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In logic , the scope of a quantifier or connective is the shortest formula in which it occurs, [ 1 ] determining the range in the formula to which the quantifier or connective is applied. [ 2 ] [ 3 ] [ 4 ] The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, [ 2 ] [ 5 ] and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope . [ 6 ] [ 7 ]
The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. [ 2 ] [ 6 ] [ 8 ] The connective with the largest scope in a formula is called its dominant connective, [ 9 ] [ 10 ] main connective , [ 6 ] [ 8 ] [ 7 ] main operator , [ 2 ] major connective , [ 4 ] or principal connective ; [ 4 ] a connective within the scope of another connective is said to be subordinate to it. [ 6 ]
For instance, in the formula ( ( ( P → Q ) ∨ ¬ Q ) ↔ ( ¬ ¬ P ∧ Q ) ) {\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))} , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧; the first ¬ is also subordinate to the ∨, but not to the →; the second ¬ is subordinate to the ∧, but not to the ∨ or the →; and the third ¬ is subordinate to the second ¬, as well as to the ∧, but not to the ∨ or the →. [ 6 ] If an order of precedence is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form ( P → Q ) ∨ ¬ Q ↔ ¬ ¬ P ∧ Q {\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q} , which some may find easier to read. [ 6 ]
The scope of a quantifier is the part of a logical expression over which the quantifier exerts control. [ 3 ] It is the shortest full sentence [ 5 ] written right after the quantifier, [ 3 ] [ 5 ] often in parentheses; [ 3 ] some authors [ 11 ] describe this as including the variable written right after the universal or existential quantifier. In the formula ∀ xP , for example, P [ 5 ] (or xP ) [ 11 ] is the scope of the quantifier ∀ x [ 5 ] (or ∀ ). [ 11 ]
This gives rise to the following definitions: [ a ] | https://en.wikipedia.org/wiki/Scope_(logic) |
In the fields of computational chemistry and molecular modelling , scoring functions are mathematical functions used to approximately predict the binding affinity between two molecules after they have been docked . Most commonly one of the molecules is a small organic compound such as a drug and the second is the drug's biological target such as a protein receptor . [ 1 ] Scoring functions have also been developed to predict the strength of intermolecular interactions between two proteins [ 2 ] or between protein and DNA . [ 3 ]
Scoring functions are widely used in drug discovery and other molecular modelling applications. These include: [ 4 ]
A potentially more reliable but much more computationally demanding alternative to scoring functions are free energy perturbation calculations. [ 8 ]
Scoring functions are normally parameterized (or trained) against a data set consisting of experimentally determined binding affinities between molecular species similar to the species that one wishes to predict.
For currently used methods aiming to predict affinities of ligands for proteins the following must first be known or predicted:
The above information yields the three-dimensional structure of the complex. Based on this structure, the scoring function can then estimate the strength of the association between the two molecules in the complex using one of the methods outlined below. Finally the scoring function itself may be used to help predict both the binding mode and the active conformation of the small molecule in the complex, or alternatively a simpler and computationally faster function may be utilized within the docking run.
There are four general classes of scoring functions: [ 9 ] [ 10 ] [ 11 ]
The first three types, force-field, empirical and knowledge-based, are commonly referred to as classical scoring functions and are characterized by assuming their contributions to binding are linearly combined. Due to this constraint, classical scoring functions are unable to take advantage of large amounts of training data. [ 35 ]
Since different scoring functions are relatively co-linear, consensus scoring functions may not improve accuracy significantly. [ 36 ] This claim went somewhat against the prevailing view in the field, since previous studies had suggested that consensus scoring was beneficial. [ 37 ]
A perfect scoring function would be able to predict the binding free energy between the ligand and its target. But in reality both the computational methods and the computational resources put restraints to this goal. So most often methods are selected that minimize the number of false positive and false negative ligands. In cases where an experimental training set of data of binding constants and structures are available a simple method has been developed to refine the scoring function used in molecular docking. [ 38 ] | https://en.wikipedia.org/wiki/Scoring_functions_for_docking |
A scorpion sting is an injury caused by the stinger of a scorpion resulting in the medical condition known as scorpionism , which may vary in severity. The anatomical part of the scorpion that delivers the sting is called a " telson ". In typical cases, scorpion stings usually result in pain, paresthesia , and variable swelling. In serious cases, scorpion stings may involve the envenomation of humans by toxic scorpions, which may result in extreme pain, serious illness, or even death depending on the toxicity of the venom. [ 1 ] [ 2 ] [ 3 ]
Most scorpion stings range in severity from minor swelling to medically significant lesions, with only a few able to cause severe allergic, neurotic or necrotic reactions. However, scorpion stings account for approximately 3,000 deaths a year worldwide. [ 4 ] The Brazilian yellow scorpion ( Tityus serrulatus ) is one species known for being especially dangerous, being responsible for most scorpion sting fatalities in South America . [ 4 ]
Scorpion stings are seen all over the world but are predominantly seen in the tropical and subtropical areas. In the Western hemisphere, these areas include Mexico , northern South America and southeast Brazil . In the Eastern hemisphere, these regions include Sub-Saharan Africa , the Middle East , and the Indian subcontinent .
The byproducts of some arthropods may be used as an aphrodisiac . Some of these arthropods whose byproduct may be used as medicines can be found in North America. [ 5 ] Across North America, the Arizona bark scorpion ( Centruroides sculpturatus) has proven to be the most venomous scorpion. While stings from this species will rarely result in death, side effects can include numbness, tingling, convulsions, difficult breathing, and occasionally, paralysis. These side effects may last up to 72 hours after injection of the venom. It is also observed that penile erection may occur after being stung. The pain of a sting from the Arizona Bark Scorpion has been compared to being struck by a bolt of lightning or electrical current. [ 5 ] These symptoms may become visible 4 to 7 minutes after envenomation.
Envenomation of a human by a scorpion may affect the sympathetic or parasympathetic systems depending on the species of scorpion. Some of the more severe side effects include respiratory distress syndrome, pulmonary edema, cardiac dysfunction, impaired hemostasis, pancreatitis, and multiple organ failure. Additionally, treatment of the sting depends on the severity of the incident, which is classified as mild, moderate, or severe. This treatment is composed of three different aspects of the sting: symptomatic measures, vital functions support, and injection antivenom . Not all envenomations result in systemic complications; only a small proportion of stings have this effect on the victim. [ 6 ]
The composition of scorpion venom consists of different compounds of varying concentrations. The compounds consist of neurotoxins, cardiotoxin, nephrotoxin, hemolytic toxin, phosphodiesterases, phospholipase, histamine, serotonin , etc. Of these different toxins, the most important and most potent one is the neurotoxin concentration. This compound has neuromuscular and neuroautonomic effects, as well as damages the surrounding local tissue. Neurotoxins work to change voltage-dependent sodium channels, resulting in prolonged neuronal and neuromuscular activity. This prolonged activity of sodium channels results in an erection. There may be nerve damage due to the stabilization of voltage-dependent sodium channels in the open conformation. This position leads to the prolonged and continuous firing of neurons in the somatic, sympathetic, and parasympathetic nervous systems. Continuous firing of neurons causes over excitation and prevents the transmission of normal nerve impulses down the axon. [ 5 ]
The venom composition of the deathstalker scorpion contains neurotoxins which are almost completely responsible for this symptom. The poison from this scorpion contain 4 components: chlorotoxin , charybdotoxin , scyllatoxin , and agitoxins . Upon injection with the venom, sacral parasympathetic nerve are stimulated causing a change in the neuronal transmission in vascular and nonvascular smooth muscles. The compound known as the vasoactive intestinal polypeptide (VIP) is the main transmitter. This polypeptide is realized from nerves found long the erectile tissue of the corpus calosum. VIP is the strongest relaxant of penile smooth muscle structure, resulting in an erection upon envenomation. This is the proposed mechanics for all scorpion of the family Buthidae, whose venom composition contains these compounds. [ 5 ]
Scorpions are nocturnal animals that typically live in deserts, mountains, caves, and under rocks. It is when they are disturbed that they attack. Scorpions that possess the ability to inject toxic venom with their sting belong to the family Buthidae . The Middle East and North Africa are home to the deadliest scorpions, belonging to the genera Buthus , Leiurus , Androctonus , and Hottentotta . In South America, the deadliest scorpion belongs to genus Tityus . In India and Mexico, the deadliest scorpions involved in scorpionism are Mesobuthus and Centruroides , respectively. [ 7 ]
Scorpions are nocturnal arachnids that have shown a seasonal pattern that is also related to climate. [ 8 ] Specifically in Central America, scorpion attacks are mostly seen during the hot months of the year, noting that in Argentina this occurs in the months of October to April. Additionally, a rainy climate may also change the frequency of scorpion incidents. Lower levels of rainfall, specifically precipitation below 30 mm/month, can be associated with fewer scorpion stings, whereas rainfall greater than 30 mm/month shows no relationship to incident rate. This could be due to potentially disruptive effects of rainfall on scorpion habitat. [ 6 ]
In Central America, most scorpion stings are mildly toxic to humans. However, Panama has reported an incidence of 52 cases per 100,000 people in 2007. Between 1998 and 2006, 28 people have died as result of scorpion stings. In Panama, the taxa of scorpions responsible for these deaths belong to the genus Tityus . This scorpion species is also found in parts of northern South America. Historically, the presence of these scorpions in Panama could be due to the closure of the Panamanian isthmus, thus allowing for the migration of the scorpions from Panama into the northern part of South America. [ 9 ] Tityus pachyurus is among the most important scorpionist species. Envenomation by this scorpion is characterized by intense local pain, that usually does not result in tissue injury. [ 6 ] Scorpions possess venom glands located at the distal extremity of their abdomen. There are currently 1,400 known species of scorpions and each possesses venom glands. However, of these 1,400 species, only 25 are known to be dangerous enough to humans to potentially cause death upon envenomation. [ 3 ] Other countries in Central America are habitat to the scorpion genus Centruroides . Species in this genus are only mildly toxic to humans even though they have ion channel-active toxins in their venom. [ 9 ] | https://en.wikipedia.org/wiki/Scorpion_sting |
In coordination chemistry , a scorpionate ligand is a tridentate (three-donor-site) ligand that binds to a central atom in a fac manner. The most popular class of scorpionates are the hydrotris(pyrazolyl)borates or Tp ligands. These were also the first to become popular. These ligands first appeared in journals in 1966 from the then little-known DuPont chemist of Ukrainian descent, Swiatoslaw Trofimenko. Trofimenko called this discovery "a new and fertile field of remarkable scope". [ 1 ] [ 2 ] [ 3 ]
The term scorpionate comes from the fact that the ligand can bind a metal with two donor sites like the pincers of a scorpion ; the third and final donor site reaches over the plane formed by the metal and the other two donor atoms to bind to the metal. The binding can be thought of as being like a scorpion grabbing the metal with two pincers before stinging it.
While many scorpionate ligands are of the Tp class, many other scorpionate ligands are known. For example, the Tm and tripodal phosphine classes have an equally good claim to be scorpionate ligands. Many of the scorpionate ligands have a central boron atom which bears a total of four groups, but it is possible to create ligands which use other central atoms.
Trofimenko's initial work in the field was with the homoscorpionates where three pyrazolyl groups are attached to a boron . Since this work a range of ligands have been reported where more than one type of metal binding group is attached to the central atom; these are the heteroscorpionates.
Many other chemists continue to explore the possibilities of scorpionate ligand alternatives, such as:
Since the work by Wilkinson and others on ferrocene a vast amount of work has been done on cyclopentadienyl complexes. It was soon understood by many organometallic chemists that a Cp ligand is isolobal to Tp. As many insights into chemistry can be obtained by the study of a series of closely related compounds (where only one feature is changed) a great deal of organometallic chemistry has been done using Tp (and more recently Tm) as a co-ligand on the metal.
The Tp, Tm, trithia-9-crown-3 (a sulfur version of a small crown ether ) and cyclopentadienyl ( Cp ) ligands related ligands and form related complexes. These ligands donate the same number of electrons to the metal, and the donor atoms are arranged in a fac manner covering a face of a polyhedron .
The Tp and Tm ligands are isolobal with the Cp ligands. For example, the Cp manganese tricarbonyl complex is a half sandwich compound , with one face of the Cp ligand binding to the metal atom. The tricarbonyl manganese complex of trithia-9-crown-3 has the three sulfur atoms binding to the metal atom in the same place as the Cp ligand and using the same sort of orbitals for the bonding.
While the geometry of the Tp ligands do not allow the formation of simple borane complexes with the metals, the geometry of the Tm ligands (and sometimes their bidentate versions Bm) are such that with late transition metals such as osmium and platinum it is possible to turn the Tm ligand inside out to form a borane to which the metal forms a dative bond .
Here is the manganese complex of Tm with ( again three carbonyls ).
The tris(pyrazolyl)borate ligand is often known as Tp to many inorganic chemists - using different pyrazoles substituted in the 3,4, and 5 positions, a range of different ligands can be formed. In this article we will group all the trispyrazolylborates together.
These compounds are usually synthesized by reacting pyrazole with alkali-metal borohydrides, such as sodium borohydride NaBH 4 , under reflux. H 2 is evolved as the borohydride is sequentially converted first to pyrazolylborate [H 3 B(C 3 N 2 H 3 )], then to bis(pyrazolyl)borate [H 2 B(C 3 N 2 H 3 ) 2 ], and finally to tris(pyrazolyl)borate [HB(C 3 N 2 H 3 ) 3 ]. Bulky pyrazolyl borates can be prepared from 3,5-disubstituted pyrazoles, such as the dimethyl derivative. These bulky pyrazolyl borates have proven especially valuable in the preparation of catalysts and models for enzyme active sites. Utilizing scorpionate ligands in the syntheses of metal catalysts may allow simpler and more accurate methods to be developed. Ligands allow for good shielding of the bound metal while strong sigma bonds between the nitrogens and the metal stabilize the metal; these attributes help scorpionate compounds with creating highly symmetrical supramolecular silver complexes and olefin polymerization (with the compound hydrotris(pyrazolyl) borate Mn).
By replacing the nitrogen donor of a Tp ligand atoms with sulfur atoms, a class of ligands known as Tm can be made. These are related to the thioureas .¹; Several research groups including Anthony F. Hill 's group [ 5 ] have been working on this ligand class. To form NaTm {Na + HB(mt) 3 − ), Methimazole and sodium borohydride are heated together.
Coordination chemistry with ruthenium , rhodium , osmium , molybdenum , tungsten , and other metals has been reported.
A range of tripodal phosphines such as HC(CH 2 PR 2 ) 3 , N(CH 2 CH 2 PPh 2 ) 3 and P(CH 2 CH 2 PMe 2 ) 3 have been reviewed. [ 6 ] The tetra amine ( tris(2-aminoethyl)amine ) can be reacted with salicylaldehyde to form a ligand which can bind with three oxygens and three nitrogens to a metal. Trispyrazolylmethane (Tpm) is another class of scorpionate ligands, notable for having identical geometry and very similar coordination chemistry to Tp with only a difference in charge between them. [ 7 ] Another variation is the Trisoxazolinylborate ligand .
Hydrotris(pyrazolyl)aluminate (Tpa) complexes have similar coordination geometries to Tp complexes, however Tpa ligands are more reactive due to the weaker Al-N and Al-H bonds, compared to B-N and B-H bonds of Tp ligands, which results in either Tpa ligand transfer, pyrazolate transfer, or hydride transfer with MX 2 (M = Mg, Mn, Fe, Co, Ni, Cu, Zn; X = Cl, Br). [ 8 ] | https://en.wikipedia.org/wiki/Scorpionate_ligand |
Scorpionism is defined as the accidental envenomation of humans by toxic scorpions . If the injection of venom in a human results in death, this is defined as scorpionism. [ 1 ] This is seen all over the world but is predominantly seen in the tropical and subtropical areas. These areas include Mexico, northern South America and southeast Brazil in the Western hemisphere. In the Eastern hemisphere, scorpionism possess a public health threat in the regions of South Africa, the Middle East, and the Indian subcontinent.
Scorpions are nocturnal animals that typically live in deserts, mountains, caves, and under rocks. It is when they are disturbed that they attack. Scorpions that possess the ability to inject poisonous venom with their sting belong to the family Buthidae. The Middle East and North Africa are home to the deadliest scorpions, belonging to the genus Buthus, Leiurus, Androctonus, and Hottentotta . In the region of South Africa, the deadliest scorpion belongs to the Tityus genus. In India and Mexico, the deadliest scorpions involved in scorpionism are Mesobuthus and Centruroides , respectively. [ 2 ]
In Central America, most scorpion stings are mildly toxic to humans, however, Panama has reported an incidence of 52 cases per 100,000 people in 2007. Between 1998 and 2006, 28 people have died as result of scorpion stings. In Panama, the taxa of scorpions responsible for these deaths belong to the genus Tityus . This scorpion species is also found in parts of northern South America. Historically, the presence of these scorpions in Panama could be due to the closure of the Panamanian isthmus, thus allowing for the migration of the scorpions from Panama into the northern part of South America. [ 3 ] Tityus pachyurus b elongs to the family of Tityus scorpions found in Panama. T. pachyurus is among the most medically important species. Envenomation by this kind of scorpion is characterized by intense local pain, that usually does not result in tissue injury. [ 4 ] Scorpions possess poison glands located at the distal extremity of their abdomen. There are currently 1400 known species of scorpions and each possess venom glands. However, of these 1400 scorpions, only 25 are known to be dangerous and can result in death upon injection of their venom. [ 1 ] Other countries in Central America are habitat to the scorpion genus Centruroides. Species in this genus are only mildly toxic to humans even though they have ion channel-active toxins in their venom. [ 3 ]
Scorpions are nocturnal arachnids that have shown a seasonal pattern that is also related to climate. [ 5 ] Specifically in Central America, scorpion attacks are mostly seen during the hot months of the year, noting that in Argentina this occurs in the months of October to April. Additionally, a rainy climate may also change the frequency of scorpion incidents. Lower levels of rainfall, specifically precipitation below 30 mm/month, can be associated with less scorpion stings, whereas rainfall greater than 30 mm/month shows no relationship to incident rate. This could be due to the disruption of the scorpion habitat due to the rain. [ 4 ]
Across North America, the Arizona bark scorpion ( Centruroides sculpturatus) has proven to be the most venomous scorpion. However, a sting from this arachnid will rarely result in death. Rather side effects of their sting include pain, numbness, tingling, convulsions, difficult breathing, and may sometimes lead to paralysis. These side effects may last up to 72 hours after injection of the venom. It is also observed that penile erection may occur after being stung. The pain of a sting from the Arizona Bark Scorpion has been compared to being struck by a bolt of lightning or electrical current. [ 6 ] These symptoms may become visible 4 to 7 minutes after injection of venom.
