficult optimization problem. [ 2 ] [ 3 ] [ 4 ]
Graduated optimization is an improvement to hill climbing that enables a hill climber to avoid settling into local optima. [ 5 ] It breaks a difficult optimization problem into a sequence of optimization problems, such that the first problem in the sequence is convex (or nearly convex), the solution to each problem gives a good starting point to the next problem in the sequence, and the last problem in the sequence is the difficult optimization problem that it ultimately seeks to solve. Often, graduated optimization gives better results than simple hill climbing. Further, when certain conditions exist, it can be shown to find an optimal solution to the final problem in the sequence. These conditions are:
It can be shown inductively that if these conditions are met, then a hill climber will arrive at the global optimum for the difficult problem. Unfortunately, it can be difficult to find a sequence of optimization problems that meet these conditions. Often, graduated optimization yields good results even when the sequence of problems cannot be proven to strictly meet all of these conditions.
Graduated optimization is commonly used in image processing for locating objects within a larger image. This problem can be made to be more convex by blurring the images. Thus, objects can be found by first searching the most-blurred image, then starting at that point and searching within a less-blurred image, and continuing in this manner until the object is located with precision in the original sharp image. The proper choice of the blurring operator depends on the geometric transformation relating the object in one image to the other. [ 6 ]
Graduated optimization can be used in manifold learning. The Manifold Sculpting algorithm, for example, uses graduated optimization to seek a manifold embedding for non-linear dimensionality reduction . [ 7 ] It gradually scales variance out of extra dimensions within a data set while optimizing in the remaining dimensions. It has also been used to calculate conditions for fractionation with tumors, [ 8 ] for object tracking in computer vision, [ 9 ] and other purposes.
A thorough review of the method and its applications can be found in. [ 4 ]
Simulated annealing is closely related to graduated optimization. Instead of smoothing the function over which it is optimizing, simulated annealing randomly perturbs the current solution by a decaying amount, which may have a similar effect. [ citation needed ] Because simulated annealing relies on random sampling to find improvements, however, its computation complexity is exponential in the number of dimensions being optimized. [ citation needed ] By contrast, graduated optimization smooths the function being optimized, so local optimization techniques that are efficient in high-dimensional space (such as gradient-based techniques, hill climbers, etc.) may still be used. | https://en.wikipedia.org/wiki/Graduated_optimization |
A graduation is a marking used to indicate points on a visual scale , which can be present on a container , a measuring device , or the axes of a line plot , usually one of many along a line or curve, each in the form of short line segments perpendicular to the line or curve. Often, some of these line segments are longer and marked with a numeral, such as every fifth or tenth graduation. The scale itself can be linear (the graduations are spaced at a constant distance apart) or nonlinear.
Linear graduation of a scale occurs mainly (but not exclusively) on straight measuring devices, such as a rule or measuring tape, using units such as inches or millimetres .
Graduations can also be spaced at varying spatial intervals, such as when using a logarithmic , for instance on a measuring cup , can vary in scale due to the container's non- cylindrical shape.
Circular graduations of a scale occur on a circular arc or limb of an instrument. In some cases, non-circular curves are graduated in instruments. A typical circular arc graduation is the division into angular measurements, such as degrees, minutes and seconds. These types of graduated markings are traditionally seen on devices ranging from compasses and clock faces to alidades found on such instruments as telescopes , theodolites , inclinometers , astrolabes , armillary spheres , and celestial spheres .
There can also be non-uniform graduations such as logarithmic or other scales such as seen on circular slide rules and graduated cylinders .
Graduations can be placed on an instrument by etching , scribing or engraving , painting , printing or other means. For durability and accuracy, etched or scribed marks are usually preferable to surface coatings such as paints and inks. Markings can be a combination of both physical marks such as a scribed line and a paint or other marking material. For example, it is common for black ink or paint to fill the grooves cut in a scribed rule. Inexpensive plastic devices can be molded and painted or molded with two or more colors of plastic used. Some rather high-quality devices can be manufactured with plastic and reveal high-precision graduations.
Graduations traditionally have been scribed into an instrument by hand with a sharp, hard tool . [ 1 ] Later developments in devices such as dividing engines allowed the process to be automated with greater precision. Modern devices can be stamped , cut on a milling machine or with a CNC machine. In the case of stamping, the master has the precision built into itself and the stamped device is as accurate as the stamping process allows. Similarly, molding of plastic can be as precise as the mold process. With proper concern for such effects as thermal expansion or contraction and shrinkage , the precision can be very high.
The US graduation style of an instrument was a Federal standard for codes used by manufacturers to quickly determine which types of scales are marked on the instrument.
Other commonly recognized styles are: [ citation needed ]
Suffix key: | https://en.wikipedia.org/wiki/Graduation_(scale) |
A graduation tower (occasionally referred to as a thorn house [ 1 ] ) is a structure, used in the production of salt, that removes water from a saline solution by evaporation , increasing its concentration of mineral salts. The tower consists of a wooden wall-like frame stuffed with bundles of brushwood (typically blackthorn ) that have to be changed every five to ten years, as they become encrusted with mineral deposits over time. [ 2 ] The salt water runs down the tower and partly evaporates. At the same time, some minerals from the solution are left behind on the brushwood twigs.
Graduation towers can be found in a number of spa towns, primarily in Germany but also Poland and Austria. The mineral-rich water droplets in the air are regarded [ by whom? ] as having beneficial health effects similar to that of breathing in sea air . [ citation needed ]
Large graduation tower complexes are located in Ciechocinek and Inowrocław , Poland. [ 2 ] Ciechocinek's entirely wooden construction was erected in the 19th century by Stanisław Staszic . The complex consists of three towers, with a total length of over 2 km. Many tourists visit it for health reasons. [ citation needed ]
With years of initial construction where available. Does not include modern indoor facilities found in some spas.
Media related to Graduation towers at Wikimedia Commons | https://en.wikipedia.org/wiki/Graduation_tower |
In fluid dynamics , the Graetz number ( Gz ) is a dimensionless number that characterizes laminar flow in a conduit. The number is defined as: [ 1 ]
where
This number is useful in determining the thermally developing flow entrance length in ducts. A Graetz number of approximately 1000 or less is the point at which flow would be considered thermally fully developed. [ 2 ]
When used in connection with mass transfer the Prandtl number is replaced by the Schmidt number , Sc, which expresses the ratio of the momentum diffusivity to the mass diffusivity.
The quantity is named after the physicist Leo Graetz .
This fluid dynamics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Graetz_number |
Graffiti is an essentially single-stroke shorthand handwriting recognition system used in PDAs based on the Palm OS . Graffiti was originally written by Palm, Inc. as the recognition system for GEOS -based devices such as HP's OmniGo 100 and 120 or the Magic Cap -line and was available as an alternate recognition system for the Apple Newton MessagePad, when NewtonOS 1.0 could not recognize handwriting very well. Graffiti also runs on the Windows Mobile platform, where it is called "Block Recognizer", and on the Symbian UIQ platform as the default recognizer and was available for Casio's Zoomer PDA.
The software is based primarily on a neography of upper-case characters that can be drawn blindly with a stylus on a touch-sensitive panel. Since the user typically cannot see the character as it is being drawn, complexities have been removed from four of the most difficult letters. "A" "F", "K" and "T" all are drawn without any need to match up a cross-stroke.
Some letters can be drawn with strokes other than the "official" ones. Two examples of these alternative strokes are the letters "V" (drawn the same only from right to left) and "X" (drawn the same as the letter "K" except reversed from right to left). These alternative strokes are frequently recognized with greater reliability.
Graffiti was developed by Jeff Hawkins , who had previously created "PalmPrint" (the character recognition system used by the Casio Zoomer [ 1 ] ) to recognize natural handwriting. [ 2 ] By using a simpler alphabet, computers could easily recognize handwriting. Hawkins believed that people would take the time to learn Graffiti just as people learn to touch-type . Hawkins recalled his insight: "And then it came to me in a flash. Touch-typing is a skill you learn ." [ 2 ]
The program was first released in 1994 for the Casio Zoomer PDA, [ 3 ] while the first device to have the program preinstalled were the Pilot 1000 and 5000 PDAs, both of which were released in 1996. [ 4 ]
Hawkins also envisioned a single area for writing letters on top of each other. [ 2 ]
Hawkins called this system "PowerPalmPrint" or P3. Other engineers at Palm revised and expanded the alphabet that Hawkins had created. Joe Sipher and Ron Marianetti created more characters and punctuation, and also designed a prototype of Graffiti that ran on a PC with a tablet peripheral . [ 2 ]
Graffiti 2, whose gestures resembled natural handwriting more, was released in 2003 as a result of the lost lawsuit from Xerox. [ 5 ]
Graffiti was also implemented on the Apple Newton . In 2008, an unauthorized version of Graffiti was introduced for iOS ( iPhone and iPad ) devices. An Android version was released in 2010 by ACCESS Co., Ltd. of Japan, which acquired the rights to Graffiti when it acquired PalmSource, Inc. in 2005. [ 6 ] The original patent expired at the end of 2016. [ 7 ]
StrokeInput, [ 8 ] is an Apple App for an additional keyboard that enables - when activated - Graffiti input for every text on iPhone or iPad.
The original Graffiti system was the subject of a lawsuit from Xerox , claiming it violated Xerox's patent relating to its Unistrokes technology ( U.S. patent 5,596,656 , granted in 1997). The Unistrokes technology was invented at the Palo Alto Research Center (PARC) by David Goldberg in 1993. [ 9 ]
Palm later appealed the original court ruling, both on the claim it violated Xerox's patent and as to the validity of the patent in the first place. An appeals court ruled in favor of Xerox with regard to the original ruling, that Palm had violated Xerox's patent, but sent the case back down to the lower court to decide whether the patent was valid to begin with. In 2004, a judge ruled in favor of Palm on the patent review, saying Xerox's patent was not valid on the basis that "prior art references to anticipate and render obvious the claim." [ 10 ] [ 11 ] [ 12 ] Xerox appealed the ruling. [ 13 ] Xerox also obtained a US$22.5 million payment from Palm for retrospective licensing fees. Palm and Xerox agreed to not sue each other for seven years over certain patents, without publicly specifying which patents. [ 14 ] | https://en.wikipedia.org/wiki/Graffiti_(Palm_OS) |
Graffiti is a computer program which makes conjectures in various subfields of mathematics (particularly graph theory ) [ 1 ] and chemistry , but can be adapted to other fields. It was written by Siemion Fajtlowicz and Ermelinda DeLaViña at the University of Houston . Research on conjectures produced by Graffiti has led to over 60 publications by other mathematicians. [ 2 ]
This article about mathematics software is a stub . You can help Wikipedia by expanding it .
This article about chemistry software is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Graffiti_(program) |
Graft-versus-tumor effect ( GvT ) appears after allogeneic hematopoietic stem cell transplantation (HSCT). The graft contains donor T cells (T lymphocytes) that can be beneficial for the recipient by eliminating residual malignant cells. [ 1 ] GvT might develop after recognizing tumor-specific or recipient-specific alloantigens. It could lead to remission or immune control of hematologic malignancies. [ 2 ] This effect applies in myeloma and lymphoid leukemias , lymphoma , multiple myeloma and possibly breast cancer . [ 3 ] It is closely linked with graft-versus-host disease (GvHD), as the underlying principle of alloimmunity is the same. CD4+CD25+ regulatory T cells (Treg) can be used to suppress GvHD without loss of beneficial GvT effect. [ 4 ] The biology of GvT response is still not fully understood but it is probable that the reaction with polymorphic minor histocompatibility antigens expressed either specifically on hematopoietic cells or more widely on a number of tissue cells or tumor-associated antigens is involved. [ 5 ] [ 6 ] This response is mediated largely by cytotoxic T lymphocytes (CTL) but it can be employed by natural killers (NK cells) as separate effectors, particularly in T-cell-depleted HLA-haploidentical HSCT. [ 6 ]
Graft-versus-leukemia (GvL) is a specific type of GvT effect. As the name of this effect indicates, GvL is a reaction against leukemic cells of the host. GvL requires genetic disparity because the effect is dependent on the alloimmunity principle. GvL is a part of the reaction of the graft against the host. Whereas graft-versus-host-disease (GvHD) has a negative impact on the host, GvL is beneficial for patients with hematopeietic malignancies. After HSC transplantation both GvL and GvHD develop. The interconnection of those two effects can be seen by comparison of leukemia relapse after HSC transplantation with development of GvHD. Patients who develop chronic or acute GvHD have lower chance of leukemia relapse. [ 7 ] When transplanting T-cell depleted stem cell transplant, GvHD can be partially prevented, but in the same time the GvL effect is also reduced, because T-cells play an important role in both of those effects. [ 8 ] The possibilities of GvL effect in the treatment of hematopoietic malignancies are limited by GvHD. The ability to induce GvL but not GvH after HSCT would be very beneficial for those patients. There are some strategies to suppress the GvHD after transplantation or to enhance GvL but none of them provide an ideal solution to this problem. For some forms of hematopoietic malignancies, for example acute myeloid leukemia (AML), the essential cells during HSCT are, beside the donor's T cells, the NK cells, which interact with KIR receptors . NK cells are within the first cells to repopulate host's bone marrow which means they play important role in the transplant engraftment. For their role in the GvL effect, their alloreactivity is required. [ 9 ] Because KIR and HLA genes are inherited independently, the ideal donor can have compatible HLA genes and KIR receptors that induce the alloreaction of NK cells at the same time. This will occur with most of the non-related donor. When transplanting HSC during AML, T-cells are usually selectively depleted to prevent GvHD while NK cells help with the GvL effect which prevent leukemia relapse. When using non-depleted T-cell transplant, cyclophosphamide is used after transplantation to prevent GvHD or transplant rejection. Other strategies currently clinically used for suppressing GvHD and enhancing GvL are for example optimization of transplant condition or donor lymphocyte infusion (DLI) after transplantation. [ 10 ] [ 11 ] However, none of those provide satisfactory universal results, thus other options are still being inspected. One of the possibilities is the use of cytokines. Granulocyte colony-stimulating factor (G-CSF) is used to mobilize HSC and mediate T cell tolerance during transplantation. G-CSF can help to enhance GvL effect and suppress GvHD by reducing levels of LPS and TNF-α. Using G-CSF also increases levels of Treg, which can also help with prevention of GvHD. Other cytokines can also be used to prevent or reduce GvHD without eliminating GvL, for example KGF, IL-11, IL-18 and IL-35. [ 11 ] | https://en.wikipedia.org/wiki/Graft-versus-tumor_effect |
graft macromolecule : A macromolecule with one or more species of block connected to the main chain as side-chains, these side-chains having constitutional or configurational features that differ from those in the main chain.
comb macromolecule : A macromolecule comprising a main chain with multiple trifunctional branch points from each of which a linear side-chain emanates.
Notes
1. If the subchains between the branch points of the main chain and the terminal subchains of the main chain are identical with respect to constitution and degree of polymerization, and the side chains are identical with respect to constitution and degree of polymerization, the macromolecule is termed a ’’regular comb macromolecule’’.
2. If at least some of the branch points are of functionality greater than three, the macromolecule may be termed a ‘’brush macromolecule’’.
In polymer chemistry , graft polymers are segmented copolymers with a linear backbone of one composite and randomly distributed branches of another composite. The picture labeled "graft polymer" shows how grafted chains of species B are covalently bonded to polymer species A. Although the side chains are structurally distinct from the main chain, the individual grafted chains may be homopolymers or copolymers.
Graft polymers have been synthesized for many decades and are especially used as impact resistant materials, thermoplastic elastomers , compatibilizers , or emulsifiers for the preparation of stable blends or alloys . One of the better-known examples of a graft polymer is a component used in high impact polystyrene , consisting of a polystyrene backbone with polybutadiene grafted chains.