Envenomation of a human by a scorpion may affect the sympathetic or parasympathetic systems depending on the species of scorpion. Some of the more severe side effects include respiratory distress syndrome, pulmonary edema, cardiac dysfunction, impaired hemostasis, pancreatitis, and multiple organ failure. Additionally, treatment of the sting depends on the severity of the incident ranking from mild, moderate, or severe. This treatment is composed of 3 different aspects of the sting: symptomatic measures, vital functions support, and injection antivenom . Not all envenomation resulting systemic complications; only a small proportion stings have this effect on the victim. [ 4 ]
The composition of scorpion venom consists of different compounds of varying concentrations. The compounds consist of neurotoxins, cardiotoxin, nephrotoxin, hemolytic toxic, phosphodiesterases, phospholipase, histamine, serotonin, etc. Of these different toxins, the most important and most potent one is the neurotoxin concentration. This compound has neuromuscular and neuroautonomic effects, as well as damages the surrounding local tissue. Neurotoxins work to change voltage- dependent sodium channels, resulting in prolonged neuronal and neuromuscular activity. This prolonged activity of sodium channels results in an erection. There may be nerve damage due to the stabilization of voltage-dependent sodium channels in the open conformation. This position leads to the prolonged and continuous firing of neurons in the somatic, sympathetic, and parasympathetic nervous systems. Continuous firing of neurons causes over excitation and prevent the transmission of normal nerve impulses down the axon. [ 6 ]
The venom composition of this scorpion contains neurotoxins which is almost completely responsible for this symptom. The poison from this scorpion contain 4 components: chlorotoxin, charybdotoxin, scyllatoxin, and agitoxins. Upon injection with the venom, sacral parasympathetic nerve are stimulated causing a change in the neuronal transmission in vascular and nonvascular smooth muscles. The compound known as the vasoactive intestinal polypeptide (VIP) is the main transmitter. This polypeptide is realized from nerves found long the erectile tissue of the corpus calosum. VIP is the strongest relaxant of penile smooth muscle structure, resulting in an erection upon envenomation. This is the proposed mechanics for all scorpion of the family Buthidae, whose venom composition contains these compounds. [ 6 ]
The byproducts of some arthropods may be used as an aphrodisiac . Aphrodisiacs are a group of medicines that may stimulate arousal or sexual desire through the use of drugs. These medicines usually help perform the sexual act and its effects may last a few hours. Some of these arthropods whose byproduct may be used as medicines can be found in North America. [ 6 ] | https://en.wikipedia.org/wiki/Scorpionism_in_Central_America |
Scotland's Rural College ( SRUC ; Scottish Gaelic : Colaiste Dhùthchail na h-Alba ) is a public land based research institution focused on agriculture and life sciences . Its history stretches back to 1899 with the establishment of the West of Scotland Agricultural College and its current organisation came into being through a merger of smaller institutions.
After the West of Scotland Agricultural College was established in 1899, the Edinburgh and East of Scotland College of Agriculture and the Aberdeen and North of Scotland College of Agriculture were both established in the early 20th century. These three colleges were merged into a single institution, the Scottish Agricultural College, in 1990. In October 2012, the Scottish Agricultural College was merged with Barony College, Elmwood College and Oatridge College to re-organise the institution as Scotland's Rural College, initialised as SRUC in preparation for it gaining the status of a university college with degree awarding powers.
SRUC has six campuses across Scotland – Aberdeen, Ayr, Barony, Elmwood, King's Buildings and Oatridge. Students study land based courses from further education to postgraduate level and degrees are currently awarded by the University of Edinburgh or the University of Glasgow depending on the course of study. Undergraduates study over a period of three terms each year during their first two years and two semesters during their third and fourth years. In addition to higher education , SRUC has a consulting division, SAC Consulting, which works with clients in agricultural businesses and associated rural industries and it also has a research division which carries out research in agriculture and life sciences.
SRUC has attracted notable botanists, chemists and agriculturists as lecturers and researchers and the institution has counted Henry Dyer , Victor Hope and Maitland Mackie amongst its academic staff. In addition to careers in agriculture and life sciences, the institution's alumni have gone on to have careers in politics, sport, the military and broadcasting.
In 1899, Glasgow and West of Scotland Technical College's agriculture department amalgamated with the Scottish Dairy Institute to form the West of Scotland Agricultural College. Originally based in Blythswood Square , Glasgow , the institution began moving to Ayrshire when in 1927 the Auchincruive estate in the parish of St Quivox near Ayr was left to the college by the late John Hannah of Girvan Mains. In 1974, the Blythswood Square site was closed. [ 2 ]
The Edinburgh and East of Scotland College of Agriculture was formed in 1901 and carried out experimental work in agriculture and animal breeding in south-east Scotland . Academic Robert Wallace helped found the college, having set up the bachelor's degree programme in agriculture during his time as a professor at the University of Edinburgh . Its main premises were in George Square , Edinburgh , and these were expanded in 1904 to a design by Thomas Purves Marwick architects. [ 3 ] The college also had experimental grounds at Pinkie Hill Farm, Inveresk . [ 4 ] In 1913, the college and the University of Edinburgh formed the joint committee on research in animal breeding which would research genetics. [ 5 ] [ 6 ]
The Aberdeen and North of Scotland College of Agriculture began in 1904 through grants from benefactors including the University of Aberdeen . Initially classes were delivered at Marischal College and these were then delivered at 581 King Street after 1969. Classes were then moved to a new teaching campus at the Craibstone Estate established in 1999, a site which the college had purchased in 1914 for research and fieldwork. [ 7 ] Nutritional physiologist John Boyd Orr , later a president of the National Peace Council and winner of the 1949 Nobel Peace Prize , ran the institution's joint committee for research into animal nutrition with the University of Aberdeen.
As technical colleges to transfer the growing scientific knowledge of agricultural issues to farmers and the general public, the three Scottish agricultural colleges were among ten central institutions noted in 1906 as providing technical instruction and sound scientific instruction meeting the "continuation class code" set of regulations drawn up in 1901. [ 8 ]
The company Scottish Agricultural Colleges was established in 1987 to provide direct management of advisory and veterinary functions of the regional colleges. In 1990, the West of Scotland Agricultural College was merged with the East of Scotland College of Agriculture and the North of Scotland College of Agriculture into the Scottish Agricultural College, a single higher education and research institution specialising in agriculture . The institution's three main divisions offered research, education and consultancy. [ 9 ] The new specialist institution was one of the largest of its type in Europe and the largest in the UK. [ 10 ] The institution offered undergraduate and postgraduate programmes from its three campuses in Ayr, Aberdeen and Edinburgh, as well as training and online study on topics including the environment, business, leisure, agriculture, horticulture and science.
Before it became a college, the 300 acre Barony estate had a varied existence. It was an elegant home, a home for the elderly, a wartime army training camp and, up until 1947, a prisoner of war camp. In 1949 Dumfries County Council education department purchased the estate with the purpose of turning it into an agricultural school. The Barony Farm School opened in 1953, with a class of 46 boys aged 14 to 15 years. Day release classes in agriculture and engineering began in 1962.
In 1972, the school became Barony Agricultural College and, over the 1970s, courses on offer expanded to include NC awards in agriculture, fish farming, forestry, countryside rangers, horticulture, animal care, veterinary nursing and equine studies. By this time, most students at the college were studying full time. A new teaching block, complete with a large sports hall, multigym and bar, was opened in 1992. The new millennium brought extra investment in animal care and veterinary nursing, an equine unit and a forestry technology centre. The dairy technology centre with a robotic milking system was opened in 2006.
Oatridge Agricultural College was established as a residential further education college specialising in agriculture and rural skills training in Ecclesmachan , West Lothian in 1969, with an initial intake of 45 residential students and 100 day students. [ 11 ] The college was local-authority owned by West Lothian District Council , having been established by a consortium of the district councils of West Lothian, Midlothian, East Lothian, Peebles, Roxburghshire, Selkirkshire and Berwickshire. [ 12 ]
The courses were initially taught in temporary accommodation on the farm site. New college buildings, workshops and accommodation were officially opened by Prince Philip, Duke of Edinburgh in 1974, and provided facilities for courses in agriculture, agricultural engineering, green keeping, horse care and forestry. [ 11 ]
Elmwood College was founded as Elmwood Agricultural and Technical College in 1972 as a rural further education institution based in Cupar , Fife . Its foundations were laid 20 years earlier when holding classes in the local school and cricket club before the education committee of Fife County Council acquired some land and erected a Nissen hut . This was followed by the purchase of Elmwood House, Gardens and Greenhouses in 1953 for £2,300. In 1956, the first day release classes in Scotland for agricultural and horticultural apprentices commenced at Elmwood Agricultural Centre. Elmwood College continued expanding during the 1960s and a new building was completed in 1972. By then Elmwood had also acquired Stratheden Hospital Farm.
Elmwood College was officially opened in 1972 by Hector Monro . [ 13 ] The Scottish Technical Education Consultative Council had made recommendations in 1967 around establishing regional farm centres. The college had started classes about twenty years earlier, with student numbers rising from 100 to around 2000 by 1972. [ 13 ] There were full-time as well as part-time courses, work based courses, and modern apprenticeships . Before the purpose-built building opened in 1972, classes had been held on borrowed premises over a few years. [ 14 ] A 350-acre farm was attached to the college and it offered courses such as hill-farming and shepherding. [ 13 ]
A college with a part-focus on golf education, Elmwood opened its own 18-hole working golf course in 1997. [ 15 ] Construction of a golf course began in 1995 with attention given to both the quality of the course and consideration of the local environment; the course was Geo Certified in 2013.
A proposal to merge the Scottish Agricultural College with Barony College, Elmwood College and Oatridge College was put to public consultation between March and May 2012. [ 16 ] Education Secretary Mike Russell voiced support for the merger in June 2012, [ 17 ] and Scotland's Rural College formally came into existence on 1 October 2012. [ 18 ] [ 19 ] [ 20 ] The work of the Scottish Agricultural College in education and training, research and development and consultancy services, would be continued by the newly-merged institution. [ 21 ] Scotland's Rural College was given the initialism SRUC upon its founding, as it would be working towards gaining the status of a university college with degree awarding powers. [ 22 ] [ 23 ] Professor Wayne Powell was appointed to serve as principal of SRUC in April 2016. [ 24 ]
In 2022, SRUC's application for degree awarding powers was approved to advance to the scrutiny stage by the Quality Assurance Agency for Higher Education advisory committee. This involves a minimum of 12 months of scrutiny, with the result of the application expected no earlier than summer 2023. [ 25 ]
SRUC has six education campuses located throughout Scotland, each offering varied land-based education courses. [ 26 ]
The Aberdeen campus is based on Craibstone Estate about 5 miles (8.0 km) outside Aberdeen in the north east of Scotland. As well as halls of residence and a library, the campus also boasts many sporting opportunities. Courses on offer in Aberdeen include agriculture, organic farming and countryside and environmental management.
The Ayr campus is shared with students from the University of the West of Scotland . The £81 million facility was opened in September 2011 and was awarded the internationally recognised BREEAM excellence rating for its environmentally friendly design. [ 27 ] As well as student accommodation, the campus has library, sporting activities and opportunities for climbing and horse riding. Courses on offer in Ayr include Agricultural Bioscience and Green Technology.
Barony campus is set in a working 260 hectares (2.6 km 2 ) estate in Dumfries and Galloway in south west Scotland. As well as the usual student facilities such as library and accommodation, the campus is home to the Scottish dairy technology centre and the Scottish Forestry Technology Centre. Courses on offer at Barony include animal care and forestry and arboriculture.
The King's Buildings campus is located on the south side of Edinburgh . This location allows students to access the University of Edinburgh 's academic and recreational facilities, with which it shares the campus. As well as libraries and cafes, the campus also has sporting amenities and bus links to the city centre. Courses on offer in Edinburgh include horticulture, applied animal science and rural resource management. The SRUC also has research facilities at the Easter Bush estate. Students studying horticulture with plantsmanship at the King's Buildings campus also study at the Royal Botanic Garden Edinburgh .
SRUC's Elmwood campus is based in Cupar, a small town in Fife approximately nine miles from St Andrews . There are three parts to Elmwood campus. The main campus is on Carslogie Road, Cupar. The second campus is at Stratheden , which is where the college's golf course is based. Cuparmuir Farm is the third campus, where most of the land-based courses are taught. As well having as a golf course, students can use badminton, table tennis, football and gym facilities. It continues teaching land based courses including conservation, greenkeeping and gamekeeping.
Situated in West Lothian , SRUC's Oatridge campus is set on a large estate which includes a working farm. As well as a student accommodation and a library, there is also a nine-hole golf course, and the campus is home to the Scottish National Equestrian Centre (SNEC). Courses on offer at Oatridge include farriery and forge work, and land-based engineering.
As a public institution , SRUC is registered as a charity under Scottish law. [ 29 ] The further education and undergraduate degree programmes at SRUC are grouped into six main departments: Agriculture and Business Management, Animal and Equine, Engineering, Science and Technology, Environment and Countryside, Horticulture and Landscape, and Sport and Tourism. Students can study taught courses which range from vocational and access level through undergraduate level, covering Higher National Certificate , Higher National Diploma and bachelor's degree courses, to postgraduate level, covering master's degree and PhD courses. Degrees are awarded by the University of Edinburgh and the University of Glasgow . Undergraduate students study over a period of three terms each year during their first two years and a period of two semesters during their third and fourth years.
The institution's consulting division, SAC Consulting, works with more than 12,000 clients in rural businesses and associated industries. The consulting arm has 24 offices located across Scotland and in the north of England, as well as eight veterinary surveillance centres. SRUC's research division operates in six research centres, and SRUC also runs eight farms for both research and educational purposes. SRUC's research division is divided into four interdisciplinary research groups, each devoted to its own focus of land-based research: Animal Health and Veterinary Science Group, Crop and Soil Systems Research Group, Future Farming Systems Research Group, and Land Economy, Environment and Society Research Group.
The institution has educated politicians Alex Fergusson , John Home Robertson , Ian Liddell-Grainger , Róisín McLaren , Hugh Roberton , Douglas Ross , Mark Ruskell , Struan Stevenson and Andy Stewart ; sportspeople Ian Barr , Thomas Muirhead , Jo Pitt and Doddie Weir ; military officers John Gilmour and William Reid VC ; and broadcaster and author Frances Tophill .
Governors of the institution have included pioneering technical educator Henry Dyer and agriculturist and Liberal Party politician Maitland Mackie . Agriculturist Victor Hope , later Governor-General of India , served as a president of the institution in the early 1930s. Chemist William Gammie Ogg , later director of the Rothamsted Experimental Station , worked as an advisory officer. Government agricultural adviser Arthur Wannop was a director of county work. Academics Ernest Shearer and Stephen John Watson successively served as principal in addition to their role as professor of agriculture at the University of Edinburgh. Margaret Farquhar , later Lord Provost of Aberdeen , had been a clerk at the institution before entering local government.
Botanists who have worked at the institution have included Green Party politician Martin Ford , Noel Farnie Robertson (who ran the partnership between the institution and the University of Edinburgh), William Gardner Smith and Edward Wyllie Fenton . Alexander Lauder and Hugh Nicol were both chemists who lectured there. Mycologist and plant pathologist R. W. G. Dennis researched oat pathology at the institution. Allen Kerr , a professor of plant pathology at University of Adelaide known for his study of crown gall , worked as an assistant mycologist and Alan Gemmell , the first professor of biology at Keele University , as an agricultural researcher. Veterinary surgeon William Christopher Miller lectured in animal hygiene and decorated Scout leader Alec Spalding MBE was an agricultural economist at the institution. Entomologist Daniel MacLagan served as head of the zoology department and William Whigham Fletcher as head of botany in Glasgow . Academic Allison Bailey worked at the institution before moving to New Zealand to become professor of farm management at Lincoln University . | https://en.wikipedia.org/wiki/Scotland's_Rural_College |
Scotobiology is the study of biology as directly and specifically affected by darkness , as opposed to photobiology , which describes the biological effects of light .
The science of scotobiology gathers together under a single descriptive heading a wide range of approaches to the study of the biology of darkness. This includes work on the effects of darkness on the behavior and metabolism of animals, plants, and microbes . Some of this work has been going on for over a century, and lays the foundation for understanding the importance of dark night skies , not only for humans but for all biological species.
The great majority of biological systems have evolved in a world of alternating day and night and have become irrevocably adapted to and dependent on the daily and seasonally changing patterns of light and darkness. Light is essential for many biological activities such as sight and photosynthesis . These are the focus of the science of photobiology. But the presence of uninterrupted periods of darkness, as well as their alternation with light, is just as important to biological behaviour. Scotobiology studies the positive responses of biological systems to the presence of darkness, and not merely the negative effects caused by the absence of light.
Many of the biological and behavioural activities of plants , animals (including birds and amphibians ), insects , and microorganisms are either adversely affected by light pollution at night or can only function effectively either during or as the consequence of nightly darkness. Such activities include foraging , breeding and social behavior in higher animals, amphibians, and insects, which are all affected in various ways if light pollution occurs in their environment. [ 1 ] These are not merely photobiological phenomena; light pollution acts by interrupting critical dark-requiring processes.
But perhaps the most important scotobiological phenomena relate to the regular periodic alternation of light and darkness. These include breeding behavior in a range of animals, the control of flowering and the induction of winter dormancy in many plants, and the operational control of the human immune system . In many of these biological processes the critical point is the length of the dark period rather than that of the light. For example, "short-day" and "long-day" plants are, in fact, "long-night" and "short-night" respectively. That is to say, plants do not measure the length of the light period, but of the dark period. [ 2 ] One consequence of artificial light pollution [ 3 ] is that even brief periods of relatively bright light during the night may prevent plants or animals (including humans) from measuring the length of the dark period, and therefore from behaving in a normal or required manner. This is a critical aspect of scotobiology, and one of the major areas in the study of the responses of biological systems to darkness.
In discussing scotobiology, it is important to remember that darkness (the absence of light) is seldom absolute. An important aspect of any scotobiological phenomenon is the level and quality ( wavelength ) of light that is below the threshold of detection for that phenomenon and in any specific organism. This important variable in scotobiological studies is not always properly noted or examined. There are substantial levels of natural light pollution at night, of which moonlight is usually the strongest. For example, plants that rely on night length to program their behaviour have the capacity to ignore full moonlight during an otherwise dark night. If this ability had not evolved, plants would not be able to respond to changing night-length for such behavioural programs as the initiation of flowering and the onset of dormancy. On the other hand, some animal behavioural patterns are strongly responsive to moonlight. It is thus most important in any scotobiological study to determine the threshold level of light that may be required to interfere with or negate the normal pattern of dark-night activity .
In 2003, at a symposium on the Ecology of the Night held in Muskoka , Canada , discussion centered around the many effects of night-time light pollution on the biology of a wide range of organisms, but it went far beyond this in describing darkness as a biological imperative for the functioning of biological systems. [ 1 ] Presentations focused on the absolute requirement of darkness for many aspects of normal behaviour and metabolism of many organisms and for the normal progression of their life cycles . Because there was no suitable term to describe the Symposium's main focus, the term scotobiology was introduced. The word is derived from the Greek scotos , σκότος, "dark," and relates to photobiology , which describes the biological effects of light (φῶς, phos ; root: φωτ-, phot- ). The term scotobiology appears not to have been used previously, although related terms such as skototropism [ 4 ] and scotophyle [ 5 ] have appeared in the literature. | https://en.wikipedia.org/wiki/Scotobiology |
Scotochromogenic bacteria develop pigment in the dark. Runyon Group II nontuberculous mycobacteria such as Mycobacterium gordonae are examples [ 1 ] [ page needed ] but the term could apply to many other organisms.
This bacteria -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scotochromogenic |
A scotophor is a material showing reversible darkening and bleaching when subjected to certain types of radiation. The name means dark bearer , in contrast to phosphor , which means light bearer . [ 1 ] Scotophors show tenebrescence (reversible photochromism ) and darken when subjected to an intense radiation such as sunlight . Minerals showing such behavior include hackmanite sodalite , spodumene and tugtupite . Some pure alkali halides also show such behavior.
Scotophors can be sensitive to light , particle radiation (e.g. electron beam – see cathodochromism ), X-rays , or other stimuli. The induced absorption bands in the material, caused by F-centers created by electron bombardment, can be returned to their non-absorbing state, usually by light and/or heating.
Scotophors sensitive to electron beam radiation can be used instead of phosphors in cathode ray tubes , for creating a light absorbing instead of light emitting image. Such displays are viewable in bright light and the image is persistent, until erased.
The image would be retained until erased by flooding the scotophor with a high-intensity infrared light or by electro-thermal heating. Using conventional deflection and raster formation circuitry, a bi-level image could be created on the membrane and retained even when power was removed from the CRT.
In Germany, scotophor tubes were developed by Telefunken as blauschrift-röhre ("dark-trace tube"). The heating mechanism was a layer of mica with transparent thin film of tungsten . When the image was to be erased, current was applied to the tungsten layer; even very dark images could be erased in 5–10 seconds. [ 2 ]
Scotophors typically require a higher-intensity electron beam to change color than phosphors need to emit light. Screens with layers of a scotophor and a phosphor are therefore possible, where the phosphor, flooded with a dedicated wide-beam low-intensity electron gun , produces backlight for the scotophor, and optionally highlights selected areas of the screen if bombarded with electrons with higher energy but still insufficient to penetrate the phosphor and change the scotophor state. [ 3 ]
The main application of scotophors was in plan position indicators , specialized military radar displays. The achievable brightness allowed projecting the image to a larger surface. [ 4 ] The ability to quickly record a persistent trace found its use in some oscilloscopes .
Potassium chloride is used as a scotophor with designation P10 in dark-trace CRTs (also called dark trace tubes , color center tubes , cathodochromic displays or scotophor tubes ), e.g. in the Skiatron . This CRT replaced the conventional light-emitting phosphor layer on the face of the tube screen with a scotophor such as potassium chloride (KCl). Potassium chloride has the property that when a crystal is struck by an electron beam , that spot would change from translucent white to a dark magenta color. [ citation needed ] By backlighting such a CRT with a white or green circular fluorescent lamp , the resulting image would appear as black information against a green background or as magenta information against a white background. A benefit, aside from the semi-permanent storage of the displayed image, is that the brightness of the resultant display is only limited by the illumination source and optics. The F-centers, however, have tendency to aggregate, and the screen needs to be heated to fully erase the image.
The image on KCl can be formed by depositing a charge of over 0.3 micro coulomb per square centimeter, by an electron beam with energy typically at 8–10 keV. The erasure can be achieved in less than a second by heating the scotophor at 150 °C. [ 3 ]
KCl was the most common scotophor used. Other halides show the same property; potassium bromide absorbs in bluish end of the spectrum, resulting in a brown trace, sodium chloride produces a trace that is colored more towards orange. [ 5 ]
Another scotophor used in dark-trace CRTs is a modified sodalite , fired in reducing atmosphere or having some chlorides substituted with sulfate ions. Its advantage against KCl is its higher writing speed, less fatigue, and the F-centers do not aggregate, therefore it is possible to substantially erase the screen with light only, without heating. [ 6 ] | https://en.wikipedia.org/wiki/Scotophor |
Scott's rule is a method to select the number of bins in a histogram . [ 1 ] Scott's rule is widely employed in data analysis software including R , [ 2 ] Python [ 3 ] and Microsoft Excel where it is the default bin selection method. [ 4 ]
For a set of n {\displaystyle n} observations x i {\displaystyle x_{i}} let f ^ ( x ) {\displaystyle {\hat {f}}(x)} be the histogram approximation of some function f ( x ) {\displaystyle f(x)} . The integrated mean squared error (IMSE) is
Where E [ ⋅ ] {\displaystyle E[\cdot ]} denotes the expectation across many independent draws of n {\displaystyle n} data points. By Taylor expanding to first order in h {\displaystyle h} , the bin width, Scott showed that the optimal width is
This formula is also the basis for the Freedman–Diaconis rule .
By taking a normal reference i.e. assuming that f ( x ) {\displaystyle f(x)} is a normal distribution , the equation for h ∗ {\displaystyle h^{*}} becomes
where σ {\displaystyle \sigma } is the standard deviation of the normal distribution and is estimated from the data. With this value of bin width Scott demonstrates that [ 5 ]
showing how quickly the histogram approximation approaches the true distribution as the number of samples increases.
Another approach developed by Terrell and Scott [ 6 ] is based on the observation that, among all densities g ( x ) {\displaystyle g(x)} defined on a compact interval , say | x | < 1 / 2 {\displaystyle |x|<1/2} , with derivatives which are absolutely continuous , the density which minimises ∫ ∞ ∞ d x ( g ( k ) ( x ) ) 2 {\displaystyle \int _{\infty }^{\infty }dx(g^{(k)}(x))^{2}} is
Using this with k = 1 {\displaystyle k=1} in the expression for h ∗ {\displaystyle h^{*}} gives an upper bound on the value of bin width which is
So, for functions satisfying the continuity conditions, at least
bins should be used. [ 7 ]
This rule is also called the oversmoothed rule [ 7 ] or the Rice rule , [ 8 ] so called because both authors worked at Rice University . The Rice rule is often reported with the factor of 2 outside the cube root, 2 ( n ) 1 / 3 {\displaystyle 2\left(n\right)^{1/3}} , and may be considered a different rule. The key difference from Scott's rule is that this rule does not assume the data is normally distributed and the bin width only depends on the number of samples, not on any properties of the data.
In general ( 2 n ) 1 / 3 {\displaystyle \left(2n\right)^{1/3}} is not an integer so ⌈ ( 2 n ) 1 / 3 ⌉ {\displaystyle \lceil \left(2n\right)^{1/3}\rceil } is used where ⌈ ⋅ ⌉ {\displaystyle \lceil \cdot \rceil } denotes the ceiling function . | https://en.wikipedia.org/wiki/Scott's_rule |
Cobalt(II) thiocyanate is an inorganic compound with the formula Co(SCN) 2 . [ 1 ] The anhydrous compound is a coordination polymer with a layered structure. The trihydrate , Co(SCN) 2 (H 2 O) 3 , is a isothiocyanate complex used in the cobalt thiocyanate test (or Scott test ) for detecting cocaine .
The structures of Co(SCN) 2 and its hydrate Co(SCN) 2 (H 2 O) 3 have been determined using X-ray crystallography . [ 1 ] Co(SCN) 2 forms infinite 2D sheets as in the mercury(II) thiocyanate structure type, where as Co(SCN) 2 (H 2 O) 3 consists of isolated tetrahedral Co(SCN) 2 (H 2 O) 2 centers and one equivalent of water of crystallization . [ 2 ]
The hydrate may be prepared by the salt metathesis reactions, such as the reaction of aqueous cobalt(II) sulfate and barium thiocyanate to produce a barium sulfate precipitate, leaving the hydrate of Co(SCN) 2 in solution: [ 2 ]
or the reaction of the hexakisacetonitrile cobalt(II) tetrafluoroborate and potassium thiocyanate , precipitating KBF 4
The anhydrate can then be prepared via addition of diethylether as an antisolvent. [ 1 ]
Scott's Test or Scott Test refers to a rapid and low-cost method of preliminary testing for cocaine . [ 3 ] It is a mixture of an acid medium and 2% cobalt(II) thiocyanate. [ 4 ] While typically used for cocaine, it will also indicate the presence of ketamine hydrochlorides , heroin , and dibucaine in amounts higher than 1 mg, as well as diltiazem and lidocaine in amounts higher than or equal to 5 mg. [ 5 ] When the cobalt thiocyanate reagent interacts with cocaine hydrochloride, the solution turns from a reddish-brown to a vibrant blue. [ 4 ]
The test has been responsible for widespread false positives and false convictions. [ 6 ] [ 7 ]
Detailed procedures for the cobalt thiocyanate test, often sold as the "morris reagent" are available. The reagent consists of 2% cobalt thiocyanate dissolved in dilute acid. [ 8 ] Glycerol is often added to stabilise the cobalt complex, ensuring it only goes blue when in contact with an analyte and not due to drying. [ 9 ]
Addition of the cobalt thiocyanate reagent to cocaine hydrochloride results in the surface of the particles turning a bright blue (faint blue for cocaine base). The solution changes back to pink upon adding some hydrochloric acid. Addition of chloroform, results in a blue organic layer for both cocaine hydrochloride and cocaine base. Diphenhydramine and lidocaine also give blue organic layers. These compounds are known false positives for cocaine. Lidocaine is commonly used to adulterate or mimic cocaine due to its local anaesthetic effect.
If the procedure is adjusted to basify the sample rather than acidifying it, the test can be used to test for ketamine hydrochloride. [ 10 ] | https://en.wikipedia.org/wiki/Scott's_test |
Scott J. Miller (born December 11, 1966) is an American organic chemist serving as Sterling Professor of Chemistry at Yale University and as the editor-in-chief of the Journal of Organic Chemistry . [ 1 ] He has been elected to the American Association for the Advancement of Science , [ 2 ] the American Academy of Arts and Sciences , [ 3 ] and the National Academy of Sciences , [ 4 ] and is known for his research into stereochemistry , asymmetric catalysis , enzymes , modification of natural products , and the synthesis and function of complex molecules. [ 1 ] [ 5 ]
Miller was born on December 11, 1966, in Buffalo, New York . [ 1 ] From 1985 to 1989, he studied at Harvard University , receiving Bachelor of Arts and Master of Arts degrees in chemistry. He continued his studies at Harvard, and received a Doctor of Philosophy in chemistry in 1994, advised by David A. Evans . He was a National Science Foundation postdoctoral fellow in the laboratory of Robert H. Grubbs at the California Institute of Technology from 1994 to 1996. Miller became an assistant professor of chemistry at Boston College in 1996, and received the National Science Foundation CAREER Award in 1999. In 2000, he was awarded a Camille Dreyfus Teacher-Scholar Award and a Sloan Research Fellowship ; he became an associate professor the next year. From 2002 until 2006, he was employed as a professor at Boston College; during his tenure, he won the Arthur C. Cope Scholar Award in 2004. [ 6 ] [ 7 ]
In 2006, Miller was appointed a professor of chemistry at Yale University , and was named Irénée du Pont Professor of Chemistry in 2008. [ 6 ] He was elected a fellow of the American Association for the Advancement of Science in 2011. [ 8 ] Miller became the editor-in-chief of the ACS Publications journal the Journal of Organic Chemistry in 2016, [ 9 ] [ 10 ] the same year he became a member of the American Academy of Arts and Sciences in recognition of his research into enzyme-mimicking peptidic catalysts. [ 3 ] In 2020, he was additionally elected to the National Academy of Sciences for his contributions to researching catalysts of molecular synthesis. [ 11 ] He became Sterling Professor of Chemistry in 2023, [ 12 ] Yale's "highest academic honor" for a professor. [ 5 ]
This biographical article about an American chemist is a stub . You can help Wikipedia by expanding it .
This article about an American scientist in academia is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scott_Miller_(chemist) |
Scott S. Reuben (born 1958) is an American anesthesiologist who falsified data heralding the benefits of the Pfizer pain medication Celebrex while downplaying its negative side effects. [ 1 ]
He was Professor of Anesthesiology and Pain Medicine at Tufts University in Boston , Massachusetts and chief of acute pain at Baystate Medical Center in Springfield, Massachusetts from February 1991 [ 2 ] until 2009 when he was sentenced to prison for healthcare fraud . Reuben was considered to be a prolific and influential researcher in pain management , and his purported findings altered the way millions of patients are treated for pain during and after orthopedic surgeries. [ 1 ] Reuben has now admitted that he never conducted any of the clinical trials on which his conclusions were based "in what may be considered the longest-running and widest-ranging cases of academic fraud." [ 3 ] Scientific American has called Reuben the medical equivalent of Bernie Madoff , the former NASDAQ chairman who was convicted of orchestrating a $65-billion Ponzi scheme . [ 1 ]
Reuben was educated at Columbia University . [ 4 ] He graduated from medical school at the State University of New York at Buffalo in 1985 and undertook his anesthesiology residency at Mount Sinai Medical Center in New York City .
Reuben fell under suspicion when Baystate conducted a routine audit in May 2008 which revealed that Reuben had not been given approval for two studies that he intended to present during the hospital's research week. On March 10, 2009 a Baystate spokeswoman announced that Reuben admitted to fabricating many of the data underlying his research. Reuben never conducted the clinical trials that he wrote about in 21 journal articles dating from at least 1996. [ 1 ] [ 3 ] In some cases, he even invented the patients. [ 5 ] Although Reuben often co-wrote papers with other researchers, Baystate found that the other researchers did not know about or participate in Reuben's studies, [ 5 ] and their names were forged on documents. [ 1 ] [ 4 ] The hospital asked the journals to retract the studies, which reported favorable results from painkillers including Pfizer Inc. 's Bextra , Celebrex and Lyrica and Merck & Co. Inc. 's Vioxx . His studies also claimed Wyeth 's antidepressant Effexor could be used as a painkiller. Pfizer gave Reuben five research grants between 2002 and 2007. He was a paid member of the company's speakers bureau, giving talks about Pfizer drugs to colleagues. [ 6 ] [ 7 ] Reuben also wrote to the Food and Drug Administration , urging the agency not to restrict the use of many of the painkillers he studied, citing his own data on their safety and effectiveness. [ 7 ]
"Doctors have been using (his) findings very widely," said Steven Shafer , editor of Anesthesia and Analgesia , a scientific journal that published ten articles identified as containing fraudulent data. [ 8 ] "His findings had a huge impact on the field." [ 6 ] He also described Reuben's actions as the biggest case of fraud in the history of anesthesiology. [ 5 ] Fellow editor Paul White believed that Reuben's fraudulent studies may have actually harmed patients due to the sale of "billions of dollars' worth of drugs" that caused slower recovery times. [ 1 ]
Reuben's work had actually come under scrutiny as early as 2007, when several anesthesiologists noticed his studies never showed negative results. [ 4 ] Greg Koski, former director of the Office for Human Research Protections , said the fraud was unusual because Reuben was able to carry it on for almost 13 years without being caught by the peer review process. [ 5 ]
All of Reuben's 21 fraudulent articles, as well as the article abstracts, are documented in the magazine Healthcare Ledger .