Graft copolymers are a branched copolymer where the components of the side chain are structurally different than that of the main chain. Graft copolymers containing a larger quantity of side chains are capable of wormlike conformation, compact molecular dimension, and notable chain end effects due to their confined and tight fit structures. [ 1 ] The preparation of graft copolymers has been around for decades. All synthesis methods can be employed to create general physical properties of graft copolymers. They can be used for materials that are impact resistant, and are often used as thermoplastics elastomers, compatibilizers or emulsifiers for the preparation of stable blends or alloys. [ 2 ] Generally, grafting methods for copolymer synthesis results in materials that are more thermostable than their homopolymer counterparts. [ 3 ] There are three methods of synthesis, grafting to, grafting from, and grafting through, that are used to construct a graft polymer. [ 4 ]
There are many different approaches to synthesizing graft copolymers. Usually they employ familiar polymerization techniques that are commonly used such as atom transfer radical polymerization (ATRP), ring-opening metathesis polymerization (ROMP), anionic and cationic polymerizations , and free radical living polymerization. Some other less common polymerization include radiation-induced polymerization, [ 5 ] ring-opening olefin metathesis polymerization, [ 6 ] polycondensation reactions, [ 7 ] and iniferter-induced polymerization. [ 8 ]
The grafting to method involves the use of a backbone chain with functional groups A that are distributed randomly along the chain. [ 9 ] The formation of the graft copolymer originates from the coupling reaction between the functional backbone and the end-groups of the branches that are reactive. These coupling reactions are made possible by modifying the backbone chemically. [ 10 ] Common reaction mechanisms used to synthesize these copolymers include free- radical polymerization , anionic polymerization , atom-transfer radical-polymerization , and living polymerization techniques.
Copolymers that are prepared with the grafting-to method often utilize anionic polymerization techniques. This method uses a coupling reaction of the electrophilic groups of the backbone polymer and the propagation site of an anionic living polymer. This method would not be possible without the generation of a backbone polymer that has reactive groups. This method has become more popular with the rise of click chemistry . A high yield chemical reaction called atom transfer nitroxide radical coupling chemistry is for the grafting-to method for polymerization.
In the grafting-from method, the macromolecular backbone is chemically modified in order to introduce active sites capable of initiating functionality. The initiating sites can be incorporated by copolymerization , can be incorporated in a post-polymerization reaction, or can already be a part of the polymer. [ 10 ] If the number of active sites along the backbone participates in the formation of one branch, then the number of chains grafted to the macromolecule can be controlled by the number of active sites. Even though the number of grafted chains can be controlled, there may be a difference in the lengths of each grafted chain due to kinetic and steric hindrance effects. [ 9 ]
Grafting from reactions have been conducted from polyethylene , polyvinylchloride , and polyisobutylene. Different techniques such as anionic grafting, cationic grafting, atom-transfer radical polymerization , and free-radical polymerization have been used in the synthesis of grafting from copolymers.
Graft copolymers that are employed with the grafting-from method are often synthesized with ATRP reactions and anionic and cationic grafting techniques.
The grafting through, also known as the macromonomer method, is one of the simpler ways of synthesizing a graft polymer with well defined side chains. [ 10 ] Typically a monomer of a lower molecular weight is copolymerized with free radicals with an acrylate functionalized macromonomer. The ratio of monomer to macromonomer molar concentrations as well as their copolymerization behavior determines the number of chains that are grafted. As the reaction proceeds, the concentrations of monomer to macromonomer change causing random placement of branches and formation of graft copolymers with different number of branches. This method allows for branches to be added heterogeneously or homogeneously based on the reactivity ratio of the terminal functional group on the macromolecular to the monomer. [ 11 ] The difference in distribution of grafts has significant effects on the physical properties of the grafted copolymer. Polyethylene , polysiloxanes and poly(ethylene oxide) are all macromonomers that have been incorporated in a polystyrene or poly(methyl acrylate) backbone.
The macromonomer (grafting through) method can be employed using any known polymerization technique. Living polymerizations give special control over the molecular weight, molecular weight distribution, and chain-end functionalization.
Graft copolymers became widely studied due to their increased number of applications like in drug delivery vehicles, surfactants , water filtration , rheology modifiers, etc. [ 12 ] It is their unique structures relative to other copolymers such as alternating, periodic, statistical, and block copolymers.
Some common applications of graft copolymers include:
High impact polystyrene (HIPS) was discovered by Charles F. Fryling in 1961. [ 19 ] HIPS is a low cost, plastic material that is easy to fabricate and often used for low strength structural applications when impact resistance, machinability, and low cost are required. Its major applications include machined prototypes, low-strength structural components, housings, and covers. [ 20 ] In order to produce the graft polymer, polybutadiene ( rubber ) or any similar elastomeric polymer is dissolved in styrene and polymerized. This reaction allows for two simultaneous polymerizations, that of styrene to polystyrene and that of the graft polymerization of styrene -rubber. [ 19 ] During commercial use, it can be prepared by graft copolymerization with additional polymer to give the product specific characteristics.
The advantages of HIPS includes: [ 20 ]
By grafting polymers onto polymer backbones, the final grafted copolymers gain new properties from their parent polymers. Specifically, cellulose graft copolymers have various different applications that are dependent on the structure of the polymer grafted onto the cellulose. [ 21 ] Some of the new properties that cellulose gains from different monomers grafted onto it include:
These properties give new application to the ungrafted cellulose polymers that include: | https://en.wikipedia.org/wiki/Graft_polymer |
Grafting or graftage [ 1 ] is a horticultural technique whereby tissues of plants are joined so as to continue their growth together. The upper part of the combined plant is called the scion ( / ˈ s aɪ ə n / ) while the lower part is called the rootstock. The success of this joining requires that the vascular tissues grow together. The natural equivalent of this process is inosculation . The technique is most commonly used in asexual propagation of commercially grown plants for the horticultural and agricultural trades. The scion is typically joined to the rootstock at the soil line; however, top work grafting may occur far above this line, leaving an understock consisting of the lower part of the trunk and the root system.
In most cases, the stock or rootstock is selected for its roots and the scion is selected for its stems , leaves , flowers , or fruits . [ 1 ] The scion contains the desired genes to be duplicated in future production by the grafted plant.
In stem grafting, a common grafting method, a shoot of a selected, desired plant cultivar is grafted onto the stock of another type. In another common form called bud grafting, a dormant side bud is grafted onto the stem of another stock plant, and when it has inosculated successfully, it is encouraged to grow by pruning off the stem of the stock plant just above the newly grafted bud.
For successful grafting to take place, the vascular cambium tissues of the stock and scion plants must be placed in contact with each other. Both tissues must be kept alive until the graft has "taken", usually a period of a few weeks . Successful grafting only requires that a vascular connection take place between the grafted tissues. Research conducted in Arabidopsis thaliana hypocotyls has shown that the connection of phloem takes place after three days of initial grafting, whereas the connection of xylem can take up to seven days. [ 2 ] Joints formed by grafting are not as strong as naturally formed joints, so a physical weak point often still occurs at the graft because only the newly formed tissues inosculate with each other. The existing structural tissue (or wood) of the stock plant does not fuse.
Precocity : The ability to induce fruitfulness without the need for completing the juvenile phase. Juvenility is the natural state through which a seedling plant must pass before it can become reproductive. In most fruiting trees, juvenility may last between 5 and 9 years, but in some tropical fruits, e.g., mangosteen , juvenility may be prolonged for up to 15 years. Grafting of mature scions onto rootstocks can result in fruiting in as little as two years.
Dwarfing : To induce dwarfing or cold tolerance or other characteristics to the scion. Most apple trees in modern orchards are grafted on to dwarf or semi-dwarf trees planted at high density. They provide more fruit per unit of land, of higher quality, and reduce the danger of accidents by harvest crews working on ladders. Care must be taken when planting dwarf or semi-dwarf trees. If such a tree is planted with the graft below the soil, then the scion portion can also grow roots and the tree will still grow to its standard size.
Ease of propagation : [ 3 ] Because the scion is difficult to propagate vegetatively by other means, such as by cuttings . In this case, cuttings of an easily rooted plant are used to provide a rootstock. In some cases, the scion may be easily propagated, but grafting may still be used because it is commercially the most cost-effective way of raising a particular type of plant.
Hybrid breeding : To speed maturity of hybrids in fruit tree breeding programs. Hybrid seedlings may take ten or more years to flower and fruit on their own roots. Grafting can reduce the time to flowering and shorten the breeding program.
Hardiness : Because the scion has weak roots or the roots of the stock plants are tolerant of difficult conditions. e.g. many Western Australian plants are sensitive to dieback on heavy soils, common in urban gardens, and are grafted onto hardier eastern Australian relatives. Grevilleas and eucalypts are examples.
Sturdiness : To provide a strong, tall trunk for certain ornamental shrubs and trees. In these cases, a graft is made at a desired height on a stock plant with a strong stem. This is used to raise 'standard' roses , which are rose bushes on a high stem, and it is also used for some ornamental trees, such as certain weeping cherries.
Disease/pest resistance : In areas where soil-borne pests or pathogens would prevent the successful planting of the desired cultivar, the use of pest/disease tolerant rootstocks allow the production from the cultivar that would be otherwise unsuccessful. A major example is the use of rootstocks in combating Phylloxera .
Pollen source : To provide pollenizers . For example, in tightly planted or badly planned apple orchards of a single variety, limbs of crab apple may be grafted at regularly spaced intervals onto trees down rows, say every fourth tree. This takes care of pollen needs at blossom time.
Repair : [ 3 ] To repair damage to the trunk of a tree that would prohibit nutrient flow, such as stripping of the bark by rodents that completely girdles the trunk. In this case a bridge graft may be used to connect tissues receiving flow from the roots to tissues above the damage that have been severed from the flow. Where a water sprout , basal shoot or sapling of the same species is growing nearby, any of these can be grafted to the area above the damage by a method called inarch grafting. These alternatives to scions must be of the correct length to span the gap of the wound.
Changing cultivars : To change the cultivar in a fruit orchard to a more profitable cultivar, called top working . It may be faster to graft a new cultivar onto existing limbs of established trees than to replant an entire orchard.
Genetic consistency : Apples are notorious for their genetic variability, even differing in multiple characteristics, such as, size, color, and flavor, of fruits located on the same tree. In the commercial farming industry, consistency is maintained by grafting a scion with desired fruit traits onto a hardy stock.
Curiosities:
Compatibility of scion and stock : Because grafting involves the joining of vascular tissues between the scion and rootstock, plants lacking vascular cambium, such as monocots , cannot normally be grafted. As a general rule, the closer two plants are genetically, the more likely the graft union will form. Genetically identical clones and intra-species plants have a high success rate for grafting. Grafting between species of the same genus is sometimes successful. Grafting has a low success rate when performed with plants in the same family but in different genera, and grafting between different families is rare. [ 4 ]
Cambium alignment and pressure : The vascular cambium of the scion and stock should be tightly pressed together and oriented in the direction of normal growth. Proper alignment and pressure encourages the tissues to join quickly, allowing nutrients and water to transfer from the stockroot to the scion. [ 5 ] : 466
Completed during appropriate stage of plant : The grafting is completed at a time when the scion and stock are capable of producing callus and other wound-response tissues. Generally, grafting is performed when the scion is dormant, as premature budding can drain the grafting site of moisture before the grafting union is properly established. Temperature greatly affects the physiological stage of plants. If the temperature is too warm, premature budding may result. Elsewise, high temperatures can slow or halt callus formation. [ 4 ]
Proper care of graft site : After grafting, it is important to nurse the grafted plant back to health for a period of time. Various grafting tapes and waxes are used to protect the scion and stock from excessive water loss. Furthermore, depending on the type of graft, twine or string is used to add structural support to the grafting site. Sometimes it is necessary to prune the site, as the rootstock may produce shoots that inhibit the growth of the scion. [ 4 ]
Cutting tools : It is a good procedure to keep the cutting tool sharp to minimize tissue damage and clean from dirt and other substances to avoid the spread of disease. A good knife for general grafting should have a blade and handle length of about 3 inches and 4 inches respectively. Specialized knives for grafting include bud-grafting knives, surgical knives, and pruning knives. Cleavers, chisels, and saws are utilized when the stock is too large to be cut otherwise.
Disinfecting tools : Treating the cutting tools with disinfectants ensures the grafting site is clear of pathogens . The most commonly used sterilizing agent is alcohol .
Graft seals : Keeps the grafting site hydrated. Good seals should be tight enough to retain moisture while, at the same time, loose enough to accommodate plant growth. Includes specialized types of clay, wax, petroleum jelly, and adhesive tape.
Tying and support materials : Adds support and pressure to the grafting site to hold the stock and scion together before the tissues join, which is especially important in herbaceous grafting. The employed material is often dampened before use to help protect the site from desiccation. Support equipment includes strips made from various substances, twine, nails, and splints. [ 6 ]
Grafting machines : Because grafting can take a lot of time and skill, grafting machines have been created. Automation is particularly popular for seedling grafting in countries such as Japan and Korea where farming land is both limited and used intensively. Certain machines can graft 800 seedlings /hr. [ 5 ] : 496
Approach grafting or inarching is used to join together plants that are otherwise difficult to join. The plants are grown close together, and then joined so that each plant has roots below and growth above the point of union. [ 7 ] Both scion and stock retain their respective parents that may or may not be removed after joining. Also used in pleaching . The graft can be successfully accomplished any time of year. [ 8 ]
Bud grafting (also called chip budding or shield budding ) uses a bud instead of a twig. [ 9 ] Grafting roses is the most common example of bud grafting. In this method a bud is removed from the parent plant, and the base of the bud is inserted beneath the bark of the stem of the stock plant from which the rest of the shoot has been cut. Any extra bud that starts growing from the stem of the stock plant is removed. Examples: roses and fruit trees like peaches.
Budwood is a stick with several buds on it that can be cut out and used for bud grafting. It is a common method of propagation for citrus trees. [ 10 ] [ 11 ] [ 12 ]
In cleft grafting a small cut is made in the stock and then the pointed end of the scion is inserted in the stock. This is best done in the early spring and is useful for joining a thin scion about 1 cm ( 3 ⁄ 8 in) diameter to a thicker branch or stock. It is best if the former has 3–5 buds and the latter is 2–7 cm ( 3 ⁄ 4 – 2 + 3 ⁄ 4 in) in diameter. The branch or stock should be split carefully down the middle to form a cleft about 3 cm ( 1 + 1 ⁄ 8 in) deep. If it is a branch that is not vertical then the cleft should be cut horizontally. The end of the scion should be cut cleanly to a long shallow wedge, preferably with a single cut for each wedge surface, and not whittled. A third cut may be made across the end of the wedge to make it straight across.
Slide the wedge into the cleft so that it is at the edge of the stock and the centre of the wedge faces are against the cambium layer between the bark and the wood. It is preferable if a second scion is inserted in a similar way into the other side of the cleft. This helps to seal off the cleft. Tape around the top of the stock to hold the scion in place and cover with grafting wax or sealing compound. This stops the cambium layers from drying out and also prevents the ingress of water into the cleft.
In whip grafting the scion and the stock are cut slanting and then joined. The grafted point is then bound with tape and covered with a soft sealant to prevent dehydration and infection by germs. The common variation is a whip and tongue graft, which is considered the most difficult to master but has the highest rate of success as it offers the most cambium contact between the scion and the stock. It is the most common graft used in preparing commercial fruit trees. It is generally used with stock less than 1.25 cm ( 1 ⁄ 2 in) diameter, with the ideal diameter closer to 1 cm ( 3 ⁄ 8 in) and the scion should be of roughly the same diameter as the stock.
The stock is cut through on one side only at a shallow angle with a sharp knife. (If the stock is a branch and not the main trunk of the rootstock then the cut surface should face outward from the centre of the tree.) The scion is similarly sliced through at an equal angle starting just below a bud, so that the bud is at the top of the cut and on the other side than the cut face.
In the whip and tongue variation, a notch is cut downwards into the sliced face of the stock and a similar cut upwards into the face of the scion cut. These act as the tongues and it requires some skill to make the cuts so that the scion and the stock marry up neatly. The elongated "Z" shape adds strength, removing the need for a companion rod in the first season (see illustration).