Tufts has since cut ties with Reuben, [ 7 ] and his Massachusetts medical license was permanently revoked after a period in which he voluntarily agreed not to practice.
On January 7, 2010 Reuben agreed to plead guilty to one count of health care fraud . Prosecutors alleged that Reuben obtained thousands of dollars in grants for research that he never performed. [ 9 ] He formally pleaded guilty on February 21, 2010 before Judge Michael Ponsor . On May 24, Ponsor sentenced him to six months in prison, followed by three years of supervised release. He was ordered to pay a $5,000 fine, forfeit $50,000 to the government and make $360,000 in restitution to pharmaceutical companies. [ 10 ] [ 11 ] The plea deal effectively ended his career as a doctor; most states will not grant medical licenses to convicted felons.
On November 16, 2011 the U.S. Food and Drug Administration (FDA) issued an order permanently debarring Reuben from assisting in drug applications. [ 12 ]
A 2009 review of systematic review articles used in evidence-based medicine found that while some reviews were no longer valid when the Reuben studies were removed, the conclusions in the majority of them remained unchanged. [ 13 ] The review found that the key Reuben claims that needed to be re-examined were "the absence of detrimental effects of coxibs on bone healing after spine surgery, the beneficial long-term outcome after preemptive administration of coxibs including an allegedly decreased incidence of chronic pain after surgery, and the analgesic efficacy of ketorolac or clonidine when added to local anesthetics for intravenous regional anesthesia." [ 13 ]
In 2010 the editorial in Anaesthesia argued that, [ 14 ]
Reuben's fabricated data may have had impact beyond systematic review conclusions because they addressed topical questions for which anaesthetists, surgeons, and patients seek answers, such as the utility of multimodal anaesthesia, or whether non-steroidal anti-inflammatory drugs (NSAIDs) influence bone healing ... Millions of people have had NSAIDs after fractures, trauma or orthopaedic surgery without problems of bone healing. The plausibility of a sizeable negative effect of NSAIDs on bone healing has to be questioned. | https://en.wikipedia.org/wiki/Scott_Reuben |
In mathematics , given two partially ordered sets P and Q , a function f : P → Q between them is Scott-continuous (named after the mathematician Dana Scott ) if it preserves all directed suprema . That is, for every directed subset D of P with supremum in P , its image has a supremum in Q , and that supremum is the image of the supremum of D , i.e. ⊔ f [ D ] = f ( ⊔ D ) {\displaystyle \sqcup f[D]=f(\sqcup D)} , where ⊔ {\displaystyle \sqcup } is the directed join. [ 1 ] When Q {\displaystyle Q} is the poset of truth values, i.e. Sierpiński space , then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets. [ 2 ]
A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins , i.e. if all directed sets D with supremum in O have non-empty intersection with O . The Scott-open subsets of a partially ordered set P form a topology on P , the Scott topology . A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology. [ 1 ]
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets. [ 3 ]
Scott-continuous functions are used in the study of models for lambda calculi [ 3 ] and the denotational semantics of computer programs.
A Scott-continuous function is always monotonic , meaning that if A ≤ P B {\displaystyle A\leq _{P}B} for A , B ⊂ P {\displaystyle A,B\subset P} , then f ( A ) ≤ Q f ( B ) {\displaystyle f(A)\leq _{Q}f(B)} .
A subset of a directed complete partial order is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets. [ 4 ]
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T 0 separation axiom ). [ 4 ] However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial. [ 4 ] The Scott-open sets form a complete lattice when ordered by inclusion . [ 5 ]
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order : x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y . The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober : the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology. [ 4 ]
The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X ) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology. [ 5 ]
For CPO , the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply . [ 6 ]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic . [ 7 ] | https://en.wikipedia.org/wiki/Scott_continuity |
In mathematics , the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott , ( Scott 1973 ). The precise statement is as follows:
Given a 3-manifold (not necessarily compact ) with finitely generated fundamental group, there is a compact three-dimensional submanifold , called the compact core or Scott core , such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable .
A simplified proof is given in ( Rubinstein & Swarup 1990 ), and a stronger uniqueness statement is proven in ( Harris & Scott 1996 ).
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scott_core_theorem |
In the mathematical fields of order and domain theory , a Scott domain is an algebraic , bounded-complete and directed-complete partial order (dcpo). They are named in honour of Dana S. Scott , who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to algebraic lattices , being different only in possibly lacking a greatest element . They are also closely related to Scott information systems , which constitute a "syntactic" representation of Scott domains.
While the term "Scott domain" is widely used with the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications.
Originally, Dana Scott demanded a complete lattice , and the Russian mathematician Yuri Yershov constructed the isomorphic structure of dcpo . But this was not recognized until after scientific communications improved after the fall of the Iron Curtain . In honour of their work, a number of mathematical papers now dub this fundamental construction a "Scott–Ershov" domain.
Formally, a non-empty partially ordered set ( D , ≤ ) {\displaystyle (D,\leq )} is called a Scott domain if the following hold:
Since the empty set certainly has some upper bound, we can conclude the existence of a least element ⊥ {\displaystyle \bot } (the supremum of the empty set) from bounded completeness.
The property of being bounded-complete is equivalent to the existence of infima of all non-empty subsets of D . It is well known that the existence of all infima implies the existence of all suprema and thus makes a partially ordered set into a complete lattice . Thus, when a top element (the infimum of the empty set) is adjoined to a Scott domain, one can conclude that:
Consequently, Scott domains are in a sense "almost" algebraic lattices. However, removing the top element from a complete lattice does not always produce a Scott domain. (Consider the complete lattice P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} . The finite subsets of N {\displaystyle \mathbb {N} } form a directed set, but have no upper bound in P ( N ) ∖ { N } {\displaystyle {\mathcal {P}}(\mathbb {N} )\setminus \{\mathbb {N} \}} .)
Scott domains become topological spaces by introducing the Scott topology .
Scott domains are intended to represent partial algebraic data , ordered by information content. An element x ∈ D {\displaystyle x\in D} is a piece of data that might not be fully defined. The statement x ≤ y {\displaystyle x\leq y} means " y {\displaystyle y} contains all the information that x {\displaystyle x} does". The bottom element is the element containing no information at all. Compact elements are the elements representing a finite amount of information.
With this interpretation we can see that the supremum ⋁ X {\displaystyle \bigvee X} of a subset X ⊆ D {\displaystyle X\subseteq D} is the element that contains all the information that any element of X {\displaystyle X} contains, but no more . Obviously such a supremum only exists (i.e., makes sense) provided X {\displaystyle X} does not contain inconsistent information; hence the domain is directed and bounded complete, but not all suprema necessarily exist. The algebraicity axiom essentially ensures that all elements get all their information from (non-strictly) lower down in the ordering; in particular, the jump from compact or "finite" to non-compact or "infinite" elements does not covertly introduce any extra information that cannot be reached at some finite stage.
On the other hand, the infimum ⋀ X {\displaystyle \bigwedge X} is the element that contains all the information that is shared by all elements of X {\displaystyle X} , and no less . If X {\displaystyle X} contains no consistent information, then its elements have no information in common and so its infimum is ⊥ {\displaystyle \bot } . In this way all non-empty infima exist, but not all infima are necessarily interesting.
This definition in terms of partial data allows an algebra to be defined as the limit of a sequence of increasingly more defined partial algebras—in other words a fixed point of an operator that adds progressively more information to the algebra. For more information, see Domain theory .
See the literature given for domain theory . | https://en.wikipedia.org/wiki/Scott_domain |
Cobalt(II) thiocyanate is an inorganic compound with the formula Co(SCN) 2 . [ 1 ] The anhydrous compound is a coordination polymer with a layered structure. The trihydrate , Co(SCN) 2 (H 2 O) 3 , is a isothiocyanate complex used in the cobalt thiocyanate test (or Scott test ) for detecting cocaine .
The structures of Co(SCN) 2 and its hydrate Co(SCN) 2 (H 2 O) 3 have been determined using X-ray crystallography . [ 1 ] Co(SCN) 2 forms infinite 2D sheets as in the mercury(II) thiocyanate structure type, where as Co(SCN) 2 (H 2 O) 3 consists of isolated tetrahedral Co(SCN) 2 (H 2 O) 2 centers and one equivalent of water of crystallization . [ 2 ]
The hydrate may be prepared by the salt metathesis reactions, such as the reaction of aqueous cobalt(II) sulfate and barium thiocyanate to produce a barium sulfate precipitate, leaving the hydrate of Co(SCN) 2 in solution: [ 2 ]
or the reaction of the hexakisacetonitrile cobalt(II) tetrafluoroborate and potassium thiocyanate , precipitating KBF 4
The anhydrate can then be prepared via addition of diethylether as an antisolvent. [ 1 ]
Scott's Test or Scott Test refers to a rapid and low-cost method of preliminary testing for cocaine . [ 3 ] It is a mixture of an acid medium and 2% cobalt(II) thiocyanate. [ 4 ] While typically used for cocaine, it will also indicate the presence of ketamine hydrochlorides , heroin , and dibucaine in amounts higher than 1 mg, as well as diltiazem and lidocaine in amounts higher than or equal to 5 mg. [ 5 ] When the cobalt thiocyanate reagent interacts with cocaine hydrochloride, the solution turns from a reddish-brown to a vibrant blue. [ 4 ]
The test has been responsible for widespread false positives and false convictions. [ 6 ] [ 7 ]
Detailed procedures for the cobalt thiocyanate test, often sold as the "morris reagent" are available. The reagent consists of 2% cobalt thiocyanate dissolved in dilute acid. [ 8 ] Glycerol is often added to stabilise the cobalt complex, ensuring it only goes blue when in contact with an analyte and not due to drying. [ 9 ]
Addition of the cobalt thiocyanate reagent to cocaine hydrochloride results in the surface of the particles turning a bright blue (faint blue for cocaine base). The solution changes back to pink upon adding some hydrochloric acid. Addition of chloroform, results in a blue organic layer for both cocaine hydrochloride and cocaine base. Diphenhydramine and lidocaine also give blue organic layers. These compounds are known false positives for cocaine. Lidocaine is commonly used to adulterate or mimic cocaine due to its local anaesthetic effect.
If the procedure is adjusted to basify the sample rather than acidifying it, the test can be used to test for ketamine hydrochloride. [ 10 ] | https://en.wikipedia.org/wiki/Scott_reagent |
The Scottish Railway Preservation Society is a registered charity , [ 1 ] whose principal objective is the preservation and advancement of railway heritage in Scotland . The society's headquarters are at Bo'ness , in central Scotland. [ 2 ]
The society undertakes conservation, restoration, repair, maintenance, and (where appropriate) demonstration operation of railway artifacts ranging from small objects to carriages, wagons and locomotives. It is also active in educational and curatorial activities including research, interpretation and outreach. [ 3 ]
The society built the Bo'ness and Kinneil Railway heritage railway [ 4 ] [ 5 ] on which the historic collection is demonstrated in action and stored.
The Society runs the Museum of Scottish Railways at its headquarters at Bo'ness. [ 6 ]
A subsidiary company, SRPS Railtours , operates excursion trains on the main line. [ 7 ] These excursion trains are mostly operated within or originating from Scotland . SRPS Railtours uses coaches from the SRPS's large fleet of preserved British Rail Mark 1 's for its excursions. Since 1970, these trains have travelled over the railway network as far as Wick and Penzance and frequently travel over the scenic West Highland Line and Kyle of Lochalsh Line . [ 8 ] | https://en.wikipedia.org/wiki/Scottish_Railway_Preservation_Society |
In mathematical logic , the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable . [ 1 ]
A set A of lambda terms is closed under beta-convertibility if for any lambda terms X and Y, if X ∈ A {\displaystyle X\in A} and X is β-equivalent to Y then Y ∈ A {\displaystyle Y\in A} . Two sets A and B of natural numbers are recursively separable if there exists a computable function f : N → { 0 , 1 } {\displaystyle f:\mathbb {N} \rightarrow \{0,1\}} such that f ( a ) = 0 {\displaystyle f(a)=0} if a ∈ A {\displaystyle a\in A} and f ( b ) = 1 {\displaystyle f(b)=1} if b ∈ B {\displaystyle b\in B} . Two sets of lambda terms are recursively separable if their corresponding sets under a Gödel numbering are recursively separable, and recursively inseparable otherwise.
The Scott–Curry theorem applies equally to sets of terms in combinatory logic with weak equality. It has parallels to Rice's theorem in computability theorem, which states that all non-trivial semantic properties of programs are undecidable.
The theorem has the immediate consequence that it is an undecidable problem to determine if two lambda terms are β-equivalent.
The proof is adapted from Barendregt in The Lambda Calculus . [ 2 ]
Let A and B be closed under beta-convertibility and let a and b be lambda term representations of elements from A and B respectively. Suppose for a contradiction that f is a lambda term representing a computable function such that f x = 0 {\displaystyle fx=0} if x ∈ A {\displaystyle x\in A} and f x = 1 {\displaystyle fx=1} if x ∈ B {\displaystyle x\in B} (where equality is β-equality). Then define G ≡ λ x . if ( zero? ( f x ) ) a b {\displaystyle G\equiv \lambda x.{\text{if}}\ ({\text{zero?}}\ (fx))ab} . Here, zero? {\displaystyle {\text{zero?}}} is true if its argument is zero and false otherwise, and if {\displaystyle {\text{if}}} is the identity so that if b x y {\displaystyle {\text{if}}\ bxy} is equal to x if b is true and y if b is false.
Then x ∈ C ⟹ G x = a {\displaystyle x\in C\implies Gx=a} and similarly, x ∉ C ⟹ G x = b {\displaystyle x\notin C\implies Gx=b} . By the Second Recursion Theorem, there is a term X which is equal to f applied to the Church numeral of its Gödel numbering, X ' . Then X ∈ C {\displaystyle X\in C} implies that X = G ( X ′ ) = b {\displaystyle X=G(X')=b} so in fact X ∉ C {\displaystyle X\notin C} . The reverse assumption X ∉ C {\displaystyle X\notin C} gives X = G ( X ′ ) = a {\displaystyle X=G(X')=a} so X ∈ C {\displaystyle X\in C} . Either way we arise at a contradiction, and so f cannot be a function which separates A and B . Hence A and B are recursively inseparable.
Dana Scott first proved the theorem in 1963. The theorem, in a slightly less general form, was independently proven by Haskell Curry . [ 3 ] It was published in Curry's 1969 paper "The undecidability of λK-conversion". [ 4 ] | https://en.wikipedia.org/wiki/Scott–Curry_theorem |
Scouring is a preparatory treatment of certain textile materials. Scouring removes soluble and insoluble impurities found in textiles as natural, added and adventitious impurities: for example, oils, waxes, fats, vegetable matter, as well as dirt. Removing these contaminants through scouring prepares the textiles for subsequent processes such as bleaching and dyeing . [ 1 ] Though a general term, "scouring" is most often used for wool . In cotton , it is synonymously called "boiling out", and in silk , and "boiling off. [ 2 ] [ 3 ] [ 4 ]
Scouring is an essential pre-treatment for the subsequent finishing stages that include bleaching, dyeing, and printing. [ 5 ] Raw and unfinished textiles contain a significant amount of impurities, both natural and foreign. It is necessary to eliminate these impurities to make the products ready for later steps in textile manufacturing . [ 6 ] For instance, fatty substances and waxy matters are the major barriers in the hydrophilicity of the natural fibers . Absorbency helps the penetration of chemicals in the stages such as dyeing and printing or finishing of the textiles . [ 7 ] These fats and waxy substances are converted into soluble salts with the help of alkali . [ 8 ] This treatment is called Saponification . [ 9 ]
Foreign matter in addition to fiber is known as "impurities." Textile fibers contain many types of impurities. e.g. :
The term "scouring" refers to the "act of cleaning with a rubbing action". [ 15 ]
Textile manufacturing was once an everyday household activity. [ 16 ] In Europe, women were often involved in textile manufacturing. They used to spin , weave , process, and finish the products they needed at home. [ 17 ] [ 18 ] [ 19 ] [ relevant? ] In the pre-industrial era, scouring (wool scouring) was a part of the fulling process of cloth making, in which the cloths were cleaned, and then milled (a thickening process). Fulling used to be done by pounding the woolen cloth with a club, or by the fuller's feet or hands. This process was associated with waulking songs , which were sung by women in the Scottish Gaelic tradition to set the pace. [ 20 ] [ 21 ] [ 22 ] [ 23 ]
Scouring agents are the cleaning agents that remove the impurities from the textiles during the scouring process. While these are now industrially-produced, scouring agents were once produced locally; lant or stale urine and lixivium , a solution of alkaline salts extracted from wood ashes , were among the earliest scouring agents. Lant, which contains ammonium carbonate , was used to scour the wool. [ 24 ] [ 25 ]
The removal of lanolin, vegetable materials and other wool contaminants before use is an example of wool scouring. [ 26 ] [ 27 ] Wool scouring is the next process after the woollen fleece of a sheep is cut off. [ 28 ] Raw wool is also known as ''Greasy wool.''
"Grease" or "yolk'' is a combined form of dried sweat, oil and fatty matter. [ 29 ] Lanolin is the major component (5-25%) of raw wool which is a waxy substance secreted by the sebaceous glands of wool-bearing animals. [ 30 ] Greasy matter varies by breed. [ 31 ] Following the cleaning process, the wool fibers possess a chemical composition of keratin . [ 32 ] [ 33 ]
Three steps comprise the complete cleaning process for wool: steeping, scouring, and rinsing. [ 35 ]
Potash and wool fat are two beneficial substances among the contaminants in wool, necessitating the development of specific cleaning techniques capable of recovering these compounds. Steeping is an alternative scouring process, In steeping system, scouring entails in parts. Wool steeping is carried out in stages such as immersing it in lukewarm water for many hours. [ 36 ] [ 37 ] When the wool includes only a little amount of yolk, the steeping method for recovering the yolk can be skipped. [ 38 ]
Scouring is the process of cleaning wool that makes it free from grease, suint (residue from perspiration), dead skin and dirt and vegetable matter present as impurities in the wool. It may consist of a simple boiling of wool in water or an industrial process of treating wool with alkalis and detergents (or soap and Sodium carbonate .) [ 39 ] [ 6 ] [ 40 ] Bath temperature is maintained (at 65 degree Celsius) to melt wool grease. [ 41 ] (Lanolin melts at a temperature of 38-44 °C.) [ 42 ]
The next treatment is carbonization , a treatment with strong acids that convert vegetable matter into carbon . [ 43 ]
Rinsing is the process of thoroughly washing the cleaned wool.
The alternative method is solvent scouring. [ 44 ]
Solvent scouring of wool replaces soap, detergent, and alkalies with a solvent liquid such as carbon tetrachloride , ether , petroleum naphtha , Chloroform , benzene , or carbon disulfide , etc. These liquids are capable of dissolving impurities but highly volatile and flammable . Hence, they need extra care in handling. [ 45 ]
In cotton, non-cellulosic substances such as waxes, lipids, pectic substances, organic acids contribute to around ten percent of the weight. [ 13 ] Cotton, in particular, has fewer impurities than wool. [ 46 ] Cotton scouring refers to removing impurities such as natural wax, pectins, and non-fibrous matter with a wetting agent and caustic soda. [ 8 ] : 25 [ 3 ] In comparison, alkaline boiling has no effect on cellulose. [ 46 ]
Cotton Pectins, waxes, proteins, mineral compounds, and ash, etc.
In discontinuous method certain machines are used such as dyeing vessels, winches, jiggers and Kier. [ 2 ] : 19, 20 [ 8 ] : 51 [ 3 ]
Kier is a large cylindrical vessel, upright, with egg shaped ends made of boilerplate that has a capacity of treating one to three tonnes of material at a time. [ 47 ]
Kier boiling and ''Boiling off'' is the scouring process that involves boiling the materials with the caustic solution in the Kier, which is an enclosed vessel, so that the fabric can boil under pressure. [ 48 ] [ 49 ] [ 50 ] Open kiers were also used with temperatures below 100 °C (at atmospheric pressure). [ 46 ]
Biotechnology in textiles is the advanced way of processing, textiles, it contributes to numerous treatments of cellulosic materials such as desizing , denim washing, biopolishing, and scouring, etc. [ 51 ]
Enzymes are helpful in bio-singeing, bio-scouring and removing impurities from cotton, which is more environmentally friendly. [ 52 ] Biopolishing is an alternative method that is an enzymetic treatment to clean the surface of cellulosic fabrics or yarns. It is also named ''Biosingeing.'' [ 53 ] [ 54 ] Pectinase enzymes, breaks down pectin , a polysaccharide found in cellulosic materials such as cotton. [ 55 ]
Silk is an animal fiber it consists 70–80% fibroin and 20–30% sericin (the gum coating the fibres). It carries impurities like dirt, oils, fats and sericin . The purpose of silk scouring is to remove the coloring matter and the gum that is a sticky substance which envelops the silk yarn. The process is also called ''degumming''. The gum contributes nearly 30 percent of the weight of unscoured silk threads. Silk is called ''boiled off'' when the gum is removed. The process includes the boiling the silk in a soap solution and rinsing it out. [ 56 ] [ 14 ]
Oil and dirt are the impurities in Synthetic materials . Certain oils and waxes are applied as lubricants during spinning or fabric manufacturing stages such as knitting or weaving. Mild detergents can remove the impurities effectively. [ 57 ]
Effluent is waste water that is thrown away in the water bodies. Industrial wastewater contaminated with scouring residues is heavily contaminated and extremely polluted. [ 58 ] [ 4 ] | https://en.wikipedia.org/wiki/Scouring_(textiles) |
In ecology , scramble competition (or complete symmetric competition or exploitation competition ) refers to a situation in which a resource is accessible to all competitors (that is, it is not monopolizable by an individual or group). However, since the particular resource is usually finite, scramble competition may lead to decreased survival rates for all competitors if the resource is used to its carrying capacity . Scramble competition is also defined as "[a] finite resource [that] is shared equally amongst the competitors so that the quantity of food per individual declines with increasing population density ". [ 1 ] A further description of scramble competition is "competition for a resource that is inadequate for the needs of all, but which is partitioned equally among contestants, so that no competitor obtains the amount it needs and all would die in extreme cases."
Researchers recognize two main forms of intraspecific competition , where members of a species are all using a shared resource in short supply. [ 2 ] These are contest competition and scramble competition. [ 1 ]
Contest competition is a form of competition where there is a winner and a loser and where resources can be attained completely or not at all. Contest competition sets up a situation where "each successful competitor obtains all resources it requires for survival or reproduction". [ 1 ] Here "contest" refers to the fact that physical action plays an active role in securing the resource. Contest competition involves resources that are stable, i.e. food or mates. Contests can be for a ritual objective such as territory or status, and losers may return to the competition another day to try again. [ 3 ]
In scramble competition resources are limited, which may lead to group member starvation . [ 3 ] Contest competition is often the result of aggressive social domains, including hierarchies or social chains . [ 4 ] Conversely, scramble competition is what occurs by accident when competitors naturally want the same resources. [ 4 ] These two forms of competition can be interwoven into one another. Some researchers have noted parallels between intraspecific behaviors of competition and cooperation. [ 4 ] These two processes can be evolutionarily adopted and they can also be accidental, which makes sense given the aggressive competition and collaborative cooperation aspects of social behavior in humans and animals. [ 4 ] To date, few studies have looked at the interplay between contest and scramble competition, despite the fact that they do not occur in isolation. There appears to be little understanding of the interface between contest competition and scramble competition in insects. [ 5 ] Much research still needs to be conducted concerning the overlap of contest and scramble competition systems. [ 5 ] Contests can arise within a scramble competition system and conversely, scramble competition "may play a role in a system characterized by interference". [ 5 ]
Population can be greatly affected by scramble competition (and contest competition). Intraspecific competition normally leads to a decline of organisms. [ 4 ] For example, the more time that an individual spends seeking food and reproduction opportunities, the less energy that organism naturally has to defend oneself against predators, resulting in a " zero-sum game ". [ 2 ] Competition is a density dependent effect, and scramble competition is no exception. Scramble competition usually involves interactions among individuals of the same species, which makes competition balanced and often leads to a decline of population growth rate as the amount of resources depletes. [ 6 ]
The Ricker Model , [ 1 ] is used to model scramble competition. It was originally formulated to study the growth of salmon populations and is given by the equation P n+1 = R(P n ) = P n e r(1-P n /k) , where P n is the population at the nth time period, r is the Malthusian growth rate , and k is the carrying capacity of the population. The Ricker model, and a few other well-known population models, can be explicitly derived from individual-level processes assuming scramble competition and a random distribution of individuals among resources. [ 7 ]
Some researchers have noted that in certain species, such as the horseshoe crab , males are most successful at mating when they are able to practice scramble competition polygyny where they do not defend their territory but rather mate and move on, thus providing the highest likelihood of species survival and reproductive prowess. [ 8 ]
There are many examples of scramble competition within the environment. For example, cows grazing in a grassland could be operating under a scramble competition. This illustration of cows eating grass is scramble competition because there are limited resources, there is only so much grass to be eaten before all the food resource is depleted. Additionally, there is no way that others can limit the amount of resources or the access to resources that the other cows receive. [ 9 ]
Another example of scramble competition is forest defoliators. If their larvae can find shelter and food then survival is possible, but when all the foliage is destroyed then the population decreases. Their synchronized life cycle increases competition for specific resources; this greatly affects their ability to receive resources including food and shelter due to the overwhelming population increase at certain times of the year. [ 6 ]
Scramble competition can also be seen with the example of red spotted newts . Charles Darwin (1871) first explored the concept of " sexual dimorphism " which states that, "most sexually dimorphic species are also the most polygynous" which would enable males to "outcompete other males through female choice, combat, or scrambles to encounter females would be favored by selection, and sexual dimorphism would result". [ 10 ] The key to red-spotted newts increased success in scramble competition is the newts enhanced or strengthened tailfins. [ 10 ]
Another example of scramble competition is the success of small beetles over large beetles. [ 11 ] While larger beetles, similar to larger animals in general, tend to win more often in contest competition, the opposite can be true in a scramble competition. [ 11 ] Specifically with beetles, scramble competition is dependent on male movement and locomotion so that the beetle that can move faster is more likely to be successful in attaining resources, mates and food. Smaller beetles fare better in scramble competition for shelter, which could one day lead to the evolutionary adaptation of smaller beetle structures for survival purposes. The flux of contest and scramble competition in this example is important to note because it truly depends on the context of each individual to determine which type of competition is most suitable.
Scramble competition also exists in lepidopterans . For example, male mourning cloak butterflies will fly around in search for widely dispersed females. [ 12 ]
Another example of scramble competition exists in Lactrodectus hesperus , the western black widow spider. There is a male-bias or skew within the sexually active population of this species, which means that females are a finite "resource". So, while no male has a monopoly over the females, the males who reach the female first will come out on top in the world of sexual reproduction. [ 13 ] | https://en.wikipedia.org/wiki/Scramble_competition |
Scramspace was a hypersonic engine research project established by the University of Queensland , Australia's Centre for Hypersonics. It was a 1.8 meter long, free-flying, hypersonic scramjet . A scramjet is fundamentally an air-breathing engine that travels at hypersonic velocities. Built in Brisbane at an estimated cost of $14 million, it took approximately 3 years to complete. [ 1 ] Scramspace was supposed to fire at a hypersonic velocity of Mach 8 or 8600 km/hour (5343 mph) [ 1 ] but the flight-test turned out to be a failure and the rocket engine and the payload plummeted in the North Sea off the coast of Norway.
Scramspace was designed and built at Brisbane, Australia. It took 3 years to build and was estimated to cost around $14 million. It was approximated to fly at around Mach 8. It was the first and the largest research project funded by the Australian Space Research Program. A number of ground-based research tests and Mach 8 flight experiments were involved to establish the research project. A number of engineers and PhD scholars were involved in the making of this project. [ 2 ]
Ground tests up to Mach 14 were performed to assess the scientific and technical parameters of the project. This was followed by flight tests up to Mach 8. [ 2 ]
The project involved five countries in partnership: Australia, Japan, Germany, Italy, and the United States. It was led by the University of Queensland's Center for Hypersonics. [ 2 ]
In August 2013, the scramjet was airlifted to Norway for a final flight test at Mach 8. The engine was fabricated to reach an altitude of about 340 km( 211.266 miles) with the help of a two- stage rocket engine.
According to the experiment, on leaving the atmosphere, the scramjet had to separate from the rocket engine and re-orient itself for reentry . The flight -sensor data had to be collected in a three-second window before the scramjet disintegrated on reentry.
However, because of some unknown issue in the first stage rocket motor, the scramjet payload could not be delivered to the correct altitude and speed in the flight test conducted on September 18, 2013. [ 1 ]
The uncrewed spacecraft with the payload and the rocket plummeted in the North Sea off the cost of Norway. [ 3 ]
The final stage of the project did not yield any hypersonic flight data. However, the ground testing, modelling and analysis were able to provide reference results for future projects. The project provided valuable insight and results pertaining to hypersonic physics , hypersonic combustion , and the performance of materials and components. It set an example for future hypersonic aircraft research. [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/Scramspace |
Scratch building is the process of building a scale model "from scratch", i.e. from raw materials, rather than building it from a commercial kit, kitbashing or buying it pre-assembled.
Scratch building is easiest if original plans of the subject exist; however, many models have been built from photographs by measuring a known object in the photograph and extrapolating the rest of the dimensions. The necessary parts are then fashioned out of a suitable material, such as wood , plastic , plaster , clay , metal , polymer clay , or even paper , and then assembled. Some purists consider a model not to be truly scratchbuilt unless all of the parts were made from raw materials. However most modellers would consider a model including commercial detail parts as scratchbuilt. Scratchbuilding a new body onto an altered ready-to-run chassis is also acceptable.
The reasons hobbyists scratchbuild may vary. Often a desired model is unavailable in kit form in the desired scale, or entirely non-existent. Sometimes the hobbyist may be dissatisfied with the accuracy or detail of kits that are available. Other times a hobbyist will opt to scratchbuild simply for the challenge. A hobbyist may also scratchbuild out of economy, as often the raw materials cost less than a packaged commercial kit.
Most hobbyists develop their skills by building kits, then progress to kitbashing , where various kits are combined to create a unique model before attempting to scratchbuild. Sometimes scratchbuilders utilize discarded parts of other models or toys, with or without modification, either in order to speed up the building process or to allow the process to continue in spite of certain parts being difficult to make. Some companies sell parts that are of little use to anyone but scratchbuilders.
Building stock, in whichever material, can be plain sheets, strips, bars, tubes, rods, or even structural shapes such as L or T girders . Stock can also be embossed or textured to replicate a certain prototype material (such as plastic sheet embossed with grooves to mimic a scale brick wall).
For example, to scratch-build a small board fence, the modeler could use plastic rod stock to form the vertical posts, then use plastic bar stock to form horizontal rails affixed to the posts, lay plastic strip stock vertically on the horizontal rails (perhaps 'distressed' with a wire brush to mimic wood grain, or by using thin strips of actual wood), trimming the top and bottoms of the strips to be even, adding details like nail holes (using a small pin), and then finishing and weathering (making a model look like it has been used via dust, dirt, stains, and wear).
There are as many ways of scratch-building models as there are modelers, but generally the modeler obtains plans of a prototype, or designs their own, transfers these plans to paper or other material as patterns (much easier since the arrival of household computer printers and copiers), and uses these patterns to cut, trim and affix the stock together to form the model. Finishing work (such as sanding and polishing, painting, weathering, distressing by forming dents and rust, dust, etc.) is done to complete the model and (hopefully) bring it to a lifelike state. | https://en.wikipedia.org/wiki/Scratch_building |
A scratch drive actuator (SDA) is a microelectromechanical system device that converts electrical energy into one-dimensional motion. [ 1 ] [ 2 ]
The actuator component can come in many shapes and sizes, depending on the fabrication method used. It can be visualised as an 'L'. The smaller end is called the 'bushing'.
The actuator sits on top of a substrate that has a thin insulating dielectric layer on top. A voltage is applied between the actuator and the substrate, and the resulting potential pulls the body of the actuator downwards. When this occurs, the brush is pushed forwards by a small amount, and energy is stored in the strained actuator. When the voltage is removed, the actuator springs back into shape while the bushing remains in its new position. By applying a pulsed voltage, the SDA can be made to move forward.
The voltage is usually applied to the actuator by means of a 'tether'. This can consist of a rigid connector or a rail which the SDA follows. [ 3 ]
The size of an SDA is typically measured on the μm scale.
This article about materials science is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scratch_drive_actuator |
Scratch hardness refers to the hardness of a material in terms of resistance to scratches and abrasion by a harder material forcefully drawn over its surface. Scratch hardness test or scratch test refers to any of a number of methods of measuring scratch hardness. Resistance to abrasion is less affected by surface variations than indentation methods . Scratch hardness is measured with a sclerometer . [ 1 ]
Attempting to scratch a surface to test a material is a very old technique. [ 2 ] The first scientific attempt to quantify materials by scratch tests was by mineralogist Friedrich Mohs in 1812 (see Mohs scale ). [ 3 ] [ 4 ] The Mohs scale is based on relative scratch hardness of different materials; with talc assigned a value of 1 and diamond assigned a value of 10. [ 5 ] Mohs's scale had two limitations: it was not linear, and most modern abrasives fall between 9 and 10.; [ 6 ] [ 7 ] so, later scientists attempted to increase resolution at the harder end of the scale.
Raymond R. Ridgway, a research engineer at the Norton Company , modified the Mohs scale by giving garnet a hardness of 10 and diamond a hardness of 15. [ 6 ] [ 8 ] Charles E. Wooddell, working at the Carborundum Company , extended the scale further by using resistance to abrasion, and extrapolating the scale based on 7 for quartz and 9 for corundum , resulting in a value of 42.4 for South American brown diamond bort . [ 9 ] [ 10 ]
There is a linear relationship between cohesive energy density (lattice energy per volume) and Wooddell wear resistance, occurring between corundum (H=9) and diamond (H=42.5). [ 11 ]
This article about materials science is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scratch_hardness |
Scratch space is space on the hard disk drive that is dedicated for storage of temporary user data, by analogy of "scratch paper." [ 1 ] [ 2 ] It is unreliable by intention and has no backup . Scratch disks may occasionally be set to erase all data at regular intervals so that the disk space is left free for future use. The management of scratch disk space is typically dynamic, occurring when needed. Its advantage is that it is faster than e.g. network filesystems.