The joint is then taped around and treated with tree-sealing compound or grafting wax. A whip graft without a tongue is less stable and may need added support.
Stub grafting is a technique that requires less stock than cleft grafting, and retains the shape of a tree. Also scions are generally of 6–8 buds in this process.
An incision is made into the branch 1 cm ( 3 ⁄ 8 in) long, then the scion is wedged and forced into the branch. The scion should be at an angle of at most 35° to the parent tree so that the crotch remains strong. The graft is covered with grafting compound.
After the graft has taken, the branch is removed and treated a few centimeters above the graft, to be fully removed when the graft is strong.
The four-flap graft (also called banana graft) is commonly used for pecans, and first became popular with this species in Oklahoma in 1975. [ 13 ] It is heralded for maximum cambium overlap, but is a complex graft. It requires similarly sized diameters for the rootstock and scion. The bark of the rootstock is sliced and peeled back in four flaps, and the hardwood is removed, looking somewhat like a peeled banana. It is a difficult graft to learn.
Awl grafting takes the least resources and the least time. It is best done by an experienced grafter, as it is possible to accidentally drive the tool too far into the stock, reducing the scion's chance of survival. Awl grafting can be done by using a screwdriver to make a slit in the bark, not penetrating the cambium layer completely. Then inset the wedged scion into the incision.
Veneer grafting, or inlay grafting, is a method used for stock larger than 2.5 centimetres (1 in) in diameter. [ 14 ] The scion is recommended to be about as thick as a pencil. Clefts are made of the same size as the scion on the side of the branch, not on top. The scion end is shaped as a wedge, inserted, and wrapped with tape to the scaffolding branches to give it more strength.
Rind grafting involves grafting a small scion onto the end of a thick stock. The thick stock is sawn off, and a ~4 cm long bark-deep cut is made parallel to the stock, from the sawn-off end down, and the bark is separated from the wood on one or both sides. The scion is shaped as a wedge, exposing cambium on both sides, and is pushed in under the back of the stock, with a flat side against the wood.
Tree branches and more often roots of the same species will sometimes naturally graft; this is called inosculation . The bark of the tree may be stripped away when the roots make physical contact with each other, exposing the vascular cambium and allowing the roots to graft together. A group of trees can share water and mineral nutrients via root grafts, which may be advantageous to weaker trees, and may also form a larger rootmass as an adaptation to promote fire resistance and regeneration as exemplified by the California black oak ( Quercus kelloggii ). [ 15 ] Additionally, grafting may protect the group from wind damages as a result of the increased mechanical stability provided by the grafting. [ 16 ] Albino redwoods use root grafting as a form of plant parasitism of normal redwoods.
A problem with root grafts is that they allow transmission of certain pathogens , such as Dutch elm disease . Inosculation also sometimes occurs where two stems on the same tree, shrub or vine make contact with each other. This is common in plants such as strawberries and potato .
Natural grafting is rarely seen in herbaceous plants as those types of plants generally have short-lived roots with little to no secondary growth in the vascular cambium. [ 16 ]
Occasionally, a so-called " graft hybrid " or more accurately graft chimera can occur where the tissues of the stock continue to grow within the scion. Such a plant can produce flowers and foliage typical of both plants as well as shoots intermediate between the two. The best-known example this is probably + Laburnocytisus 'Adamii' , a graft hybrid between Laburnum and Cytisus , which originated in a nursery near Paris, France , in 1825. This small tree bears yellow flowers typical of Laburnum anagyroides , purple flowers typical of Cytisus purpureus and curious coppery-pink flowers that show characteristics of both "parents". Many species of cactus can also produce graft chimeras under the right conditions although they are often created unintentionally and such results are often hard to replicate.
Grafting has been important in flowering research. Leaves or shoots from plants induced to flower can be grafted onto uninduced plants and transmit a floral stimulus that induces them to flower. [ 17 ]
The transmission of plant viruses has been studied using grafting. Virus indexing involves grafting a symptomless plant that is suspected of carrying a virus onto an indicator plant that is very susceptible to the virus.
Grafting can transfer chloroplasts (plant organelles that can conduct photosynthesis ), mitochondrial DNA and the entire cell nucleus containing the genome to potentially make a new species making grafting a form of natural genetic engineering . [ 18 ]
White spruce can be grafted with consistent success by using 8–10 cm (3–4 in) scions of current growth on thrifty 4- to 5-year-old rootstock (Nienstaedt and Teich 1972). [ 19 ] Before greenhouse grafting, rootstocks should be potted in late spring, allowed to make seasonal growth, then subjected to a period of chilling outdoors, or for about 8 weeks in a cool room at 2 °C (Nienstaedt 1966). [ 20 ]
A method of grafting white spruce of seed-bearing age during the time of seed harvest in the fall was developed by Nienstaedt et al. (1958). [ 21 ] Scions of white spruce of 2 ages of wood from 30- to 60-year-old trees were collected in the fall and grafted by 3 methods on potted stock to which different day-length treatments had been applied prior to grafting. The grafted stock were given long-day and natural-day treatments. Survival was 70% to 100% and showed effects of rootstock and post-grafting treatments in only a few cases. Photoperiod and temperature treatments after grafting, however, had considerable effect on scion activity and total growth. The best post-grafting treatment was 4 weeks of long-day treatment followed by 2 weeks of short-day treatment, then 8 weeks of chilling, and finally long-day treatment.
Since grafts of white spruce put on relatively little growth in the 2 years after grafting, techniques for accelerating the early growth were studied by Greenwood (1988) [ 22 ] and others. The cultural regimes used to promote one additional growth cycle in one year involve manipulation of day length and the use of cold storage to satisfy chilling requirements. Greenwood took dormant potted grafts into the greenhouse in early January then gradually raised the temperature during the course of a week until the minimum temperature rose to 15 °C. Photoperiod was increased to 18 hours using incandescent lighting. In this technique, grafts are grown until elongation has been completed, normally by mid-March. Soluble 10-52-10 fertilizer is applied at both ends of the growth cycle and 20-20-20 during the cycle, with irrigation as needed. When growth elongation is complete, day length is reduced to 8 hours using a blackout curtain. Budset follows, and the grafts are held in the greenhouse until mid-May. Grafts are then moved into a cooler at 4 °C for 1000 hours, after which they are moved to a shade frame where they grow normally, with applications of fertilizer and irrigation as in the first cycle. Grafts are moved into cold frames or unheated greenhouse in September until January. Flower induction treatments are begun on grafts that have reached a minimum length of 1.0 m. Repotting from an initial pot size of 4.5 litre to 16 litre containers with a 2:1:1 soil mix of peat moss, loam, and aggregate.
In one of the first accelerated growth experiments, white spruce grafts made in January and February that would normally elongate shortly after grafting, set bud, and remain in that condition until the following spring, were refrigerated for 500, 1000, or 1500 hours beginning in mid-July, and a non-refrigerated control was held in the nursery. [ 22 ] After completion of the cold treatment, the grafts were moved into the greenhouse with an 18-hour photoperiod until late October. Height increment was significantly (P 0.01) influenced by cold treatment. Best results were given by the 1000-hour treatment. [ 22 ]
The refrigeration (cold treatment) phase was subsequently shown to be effective when applied 2 months earlier with proper handling and use of blackout curtains, which allows the second growth cycle to be completed in time to satisfy dormancy requirements before January (Greenwood et al. 1988). [ 22 ]
Grafting is often done for non- woody and vegetable plants ( tomato , cucumber , eggplant and watermelon ). [ 23 ] Tomato grafting is very popular in Asia and Europe, and is gaining popularity in the United States. The main advantage of grafting is for disease-resistant rootstocks. Researchers in Japan developed automated processes using grafting robots as early as 1987. [ 24 ] [ 25 ] [ 26 ] Plastic tubing can be used to prevent desiccation and support the healing at the graft/scion interface. [ 27 ]
As humans began to domesticate plants and animals, horticultural techniques that could reliably propagate the desired qualities of long-lived woody plants needed to be developed. Although grafting is not specifically mentioned in the Hebrew Bible , it is claimed that ancient Biblical text hints at the practice of grafting. For example, Leviticus 19:19 states "[the Hebrew people] shalt not sow their field with mingled seed" ( King James Bible ). Some scholars believe the phrase mingled seeds includes grafting, [ citation needed ] although this interpretation remains contentious among scholars.
Grafting is also mentioned in the New Testament . In Romans 11 , starting at verse 17, there is a discussion about the grafting of wild olive trees concerning the relationship between Jews and Gentiles . [ 28 ] [ 29 ]
By 500 BCE grafting was well established and practiced in the region as the Mishna describes grafting as a commonplace technique used to grow grapevines. [ 30 ]
Evidence for grafting in China is found in Jia Sixie's 6th century CE agricultural treatise Qimin Yaoshu (Essential Skills for the Common People) . [ 31 ] It discusses grafting pear twigs onto crab apple, jujube and pomegranate stock (domesticated apples had not yet arrived in China), as well as grafting persimmons. The Qimin yaoshu refers to older texts that referred to grafting, but those works are missing.
In Greece, a medical record written in 424 BCE contains the first direct reference to grafting. The title of the work is On the Nature of the Child and is thought to be written by a follower of Hippocrates . The language of the author suggests that grafting appeared centuries before this period.
In Rome, Cato the Elder wrote the oldest surviving Latin text in 160 BCE: the De Agri Cultura , and provided extensive instruction about several grafting methods. [ 32 ] Other authors in the region would write about grafting in the following years, however, the publications often featured fallacious scion-stock combinations. [ citation needed ]
Creating lavishly flourished gardens would be a common form of competition among medieval Islamic leaders at the time. Because the region would receive an influx of foreign ornamentals to decorate these gardens, grafting was used much during this period. [ 30 ]
After the fall of the Roman Empire , grafting kept being practiced in Christian monasteries and regained popular appeal among lay people during the Renaissance . The invention of the printing press inspired a number of authors to publish books on gardening that included information on grafting. One example, A New Orchard and Garden: Or, the Best Way for Planting, Graffing, and to Make Any Ground Good for a Rich Orchard, Particularly in the North , was written by William Lawson in 1618. While the book contains practical grafting techniques, some even still used today, it suffers from exaggerated claims of scion-stock compatibility typical of this period. [ citation needed ]
While grafting continued to grow in Europe during the eighteenth century, it was considered unnecessary in the United States as the produce from fruit trees was largely used either to make cider or feed hogs. [ 30 ]
Beginning in 1864, and without warning, grapevines across France began to sharply decline. Thanks to the efforts of scientists such as C. V. Riley and J. E. Planchon , the culprit was identified to be phylloxera , an insect that infests the roots of vines and causes fungal infections. Initially, farmers unsuccessfully attempted to contain the pest by removing and burning affected vines. When it was discovered that phylloxera was an invasive species introduced from North America, some suggested importing rootstock from the region as the North American vines were resistant to the pest. Others, opposed to the idea, argued that American rootstocks would imbue the French grapes with an undesirable taste; they instead preferred to inject the soil with expensive pesticides. Ultimately, grafting French vines onto American rootstocks became prevalent throughout the region, creating new grafting techniques and machines. American rootstocks had trouble adapting to the high soil pH value of some regions in France so the final solution to the pandemic was to hybridize the American and French variants. [ 30 ] | https://en.wikipedia.org/wiki/Grafting |
Graham's law of effusion (also called Graham's law of diffusion ) was formulated by Scottish physical chemist Thomas Graham in 1848. [ 1 ] Graham found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the molar mass of its particles . [ 1 ] This formula is stated as:
where:
Graham's law states that the rate of diffusion or of effusion of a gas is inversely proportional to the square root of its molecular weight. Thus, if the molecular weight of one gas is four times that of another, it would diffuse through a porous plug or escape through a small pinhole in a vessel at half the rate of the other (heavier gases diffuse more slowly). A complete theoretical explanation of Graham's law was provided years later by the kinetic theory of gases . Graham's law provides a basis for separating isotopes by diffusion—a method that came to play a crucial role in the development of the atomic bomb. [ 2 ]
Graham's law is most accurate for molecular effusion which involves the movement of one gas at a time through a hole. It is only approximate for diffusion of one gas in another or in air, as these processes involve the movement of more than one gas. [ 2 ]
In the same conditions of temperature and pressure, the molar mass is proportional to the mass density . Therefore, the rates of diffusion of different gases are inversely proportional to the square roots of their mass densities:
where:
First Example: Let gas 1 be H 2 and gas 2 be O 2 . (This example is solving for the ratio between the rates of the two gases)
Therefore, hydrogen molecules effuse four times faster than those of oxygen. [ 1 ]
Graham's law can also be used to find the approximate molecular weight of a gas if one gas is a known species, and if there is a specific ratio between the rates of two gases (such as in the previous example). The equation can be solved for the unknown molecular weight.
Graham's law was the basis for separating uranium-235 from uranium-238 found in natural uraninite (uranium ore) during the Manhattan Project to build the first atomic bomb. The United States government built a gaseous diffusion plant at the Clinton Engineer Works in Oak Ridge, Tennessee , at the cost of $479 million (equivalent to $6.44 billion in 2023). In this plant, uranium from uranium ore was first converted to uranium hexafluoride and then forced repeatedly to diffuse through porous barriers, each time becoming a little more enriched in the slightly lighter uranium-235 isotope. [ 2 ]
Second Example: An unknown gas diffuses 0.25 times as fast as He. What is the molar mass of the unknown gas?
Using the formula of gaseous diffusion, we can set up this equation.
Which is the same as the following because the problem states that the rate of diffusion of the unknown gas relative to the helium gas is 0.25.
Rearranging the equation results in
Graham's research on the diffusion of gases was triggered by his reading about the observations of German chemist Johann Döbereiner that hydrogen gas diffused out of a small crack in a glass bottle faster than the surrounding air diffused in to replace it. Graham measured the rate of diffusion of gases through plaster plugs, through very fine tubes, and through small orifices. In this way he slowed down the process so that it could be studied quantitatively. He first stated in 1831 that the rate of effusion of a gas is inversely proportional to the square root of its density, and later in 1848 showed that this rate is inversely proportional to the square root of the molar mass. [ 1 ] Graham went on to study the diffusion of substances in solution and in the process made the discovery that some apparent solutions actually are suspensions of particles too large to pass through a parchment filter. He termed these materials colloids , a term that has come to denote an important class of finely divided materials. [ 3 ]
Around the time Graham did his work, the concept of molecular weight was being established largely through the measurements of gases. Daniel Bernoulli suggested in 1738 in his book Hydrodynamica that heat increases in proportion to the velocity, and thus kinetic energy, of gas particles. Italian physicist Amedeo Avogadro also suggested in 1811 that equal volumes of different gases contain equal numbers of molecules. Thus, the relative molecular weights of two gases are equal to the ratio of weights of equal volumes of the gases. Avogadro's insight together with other studies of gas behaviour provided a basis for later theoretical work by Scottish physicist James Clerk Maxwell to explain the properties of gases as collections of small particles moving through largely empty space. [ 4 ]
Perhaps the greatest success of the kinetic theory of gases, as it came to be called, was the discovery that for gases, the temperature as measured on the Kelvin (absolute) temperature scale is directly proportional to the average kinetic energy of the gas molecules. Graham's law for diffusion could thus be understood as a consequence of the molecular kinetic energies being equal at the same temperature. [ 5 ]
The rationale of the above can be summed up as follows:
Kinetic energy of each type of particle (in this example, Hydrogen and Oxygen, as above) within the system is equal, as defined by thermodynamic temperature :
Which can be simplified and rearranged to:
or:
Ergo, when constraining the system to the passage of particles through an area, Graham's law appears as written at the start of this article. | https://en.wikipedia.org/wiki/Graham's_law |
Graham Charles Wood FRS (6 February 1934—4 November 2016) was an English corrosion scientist. [ 1 ]
Born in Farnborough, he went on to study metallurgy at Cambridge . [ 2 ] Following postdoctoral work at Cambridge, he moved to Manchester, where his career in corrosion science would be based, starting at the Manchester College of Science and Technology (now the University of Manchester ) where he joined the Department of Chemical Engineering as a lecturer in corrosion science. In 1972 he established and led the Corrosion and Protection Centre at UMIST ( University of Manchester Institute of Science and Technology ) as professor of corrosion science. In 1973 he helped to establish a consulting organisation, Corrosion and Protection Centre Industrial Service (CAPCIS – now part of Intertek ).