Scratch space is commonly used in scientific computing workstations, and in graphic design programs such as Adobe Photoshop . It is used when programs need to use more data than can be stored in system RAM . A common error in that program is "scratch disks full", which occurs when one has the scratch disks configured to be on the boot drive. Many computer users gradually fill up their primary hard drive with permanent data, slowly reducing the amount of space the scratch disk may take up.
Partitioning off a significant fraction of the boot hard drive and leaving that space empty will ensure a reliable scratch disk. Hard drive space, on a per-gigabyte basis, is far cheaper than RAM, though performs far slower. Although dedicating a separate physical drive from the main operating system and software can improve performance, a scratch disk will not match RAM for speed. | https://en.wikipedia.org/wiki/Scratch_space |
Scratching , sometimes referred to as scrubbing , is a DJ and turntablist technique of moving a vinyl record back and forth on a turntable to produce percussive or rhythmic sounds. A crossfader on a DJ mixer may be used to fade between two records simultaneously.
While scratching is most associated with hip hop music , where it emerged in the mid-1970s, from the 1990s it has been used in some styles of EDM like techno , trip hop , and house music and rock music such as rap rock , rap metal , rapcore, and nu metal . In hip hop culture, scratching is one of the measures of a DJ's skills. DJs compete in scratching competitions at the DMC World DJ Championships and IDA (International DJ Association), formerly known as ITF (International Turntablist Federation). At scratching competitions, DJs can use only scratch-oriented gear (turntables, DJ mixer, digital vinyl systems or vinyl records only). In recorded hip hop songs, scratched "hooks" often use portions of other songs.
A rudimentary form of turntable manipulation that is related to scratching was developed in the late 1940s by radio music program hosts, disc jockeys (DJs), or the radio program producers who did their own technical operation as audio console operators. It was known as back-cueing, and was used to find the very beginning of the start of a song (i.e., the cue point) on a vinyl record groove. [ 1 ] This was done to permit the operator to back the disc up (rotate the record or the turntable platter itself counter-clockwise) in order to permit the turntable to be switched on, and come up to full speed without ruining the first few bars of music with the "wow" of incorrect, unnaturally slow-speed playing. This permitted the announcer to time their remarks, and start the turntable in time for when they wanted the music on the record to begin.
Back cueing was a basic skill that all radio production staff needed to learn, and the dynamics of it were unique to the brand of professional turntable in use at a given radio station. The older, larger and heavier turntables needed a 180-degree backward rotation to allow for run up to full speed; some of the newer 1950s models used aluminum platters and cloth-backed rubber mats which required a third of a rotational turn or less to achieve full speed when the song began. All this was done in order to present a music show on air with the least amount of silence ("alive air") between music, the announcer's patter and recorded advertising commercials. The rationale was that any "dead air" on a radio station was likely to prompt a listener to switch stations, so announcers and program directors instructed DJs and announcers to provide a continuous, seamless stream of sound–from music to an announcer to a pre-recorded commercial, to a "jingle" (radio station theme song), and then immediately back to more music.
Back-cueing was a key function in delivering this seamless stream of music. Radio personnel demanded robust equipment and manufacturers developed special tonearms, styli, cartridges and lightweight turntables to meet these demands.
Modern scratching techniques were made possible by the invention of direct-drive turntables , which led to the emergence of turntablism . Early belt-drive turntables were unsuitable for scratching since they had a slow start-up time, and they were prone to wear and tear and breakage, [ 2 ] as the belt would break from backspinning or scratching. [ 3 ] The first direct-drive turntable was invented by Shuichi Obata, an engineer at Matsushita (now Panasonic ), [ 4 ] based in Osaka , Japan . [ 2 ] It eliminated belts, and instead employed a motor to directly drive a platter on which a vinyl record rests. [ 5 ] In 1969, Matsushita released it as the SP-10 , [ 5 ] the first direct-drive turntable on the market, [ 6 ] and the first in their influential Technics series of turntables. [ 5 ]
In the 1970s, hip hop musicians and club DJs began to use this specialized turntable equipment to move the record back and forth, creating percussive sounds and effects–"scratching"–to entertain their dance floor audiences. Whereas the 1940s–1960s radio DJs had used back-cueing while listening to the sounds through their headphones, without the audience hearing, with scratching, the DJ intentionally lets the audience hear the sounds that are being created by manipulating the record on the turntable, by directing the output from the turntable to a sound reinforcement system so that the audience can hear the sounds. Scratching was developed by early hip hop DJs from New York City such as Grand Wizzard Theodore , who described scratching as, " nothing but the back-cueing that you hear in your ear before you push it [the recorded sound] out to the crowd." [ 7 ] He developed the technique when experimenting with the Technics SL-1200 , a direct-drive turntable released by Matsushita in 1972 when he found that the motor would continue to spin at the correct RPM even if the DJ wiggled the record back and forth on the platter. Afrika Bambaataa made a similar discovery with the SL-1200 in the 1970s. The Technics SL-1200 went on to become the most widely used turntable for the next several decades. [ 8 ]
Jamaican-born DJ Kool Herc , who immigrated to New York City, influenced the early development of scratching. Kool Herc developed break-beat DJing , where the breaks of funk songs—being the most danceable part, often featuring percussion —were isolated and repeated for the purpose of all-night dance parties. [ 9 ] He was influenced by Jamaican dub music, [ 10 ] and developed his turntable techniques using the Technics SL-1100, released in 1971, due to its strong motor, durability, and fidelity. [ 5 ]
Although previous artists such as writer and poet William S. Burroughs had experimented with the idea of manipulating a reel-to-reel tape manually to make sounds, as with his 1950s recording, "Sound Piece"), vinyl scratching as an element of hip hop pioneered the idea of making the sound an integral and rhythmic part of music instead of an uncontrolled noise. Scratching is related to "scrubbing" (in terms of audio editing and production) when the reels of an open reel-to-reel tape deck (typically 1/4 inch magnetic audiotape) are gently rotated back and forth while the playback head is live and amplified, to isolate a specific spot on the tape where an editing "cut" is to be made. Today, both scratching and scrubbing can be done on digital audio workstations (DAWs) which are equipped for these techniques.
Christian Marclay was one of the earliest musicians to scratch outside hip hop. In the mid-1970s, Marclay used gramophone records and turntables as musical instruments to create sound collages . He developed his turntable sounds independently of hip hop DJs. Although he is little-known to mainstream audiences, Marclay has been described as "the most influential turntable figure outside hip hop" [ 11 ] and the "unwitting inventor of turntablism ." [ 12 ]
In 1981 Grandmaster Flash released the song " The Adventures of Grandmaster Flash on the Wheels of Steel " which is notable for its use of many DJ scratching techniques. It was the first commercial recording produced entirely using turntables. In 1982, Malcolm McLaren & the World's Famous Supreme Team released a single " Buffalo Gals ", juxtaposing extensive scratching with calls from square dancing , and, in 1983, the EP, D'ya Like Scratchin'? , which is entirely focused on scratching. Another 1983 release to prominently feature scratching is Herbie Hancock 's Grammy Award -winning single " Rockit ". This song was also performed live at the 1984 Grammy Awards , and in the documentary film Scratch , the performance is cited by many 1980s-era DJs as their first exposure to scratching. The Street Sounds Electro compilation series which started in 1983 is also notable for early examples of scratching. Also, a notable piece was "For A Few Dollars More" by Bill Laswell - Michael Beinhorn band Material , released on 12" single in Japan and containing scratch performed by Grand Mixer DXT , another pioneer of scratching.
Most scratches are produced by rotating a vinyl record on a direct drive turntable rapidly back and forth with the hand with the stylus ("needle") in the record's groove. This produces the distinctive sound that has come to be one of the most recognizable features of hip hop music. [ 13 ] Over time with excessive scratching, the stylus will cause what is referred to as "cue burn", or "record burn". [ citation needed ]
The basic equipment setup for scratching includes two turntables and a DJ mixer , which is a small mixer that has a crossfader and cue buttons to allow the DJ to cue up new music in their headphones without the audience hearing. [ citation needed ] When scratching, this crossfader is utilized in conjunction with the scratching hand that is manipulating the record platter. The hand manipulating the crossfader is used to cut in and out of the record's sound. [ citation needed ]
Using a digital vinyl system (DVS) consists of playing vinyl discs on turntables whose contents are a timecode signal instead of a real music record.
There is not a single standard of DVS, so each form of DJ software has its own settings. Some DJ software such as Traktor Scratch Pro or Serato Scratch Live supports only the audio interface sold with their software, requiring multiple interfaces for one computer to run multiple programs.
Some digital vinyl systems software include:
While some turntablists consider the only true scratching media to be the vinyl disc, there are other ways to scratch, such as:
Sounds that are frequently scratched include but are not limited to drum beats, horn stabs , spoken word samples , and vocals/lyrics from other songs. Any sound recorded to vinyl can be used, and CD players providing a turntable-like interface allow DJs to scratch not only material that was never released on vinyl, but also field recordings and samples from television and movies that have been burned to CD-R . Some DJs and anonymous collectors release 12-inch singles called battle records that include trademark, novel or hard-to-find scratch "fodder" (material). The most recognizable samples used for scratching are the "Ahh" and "Fresh" samples, which originate from the song " Change the Beat " by Fab 5 Freddy .
There are many scratching techniques, which differ in how the movements of the record are combined with opening and closing the crossfader (or another fader or switch, such as a kill switch , where "open" means that the signal is audible, and "closed" means that the signal is inaudible). This terminology is not unique; the following discussion, however, is consistent with the terminology used by DJ QBert on his Do It Yourself Scratching DVD.
More complex combinations can be generated by grouping elementary crossfader motions (such as the open, close, and tap) into three and four-move sequences. [ 17 ] Closing and tapping motions can be followed by opens and taps, and opens can be followed by closes only.
Note that some sequences of motions ultimately change the direction of the switch, whereas others end in a position such that they can be repeated immediately without having to reset the position of the switch. Sequences that change the direction of the switch can be dovetailed with sequences that change it in the opposite directions to produce repeating patterns, or can be used to transition between open and closed crossfader techniques, such as chirps/flares and transforms, respectively. [ 18 ] These crossfader sequences are frequently combined with orbits and tears to produce combination scratches, such as the aquaman scratch, which goes "close-tap-open". [ 19 ]
While scratching is becoming more and more popular in pop music, particularly with the crossover success of pop-hip hop tracks in the 2010s, sophisticated scratching and other expert turntablism techniques are still predominantly an underground style developed by the DJ subculture. The Invisibl Skratch Piklz from San Francisco focuses on scratching. In 1994, the group was formed by DJs Q-Bert , Disk & Shortkut and later Mix Master Mike . In July 2000, San Francisco 's Yerba Buena Center for the Arts held Skratchcon2000, the first DJ Skratch forum that provided "the education and development of skratch music literacy". In 2001, Thud Rumble became an independent company that works with DJ artists to produce and distribute scratch records. [ citation needed ]
In 2004, Scratch Magazine , one of the first publications about hip hop DJs and record producers , released its debut issue, following in the footsteps of the lesser-known Tablist magazine. Pedestrian is a UK arts organisation that runs Urban Music Mentors workshops led by DJs. At these workshops, DJs teach youth how to create beats, use turntables to create mixes, act as an MC at events, and perform club sets.
Scratching has been incorporated into a number of other musical genres, including pop , rock , jazz , some subgenres of heavy metal (notably nu metal ) and some contemporary and avant-garde classical music performances. For recording use, samplers are often used instead of physically scratching a vinyl record.
DJ Product©1969, formerly of the rap rock band Hed PE , recalled that the punk rock band the Vandals was the first rock band he remembered seeing use turntable scratching. [ 20 ] Product©1969 also recalled the early rap metal band Proper Grounds, which was signed to Madonna 's Maverick Records , as being another one of the first rock bands to utilize scratching in their music. [ 20 ]
Guitarist Tom Morello , known for his work with Rage Against the Machine and Audioslave , has performed guitar solos that imitate scratching by using the kill switch on his guitar. Perhaps the best-known example is " Bulls on Parade ", in which he creates scratch-like rhythmic sounds by rubbing the strings over the pick-ups while using the pickup selector switch as a crossfader .
Since the 1990s, scratching has been used in a variety of popular music genres such as nu metal , exemplified by Linkin Park , Slipknot and Limp Bizkit . It has also been used by artists in pop music (e.g. Nelly Furtado ) and alternative rock (e.g. Incubus ). Scratching is also popular in various electronic music styles, such as techno . | https://en.wikipedia.org/wiki/Scratching |
Scream Tracker is a tracker (an integrated multi-track step sequencer and sampler as a software application). It was created by Psi (Sami Tammilehto), one of the founders of the Finnish demogroup Future Crew . [ 1 ] [ 2 ] It was written in C and assembly language .
The first version (1.0) had monophonic 4-bit output via the PC speaker , as well as 8-bit output via Covox 's Speech Thing (a digital-to-analog converter using the parallel port ) or a Sound Blaster 1.x card .
The first popular version of Scream Tracker, version 2.2 , was published in 1990. [ 3 ] Versions prior to 3.0 created STM (Scream Tracker Module ) files, while versions 3.0 and above used the S3M (ScreamTracker 3 Module) format.
As of version 3.0 , Scream Tracker supports up to 99 8-bit samples, 32 channels, 100 patterns, and 256 order positions. It can also handle up to 9 FM -synthesis channels on sound cards using the popular OPL2 / 3 / 4 chipsets, and, unusually, can play PCM samples and FM instruments at the same time. There are channels referred to as R1..8, L1..8 and A1..9 to be assigned to those 32 ones, which gives an effective amount of only 25 channels. 16-position free panning is available using the S8x command, but only on the Gravis Ultrasound . The usage of the A channels requires the presence of an AdLib-compatible card either by itself or alongside another sound card.
The last version of Scream Tracker was 3.21 , released in 1994, placing it in competition with FastTracker 2 . It was the precursor of the PC tracking scene and its interface inspired newer trackers like Impulse Tracker . [ 4 ] [ 5 ] Various other trackers (such as Impulse Tracker or OpenMPT ) adopted the use of the Scream Tracker's S3M format. | https://en.wikipedia.org/wiki/Scream_Tracker |
" Screaming Jelly Babies " ( British English ), also known as " Growling Gummy Bears " ( American and Canadian English ), is a classroom chemistry demonstration in which a piece of candy bursts loudly into flame when dropped into potassium chlorate . [ 1 ] The experiment is practiced in schools around the world and is often used at open evenings to show the more engaging and entertaining aspects of science in secondary education settings. [ 2 ] [ 3 ]
The experiment shows the amount of energy there is in a piece of candy. Jelly babies [ 4 ] or gummy bears [ 5 ] are often used for theatrics. Potassium chlorate, a strong oxidising agent, rapidly oxidises the sugar in the candy causing it to burst into flames. The reaction produces a "screaming" sound as rapidly expanding gases are emitted from the test tube. [ 6 ] The aroma of caramel is given off. [ 7 ] Other carbohydrate or hydrocarbon containing substances can be dropped into test tubes of molten chlorate to produce similar results. [ 8 ] [ 9 ]
4 KClO 3 (s) + C 12 H 22 O 11 (s) + 6 O 2 (g) → 4 KCl + 12 CO 2 (g) + 11 H 2 O (g)
The solid potassium chlorate is melted into a liquid.
KClO 3 (s) + energy → K + ClO 3 − (l)
The liquid potassium chlorate decomposes into potassium perchlorate and potassium chloride .
4 KClO 3 → KCl + 3 KClO 4
The potassium perchlorate decomposes into potassium chloride and oxygen .
KClO 4 → KCl + 2 O 2
The sugar in the candy reacts with oxygen , forming water and carbon dioxide . The reaction is exothermic and produces heat, smoke, and fire.
C 12 H 22 O 11 (s) + 12 O 2 (g) → 12 CO 2 (g) + 11 H 2 O (g) + energy. [ 10 ]
Care should be taken in performing this experiment, which should only be attempted by a professional. Potassium chlorate is a strong oxidizer and can cause fire or explosions. It is toxic by inhalation or ingestion and is hazardous to aquatic environments. [ 11 ] Reagent grade potassium chlorate should be used. Upon completion of the demonstration, all chemicals should be disposed of in designated chemical waste containers to prevent harm to people or the environment. [ 7 ]
All participants in the experiment should wear personal protective equipment , including eye protection, and should stand a safe distance away from the demonstration. [ 12 ] A face-shield and heat resistant gloves should be worn by the person adding the jelly baby to the molten potassium chlorate . [ 12 ]
Deviation from the experiment is not recommended, and has been linked with accidents. [ 12 ] Candy with low moisture content or high surface area may cause explosions. [ 12 ] | https://en.wikipedia.org/wiki/Screaming_jelly_babies |
Screed has three meanings in building construction:
In the United States, a person called a concrete finisher performs the process of screeding, which is the process of cutting off excess wet concrete to bring the top surface of a slab to the proper grade and smoothness. A power concrete screed has a gasoline motor attached, which helps smooth and vibrate concrete as it is flattened. After the concrete is flattened it is smoothed with a concrete float or power trowel . A concrete floor is sometimes called a solid ground floor .
A plasterer also may use a screed to level a wall or ceiling surface in plasterwork .
This sense of screed has been extended to asphalt paving where a free floating screed is part of a machine that spreads the paving material.