Graham served terms as President of the Corrosion and Protection Association and the Institution of Corrosion Science and Technology ( Institute of Corrosion ), and chaired the National Council for Corrosion Societies. He also served as a UK representative on the International Corrosion Council for 15 years, for which he also served as vice-chair and chair.
At UMIST, he held several administerial roles, including vice-principal for academic development, dean, and pro-vice chancellor.
He was elected Fellow of the Royal Academy of Engineering in 1990, and Fellow of the Royal Society in 1997. [ 1 ] | https://en.wikipedia.org/wiki/Graham_Charles_Wood |
William Graham Richards (1 October 1939 – 11 February 2025) was an English chemist and Emeritus Fellow of Brasenose College, Oxford . [ 3 ] He served as head of the department of chemistry at the University of Oxford from 1997 to 2006. [ 4 ]
Richards was a pioneer in the field of computer-aided molecular design, in particular its application to the pharmaceuticals industry. He was the founding scientist of Oxford Molecular Ltd., [ 5 ] and introduced a novel model for the funding of research at Oxford University, which has been copied elsewhere. [ 6 ] [ 7 ] Richards was one of the scientific co-founders of Oxford Molecular Limited (OMG). The company developed software for modelling of small molecules and proteins, and drug design. Benefiting from the economic and legal changes, the company was floated on the London Stock Exchange in 1992, making the university £10 million. The company was worth £450 million at its peak but was eventually sold for £70 million. Richards published more than 300 scientific papers, [ 8 ] including 15 books.
Graham Richards was born on 1 October 1939 in Hoylake , England, to Percy Richards and Gwendoline Julia Richards (née Evans). [ 9 ] [ 2 ] Both parents were of Welsh extraction. [ 2 ] Richards was educated at Birkenhead School . [ 10 ] He won a scholarship to Brasenose College, Oxford , starting his studies there in 1958. [ 10 ]
He received his bachelor's degree in Chemistry with first class honours from the University of Oxford [ 4 ] in 1961. [ citation needed ] He then studied the electronic spectroscopy of diatomic molecules with Richard F. Barrow, [ 4 ] earning his Master of Arts and Doctor of Philosophy [ 1 ] degrees from the University of Oxford in 1964. [ citation needed ]
After his DPhil, Richards continued his spectroscopic work with fellowships in Oxford (ICI Research Fellowship, Junior Research Fellowship at Balliol College) and Paris, France (Centre de Mécanique Ondulatoire Appliquée). [ 11 ]
He soon returned to Oxford as a research fellow at Balliol College, Oxford (1964–1966). He was promoted to a lecturer at Oxford University (1966–1994), to reader (1994–1996), and to professor (1996–2007). He served as chairman of the chemistry department from 1997 to 2006. [ 4 ] Richards celebrated his formal retirement from the University of Oxford on 18 May 2007. [ 12 ] He was [ when? ] an Emeritus Fellow of Brasenose College. [ 3 ]
In the fourth year of his degree course Richards' research project led him to using Oxford's Ferranti Mercury computer to solve integrals . [ 2 ] During a fellowship year in France at Centre de Mécanique Ondulatoire Appliquée, he was able to use more powerful computers. [ 13 ] Returning to Oxford, he worked on ab initio computations and applied computational techniques to solving quantum mechanical problems in theoretical chemistry, in particular studying spin-orbit coupling . [ 5 ]
His influential paper Third age of quantum chemistry (1979) marked the development of computational techniques for theoretical analysis whose precision equaled or surpassed experimental results. [ 14 ] [ 15 ] [ 16 ]
"The work represents perhaps a near perfect instance of theory being in harmony with experiment, each aspect vital to the other and the combination much more than the sum of the separate parts." – Graham Richards [ 16 ] [ 14 ]
Richards saw the potential to apply computer techniques for examining the structure and properties of compounds in the area of pharmaceutical applications. He became a pioneer in the field of computer-aided molecular design. He was the first to produce coloured images modelling molecular structure graphically, [ 2 ] and introduced many of the techniques now widely used in academia and industry. [ 5 ]
In 1982, Richards became a founding member of the Molecular Graphics Society (now the Molecular Graphics and Modelling Society, MGMS). [ 17 ] The society started the Journal of Molecular Graphics in 1983. He served as the editor-in-chief of the journal from 1984 to 1996. The journal's name changed to Journal of Molecular Graphics and Modelling in 1997. [ 18 ]
In 1989, Richards was the scientific co-founder (with Tony Marchington , David Ricketts, James Hiddleston , and Anthony Rees ) of Oxford Molecular Limited. The company developed software for modelling of small molecules and proteins, and drug design. [ 19 ]
The company was possible in part because of economic and legal changes under the government of Margaret Thatcher that enabled British universities to become involved with venture capital and technology transfer . As Oxford Molecular Group, Ltd. (OMG) the company was floated on the London Stock Exchange in 1992, making the university £10 million. The company was worth £450 million at its peak but was eventually sold for £70 million. [ 2 ] [ 20 ] It was one of several companies that combined to form Accelrys in 2001. [ 21 ]
Richards was instrumental in raising £64 million to fund a new laboratory for Oxford University through an innovative funding approach. £20 million worth of funding began with an "unusual collaboration" between Richards and David Norwood . [ 6 ] Norwood then arranged for Beeson-Gregory to provide £20 million in exchange for half the University's equity share of any spin-out companies emanating from the Chemistry Department for 15 years. In 2003, Beeson-Gregory and Evolution Group merged, later creating a subsidiary, IP2IPO ("Intellectual property to initial public offering"). [ 6 ] [ 22 ] Richards became a non-executive director of IP2IPO in 2001, and non-executive chairman of IP2IPO in 2004. [ 20 ]
Through this arrangement the Chemistry Department has contributed over £100 million to the University of Oxford. [ 10 ]
Richards served as a director of ISIS Innovation Ltd., the University of Oxford's technology transfer company. [ 8 ] It became Oxford University Innovation as of June 2016. [ 23 ] It has brought around 60 spin-out companies into existence. [ 2 ]
The Financial Times has described the approach as "the way universities should be financed in the future". [ 22 ] : 188
Richards also introduced the use of distributed computing in pharmaceutical design. Started in 2000, his Screensaver Lifesaver project exploited idle time on more than 3.5 million personal computers in over 200 countries, whose owners agreed to be involved and downloaded the project's screensaver . Using idle time from these computers, the project's software created a virtual supercomputer that screened billions of compounds against protein targets, searching for possible drug treatments for cancer , anthrax and smallpox . [ 5 ] [ 24 ] [ 25 ] The project involved collaboration between Intel , United Devices , and the Centre for Computational Drug Discovery at the University of Oxford, [ 26 ] headed by Richards [ 8 ] and funded by the National Foundation for Cancer Research (NFCR). [ 26 ]
Richards formed the spin-out company InhibOx Ltd. in 2001. [ 27 ] [ 10 ] InhibOx applied cloud computing techniques to computational chemistry and drug discovery, and developed a searchable database of small-molecules called Scopius. [ 28 ] In 2002, Richards donated his shares, twenty-five per cent of the company, to the National Foundation for Cancer Research. [ 29 ] [ 24 ] In 2017, InhibOx relaunched as Oxford Drug Design Ltd., [ 30 ] with a new focus on antibiotic discovery. [ 27 ]
As of 2011 [update] Richards joined the Science Advisory Panel of Oxford Medical Diagnostics. [ 8 ] He was a non-executive director of IP Group plc, having also served as its chairman. [ 5 ] [ 31 ]
Richards married his first wife, Jessamy Kershaw, on 12 December 1970. She died of cancer in November 1988. On 5 October 1996, Richards married Mary Elizabeth Phillips, director of research planning at University College London . He had two sons and three stepchildren. [ 2 ] Richards died on 11 February 2025, at the age of 85. [ 32 ]
Richards was a council member of the Royal Society of Chemistry and of the Royal Institution , and a Fellow of the Royal Society , [ 5 ] and was appointed Commander of the Order of the British Empire (CBE) in the 2001 Birthday Honours for services to Chemistry. [ 34 ] The Times Higher Education Supplement (2006) considered Richards to be one of twelve academic "super-earners" in the United Kingdom. [ 2 ] [ 35 ] Times magazine's first Eureka issue (2010) included Richards in its list of the top 100 British scientists. [ 36 ] Richards' work has been acknowledged through a number of more formal awards and honours, including: | https://en.wikipedia.org/wiki/Graham_Richards |
In organic chemistry , the Graham reaction is an oxidation reaction that converts an amidine into a diazirine using a hypohalite reagent. The halide of the hypohalite oxidant, or another similar anionic additive to the reaction, is retained as a substituent on the diazirine product. The reaction was first reported in 1965. [ 1 ] Various reaction mechanisms have been proposed. [ 2 ] [ 3 ]
Amidine substrates for the reaction can easily be formed from the corresponding nitriles via the Pinner reaction . The halide substituent in the diazirine product can be displaced by a various nucleophiles . [ 4 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Graham_reaction |
In graph theory , the Graham–Pollak theorem states that the edges of an n {\displaystyle n} -vertex complete graph cannot be partitioned into fewer than n − 1 {\displaystyle n-1} complete bipartite graphs . [ 1 ] It was first published by Ronald Graham and Henry O. Pollak in two papers in 1971 and 1972 (crediting Hans Witsenhausen for a key lemma), in connection with an application to telephone switching circuitry. [ 2 ] [ 3 ]
The theorem has since become well known and repeatedly studied and generalized in graph theory, in part because of its elegant proof using techniques from algebraic graph theory . [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] More strongly, Aigner & Ziegler (2018) write that all proofs are somehow based on linear algebra : "no combinatorial proof for this result is known". [ 1 ]
A partition into exactly n − 1 {\displaystyle n-1} complete bipartite graphs is easy to obtain: just order the vertices, and for each vertex except the last, form a star connecting it to all later vertices in the ordering. [ 1 ] Other partitions are also possible.
The proof of the Graham–Pollak theorem described by Aigner & Ziegler (2018) (following Tverberg 1982 ) defines a real variable x i {\displaystyle x_{i}} for each vertex v i ∈ V {\displaystyle v_{i}\in V} , where V {\displaystyle V} denotes the set of all vertices in the graph. Let the left sides and right sides of the k {\displaystyle k} th bipartite graph be denoted L k {\displaystyle L_{k}} and R k {\displaystyle R_{k}} , respectively and for any set S {\displaystyle S} of vertices define X ( S ) {\displaystyle X(S)} to be the sum of variables for vertices in S {\displaystyle S} :
Then, in terms of this notation, the fact that the bipartite graphs partition the edges of the complete graph can be expressed as the equation
Now consider the system of linear equations that sets X ( V ) = 0 {\displaystyle X(V)=0} and X ( L k ) = 0 {\displaystyle X(L_{k})=0} for each k {\displaystyle k} . Any solution to this system of equations would also obey the nonlinear equations
But a sum of squares of real variables can only be zero if all the individual variables are zero, the trivial solution to the system of linear equations.
If there were fewer than n − 1 {\displaystyle n-1} complete bipartite graphs, the system of equations would have fewer than n {\displaystyle n} equations in n {\displaystyle n} unknowns and would have a nontrivial solution, a contradiction. So the number of complete bipartite graphs must be at least n − 1 {\displaystyle n-1} . [ 1 ] [ 4 ]
Graham and Pollak study a more general graph labeling problem, in which the vertices of a graph should be labeled with equal-length strings of the characters "0", "1", and "✶", in such a way that the distance between any two vertices equals the number of string positions where one vertex is labeled with a 0 and the other is labeled with a 1. A labeling like this with no "✶" characters would give an isometric embedding into a hypercube , something that is only possible for graphs that are partial cubes , and in one of their papers Graham and Pollak call a labeling that allows "✶" characters an embedding into a "squashed cube". [ 3 ]
For each position of the label strings, one can define a complete bipartite graph in which one side of the bipartition consists of the vertices labeled with 0 in that position and the other side consists of the vertices labeled with 1, omitting the vertices labeled "✶". For the complete graph, every two vertices are at distance one from each other, so every edge must belong to exactly one of these complete bipartite graphs. In this way, a labeling of this type for the complete graph corresponds to a partition of its edges into complete bipartite graphs,
with the lengths of the labels corresponding to the number of graphs in the partition. [ 3 ]
Noga Alon , Michael Saks , and Paul Seymour formulated a conjecture in the early 1990s that, if true, would significantly generalize the Graham–Pollak theorem: they conjectured that, whenever a graph of chromatic number k + 1 {\displaystyle k+1} has its edges partitioned into complete bipartite subgraphs, at least k {\displaystyle k} subgraphs are needed. Equivalently, their conjecture states that edge-disjoint unions of k {\displaystyle k} complete bipartite graphs can always be colored with at most k + 1 {\displaystyle k+1} colors. The conjecture was disproved by Huang and Sudakov in 2012, who constructed families of graphs formed as edge-disjoint unions of k {\displaystyle k} complete bipartite graphs that require Ω ( k 6 / 5 ) {\displaystyle \Omega (k^{6/5})} colors. [ 9 ] More strongly, the number of colors can be as large as exp log 2 − o ( 1 ) k {\displaystyle \exp \log ^{2-o(1)}k} , tight up to the o ( 1 ) {\displaystyle o(1)} term in the exponent. [ 10 ]
The biclique partition problem takes as input an arbitrary undirected graph, and asks for a partition of its edges into a minimum number of complete bipartite graphs. It is NP-hard , but fixed-parameter tractable . The best approximation algorithm known for the problem has an approximation ratio of O ( n / log n ) {\displaystyle O(n/\log n)} . [ 11 ] [ 12 ] | https://en.wikipedia.org/wiki/Graham–Pollak_theorem |
GRAIL, Inc. is an American biotechnology company based in Menlo Park, California . It was previously a subsidiary of Illumina started as a startup seeking to develop an early cancer screening test for people who do not have symptoms. [ 4 ] Grail was spun-out from Illumina on June 24, 2024.
Their liquid biopsy (also called multi-cancer early detection test [ 5 ] ) was launched in June 2021 and is called the Galleri test. Promoted as groundbreaking, the test performed poorly in testing and Grail has subsequently faced discontent and legal action from investors claiming they have been misled about the test's potential. [ 6 ]
Illumina's own research showed that repeatedly sequencing DNA in the bloodstream made it possible to detect floating bits of DNA from cancer cells more accurately. [ 7 ] It initially aimed to recruit greater than 100,000 people into its clinical trials in order to accumulate the sizeable data required to detect and interpret cancer biomarkers . [ 8 ]
Grail calls its liquid biopsy for early cancer the Galleri test or the Galleri multi-cancer early detection (MCED) test, one of three multi-cancer screening tests which was under investigation as of November 2020. [ 9 ] [ 10 ]
Despite Grail promoting the test as a "groundbreaking and potentially life-saving advance", the results of early trials were poor. [ 6 ] A subsequent large-scale NHS England trial has been described by experts as overhyped and unethical. Grail is facing discontent and legal action from investors who suspect that its cancer testing claims misled them. [ 6 ]
Grail began as a San Francisco biotechnology and pharmaceutical startup company in 2015, the parent company being Illumina of San Diego, which produces most of the DNA sequencing machines that scientists use to study human biology and diagnose rare genetic diseases. [ 7 ] [ 11 ] [ 12 ] [ 13 ] Richard Klausner , then chief medical officer at Illumina and former director of the National Cancer Institute , advocated for the new business. According to the San Francisco Business Times , he correctly predicted how DNA sequencing technology would make it possible to detect evidence of a tumor from a blood sample. [ 14 ] He also joined Grail's board of directors. [ 15 ] According to Forbes in 2017, 20% of Grail's profits are kept by Illumina. [ 7 ]
In September 2020, Illumina announced an agreement to purchase Grail outright for $7.1 billion. [ 16 ]
On November 27, 2020, Grail announced a commercial partnership with the National Health Service (England) (NHS), to trial the Galleri test, reporting in 2026.
In March 2021, the Federal Trade Commission (FTC) sued to block the vertical merger . [ 17 ] [ 18 ] In September 2022, an administrative judge ruled against the FTC's position on antitrust grounds. [ 19 ]
In June 2023, Grail disclosed that letters were mailed to 408 patients incorrectly informing them that they may have cancer. The company blamed the incident on PWNHealth, saying that it was due to a software configuration issue, not due to incorrect Galleri test results. [ 20 ] [ 21 ]
In July 2023, it was reported that three separate lawsuits were filed against Grail by former female employees; the allegations included that Grail created a "frat house” culture and a “sexually charged, hostile work environment”. [ 22 ]
In October 2023, the European Commission ordered Grail to be divested from Illumina within the next twelve months. [ 23 ] The European Commission (EC) has since approved Illumina's divestment plan for separating from Grail. Illumina has set a goal of finalizing the divestment terms by the end of the second quarter of 2024. In April 2024, the EC approved Illumina's plan, allowing Illumina to explore either a trade sale or a capital markets transaction (spin-off) to divest Grail. In May 2024, Illumina publicly filed a Form 10 registration statement with the U.S. SEC, a necessary step for a potential capital markets separation of Grail. If a capital markets transaction occurs, Illumina must capitalize Grail with around $1 billion to fund 2.5 years of operations per the EC's divestment plan. [ 24 ] [ 25 ] [ 26 ] The spinoff was completed in June 2024, with Grail trading on the Nasdaq with ticker symbol GRAL and Illumina retaining a 14.5% share of Grail. [ 1 ] | https://en.wikipedia.org/wiki/Grail_(company) |
In materials science , a grain boundary is the interface between two grains, or crystallites , in a polycrystalline material. Grain boundaries are two-dimensional defects in the crystal structure , and tend to decrease the electrical and thermal conductivity of the material. Most grain boundaries are preferred sites for the onset of corrosion [ 1 ] and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep . [ 2 ] On the other hand, grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve mechanical strength, as described by the Hall–Petch relationship.
It is convenient to categorize grain boundaries according to the extent of misorientation between the two grains. Low-angle grain boundaries ( LAGB ) or subgrain boundaries are those with a misorientation less than about 15 degrees. [ 3 ] Generally speaking they are composed of an array of dislocations and their properties and structure are a function of the misorientation. In contrast the properties of high-angle grain boundaries , whose misorientation is greater than about 15 degrees (the transition angle varies from 10 to 15 degrees depending on the material), are normally found to be independent of the misorientation. However, there are 'special boundaries' at particular orientations whose interfacial energies are markedly lower than those of general high-angle grain boundaries.
The simplest boundary is that of a tilt boundary where the rotation axis is parallel to the boundary plane. This boundary can be conceived as forming from a single, contiguous crystallite or grain which is gradually bent by some external force. The energy associated with the elastic bending of the lattice can be reduced by inserting a dislocation, which is essentially a half-plane of atoms that act like a wedge, that creates a permanent misorientation between the two sides. As the grain is bent further, more and more dislocations must be introduced to accommodate the deformation resulting in a growing wall of dislocations – a low-angle boundary. The grain can now be considered to have split into two sub-grains of related crystallography but notably different orientations.
An alternative is a twist boundary where the misorientation occurs around an axis that is perpendicular to the boundary plane. This type of boundary incorporates two sets of screw dislocations . If the Burgers vectors of the dislocations are orthogonal, then the dislocations do not strongly interact and form a square network. In other cases, the dislocations may interact to form a more complex hexagonal structure.
These concepts of tilt and twist boundaries represent somewhat idealized cases. The majority of boundaries are of a mixed type, containing dislocations of different types and Burgers vectors, in order to create the best fit between the neighboring grains.
If the dislocations in the boundary remain isolated and distinct, the boundary can be considered to be low-angle. If deformation continues, the density of dislocations will increase and so reduce the spacing between neighboring dislocations. Eventually, the cores of the dislocations will begin to overlap and the ordered nature of the boundary will begin to break down. At this point the boundary can be considered to be high-angle and the original grain to have separated into two entirely separate grains.
In comparison to low-angle grain boundaries, high-angle boundaries are considerably more disordered, with large areas of poor fit and a comparatively open structure. Indeed, they were originally thought to be some form of amorphous or even liquid layer between the grains. However, this model could not explain the observed strength of grain boundaries and, after the invention of electron microscopy , direct evidence of the grain structure meant the hypothesis had to be discarded. It is now accepted that a boundary consists of structural units which depend on both the misorientation of the two grains and the plane of the interface. The types of structural unit that exist can be related to the concept of the coincidence site lattice , in which repeated units are formed from points where the two misoriented \
In coincident site lattice (CSL) theory, the degree of fit (Σ) between the structures of the two grains is described by the reciprocal of the ratio of coincidence sites to the total number of sites. [ 4 ]
In this framework, it is possible to draw the lattice for the two grains and count the number of atoms that are shared (coincidence sites), and the total number of atoms on the boundary (total number of site). For example, when Σ=3 there will be one atom of each three that will be shared between the two lattices. Thus a boundary with high Σ might be expected to have a higher energy than one with low Σ. Low-angle boundaries, where the distortion is entirely accommodated by dislocations, are Σ1. Some other low-Σ boundaries have special properties, especially when the boundary plane is one that contains a high density of coincident sites. Examples include coherent twin boundaries (e.g., Σ3) and high-mobility boundaries in FCC materials (e.g., Σ7). Deviations from the ideal CSL orientation may be accommodated by local atomic relaxation or the inclusion of dislocations at the boundary.
A boundary can be described by the orientation of the boundary to the two grains and the 3-D rotation required to bring the grains into coincidence. Thus a boundary has 5 macroscopic degrees of freedom . However, it is common to describe a boundary only as the orientation relationship of the neighbouring grains. Generally, the convenience of ignoring the boundary plane orientation, which is very difficult to determine, outweighs the reduced information.
The relative orientation of the two grains is described using the rotation matrix :
Using this system the rotation angle θ is:
while the direction [uvw] of the rotation axis is:
The nature of the crystallography involved limits the misorientation of the boundary. A completely random polycrystal, with no texture, thus has a characteristic distribution of boundary misorientations (see figure). However, such cases are rare and most materials will deviate from this ideal to a greater or lesser degree.
The energy of a low-angle boundary is dependent on the degree of misorientation between the neighbouring grains up to the transition to high-angle status. In the case of simple tilt boundaries the energy of a boundary made up of dislocations with Burgers vector b and spacing h is predicted by the Read–Shockley equation :
where:
with G {\displaystyle G} is the shear modulus , ν {\displaystyle \nu } is Poisson's ratio , and r 0 {\displaystyle r_{0}} is the radius of the dislocation core. It can be seen that as the energy of the boundary increases the energy per dislocation decreases. Thus there is a driving force to produce fewer, more misoriented boundaries (i.e., grain growth ).
The situation in high-angle boundaries is more complex. Although theory predicts that the energy will be a minimum for ideal CSL configurations, with deviations requiring dislocations and other energetic features, empirical measurements suggest the relationship is more complicated. Some predicted troughs in energy are found as expected while others missing or substantially reduced.
Surveys of the available experimental data have indicated that simple relationships such as low Σ {\displaystyle \Sigma } are misleading:
It is concluded that no general and useful criterion for low energy can be enshrined in a simple geometric framework. Any understanding of the variations of interfacial energy must take account of the atomic structure and the details of the bonding at the interface. [ 5 ]
The excess volume is another important property in the characterization of grain boundaries. Excess volume was first proposed by Bishop in a private communication to Aaron and Bolling in 1972. [ 6 ] It describes how much expansion is induced by the presence of a GB and is thought that the degree and susceptibility of segregation is directly proportional to this. Despite the name the excess volume is actually a change in length, this is because of the 2D nature of GBs the length of interest is the expansion normal to the GB plane. The excess volume ( δ V {\displaystyle \delta V} ) is defined in the following way,
at constant temperature T {\displaystyle T} , pressure p {\displaystyle p} and number of atoms n i {\displaystyle n_{i}} . Although a rough linear relationship between GB energy and excess volume exists the orientations where this relationship is violated can behave significantly differently affecting mechanical and electrical properties. [ 7 ]
Experimental techniques have been developed which directly probe the excess volume and have been used to explore the properties of nanocrystalline copper and nickel . [ 8 ] [ 9 ] Theoretical methods have also been developed [ 10 ] and are in good agreement. A key observation is that there is an inverse relationship with the bulk modulus meaning that the larger the bulk modulus (the ability to compress a material) the smaller the excess volume will be, there is also direct relationship with the lattice constant, this provides methodology to find materials with a desirable excess volume for a specific application.
The movement of grain boundaries (HAGB) has implications for recrystallization and grain growth while subgrain boundary (LAGB) movement strongly influences recovery and the nucleation of recrystallization.
A boundary moves due to a pressure acting on it. It is generally assumed that the velocity is directly proportional to the pressure with the constant of proportionality being the mobility of the boundary. The mobility is strongly temperature dependent and often follows an Arrhenius type relationship :
The apparent activation energy (Q) may be related to the thermally activated atomistic processes that occur during boundary movement. However, there are several proposed mechanisms where the mobility will depend on the driving pressure and the assumed proportionality may break down.
It is generally accepted that the mobility of low-angle boundaries is much lower than that of high-angle boundaries. The following observations appear to hold true over a range of conditions:
Since low-angle boundaries are composed of arrays of dislocations and their movement may be related to dislocation theory. The most likely mechanism, given the experimental data, is that of dislocation climb, rate limited by the diffusion of solute in the bulk. [ 11 ]
The movement of high-angle boundaries occurs by the transfer of atoms between the neighbouring grains. The ease with which this can occur will depend on the structure of the boundary, itself dependent on the crystallography of the grains involved, impurity atoms and the temperature. It is possible that some form of diffusionless mechanism (akin to diffusionless phase transformations such as martensite ) may operate in certain conditions. Some defects in the boundary, such as steps and ledges, may also offer alternative mechanisms for atomic transfer.
Since a high-angle boundary is imperfectly packed compared to the normal lattice it has some amount of free space or free volume where solute atoms may possess a lower energy. As a result, a boundary may be associated with a solute atmosphere that will retard its movement. Only at higher velocities will the boundary be able to break free of its atmosphere and resume normal motion.
Both low- and high-angle boundaries are retarded by the presence of particles via the so-called Zener pinning effect. This effect is often exploited in commercial alloys to minimise or prevent recrystallization or grain growth during heat-treatment .
Grain boundaries are the preferential site for segregation of impurities, which may form a thin layer with a different composition from the bulk and a variety of atomic structures that are distinct from the abutting crystalline phases. For example, a thin layer of silica, which also contains impurity cations, is often present in silicon nitride.
Grain boundary complexions were introduced by Ming Tang, Rowland Cannon, and W. Craig Carter in 2006. [ 12 ]
These grain boundary phases are thermodynamically stable and can be considered as quasi-two-dimensional phase, which may undergo to transition, similar to those of bulk phases. In this case structure and chemistry abrupt changes are possible at a critical value of a thermodynamic parameter like temperature or pressure. [ 13 ] This may strongly affect the macroscopic properties of the material, for example the electrical resistance or creep rates. [ 14 ] Grain boundaries can be analyzed using equilibrium thermodynamics but cannot be considered as phases, because they do not satisfy Gibbs' definition: they are inhomogeneous, may have a gradient of structure, composition or properties. For this reasons they are defined as complexion: an interfacial material or stata that is in thermodynamic equilibrium with its abutting phases, with a finite and stable thickness (that is typically 2–20 Å). A complexion need the abutting phase to exist and its composition and structure need to be different from the abutting phase. Contrary to bulk phases, complexions also depend on the abutting phase. For example, silica rich amorphous layer present in Si 3 N 3 , is about 10 Å thick, but for special boundaries this equilibrium thickness is zero. [ 15 ] Complexion can be grouped in 6 categories, according to their thickness: monolayer, bilayer, trilayer, nanolayer (with equilibrium thickness between 1 and 2 nm) and wetting. In the first cases the thickness of the layer will be constant; if extra material is present it will segregate at multiple grain junction, while in the last case there is no equilibrium thickness and this is determined by the amount of secondary phase present in the material. One example of grain boundary complexion transition is the passage from dry boundary to biltilayer in Au-doped Si, which is produced by the increase of Au. [ 16 ]
Grain boundaries can cause failure mechanically by embrittlement through solute segregation (see Hinkley Point A nuclear power station ) but they also can detrimentally affect the electronic properties. In metal oxides it has been shown theoretically that at the grain boundaries in Al 2 O 3 and MgO the insulating properties can be significantly diminished. [ 17 ] Using density functional theory computer simulations of grain boundaries have shown that the band gap can be reduced by up to 45%. [ 18 ] In the case of metals grain boundaries increase the resistivity as the size of the grains relative to the mean free path of other scatters becomes significant. [ 19 ]
It is known that most materials are polycrystalline and contain grain boundaries and that grain boundaries can act as sinks and transport pathways for point defects. However experimentally and theoretically determining what effect point defects have on a system is difficult. [ 20 ] [ 21 ] [ 22 ] Interesting examples of the complications of how point defects behave has been manifested in the temperature dependence of the Seebeck effect. [ 23 ] In addition the dielectric and piezoelectric response can be altered by the distribution of point defects near grain boundaries. [ 24 ] Mechanical properties can also be significantly influenced with properties such as the bulk modulus and damping being influenced by changes to the distribution of point defects within a material. [ 25 ] [ 26 ] It has also been found that the Kondo effect within graphene can be tuned due to a complex relationship between grain boundaries and point defects. [ 27 ] Recent theoretical calculations have revealed that point defects can be extremely favourable near certain grain boundary types and significantly affect the electronic properties with a reduction in the band gap. [ 28 ]
There has been a significant amount of work experimentally to observe both the structure and measure the properties of grain boundaries but the five dimensional degrees of freedom of grain boundaries within complex polycrystalline networks has not yet been fully understood and thus there is currently no method to control the structure and properties of most metals and alloys with atomic precision. [ 29 ] Part of the problem is related to the fact that much of the theoretical work to understand grain boundaries is based upon construction of bicrystal (two) grains which do not represent the network of grains typically found in a real system and the use of classical force fields such as the embedded atom method often do not describe the physics near the grains correctly and density functional theory could be required to give realistic insights. Accurate modelling of grain boundaries both in terms of structure and atomic interactions could have the effect of improving engineering which could reduce waste and increase efficiency in terms of material usage and performance. From a computational point of view much of the research on grain boundaries has focused on bi-crystal systems, these are systems which only consider two grain boundaries. There has been recent work which has made use of novel grain evolution models which show that there are substantial differences in the material properties associated with whether curved or planar grains are present. [ 30 ] | https://en.wikipedia.org/wiki/Grain_boundary |
Grain boundary sliding (GBS) is a material deformation mechanism where grains slide against each other. This occurs in polycrystalline material under external stress at high homologous temperature (above ~0.4 [ 1 ] ) and low strain rate and is intertwined with creep . Homologous temperature describes the operating temperature relative to the melting temperature of the material. There are mainly two types of grain boundary sliding: Rachinger sliding, [ 2 ] and Lifshitz sliding. [ 3 ] Grain boundary sliding usually occurs as a combination of both types of sliding. Boundary shape often determines the rate and extent of grain boundary sliding. [ 4 ]
Grain boundary sliding is a motion to prevent intergranular cracks from forming. Keep in mind that at high temperatures, many processes are underway, and grain boundary sliding is only one of the processes happening. Therefore it is not surprising that Nabarro Herring and Coble creep is dependent on grain boundary sliding. During high temperature creep, wavy grain boundaries are often observed. We can simulate this type of boundary with a sinusoidal curve, with amplitude h and wavelength λ. Steady-state creep rate increases with rising λ/h ratios. At high λ and high homologous temperatures, grain boundary sliding is controlled by lattice diffusion (Nabarro-Herring mechanism). On the other hand, it will be controlled by grain boundary diffusion (Coble Creep). Additionally, when λ/h ratios are high, it may impede diffusional flow, therefore diffusional voids may form, which leads to fracture in creep. [ 5 ] [ 6 ]
Many people have developed estimations for the contribution of grain boundary sliding to the total strain experienced by various groups of materials, such as metals, ceramics, and geological materials. Grain boundary sliding contributes a significant amount of strain, especially for fine grain materials and high temperatures. [ 1 ] It has been shown that Lifshitz grain boundary sliding contributes about 50-60% of strain in Nabarro–Herring diffusion creep. [ 7 ] This mechanism is the primary cause of ceramic failure at high temperatures due to the formation of glassy phases at their grain boundaries. [ 8 ]
Rachinger sliding is purely elastic; the grains retain most of their original shape. [ 7 ] The internal stress will build up as grains slide until the stress balances out with the external applied stress. For example, when a uniaxial tensile stress is applied on a sample, grains move to accommodate the elongation and the number of grains along the direction of applied stress increases.