A weep screed or sill screed is a screed rail which has drainage holes to allow moisture which penetrated an exterior plaster or stucco coating to drain through the screed. [ 4 ]
Flowing screeds are made from inert fillers such as sand, with a binder system based on cement or often calcium sulphate . Flow screeds are often preferred to traditional screeds as they are easier and faster to install and provide a similar finish. Flow screed is often used in combination with underfloor heating installation.
Liquid flow screed is self-levelling. No vibration is necessary to remove bubbles and densify the liquid mass.
Due to the easy consolidation thickness can sometimes be reduced in comparison to conventional screeds. This minimises heat storage leading to a floor that reacts quickly to user requirement hence raising the efficacy of underfloor heating.
A development in the UK is the delivery, mixing, and pumping of screed from a single vehicle. Where previously screed jobs required a separate pump to administer the screed, these new machines can now administer the screed directly from the mixing pan to the floor at a range of up to 60 meters. For example, the material called granolithic . | https://en.wikipedia.org/wiki/Screed |
Screen-printed electrodes (SPEs) are electrochemical measurement devices that are manufactured by printing different types of ink on plastic or ceramic substrates, allowing quick in-situ analysis with high reproducibility, sensitivity and accuracy. The composition of the different inks ( carbon , silver , gold , platinum ) used in the manufacture of the electrode determines its selectivity and sensitivity. This fact allows the analyst to design the most optimal device according to its purpose. [ 1 ]
The evolution of these electrochemical cells arises from the need to reduce the size of the devices, that implies a decrease of the sample volume required in each experiment. In addition, the development of SPEs has enable the reduction of the production costs. [ 1 ] [ 2 ] [ 3 ]
One of the principal advantages is the possibility of modifying the screen-printed electrodes, modifying the composition of its inks by adding different metals, enzymes , complexing agents, polymers , etc., which is useful for the preparation of multitude electrochemical analyses. [ 1 ] [ 3 ]
Screen printing is one of the oldest methods of reproduction. The screen-printed electrodes (SPEs) are presented as a single device in which there are three different electrodes : [ 4 ]
The three electrodes could be printed on different types of substrates (plastic or ceramic ) and could be manufactured with a great variety of inks . [ 2 ] [ 3 ] The most common inks are those composed of silver and carbon , however, they can be based on other metals such as platinum , gold , palladium or copper . In addition, the electrodes can be modified with enzymes , metallic nanoparticles , carbon nanotubes , polymers or complexing agents. [ 2 ] [ 3 ] The electrode ink composition is chosen according to the final application and the selectivity and sensitivity required for the analysis. [ 2 ] [ 5 ] [ 6 ]
The electrode manufacturing process involves the sequential deposition of different layers of conductive and/or insulating inks on the substrates of interest. The process consists of several stages: [ 5 ]
On the other hand, as mentioned above, the most commonly used inks are silver and carbon , therefore, their printing and manufacturing characteristics should be highlighted:
Screen-printed electrodes offer several advantages such as low cost, flexibility of their design, great reproducibility of the process and of the electrodes obtained, the possibility of manufacturing them with different materials and the wide capacity of modification of the work surface. Another advantage is the possibility of connection to a portable instrumentation allowing the in-situ determination of specific analytes. In addition, screen-printed electrodes avoid tedious cleaning processes. [ 2 ] [ 5 ]
Currently, they are used as a support to produce portable electrochemical biosensors for environmental analysis. Some applications are: [ 9 ]
On the other hand, a correct manufacturing process is important to avoid low reproducibilities, to encourage mineral binders or insulating polymers that achieve a high resistance of SPE, and to use inks that do not significantly affect the kinetics of the reactions that take place. In manufacturing, surface treatments are used to remove organic contaminants from the ink. This improves their electrochemical properties by increasing the surface roughness. [ 3 ] | https://en.wikipedia.org/wiki/Screen-printed_electrodes |
A screen filter is a type of water purification using a rigid or flexible screen to separate sand and other fine particles out of water for irrigation or industrial applications. [ 1 ] These are generally not recommended for filtering out organic matter such as algae , since these types of contaminants can be extruded into spaghetti -like strings through the filter if enough pressure drop occurs across the filter surface. Typical screen materials include stainless steel ( mesh ), polypropylene , nylon and polyester . [ citation needed ]
Self-cleaning screen filters incorporate an automatic backwash cycle to overcome these limitations. [ which? ] Backwash cycles are far more frequent when compared to a media filter with similar capacity, and each backwash requires far less water to perform. Their ability to quickly remove contaminants from water before they leach their nutrients make such filters popular choices for recirculating aquaculture systems. They have also become popular in closed loop industrial systems such as cooling tower, heat exchanger, and other equipment protection applications. Similar devices with larger openings designed only to keep out large objects are called strainers . [ citation needed ]
Stainless-steel strainers are used in industrial, municipal, and irrigation applications, and can be designed for very high flow rates. When paired with a controller and flush valve, a strainer can be fully automated. Suspended particles collect on the inside of the screen, and the flush valve opens to expel the buildup. This eliminates the need for manual cleaning of the strainer element. [ citation needed ]
This water supply –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Screen_filter |
A screen hotspot , in computing , provides a special area on the display screen of a computer for hyperlinking or for other GUI -based activity (such as re-direction , pop-up display, macro execution, etc.).
Hotspots may not look visually distinct; however, a mouseover operation over elements such as hyperlinks, buttons or idle windows will often reveal them by changing the shape of the pointer .
The corners and edges of the whole screen may also act as hotspots. Some screen savers under DOS and older versions of Windows can be configured to be activated immediately (that is, without waiting for some period of inactivity to time out) or to never activate the screen saver mode at all (even after timeout) when the mouse is moved into a particular (often configurable) corner of the screen.
Hotspots are used extensively in Windows 8 , where they are referred to as "hot corners". [ 1 ]
According to Fitts's law , which predicts the time it takes to reach a target area, moving mouse and trackball pointers to those spots is easy and fast. As the pointer usually stops when reaching a screen edge, the size of those spots can be considered of virtual infinite size, so the hot corners and edges can be reached quickly by throwing the pointer toward the edges. [ 2 ] [ 3 ]
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Screen_hotspot |
Screen/Scroll centrifuge is a filtering or screen centrifuge which is also known as worm screen or conveyor discharge centrifuge. This centrifuge was first introduced in the midst of 19th century. After developing new technologies over the decades, it is now one of the widely used processes in many industries for the separation of crystalline, granular or fibrous materials from a solid-liquid mixture. Also, this process is considered to dry the solid material. This process has been some of the most frequently seen within, especially, coal preparation industry . Moreover, it can be found in other industries such as chemical, environmental, food and other mining fields.
Screen scroll centrifuge is a filtering centrifuge which separates solids and liquid from a solid-liquid mixture. This type of centrifuge is commonly used with a continuous process in which slurry containing both solid and liquid is continuously fed into and continuously discharged from the centrifuge. In a typical screen scroll centrifuge, the basic principle is that entering feed is separated into liquid and solids as two products. The feed is transported from small to larger diameter end of frustoconical basket by the inclination of the screen basket and slightly different speed of the scraper worm. [ 1 ] The solid material retained on the screen is moved along the cone via an internal screw conveyor while the liquid output is obtained due to centrifugal force causes the feed slurry to pass through the screen openings. [ 2 ] Furthermore, screen scroll centrifuge may rotate either in horizontal or vertical position.
The use of screen scroll centrifuge has been seen in numerous process engineering industries. One of the most noticeable applications is within coal preparation industry . In addition to that, this centrifuge is also employed in the dewatering of potash, gilsonite , in salt processes and in dewatering various sands. [ 3 ] Moreover, it is also designed for use in the food processing industry, for instant, dairy production, and cocoa butter equivalents and other confectionery fats. [ 4 ]
Screen scroll centrifuges, which are also known as worm screen or the conveyor discharge, instigate the solids to move along the cone through an internal screw conveyor. The conveyor in the centrifuge spins at a differential speed to the conical screen and centrifugal forces approximately 1800 g - 2600 g facilitate reasonable throughputs. [ 2 ] [ 5 ] Some of the screen scroll centrifuges are available with up to four separate stages for improved performance. The first stage is used to de-liquor the feed which is followed by a washing stage, with the final stage being used for drying. In an advanced screen scroll centrifuge with four stages, two separate washes are employed in order to segregate the wash liquors. [ 5 ]
The two most common types of screen/scroll centrifuge used in many industrial applications are vertical screen/scroll centrifuge and horizontal screen/scroll centrifuge.
Vertical screen scroll is built with the main components of screen, scroll, basket, housing, and helical screw. Feed containing liquid and solid materials is introduced into vertical screen scroll centrifuge from the top. This is sped up by centrifugal acceleration produced from the rotating parts contacted. As such, centrifugal force slings liquids through the openings, while solids are held on the screen surface as they cannot pass through because of granular particles larger than the screen pores or due to agglomeration. Movement of solids across the screen surface is manipulated by flights. Liquids that have gone through screen are obtained and discharged through effluent outlet from the side of machine, while solids collected from the screen fall by gravity through the bottom discharge of the machine. [ 3 ]
Some of the available vertical screen scroll centrifuges are CMI model EBR and CMI model EBW which are manufactured by Centrifugal & Mechanical Industries (CMI). The former can dewater coarser particles size ranging from 1.5 in to 28 mesh whereas the latter can dewater finer particles size ranging from 1 mm to 150 mesh. [ 3 ]
Similar to a vertical screen scroll centrifuge, a horizontal screen scroll centrifuge is constructed of several main parts: screen, scroll, basket, housing, and helical screw. The screen and the basket with frustoconical geometry are assembled into the housing in a horizontal axis. Inside the frustoconical structure there is a tubular wall. Inside the tubular wall there is a cylinder of helical screw which flight on scroll pass. The tubular wall will have a slightly different angular speed to the helical screw. [ 6 ]
The solid-liquid mixture is fed into the closed rearward portion of the scroll. The rotation movement of the scroll, screen, and basket allows the liquid to pass through from the openings on the screen (via centrifugal force). The solid remains will be separated according to size due to the difference of the angular velocity of the helical screw and the basket. The helical screw pushes the solid material to be discharged to the forward end of the scroll. The processing time depends on helical screw pitch and the angular velocity difference. It may also be influenced by the design of the scroll feed opening. The solid particles exiting are usually collected via a conveyor in the collection unit. [ 6 ]
The performance and output efficiency of the screen scroll centrifuge can be affected by several factors, such as particle size and feed concentration, flow rate of feed and screen mesh size of the centrifuge.
Particle size in the feed is one of the most important parameters to be taken into account since the choice of slot and screen holes size of screen scroll centrifuge or different types of process depends on feed contents. Non-uniform particles size in the feed can cause partial blockage on the screen due to the small size solids blocking the holes besides normal and larger particles. So, liquids flow over the screen instead of passing through it. As such, it requires higher solids content in the feed in order to obtain good and reasonable results - normally greater than 15% and up to 60% w/w. [ 5 ] Nevertheless, the flow rate of the feed can be monitored to overcome this setback. Another possible method is to carry out pre-treatment on the feed to be used for screen scroll centrifuge, for example, by the filtration process. Particle size, thereafter, can be analysed and the selection of particular screen size can be determined. However, it increases the total operating cost.
Typical operating range of particle size and feed concentration for screen scroll centrifuges are 100 – 20,000 μm and 3 – 90% mass of the solids in the feed. [ 7 ] In general, slot and screen holes size range 40 - 200 μm with open areas from 5 - 15%. [ 5 ] Nevertheless, recent products are claimed to be able to handle the particle size as low as 50 μm. [ 3 ] Screens are generally metallic foil or wedge wire and more recently metallic and composite screens perforated with micro- waterjet cutting.
As mentioned in the previous section, feed flow rate is one of the crucial parameters to be controlled to achieve highly efficient output. Centrifuge performance is sensitive to feed flow rate. Even though increasing the feed flow rate can prevent from blocking the screens, it is mentioned that wetter solids is achieved. [ 8 ] This is due to increase in hydraulic load on the centrifuge when higher feed rate is applied, while differential rotation speed between the cone and scroll, and retention time within dewatering zone of the basket are fixed. In addition, higher feed rate leads to a surge in the effective thickness of the bed since it is dragged down by the scroll.
The material variations for constructing and the design of main components of centrifuge such as the screen plate, helical screw and basket could actually improve the longer life term of the machine. Another important factor is the conical basket size and its angle within the centrifuge. Different basket size and angle between basket and helical screw can vary the angular speed; as a result, the quality of the product is affected. Moreover, the shape of the helical screw is also important since it optimizes the transportation of cake. [ 3 ] A selection of typical screen scroll centrifuge with different basket sizes found in the market is presented in the following Table 1. The helical scroll and conical basket sections are commonly built at the angle of 10°, 15° and 20°. [ 5 ]
Table 1 A selection of screen scroll centrifuge sizes
The screen scroll centrifuge has an advantage of having a driven scroll helical conveyor which gives a small differential speed relative to the conical basket. The helical conveyor is installed in the centrifuge to control the transport of the incoming feed, allowing the residence time of the solids in the basket to be increased giving enhanced process performance. [ 5 ] Moreover, the helical conveyor and conical basket sections are designed in certain angle of 10°, 15° and 20° being common such that solid particles are dragged on the conveyor along the cone towards the discharge point. As a result, there is no formation of even solids layer but form piles of triangular section in front of the blades of the conveyor. The residence time within screen scroll centrifuge is typically about 4 to 15 seconds which is longer than normal simpler conical basket centrifuge. This permits a sufficient interaction time between wash liquids and cake. However, the presence of the conveyor causes crystals breakage and abrasion problem as well as the formation of uneven solids layer which can lead to poor washing. This can be controlled by conveyor speed. [ 5 ]
TEMA engineers, specialist in centrifuges, claims that horizontal screen scroll centrifuge can achieve higher overall recovery of fines up to 99% can be achieved, combining with very low product moisture. Furthermore, it is recommended that operating with the feed containing more than 40% solids with minimal size of 100 μm achieve the best results. [ 9 ] The use of the screen scroll centrifuge with horizontal orientation is more economical as its capacity is 40% more tonnage than that of vertical orientation of the same size for the same energy cost. In addition, maintenance of the horizontal screen scroll centrifuge can be carried out easily since total disassembly is not needed. [ 10 ] Nowadays, screen scroll centrifuges are equipped with CIP-cleaning system for the purpose of self-cleaning within the centrifuge.