Lifshitz sliding only occurs with Nabarro–Herring and Coble creep. [ 7 ] The sliding motion is accommodated by the diffusion of vacancies from induced stresses and the grain shape changes during the process. For example, when a uniaxial tensile stress is applied, diffusion will occur within grains and the grain will elongate in the same direction as the applied stress. There will not be an increase in number of grains along the direction of applied stress.
When polycrystalline grains slide relative to each other, there must be simultaneous mechanisms that allow for this sliding to occur without the overlapping of grains (which would be physically impossible). [ 10 ] Various accommodation mechanisms have been proposed to account for this issue.
Grain boundary sliding accommodated by diffusional flow:
Grain boundary sliding accommodated by diffusional flow takes place by grain-switching while preserving grain shape. This type of mechanism is synonymous to Nabarro Herring and Coble creep but describes the grain at superplastic conditions. This concept was originally proposed by Ashby and Verral. During grain switching, we can describe the process through three steps: a) Initial state b) Intermediate stage c) Final state. During the intermediate stage, there must first be an applied stress exceeding the “threshold” stress so that there is an increase in grain boundary area which is provided by the diffusional flow that occurs once the threshold stress is achieved. Under the assumption that the applied stress is much greater than the threshold stress, the strain rate is greater than conventional diffusional creep. The reason for this is that for grain switching diffusion, the distance is about 1/7 the distance of diffusional creep and there are two more paths to grain switching in comparison with diffusional creep. Thus, this will lead to about an order magnitude higher strain rate than diffusional creep.
Grain boundary sliding accommodated by dislocation flow:
At superplastic temperature, strain rate and stress conditions, dislocations are really observed because they are quickly emitted and absorbed at grain boundaries. However, careful studies have been conducted to verify that dislocations are indeed emitted during superplastic deformation. During dislocation flow, the shape of the grain must be ensured to not change. Based on models of super plasticity, transitioning from dislocation creep to super plasticity occurs when the sub grain size is less than the grain size. The sub grain size: often denoted as d’ can be described in the equation below:
d’/b =10G/𝜏, Where it has an inverse relationship with shear stress. [ 12 ]
Generally speaking, the minimum creep rate for diffusion can be expressed as: [ 13 ] [ 7 ]
where the terms are defined as follows:
In the case where this minimum creep rate is controlled by grain boundary sliding, the exponents become p = 1 {\displaystyle p=1} , n = 2 {\displaystyle n=2} , and the diffusion coefficient D {\displaystyle D} becomes D L {\displaystyle D_{L}} (the lattice diffusion coefficient). [ 13 ] [ 7 ] Thus, the minimum creep rate becomes:
The total strain under creep conditions can be denoted as εt , where:
ε t = ε g + ε gbs +ε dc
ε g = Strain associated with intragranular dislocation processes
ε gbs = Strain due to Rachinger GBS associated with intragranular sliding
ε dc = Strain due to Lifshitz GBS associated with diffusion creep
During practice, experiments are normally performed in conditions where creep is negligible, therefore equation 1 will reduce to:
ε t = ε g + ε gbs
Therefore the contribution of GBS to the total strain can be denoted as:
Ⲝ = ε gbs / ε t
First, we need to illustrate the three perpendicular displacement vectors: u, v, and w, with a grain boundary sliding vector: s. It can be imagined as the w displacement vector coming out of the plane. While the v and u vectors are in the plane. The displacement vector u is also the tensile stress direction. The sliding contribution may be estimated by individual measurements of εgbs through these displacement vectors. We can further define the angle at the u v plane of displacements as Ѱ, and the angle between the u w planes as Θ. u can then be related by the tangents of these angles through the equation:
U = vtan Ѱ + wtanΘ
A common and easier way in practice is to use interferometry to measure fringes along the v displacement axis. The sliding strain is then given by:
εgbs = k’’nr vr
Where k’’ is constant, nr is the number of measurements, and vr is the average of n measurements.
Thus we can calculate the percentage of GBS strain. [ 14 ]
Grain boundary scattering has been observed experimentally using various microscopy techniques. It was first observed in NaCl and MgO bicrystals in 1962 by Adams and Murray. [ 15 ] By scratching the surface of their samples with a marker line, they were able to observe an offset of that line at the grain boundary as a result of adjacent grains sliding with respect to each other. Subsequently this was observed in other systems as well including in Zn-Al alloys using electron microscopy, [ 16 ] and octachloropropane using in situ techniques. [ 10 ]
Nano-crystalline materials, or nanomaterials, have fine grains which helps suppress lattice creep. This is beneficial for relatively low temperature operations as it impedes dislocations motion or diffusion due to high volume fraction of grain boundaries. However, fine grains are undesirable at high temperature due to the increased probability of grain boundary sliding. [ 17 ]
Grain shape plays a large role in determining the sliding rate and extent. Thus, by controlling the grain size and shape, the amount of grain boundary sliding can be limited. Generally, materials with coarser grains are preferred, as the material will have less grain boundaries. Ideally, single crystals will completely suppress this mechanism as the sample will not have any grain boundaries.
Another method is to reinforce grain boundaries by adding precipitates. Small precipitates located at grain boundaries can pin grain boundaries and prevent grains from sliding against each other. However, not all precipitates are desirable at boundaries. Large precipitates may have the opposite effect on grain boundary pinning as it allows more gaps or vacancies between grains to accommodate the precipitates, which reduces the pinning effect.
The application of high-strength steel is ubiquitous in the engineering world today. To provide a substantial engineering basis for real-world construction, the modeling of high-strength steel is very important.
By inputting parameters such as elastic modulus, yield strength, Poisson’s ratio, and specific heat of high strength steel from two temperatures, we can derive the related GBS energy as a function of temperature and thus its yield strength as a function of temperature. [ 18 ]
The superplastic forming technique is a technique where materials are deformed beyond the yield stress to form a complex shaped lightweight construction. This phenomenon is possible through grain boundary sliding that is enabled by dislocation slip/creep and diffusional creep.
An example would be for commercial fine-grained Al-Mg alloys, unusually weak grain boundary sliding is observed during the initial stage of superplastic deformation. Through a tensile test, grains were elongated along the tensile direction to 50~70%. The deformation was orchestrated by increased precipitation depletion zone fractions, particle segregation on the longitudinal grain boundaries, dislocation activity, and subgrains. Increased Mg content leads to increased GBS. Increasing Mg content from 4.8 to 6.5~7.6% aids grain size stability during the increased temperature process, simplified the GBS and decreased diffusion creep contribution, and increased the failure strain from 300% to 430%. [ 19 ]
The operation temperature for tungsten filaments used in incandescent lightbulbs is around 2000K to 3200K which is near the melting point of tungsten (T m = 3695 K). [ 20 ] As lightbulbs are expected to operate for long periods of time at a homologous temperature up to 0.8, understanding and preventing creep mechanism is crucial to extending their life expectancy.
Researchers found that the predominant mechanism for failure in these tungsten filaments was grain boundary sliding accommodated by diffusional creep. [ 21 ] This is because tungsten filaments, being as thin as they are, typically consist of only a handful of elongated grains. In fact there is usually less than one grain boundary per turn in a tungsten coil. [ 21 ] This elongated grain structure is generally called a bamboo structure, as the grains look similar to the internodes of bamboo stalks. During operation, the tungsten wire is stressed under the load of its own weight and because of the diffusion that can occur at high temperatures, grains begin to rotate and slide. This stress, because of variations in the filament, causes the filament to sag nonuniformly, which ultimately introduces further torque on the filament. [ 21 ] It is this sagging that inevitably results in a rupture of the filament, rendering the incandescent lightbulb useless. The typical lifetime for these single coil filaments is approximately 440 hours. [ 21 ]
To combat this grain boundary sliding, researchers began to dope the tungsten filament with aluminum, silicon and most importantly potassium. This composite material (AKS tungsten) is unique as it is composed of potassium and tungsten, which are non-alloying. [ 22 ] This feature of potassium results in nanosized bubbles of either liquid or gaseous potassium being distributed throughout the filament after proper manufacturing. [ 22 ] These bubbles interact with all defects in the filament pinning dislocations and most importantly grain boundaries. Pinning these grain boundaries, even at high temperatures, drastically reduces grain boundary sliding. This reduction in grain boundary sliding earned these filaments the title of "non-sag filaments" as they would no longer bow under their own weight. [ 22 ] Thus, this initially counter-intuitive approach to strengthening tungsten filaments began to be widely used in almost every incandescent lightbulb to greatly increase their lifetime. | https://en.wikipedia.org/wiki/Grain_boundary_sliding |
In materials science , grain-boundary strengthening (or Hall–Petch strengthening ) is a method of strengthening materials by changing their average crystallite (grain) size. It is based on the observation that grain boundaries are insurmountable borders for dislocations and that the number of dislocations within a grain has an effect on how stress builds up in the adjacent grain, which will eventually activate dislocation sources and thus enabling deformation in the neighbouring grain as well. By changing grain size, one can influence the number of dislocations piled up at the grain boundary and yield strength . For example, heat treatment after plastic deformation and changing the rate of solidification are ways to alter grain size. [ 1 ]
In grain-boundary strengthening, the grain boundaries act as pinning points impeding further dislocation propagation. Since the lattice structure of adjacent grains differs in orientation, it requires more energy for a dislocation to change directions and move into the adjacent grain. The grain boundary is also much more disordered than inside the grain, which also prevents the dislocations from moving in a continuous slip plane. Impeding this dislocation movement will hinder the onset of plasticity and hence increase the yield strength of the material.
Under an applied stress, existing dislocations and dislocations generated by Frank–Read sources will move through a crystalline lattice until encountering a grain boundary, where the large atomic mismatch between different grains creates a repulsive stress field to oppose continued dislocation motion. As more dislocations propagate to this boundary, dislocation 'pile up' occurs as a cluster of dislocations are unable to move past the boundary. As dislocations generate repulsive stress fields, each successive dislocation will apply a repulsive force to the dislocation incident with the grain boundary. These repulsive forces act as a driving force to reduce the energetic barrier for diffusion across the boundary, such that additional pile up causes dislocation diffusion across the grain boundary, allowing further deformation in the material. Decreasing grain size decreases the amount of possible pile up at the boundary, increasing the amount of applied stress necessary to move a dislocation across a grain boundary. The higher the applied stress needed to move the dislocation, the higher the yield strength. Thus, there is then an inverse relationship between grain size and yield strength, as demonstrated by the Hall–Petch equation. However, when there is a large direction change in the orientation of the two adjacent grains, the dislocation may not necessarily move from one grain to the other but instead create a new source of dislocation in the adjacent grain. The theory remains the same that more grain boundaries create more opposition to dislocation movement and in turn strengthens the material.
There is a limit to this mode of strengthening, as infinitely strong materials do not exist. Grain sizes can range from about 100 μm (0.0039 in) (large grains) to 1 μm (3.9 × 10 −5 in) (small grains). Lower than this, the size of dislocations begins to approach the size of the grains. At a grain size of about 10 nm (3.9 × 10 −7 in), [ 2 ] only one or two dislocations can fit inside a grain (see Figure 1 above). This scheme prohibits dislocation pile-up and instead results in grain boundary diffusion . The lattice resolves the applied stress by grain boundary sliding, resulting in a decrease in the material's yield strength.
To understand the mechanism of grain boundary strengthening one must understand the nature of dislocation-dislocation interactions. Dislocations create a stress field around them given by:
where G is the material's shear modulus , b is the Burgers vector , and r is the distance from the dislocation. If the dislocations are in the right alignment with respect to each other, the local stress fields they create will repel each other. This helps dislocation movement along grains and across grain boundaries. Hence, the more dislocations are present in a grain, the greater the stress field felt by a dislocation near a grain boundary:
Interphase boundaries can also contribute to grain boundary strengthening, particularly in composite materials and precipitation-hardened alloys. Coherent IPBs, in particular, can provide additional barriers to dislocation motion, similar to grain boundaries. In contrast, non-coherent IPBs and partially coherent IPBs can act as sources of dislocations, which can lead to localized deformation and affect the mechanical properties of the material. [ 3 ]
A subgrain is a part of the grain that is only slightly disoriented from other parts of the grain. [ 4 ] Current research is being done to see the effect of subgrain strengthening in materials. Depending on the processing of the material, subgrains can form within the grains of the material. For example, when Fe-based material is ball-milled for long periods of time (e.g. 100+ hours), subgrains of 60–90 nm are formed. It has been shown that the higher the density of the subgrains, the higher the yield stress of the material due to the increased subgrain boundary. The strength of the metal was found to vary reciprocally with the size of the subgrain, which is analogous to the Hall–Petch equation. The subgrain boundary strengthening also has a breakdown point of around a subgrain size of 0.1 μm, which is the size where any subgrains smaller than that size would decrease yield strength. [ 5 ]
Coherent grain boundaries are those in which the crystal lattice of adjacent grains is continuous across the boundary. In other words, the crystallographic orientation of the grains on either side of the boundary is related by a small rotation or translation. Coherent grain boundaries are typically observed in materials with small grain sizes or in highly ordered materials such as single crystals. Because the crystal lattice is continuous across the boundary, there are no defects or dislocations associated with coherent grain boundaries. As a result, they do not act as barriers to the motion of dislocations and have little effect on the strength of a material. However, they can still affect other properties such as diffusion and grain growth. [ 3 ]
When solid solutions become supersaturated and precipitation occurs, tiny particles are formed. These particles typically have interphase boundaries that match up with the matrix, despite differences in interatomic spacing between the particle and the matrix. This creates a coherency strain, which causes distortion. Dislocations respond to the stress field of a coherent particle in a way similar to how they interact with solute atoms of different sizes. It is worth noting that the interfacial energy can also influence the kinetics of phase transformations and precipitation processes. For instance, the energy associated with a strained coherent interface can reach a critical level as the precipitate grows, leading to a transition from a coherent to a disordered (non-coherent) interface. This transition occurs when the energy associated with maintaining the coherency becomes too high, and the system seeks a lower energy configuration. This happens when particle dispersion is introduced into a matrix. Dislocations pass through small particles and bow between large particles or particles with disordered interphase boundaries. The predominant slip mechanism determines the contribution to strength, which depends on factors such as particle size and volume fraction. [ 3 ] [ 6 ] [ 7 ]
A partially coherent interphase boundary is an intermediate type of IPB that lies between the completely coherent and non-coherent IPBs. In this type of boundary, there is a partial match between the atomic arrangements of the particle and the matrix, but not a perfect match. As a result, coherency strains are partially relieved, but not completely eliminated. The periodic introduction of dislocations along the boundary plays a key role in partially relieving the coherency strains. These dislocations act as periodic defects that accommodate the lattice mismatch between the particle and the matrix. The dislocations can be introduced during the precipitation process or during subsequent annealing treatments. [ 3 ]
Incoherent grain boundaries are those in which there is a significant mismatch in crystallographic orientation between adjacent grains. This results in a discontinuity in the crystal lattice across the boundary, and the formation of a variety of defects such as dislocations, stacking faults, and grain boundary ledges.The presence of these defects creates a barrier to the motion of dislocations and leads to a strengthening effect. This effect is more pronounced in materials with smaller grain sizes, as there are more grain boundaries to impede dislocation motion. In addition to the barrier effect, incoherent grain boundaries can also act as sources and sinks for dislocations. This can lead to localized plastic deformation and affect the overall mechanical response of a material. [ 6 ]
When small particles are formed through precipitation from supersaturated solid solutions, their interphase boundaries may not be coherent with the matrix. In such cases, the atomic bonds do not match up across the interface and there is a misfit between the particle and the matrix. This misfit gives rise to a non-coherency strain, which can cause the formation of dislocations at the grain boundary. As a result, the properties of the small particle can be different from those of the matrix. The size at which non-coherent grain boundaries form depends on the lattice misfit and the interfacial energy. [ 3 ] [ 7 ]
Understanding the interfacial energy of materials with different types of interphase boundaries (IPBs) provides valuable insights into several aspects of their behavior, including thermodynamic stability, deformation behavior, and phase evolution.
Interfacial energy affects the mechanisms of grain boundary sliding and dislocation transmission. Higher interfacial energy promotes greater resistance to grain boundary sliding, as the higher energy barriers inhibit the relative movement of adjacent grains. Additionally, dislocations that encounter grain boundaries can either transmit across the boundary or be reflected back into the same grain. The interfacial energy influences the likelihood of dislocation transmission, with higher interfacial energy barriers impeding dislocation motion and enhancing grain boundary strengthening. [ 8 ]
High-angle grain boundaries, which have large misorientations between adjacent grains, tend to have higher interfacial energy and are more effective in impeding dislocation motion. In contrast, low-angle grain boundaries with small misorientations and lower interfacial energy may allow for easier dislocation transmission and exhibit weaker grain boundary strengthening effects. [ 9 ]
Grain boundary engineering involves manipulating the grain boundary structure and energy to enhance mechanical properties. By controlling the interfacial energy, it is possible to engineer materials with desirable grain boundary characteristics, such as increased interfacial area, higher grain boundary density, or specific grain boundary types. [ 10 ]
Introducing alloying elements into the material can alter the interfacial energy of grain boundaries. Alloying can result in segregation of solute atoms at the grain boundaries, which can modify the atomic arrangements and bonding, and thereby influence the interfacial energy. [ 10 ]
Applying surface treatments or coatings can modify the interfacial energy of grain boundaries. Surface modification techniques, such as chemical treatments or deposition of thin films, can alter the surface energy and consequently affect the grain boundary energy. [ 10 ]
Thermal treatments can be employed to modify the interfacial energy of grain boundaries. Annealing at specific temperatures and durations can induce atomic rearrangements, diffusion, and stress relaxation at the grain boundaries, leading to changes in the interfacial energy. [ 10 ]
Once the interfacial energy is controlled, grain boundaries can be manipulated to enhance their strengthening effects.
Applying severe plastic deformation techniques, such as equal-channel angular pressing (ECAP) or high-pressure torsion (HPT), can lead to grain refinement and the creation of new grain boundaries with tailored characteristics. These refined grain structures can exhibit a high density of grain boundaries, including high-angle boundaries, which can contribute to enhanced grain boundary strengthening. [ 10 ]
Utilizing specific thermomechanical processing routes, such as rolling, forging, or extrusion, can result in the creation of a desired texture and the development of specific grain boundary structures. These processing routes can promote the formation of specific grain boundary types and orientations, leading to improved grain boundary strengthening. [ 10 ]
There is an inverse relationship between delta yield strength and grain size to some power, x .
where k is the strengthening coefficient and both k and x are material specific. Assuming a narrow monodisperse grain size distribution in a polycrystalline material, the smaller the grain size, the smaller the repulsion stress felt by a grain boundary dislocation and the higher the applied stress needed to propagate dislocations through the material.
The relation between yield stress and grain size is described mathematically by the Hall–Petch equation: [ 12 ]
where σ y is the yield stress, σ 0 is a materials constant for the starting stress for dislocation movement (or the resistance of the lattice to dislocation motion), k y is the strengthening coefficient (a constant specific to each material), and d is the average grain diameter. It is important to note that the H-P relationship is an empirical fit to experimental data, and that the notion that a pileup length of half the grain diameter causes a critical stress for transmission to or generation in an adjacent grain has not been verified by actual observation in the microstructure.
Theoretically, a material could be made infinitely strong if the grains are made infinitely small. This is impossible though, because the lower limit of grain size is a single unit cell of the material. Even then, if the grains of a material are the size of a single unit cell, then the material is in fact amorphous, not crystalline, since there is no long range order, and dislocations can not be defined in an amorphous material. It has been observed experimentally that the microstructure with the highest yield strength is a grain size of about 10 nm (3.9 × 10 −7 in), because grains smaller than this undergo another yielding mechanism, grain boundary sliding . [ 2 ] Producing engineering materials with this ideal grain size is difficult because only thin films can be reliably produced with grains of this size. In materials having a bi-disperse grain size distribution, for example those exhibiting abnormal grain growth , hardening mechanisms do not strictly follow the Hall–Petch relationship and divergent behavior is observed.
In the early 1950s two groundbreaking series of papers were written independently on the relationship between grain boundaries and strength.
In 1951, while at the University of Sheffield, E. O. Hall wrote three papers which appeared in volume 64 of the Proceedings of the Physical Society . In his third paper, Hall [ 13 ] showed that the length of slip bands or crack lengths correspond to grain sizes and thus a relationship could be established between the two. Hall concentrated on the yielding properties of mild steels .
Based on his experimental work carried out in 1946–1949, N. J. Petch of the University of Leeds , England published a paper in 1953 independent from Hall's. Petch's paper [ 14 ] concentrated more on brittle fracture . By measuring the variation in cleavage strength with respect to ferritic grain size at very low temperatures, Petch found a relationship exact to that of Hall's. Thus this important relationship is named after both Hall and Petch.
The Hall–Petch relation predicts that as the grain size decreases the yield strength increases. The Hall–Petch relation was experimentally found to be an effective model for materials with grain sizes ranging from 1 millimeter to 1 micrometer. Consequently, it was believed that if average grain size could be decreased even further to the nanometer length scale the yield strength would increase as well. However, experiments on many nanocrystalline materials demonstrated that if the grains reached a small enough size, the critical grain size which is typically around 10 nm (3.9 × 10 −7 in), the yield strength would either remain constant or decrease with decreasing grains size. [ 15 ] [ 16 ] This phenomenon has been termed the reverse or inverse Hall–Petch relation. A number of different mechanisms have been proposed for this relation. As suggested by Carlton et al. , they fall into four categories: (1) dislocation-based, (2) diffusion-based, (3) grain-boundary shearing-based, (4) two-phase-based. [ 17 ]
There have been several works done to investigate the mechanism behind the inverse Hall–Petch relationship on numerous materials. In Han’s work, [ 18 ] a series of molecular dynamics simulations were done to investigate the effect of grain size on the mechanical properties of nanocrystalline graphene under uniaxial tensile loading, with random shapes and random orientations of graphene rings. The simulation was run at grain sizes of nm and at room temperature. It was found that in the grain size of range 3.1 nm to 40 nm, inverse Hall–Petch relationship was observed. This is because when the grain size decreases at nm scale, there is an increase in the density of grain boundary junctions which serves as a source of crack growth or weak bonding. However, it was also observed that at grain size below 3.1 nm, a pseudo Hall–Petch relationship was observed, which results an increase in strength. This is due to a decrease in stress concentration of grain boundary junctions and also due to the stress distribution of 5-7 defects along the grain boundary where the compressive and tensile stress are produced by the pentagon and heptagon rings, etc. Chen at al. [ 19 ] have done research on the inverse HallPetch relations of high-entropy CoNiFeAl x Cu 1– x alloys. In the work, polycrystalline models of FCC structured CoNiFeAl 0.3 Cu 0.7 with grain sizes ranging from 7.2 nm to 18.8 nm were constructed to perform uniaxial compression using molecular dynamic simulations. All compression simulations were done after setting the periodic boundary conditions across the three orthogonal directions. It was found that when the grain size is below 12.1 nm the inverse Hall–Petch relation was observed. This is because as the grain size decreases partial dislocations become less prominent and so as deformation twinning. Instead, it was observed that there is a change in the grain orientation and migration of grain boundaries and thus cause the growth and shrinkage of neighboring grains. These are the mechanisms for inverse Hall–Petch relations. Sheinerman et al. [ 20 ] also studied inverse Hall–Petch relation for nanocrystalline ceramics. It was found that the critical grain size for the transition from direct Hall–Petch to inverse Hall–Petch fundamentally depends on the activation energy of grain boundary sliding. This is because in direct Hall–Petch the dominant deformation mechanism is intragrain dislocation motion while in inverse Hall–Petch the dominant mechanism is grain boundary sliding. It was concluded that by plotting both the volume fraction of grain boundary sliding and volume fraction of intragrain dislocation motion as a function of grain size, the critical grain size could be found where the two curves cross.
Other explanations that have been proposed to rationalize the apparent softening of metals with nanosized grains include poor sample quality and the suppression of dislocation pileups. [ 21 ]
The pileup of dislocations at grain boundaries is a hallmark mechanism of the Hall–Petch relationship. Once grain sizes drop below the equilibrium distance between dislocations, though, this relationship should no longer be valid. Nevertheless, it is not entirely clear what exactly the dependency of yield stress should be on grain sizes below this point.
Grain refinement, also known as inoculation , [ 22 ] is the set of techniques used to implement grain boundary strengthening in metallurgy . The specific techniques and corresponding mechanisms will vary based on what materials are being considered.
One method for controlling grain size in aluminum alloys is by introducing particles to serve as nucleants, such as Al–5%Ti. Grains will grow via heterogeneous nucleation ; that is, for a given degree of undercooling beneath the melting temperature, aluminum particles in the melt will nucleate on the surface of the added particles. Grains will grow in the form of dendrites growing radially away from the surface of the nucleant. Solute particles can then be added (called grain refiners) which limit the growth of dendrites, leading to grain refinement. [ 23 ] Al-Ti-B alloys are the most common grain refiner for Al alloys; [ 24 ] however, novel refiners such as Al 3 Sc have been suggested.
One common technique is to induce a very small fraction of the melt to solidify at a much higher temperature than the rest; this will generate seed crystals that act as a template when the rest of the material falls to its (lower) melting temperature and begins to solidify. Since a huge number of minuscule seed crystals are present, a nearly equal number of crystallites result, and the size of any one grain is limited. | https://en.wikipedia.org/wiki/Grain_boundary_strengthening |
A grain elevator or grain terminal is a facility designed to stockpile or store grain. In the grain trade , the term "grain elevator" also describes a tower containing a bucket elevator or a pneumatic conveyor , which scoops up grain from a lower level and deposits it in a silo or other storage facility.
In most cases, the term "grain elevator" also describes the entire elevator complex, including receiving and testing offices, weighbridges , and storage facilities. It may also mean organizations that operate or control several individual elevators, in different locations. In Australia, the term describes only the lifting mechanism.
Before the advent of the grain elevator, grain was usually handled in bags rather than in bulk (large quantities of loose grain). The Dart elevator was a major innovation—it was invented by Joseph Dart , a merchant, and Robert Dunbar , an engineer, in 1842, in Buffalo, New York . Using the steam-powered flour mills of Oliver Evans as their model, they invented the marine leg, which scooped loose grain out of the hulls of ships and elevated it to the top of a marine tower. [ 1 ]
Early grain elevators and bins were often built of framed or cribbed wood, and were prone to fire. In 1899 Frank H. Peavey "The Elevator King' along with Charles F. Haglin , invented the modern grain elevator. The first Peavey-Haglin Experimental Concrete Grain Elevator still stands today in St. Louis Park, Minnesota. The Peavey invented elevator was the first cylindrical concrete grain elevator in the world and is now widely used across Canada and the US.
Grain elevator bins, tanks, and silos are now usually made of steel or reinforced concrete. Bucket elevators are used to lift grain to a distributor or consignor, from which it falls through spouts and/or conveyors and into one or more bins, silos, or tanks in a facility. When desired, silos, bins, and tanks are emptied by gravity flow, sweep augers , and conveyors. As grain is emptied from bins, tanks, and silos, it is conveyed, blended, and weighted into trucks, railroad cars , or barges for shipment.
In Australian English , the term "grain elevator" is reserved for elevator towers, while a receival and storage building or complex is distinguished by the formal term "receival point" or as a "wheat bin" or "silo". Large-scale grain receival, storage, and logistics operations are known in Australia as bulk handling.
In Canada, the term "grain elevator" is used to refer to a place where farmers sell grain into the global grain distribution system, and/or a place where the grain is moved into rail cars or ocean-going ships for transport. Specifically, several types of grain elevators are defined under Canadian law, in the Canadian Grain Act , section 2. [ 2 ]
Both necessity and the prospect of making money gave birth to the steam-powered grain elevator in Buffalo, New York , in 1843. Due to the completion of the Erie Canal in 1825, Buffalo enjoyed a unique position in American geography. It stood at the intersection of two great all-water routes; one extended from New York Harbor , up the Hudson River to Albany , and beyond it, the Port of Buffalo; the other comprised the Great Lakes , which could theoretically take boaters in any direction they wished to go (north to Canada , west to Michigan or Wisconsin , south to Toledo and Cleveland , or east to the Atlantic Ocean ). All through the 1830s, Buffalo benefited tremendously from its position. In particular, it was the recipient of most of the increasing quantities of grain (mostly wheat) that was being grown on farms in Ohio and Indiana , and shipped on Lake Erie for trans-shipment to the Erie Canal. If Buffalo had not been there, or when things got backed up there, that grain would have been loaded onto boats at Cincinnati and shipped down the Mississippi River to New Orleans . [ 1 ]
By 1842, Buffalo's port facilities clearly had become antiquated. They still relied upon techniques that had been in use since the European Middle Ages ; work teams of stevedores use block and tackles and their own backs to unload or load each sack of grain that had been stored ashore or in the boat's hull. Several days, sometimes even a week, were needed to serve a single grain-laden boat. Grain shipments were going down the Mississippi River, not over the Great Lakes/Erie Canal system.
A merchant named Joseph Dart Jr., is generally credited as being the one who adapted Oliver Evans ' grain elevator (originally a manufacturing device) for use in a commercial framework (the trans-shipment of grain in bulk from lakers to canal boats), but the actual design and construction of the world's first steam-powered "grain storage and transfer warehouse" was executed by an engineer named Robert Dunbar. Thanks to the historic Dart's Elevator (operational on 1 June 1843), which worked almost seven times faster than its nonmechanized predecessors, Buffalo was able to keep pace with—and thus further stimulate—the rapid growth of American agricultural production in the 1840s and 1850s, but especially after the Civil War , with the coming of the railroads . [ 1 ]
The world's second and third grain elevators were built in Toledo, Ohio, and Brooklyn , New York, in 1847. These fledgling American cities were connected through an emerging international grain trade of unprecedented proportions. Grain shipments from farms in Ohio were loaded onto ships by elevators at Toledo; these ships were unloaded by elevators at Buffalo that shipped their grain to canal boats (and, later, rail cars), which were unloaded by elevators in Brooklyn, where the grain was either distributed to East Coast flour mills or loaded for further shipment to England , the Netherlands , or Germany . This eastern flow of grain, though, was matched by an equally important flow of people and capital in the opposite direction, that is, from east to west. Because of the money to be made in grain production, and of course, because of the existence of an all-water route to get there, increasing numbers of immigrants in Brooklyn came to Ohio, Indiana, and Illinois to become farmers. More farmers meant more prairies turned into farmlands, which in turn meant increased grain production, which of course meant that more grain elevators would have to be built in places such as Toledo, Buffalo, and Brooklyn (and Cleveland, Chicago , and Duluth ). Through this loop of productivity set in motion by the invention of the grain elevator, the United States became a major international producer of wheat, corn, and oats. [ 1 ]
In the early 20th century, concern arose about monopolistic practices in the grain elevator industry, leading to testimony before the Interstate Commerce Commission in 1906. [ 3 ] This led to several grain elevators being burned down in Nebraska, allegedly in protest. [ 3 ]
Today, grain elevators are a common sight in the grain-growing areas of the world, such as the North American prairies . Larger terminal elevators are found at distribution centers, such as Chicago and Thunder Bay , Ontario, where grain is sent for processing, or loaded aboard trains or ships to go further afield.
Buffalo, New York, the world's largest grain port from the 1850s until the first half of the 20th century, once had the United States' largest capacity for the storage of grain in over 30 concrete grain elevators located along the inner and outer harbors. While several are still in productive use, many of those that remain are presently idle. In a nascent trend, some of the city's inactive capacity has recently come back online, with an ethanol plant started in 2007 using one of the previously mothballed elevators to store corn. In the early 20th century, Buffalo's grain elevators inspired modernist architects such as Le Corbusier , who exclaimed, "The first fruits of the new age!" when he first saw them. Buffalo's grain elevators have been documented for the Historic American Engineering Record and added to the National Register of Historic Places . Currently, Enid, Oklahoma , holds the title of most grain storage capacity in the United States.
In farming communities, each town had one or more small grain elevators that served the local growers. The classic grain elevator was constructed with wooden cribbing and had nine or more larger square or rectangular bins arranged in 3 × 3 or 3 × 4 or 4 × 4 or more patterns. Wooden-cribbed elevators usually had a driveway with truck scale and office on one side, a rail line on the other side, and additional grain-storage annex bins on either side.
In more recent times with improved transportation, centralized and much larger elevators serve many farms. Some of them are quite large. Two elevators in Kansas (one in Hutchinson and one in Wichita ) are half a mile long. The loss of the grain elevators from small towns is often considered a great change in their identity, and efforts to preserve them as heritage structures are made. At the same time, many larger grain farms have their own grain-handling facilities for storage and loading onto trucks.
Elevator operators buy grain from farmers, either for cash or at a contracted price, and then sell futures contracts for the same quantity of grain, usually each day. They profit through the narrowing "basis", that is, the difference between the local cash price, and the futures price, that occurs at certain times of the year.
Before economical truck transportation was available, grain elevator operators sometimes used their purchasing power to control prices. This was especially easy, since farmers often had only one elevator within a reasonable distance of their farms. This led some governments to take over the administration of grain elevators. An example of this is the Saskatchewan Wheat Pool . For the same reason, many elevators were purchased by cooperatives .
A recent problem with grain elevators is the need to provide separate storage for ordinary and genetically modified grain to reduce the risk of accidental mixing of the two.
In the past, grain elevators sometimes experienced silo explosions . Fine powder from the millions of grains passing through the facility would accumulate and mix with the oxygen in the air. A spark could spread from one floating particle to the other, creating a chain reaction that would destroy the entire structure. (This dispersed-fuel explosion is the mechanism behind fuel-air bombs .) To prevent this, elevators have very rigorous rules against smoking or any other open flame. Many elevators also have various devices installed to maximize ventilation, safeguards against overheating in belt conveyors, legs, bearings, and explosion-proof electrical devices such as electric motors , switches, and lighting .
Grain elevators in small Canadian communities often had the name of the community painted on two sides of the elevator in large block letters, with the name of the elevator operator emblazoned on the other two sides. This made identification of the community easier for rail operators (and incidentally, for lost drivers and pilots). The old community name often remained on an elevator long after the town had either disappeared or been amalgamated into another community; the grain elevator at Ellerslie, Alberta , remained marked with its old community name until it was demolished, which took place more than 20 years after the village had been annexed by Edmonton .
One of the major historical trends in the grain trade has been the closure of many smaller elevators and the consolidation of the grain trade to fewer places and among fewer companies. For example, in 1961, 1,642 "country elevators" (the smallest type) were in Alberta, holding 3,452,240 tonnes (3,805,440 short tons) of grain. By 2010, only 79 "primary elevators" (as they are now known) remained, holding 1,613,960 tonnes (1,779,090 short tons).
Despite this consolidation, overall storage capacity has increased in many places. In 2017, the United States had 0.88 cubic kilometres (25 billion US bushels) of storage capacity, a growth of 25% over the previous decade. [ 5 ]
The city of Buffalo is not only the birthplace of the modern grain elevator, but also has the world's largest number of extant examples. [ 6 ] A number of the city's historic elevators are clustered along "Elevator Alley", a narrow stretch of the Buffalo River immediately adjacent to the harbor. The alley runs under Ohio Street and along Childs Street in the city's First Ward neighborhood. [ 7 ]
In Canada, the term "elevator row" refers to a row of four or more wood-crib prairie grain elevators.
In the early pioneer days of Western Canada 's prairie towns, when a good farming spot was settled, many people wanted to make money by building their own grain elevators. This brought in droves of private grain companies. Towns boasted dozens of elevator companies, which all stood in a row along the railway tracks. If a town were lucky enough to have two railways, it was to be known as the next Montreal . Many elevator rows had two or more elevators of the same company. Small towns bragged of their large elevator rows in promotional pamphlets to attract settlers. With so much competition in the 1920s, consolidation began almost immediately, and many small companies were merged or absorbed into larger companies.
In the mid-1990s, with the cost of grain so low, many private elevator companies once again had to merge, this time causing thousands of "prairie sentinels" to be torn down. Because so many grain elevators have been torn down, Canada has only two surviving elevator rows; one located in Inglis, Manitoba , and the other in Warner, Alberta . The Inglis Grain Elevators National Historic Site has been protected as a National Historic Sites of Canada . The Warner elevator row is, as of 2019, not designated a historic site, and is still in use as commercial grain elevators.
All companies operating elevators in Canada are licensed by the Canadian Grain Commission . [ 8 ]
This is a list of grain elevators that are either in the process of becoming heritage sites or museums , or have been preserved for future generations.
The Manchester Ship Canal grain elevator was completed in 1898. It had a capacity of 40,000 tons and its automatic conveying and spouting system could distribute grain into 226 bins. [ 33 ]
Given a large enough suspension of combustible flour or grain dust in the air, a significant explosion can occur. The 1878 explosion of the Washburn "A" Mill in Minneapolis, Minnesota , killed 18, leveled two nearby mills, damaged many others, and caused a destructive fire that gutted much of the nearby milling district. (The Washburn "A" mill was later rebuilt and continued to be used until 1965.) Another example occurred in 1998, when the DeBruce grain elevator in Wichita, Kansas , exploded and killed seven people. [ 37 ] An explosion on October 29, 2011, at the Bartlett Grain Company in Atchison, Kansas , killed six people. Two more men received severe burns, but the remaining four were not hurt. [ 38 ]
Almost any finely divided organic substance becomes an explosive material when dispersed as an air suspension; hence, a very fine flour is dangerously explosive in air suspension. This poses a significant risk when milling grain to produce flour, so mills go to great lengths to remove sources of sparks. These measures include carefully sifting the grain before it is milled or ground to remove stones, which could strike sparks from the millstones, and the use of magnets to remove metallic debris able to strike sparks.
The earliest recorded flour explosion took place in an Italian mill in 1785, but many have occurred since. These two references give numbers of recorded flour and dust explosions in the United States in 1994: [ 39 ] and 1997 [ 40 ] In the ten-year period up to and including 1997, there were 129 explosions.
Canadian Prairie grain elevators were the subjects of the National Film Board of Canada documentaries Grain Elevator [ 41 ] and Death of a Skyline . [ 42 ]
During the sixth season of the History Channel series Ax Men , one of the featured crews takes on the job of dismantling the Globe Elevator in Wisconsin. This structure was the largest grain-storage facility in the world when it was built in the 1880s. | https://en.wikipedia.org/wiki/Grain_elevator_explosion |
In materials science , grain growth is the increase in size of grains ( crystallites ) in a material at high temperature. This occurs when recovery and recrystallisation are complete and further reduction in the internal energy can only be achieved by reducing the total area of grain boundary . The term is commonly used in metallurgy but is also used in reference to ceramics and minerals . The behaviors of grain growth is analogous to the coarsening behaviors of grains, which implied that both of grain growth and coarsening may be dominated by the same physical mechanism.
The practical performances of polycrystalline materials are strongly affected by the formed microstructure inside, which is mostly dominated by grain growth behaviors. For example, most materials exhibit the Hall–Petch effect at room-temperature and so display a higher yield stress when the grain size is reduced (assuming abnormal grain growth has not taken place). At high temperatures the opposite is true since the open, disordered nature of grain boundaries means that vacancies can diffuse more rapidly down boundaries leading to more rapid Coble creep . Since boundaries are regions of high energy they make excellent sites for the nucleation of precipitates and other second-phases e.g. Mg–Si–Cu phases in some aluminium alloys or martensite platlets [ check spelling ] in steel. Depending on the second phase in question this may have positive or negative effects.
Grain growth has long been studied primarily by the examination of sectioned, polished and etched samples under the optical microscope . Although such methods enabled the collection of a great deal of empirical evidence, particularly with regard to factors such as temperature or composition , the lack of crystallographic information limited the development of an understanding of the fundamental physics . Nevertheless, the following became well-established features of grain growth:
The boundary between one grain and its neighbour ( grain boundary ) is a defect in the crystal structure and so it is associated with a certain amount of energy. As a result, there is a thermodynamic driving force for the total area of boundary to be reduced. If the grain size increases, accompanied by a reduction in the actual number of grains per volume, then the total area of grain boundary will be reduced.
In the classic theory, the local velocity of a grain boundary at any point is proportional to the local curvature of the grain boundary, i.e.:
v = M σ κ {\displaystyle v=M\sigma \kappa } ,
where v {\displaystyle v} is the velocity of grain boundary, M {\displaystyle M} is grain boundary mobility (generally depends on orientation of two grains), σ {\displaystyle \sigma } is the grain boundary energy and κ {\displaystyle \kappa } is the sum of the two principal surface curvatures. For example, shrinkage velocity of a spherical grain embedded inside another grain is
v = M σ 2 R {\displaystyle v=M\sigma {\frac {2}{R}}} ,
where R {\displaystyle R} is radius of the sphere. This driving pressure is very similar in nature to the Laplace pressure that occurs in foams.
In comparison to phase transformations the energy available to drive grain growth is very low and so it tends to occur at much slower rates and is easily slowed by the presence of second phase particles or solute atoms in the structure.
Recently, in contrast to the classic linear relation between grain boundary velocity and curvature, grain boundary velocity and curvature are observed to be not correlated in Ni polycrystals, [ 3 ] which conflicting results has been revealed and be theoretically interpreted by a general model of grain boundary (GB) migration in the previous literature. [ 4 ] [ 5 ] According to the general GB migration model, the classical linear relation can only be used in a specical case.
Development of theoretical models describing grain growth is an active field of research. Many models have been proposed for grain growth, but no theory has yet been put forth that has been independently validated to apply across the full range of conditions and many questions remain open. [ 6 ] By no means is the following a comprehensive review. One recent theory of grain growth posits that normal grain growth only occurs in the polycrystalline systems with grain boundaries which have undergone roughening transitions, and abnormal and/or stagnant grain growth can only occur in the polycrystalline systems with non-zero GB (grain boundary) step free energy of grains. [ 7 ] Other models explaining grain coarsening assert that disconnections are responsible for the motion of grain boundaries, and provide limited experimental evidence suggesting that they govern grain boundary migration and grain growth behavior. [ 8 ] Other models have indicated that triple junctions play an important role in determining the grain growth behavior in many systems. [ 9 ]
Ideal grain growth is a special case of normal grain growth where boundary motion is driven only by local curvature of the grain boundary. It results in the reduction of the total amount of grain boundary surface area i.e. total energy of the system. Additional contributions to the driving force by e.g. elastic strains or temperature gradients are neglected. If it holds that the rate of growth is proportional to the driving force and that the driving force is proportional to the total amount of grain boundary energy, then it can be shown that the time t required to reach a given grain size is approximated by the equation
d 2 − d 0 2 = k t {\displaystyle d^{2}-{d_{0}}^{2}=kt\,\!}
where d 0 is the initial grain size, d is the final grain size and k is a temperature dependent constant given by an exponential law:
k = k 0 exp ( − Q R T ) {\displaystyle k=k_{0}\exp \left({\frac {-Q}{RT}}\right)\,\!}
where k 0 is a constant, T is the absolute temperature and Q is the activation energy for boundary mobility. Theoretically, the activation energy for boundary mobility should equal that for self-diffusion but this is often found not to be the case.
In general these equations are found to hold for ultra-high purity materials but rapidly fail when even tiny concentrations of solute are introduced.
An old-standing topic in grain growth is the evolution of the grains size distribution. Inspired by the work of Lifshitz and Slyozov on Ostwald ripening , Hillert has suggested that in a normal grain growth process the size distribution function must converge to a self-similar solution, i.e. it becomes invariant when the grain size is scaled with a characteristic length of the system R c r {\displaystyle R_{cr}} that is proportional to the average grain size ⟨ R ⟩ {\displaystyle \langle R\rangle } .
Several simulation studies, however, have shown that the size distribution deviates from the Hillert's self-similar solution. [ 11 ] Hence a search for a new possible self-similar solution was initiated that indeed led to a new class of self-similar distribution functions. [ 12 ] [ 13 ] [ 14 ] Large-scale phase field simulations have shown that there is indeed a self-similar behavior possible within the new distribution functions. It was shown that the origin of the deviation from Hillert's distribution is indeed the geometry of grains specially when they are shrinking. [ 15 ]
In common with recovery and recrystallisation , growth phenomena can be separated into continuous and discontinuous mechanisms. In the former the microstructure evolves from state A to B (in this case the grains get larger) in a uniform manner. In the latter, the changes occur heterogeneously and specific transformed and untransformed regions may be identified. Abnormal or discontinuous grain growth is characterised by a subset of grains growing at a high rate and at the expense of their neighbours and tends to result in a microstructure dominated by a few very large grains. In order for this to occur the subset of grains must possess some advantage over their competitors such as a high grain boundary energy, locally high grain boundary mobility, favourable texture or lower local second-phase particle density. [ 16 ]
If there are additional factors preventing boundary movement, such as Zener pinning by particles, then the grain size may be restricted to a much lower value than might otherwise be expected. This is an important industrial mechanism in preventing the softening of materials at high temperature.
Certain materials especially refractories which are processed at high temperatures end up with excessively large grain size and poor mechanical properties at room temperature. To mitigate this problem in a common sintering procedure, a variety of dopants are often used to inhibit grain growth. | https://en.wikipedia.org/wiki/Grain_growth |
The grain per gallon ( gpg ) is a unit of water hardness defined as 1 grain (64.8 milligrams) of calcium carbonate dissolved in 1 US gallon of water (3.785412 L ). It translates into 1 part in about 58,000 parts of water or 17.1 parts per million (ppm). Also called Clark degree (in terms of an imperial gallon ).
Calcium and magnesium ions present as sulfates, chlorides, carbonates and bicarbonates cause water to be hard. Water chemists measure water impurities in parts per million (ppm). For understandability, hardness ordinarily is expressed in grains of hardness per gallon of water (gpg). The two systems can be converted mathematically.
According to the Water Quality Association: | https://en.wikipedia.org/wiki/Grain_per_gallon |
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