On the other hand, it has a downside of possible blockage to the screen due to the feed slurry containing small crystals besides large and normal solids crystals. Consequently, this causes the screen to become less permeable so the liquids flow over the screen rather than passing through the screen mesh. This problem, however, can be overcome by reducing the flow rate of feed. [ 5 ]
The basket, helical screw, screen filter, and other parts are designed to meet up the process input and certain performance. Most of the parts are made from metal to be able to handle the separation process. The bigger the bowl could contain more input but at the same time could increase the process and residence time. The helical screw is made to be able to hold and move the particle around to be able to control the cake movement. The screen filter is made to be able to sieve the particle and the water. The cleaning process for this type of machine could be difficult compare to other separation model. The design mostly being optimized with low maintenance feature and provided with good sealing to prevent the leaking and breakup of the construction. [ 3 ] [ 8 ]
After removing liquids from the slurry to form a cake of solids in the centrifuge, further or post-treatment is required to completely dry the solids. Drying is the most common process used in the industry. Another post-treatment system is to treat the products with another stage of deliquoring process. [ 11 ]
The modern screen/scroll centrifuge has been modified in several ways from the original design: | https://en.wikipedia.org/wiki/Screen_scroll_centrifuge |
SCP-ISM , or screened Coulomb potentials implicit solvent model , is a continuum approximation of solvent effects for use in computer simulations of biological macromolecules , such as proteins and nucleic acids , usually within the framework of molecular dynamics . [ 1 ] It is based on the classic theory of polar liquids , as developed by Peter Debye and corrected by Lars Onsager to incorporate reaction field effects. [ 1 ] The model can be combined with quantum chemical calculations to formally derive a continuum model of solvent effects suitable for computer simulations of small and large molecular systems. The model is included in the CHARMM molecular mechanics code. [ 2 ]
This simulation software article 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/Screened_Coulomb_potentials_implicit_solvent_model |
Within the environmental sciences , screening broadly refers to a set of analytical techniques used to monitor levels of potentially hazardous organic compounds in the environment, particularly in tandem with mass spectrometry techniques. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Such screening techniques are typically classified as either targeted, where compounds of interest are chosen before the analysis begins, or non-targeted, where compounds of interest are chosen at a later stage of the analysis. These two techniques can be organized into at least three approaches: target screening , using reference standards that are analogous to the target compound; suspect screening , which uses a library of cataloged data such as exact mass , isotope patterns , and chromatographic retention times in lieu of reference standards; and non-target screening , using no pre-existing knowledge for comparison before analysis. [ 1 ] [ 2 ] [ 3 ] [ 5 ] As such, target screening is most useful when monitoring the presence of specific organic compounds—particularly for regulatory purposes—which requires higher selectivity and sensitivity . When the number of detected compounds and associated metabolites needs to be maximized for discovering new or emerging environmental trends or biomarkers for disease, a more non-targeted approach has traditionally been used. [ 4 ] [ 5 ] [ 6 ] [ 7 ] However, the rapid improvement of mass spectrometers into more high-resolution forms, with increased sensitivity, has made suspect and non-target screening more attractive, either as stand-alone approaches or in conjunction with more targeted methods. [ 1 ] [ 2 ] [ 5 ] [ 6 ] [ 8 ]
Mass spectrometry methods are generally used for analysis of environmental contaminant monitoring, particularly in aquatic environments (though they can be applied in non-aquatic environments, such as with screening pesticides on plant matter [ 9 ] ), paired with chromatography for separation . [ 2 ] [ 4 ] [ 10 ] For target screening, this means using gas chromatography–mass spectrometry (GC-MS) or liquid chromatography–mass spectrometry (LC-MS) methods "that use single reaction monitoring (SIM) or selected reaction monitoring (SRM) modes." [ 4 ] However, for suspect and non-target screening, these methods are inadequate due to recording only a limited number of compounds and insufficient useful information can be determined about unknown compounds, particularly given the dearth of LC-MS comparison libraries. [ 4 ] For those non-targeted screening approaches, high-resolution mass spectrometry and high mass accuracy chromatography techniques are required. Combinations of quadrupole , time-of-flight , ion trap , and orbitrap mass spectrometry analyzers have emerged, along with high-performance liquid chromatography (and ultra-high-performance liquid chromatography), to more rapidly and effectively tackle suspect and non-target screening. [ 2 ] [ 6 ] [ 4 ] [ 10 ]
Target screening or analysis is useful when looking for a short list of predetermined organic compounds in a sample, while ignoring other compounds that may be present. Reference standards that align with the predetermined compounds are available and used to compare attributes such as chromatographic retention time, fragmentation pattern , and isotopic pattern. [ 10 ] The workflow for target screening requires the optimization of sample extraction, sample clean-up, and instrumentation methods to those predetermined compounds in order to achieve "a specific and accurate measurement." [ 2 ] Most analytical results will be quantitative in nature, given the narrow focus of screening. [ 2 ] [ 3 ] As such, targeted approaches have traditionally been used in regulatory monitoring schemes. [ 11 ] The downside, however, is that many hazardous organic compounds are not covered by environmental monitoring regulation and thus not specifically targeted, [ 8 ] and the approach is not generally adept for rapid response approaches to providing early warning of contamination events. [ 11 ]
Suspect screening is useful when looking for one or more suspected compounds with known structures in a sample, but reference standards are unavailable or don't exist. In this case, user-built databases containing information such as mass accuracy, retention time, isotopic patterns, and other structure information for the suspected compounds are consulted, filtered, and compared against the results of high-resolution mass spectrometry analyses using SRM or full scans. [ 3 ] The structure of the suspected compounds are then elucidated based on that information, ideally confirmed with authentic reference standards. [ 2 ] [ 3 ] Compared to targeted screening, the initial work performed in suspect screening is largely qualitative, with more quantitative work to potentially follow in a more targeted approach. [ 10 ] Aside from being able to analyze for more compounds, an additional benefit of this approach is that retrospective analysis, even years later, is enabled without reanalyzing the sample. [ 4 ] [ 6 ] A downside to the suspect approach is the complexity involved, including not only with data analysis (e.g., using in silico fragmentation software [ 10 ] [ 8 ] [ 6 ] ) but also carefully developing suspect screening lists and choosing databases. [ 8 ]
Non-target screening is useful when needing to investigate the presences of all the organic compounds within a sample. In this case, since no information is known about the compounds contained in the sample, no reference standard can be used for comparison, at least initially, overall making non-target screening one of the most challenging approaches. Rather, a full automated scan with mass filtering, peak detection, and other characteristics is used to make initial compound detection. Then elemental composition of detected compounds is deduced using accurate mass of the ions. Database searches can be performed to get a lock on what the most plausible structures are given the elemental composition. [ 4 ] [ 10 ] Like suspect screening, the initial work performed in non-target screening is largely qualitative, with more quantitative work to potentially follow. Similar to suspect screening, the downside to a fully non-targeted approach is the data-intensive nature of the processes, requiring multivariate statistical models, and the wide variety of data processing workflows used by researchers further complicates evaluation of method performance of those data analysis processes. [ 6 ] | https://en.wikipedia.org/wiki/Screening_(environmental) |
A screening information dataset ( SIDS ) is a study of the hazards associated with a particular chemical substance or group of related substances, prepared under the auspices of the Organisation for Economic Co-operation and Development (OECD). [ 1 ] The substances studied are high production volume (HPV) chemicals , which are manufactured or imported in quantities of more than 1000 tonnes per year for any single OECD market.
The list of HPV chemicals is prepared by the OECD Secretariat and updated regularly. As of 2004, 4,843 chemicals were on the list. [ 2 ] Of these, roughly 1000 have been prioritised for special attention, and SIDS are prepared for these chemicals, [ 3 ] usually by an official agency in one of the OECD member countries with the collaboration of the UN International Programme on Chemical Safety (IPCS).
The procedures for investigating the risks of an HPV chemical are described in the OECD Manual for Investigation of HPV Chemicals . [ 4 ] The initial stage is the collection of existing information (either published or supplied by manufacturers) on the chemical. If the existing information is insufficient to make an assessment of the risks, the chemical may be tested at this stage to collect more data. The initial report of the investigation is discussed at a SIDS initial assessment meeting (SIAM), which includes: [ 3 ]
The SIAM can either accept the draft report or call for revisions (including further testing). Once the comments and discussion of the SIAM have been taken into account, the report is published by the United Nations Environment Programme (UNEP).
The possibility of new testing to complete the study is what distinguishes SIDS reports from similar studies such as Concise International Chemical Assessment Documents (CICADs). In this sense, SIDS are similar to European Union Risk Assessment Reports (RARs). The distinction is that the SIDS programme is specifically aimed at HPV chemicals, while the chemicals selected for EU RARs are chosen more on the basis of a hazard profile, so include chemicals with much lower production volumes. | https://en.wikipedia.org/wiki/Screening_information_dataset |
A screw axis ( helical axis or twist axis ) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis. [ 1 ] [ 2 ]
Plücker coordinates are used to locate a screw axis in space , and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position. The special case when the first vector is zero is interpreted as a pure translation in the direction of the second vector. A screw axis is associated with each pair of vectors in the algebra of screws, also known as screw theory . [ 3 ]
The spatial movement of a body can be represented by a continuous set of displacements. Because each of these displacements has a screw axis, the movement has an associated ruled surface known as a screw surface . This surface is not the same as the axode , which is traced by the instantaneous screw axes of the movement of a body. The instantaneous screw axis, or 'instantaneous helical axis' (IHA), is the axis of the helicoidal field generated by the velocities of every point in a moving body.
When a spatial displacement specializes to a planar displacement, the screw axis becomes the displacement pole , and the instantaneous screw axis becomes the velocity pole , or instantaneous center of rotation , also called an instant center . The term centro is also used for a velocity pole, and the locus of these points for a planar movement is called a centrode . [ 4 ]
The proof that a spatial displacement can be decomposed into a rotation around, and translation along, a line in space is attributed to Michel Chasles in 1830. [ 5 ] Recently the work of Giulio Mozzi has been identified as presenting a similar result in 1763. [ 6 ] [ 7 ]
A screw displacement (also screw operation or rotary translation ) is the composition of a rotation by an angle φ about an axis (called the screw axis ) with a translation by a distance d along this axis. A positive rotation direction usually means one that corresponds to the translation direction by the right-hand rule . This means that if the rotation is clockwise, the displacement is away from the viewer. Except for φ = 180°, we have to distinguish a screw displacement from its mirror image . Unlike for rotations, a righthand and lefthand screw operation generate different groups.
The combination of a rotation about an axis and a translation in a direction perpendicular to that axis is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation, a pure rotation and the identity. Together these are all the direct isometries in 3D .
In crystallography , a screw axis symmetry is a combination of rotation about an axis and a translation parallel to that axis which leaves a crystal unchanged. If φ = 360° / n for some positive integer n , then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw displacement.
Applicable for space groups is a rotation by 360° / n about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n . This multiple is indicated by a subscript. So, 6 3 is a rotation of 60° combined with a translation of one half of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 2 1 , 3 1 , 4 1 , 4 2 , 6 1 , 6 2 , and 6 3 , and the enantiomorphous 3 2 , 4 3 , 6 4 , and 6 5 . [ 8 ]
Considering a screw axis n m , if g is the greatest common divisor of n and m , then there is also a g -fold rotation axis. When n / g screw operations have been performed, the displacement will be m / g , which since it is a whole number means one has moved to an equivalent point in the lattice, while carrying out a rotation by 360° / g . So 4 2 , 6 2 and 6 4 create two-fold rotation axes, while 6 3 creates a three-fold axis.
A non-discrete screw axis isometry group contains all combinations of a rotation about some axis and a proportional translation along the axis (in rifling , the constant of proportionality is called the twist rate ); in general this is combined with k -fold rotational isometries about the same axis ( k ≥ 1); the set of images of a point under the isometries is a k -fold helix ; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k -fold helix of such axes.
Let D : R 3 → R 3 be an orientation-preserving rigid motion of R 3 . The set of these transformations is a subgroup of Euclidean motions known as the special Euclidean group SE(3). These rigid motions are defined by transformations of x in R 3 given by
consisting of a three-dimensional rotation A followed by a translation by the vector d .
A three-dimensional rotation A has a unique axis that defines a line L . Let the unit vector along this line be S so that the translation vector d can be resolved into a sum of two vectors, one parallel and one perpendicular to the axis L , that is,
In this case, the rigid motion takes the form
Now, the orientation preserving rigid motion D * = A ( x ) + d ⊥ transforms all the points of R 3 so that they remain in planes perpendicular to L . For a rigid motion of this type there is a unique point c in the plane P perpendicular to L through 0 , such that
The point C can be calculated as
because d ⊥ does not have a component in the direction of the axis of A .
A rigid motion D * with a fixed point must be a rotation of around the axis L c through the point c . Therefore, the rigid motion
consists of a rotation about the line L c followed by a translation by the vector d L in the direction of the line L c .
Conclusion: every rigid motion of R 3 is the result of a rotation of R 3 about a line L c followed by a translation in the direction of the line. The combination of a rotation about a line and translation along the line is called a screw motion.
A point C on the screw axis satisfies the equation: [ 9 ]
Solve this equation for C using Cayley's formula for a rotation matrix
where [B] is the skew-symmetric matrix constructed from Rodrigues' vector
such that
Use this form of the rotation A to obtain
which becomes
This equation can be solved for C on the screw axis P (t) to obtain,
The screw axis P (t) = C + t S of this spatial displacement has the Plücker coordinates S = ( S , C × S ) . [ 9 ]
The screw axis appears in the dual quaternion formulation of a spatial displacement D = ([A], d ) . The dual quaternion is constructed from the dual vector S = ( S , V ) defining the screw axis and the dual angle ( φ , d ) , where φ is the rotation about and d the slide along this axis, which defines the displacement D to obtain,
A spatial displacement of points q represented as a vector quaternion can be defined using quaternions as the mapping
where d is translation vector quaternion and S is a unit quaternion, also called a versor , given by
that defines a rotation by 2 θ around an axis S .
In the proper Euclidean group E + (3) a rotation may be conjugated with a translation to move it to a parallel rotation axis. Such a conjugation, using quaternion homographies , produces the appropriate screw axis to express the given spatial displacement as a screw displacement, in accord with Chasles’ theorem .
The instantaneous motion of a rigid body may be the combination of rotation about an axis (the screw axis) and a translation along that axis. This screw move is characterized by the velocity vector for the translation and the angular velocity vector in the same or opposite direction. If these two vectors are constant and along one of the principal axes of the body, no external forces are needed for this motion (moving and spinning]]). As an example, if gravity and drag are ignored, this is the motion of a bullet fired from a rifled gun .
This parameter is often used in biomechanics , when describing the motion of joints of the body. For any period of time, joint motion can be seen as the movement of a single point on one articulating surface with respect to the adjacent surface (usually distal with respect to proximal ). The total translation and rotations along the path of motion can be defined as the time integrals of the instantaneous translation and rotation velocities at the IHA for a given reference time. [ 10 ]
In any single plane , the path formed by the locations of the moving instantaneous axis of rotation (IAR) is known as the 'centroid', and is used in the description of joint motion. | https://en.wikipedia.org/wiki/Screw_axis |
A screw conveyor or auger conveyor is a mechanism that uses a rotating helical screw blade, called a " flighting ", usually within a tube, to move liquid or granular materials. They are used in many bulk handling industries. Screw conveyors in modern industry are often used horizontally or at a slight incline as an efficient way to move semi-solid materials, including food waste , wood chips , aggregates, cereal grains , animal feed , boiler ash, meat , bone meal, municipal solid waste , and many others. The first type of screw conveyor was the Archimedes' screw , used since ancient times to pump irrigation water. [ 1 ]
They usually consist of a trough or tube containing either a spiral blade coiled around a shaft, driven at one end and held at the other, or a " shaftless spiral ", driven at one end and free at the other. The rate of volume transfer is proportional to the rotation rate of the shaft. In industrial control applications, the device is often used as a variable rate feeder by varying the rotation rate of the shaft to deliver a measured rate or quantity of material into a process. [ 2 ]
Screw conveyors can be operated with the flow of material inclined upward. When space allows, this is a very economical method of elevating and conveying. As the angle of inclination increases, the capacity of a given unit rapidly decreases.
The rotating part of the conveyor is sometimes called simply an auger .
The "grain auger" is used in agriculture to move grain from trucks , grain carts , or grain trailers into grain storage bins (from where it is later removed by gravity chutes at the bottom). A grain auger may be powered by an electric motor; a tractor , through the power take-off ; or sometimes an internal combustion engine mounted on the auger. The helical rotates inside a long metal tube, moving the grain upwards. On the lower end, a hopper receives grain from the truck or grain cart. A chute on the upper end guides the grain into the destination location. [ 2 ]
The modern grain auger of today's farming communities was invented by Peter Pakosh . His grain mover employed a screw-type auger with a minimum of moving parts, a totally new application for this specific use. At Massey Harris (later Massey Ferguson), young Pakosh approached the design department in the 1940s with his auger idea, but was scolded and told that his idea was unimaginable and that once the auger aged and bent that the metal on metal would, according to a head Massey designer, "start fires all across Canada ". [ 3 ] Pakosh, however, went on to design and build a first prototype auger in 1945, and 8 years later start selling tens of thousands under the ' Versatile ' name, making it the standard for modern grain augers.
A specialized form of grain auger is used to transfer grain into a seed drill and is usually quite a lot smaller in both length and diameter than the augers used to transfer grain to or from a truck, grain cart or bin. This type of auger is known as a "drill fill". Grain augers with a small diameter, regardless of the use they are put to, are often called "pencil augers".
Centerless augers are particularly popular in industrial animal farming facilities, where the primary application is distributing animal feed from a central storage location to individual or group feeding devices. The flexible nature of the auger wire allows feed or other materials to change elevation and move at angles. The first centerless auger was patented by Eldon Hostetler and Chore-Time Equipment in the context of this application. [ 4 ] [ 5 ]
Various other applications of the screw or auger conveyor include its use in snowblowers , to move snow towards an impeller , where it is thrown into the discharge chute. Combine harvesters use both enclosed and open augers to move the unthreshed crop into the threshing mechanism and to move the grain into and out of the machine's hopper. Ice resurfacers use augers to remove loose ice particles from the surface of the ice. An auger is also a central component of an injection molding machine . An auger is used in some rubbish compactors to push the rubbish into a lowered plate at one end for compaction.
Augers are also present in food processing. They are a tool of choice in powder processing when it comes to conveyor does precisely bulk solids (powders, pellets...). [ 6 ] In a conventional meat grinder, chunks of meat are led by the auger through a spinning blade and a holed plate. This method emulsifies the fat in beef to soften hamburger patties and is also used to produce a wide variety of sausages and loaves. Augers are also used to force food products through dies to produce pellets. These are then processed further to produce products such as bran flakes.
Augers are also used in oil fields as a method of transporting rock cuttings away from the shakers to skips. Augers are also used in some types of pellet stoves and barbecue grills, to move fuel from a storage hopper into the firebox in a controlled manner. Augers are often used in machining , wherein the machine tools may include an auger to direct the swarf (scrap metal or plastic) away from the workpiece.
Screw conveyors can also be found in wastewater treatment plants to evacuate solid waste from the treatment process.
The amphibious infantry fighting vehicle BMP-3 uses an auger-type propulsion unit in water.
The Olds elevator is a variant of a screw conveyor developed by Australian engineer Peter Olds in 2002. [ 7 ] [ 8 ] Rather than rotate a central screw blade, a stationary screw is contained within a rotating casing that scoops surrounding material into its base. [ 9 ] Following similar principles to the conventional screw conveyor, the Olds elevator can lift bulk materials efficiently. Since its invention, it has been assessed as a viable system for industrial uses by a number of academics. [ 10 ] | https://en.wikipedia.org/wiki/Screw_conveyor |
Screw theory is the algebraic calculation of pairs of vectors , also known as dual vectors [ 1 ] – such as angular and linear velocity , or forces and moments – that arise in the kinematics and dynamics of rigid bodies . [ 2 ] [ 3 ]
Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics , where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors . [ 4 ]
Important theorems of screw theory include: the transfer principle proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws; [ 1 ] Chasles' theorem proves that any change between two rigid object poses can be performed by a single screw; Poinsot's theorem proves that rotations about a rigid object's major and minor – but not intermediate – axes are stable.
Screw theory is an important tool in robot mechanics, [ 5 ] [ 6 ] [ 7 ] [ 8 ] mechanical design, computational geometry and multibody dynamics .
This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions . [ 9 ] Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots). [ 10 ]
A spatial displacement of a rigid body can be defined by a rotation about a line and a translation along the same line, called a screw motion . This is known as Chasles' theorem . The six parameters that define a screw motion are the four independent components of the Plücker vector that defines the screw axis, together with the rotation angle about and linear slide along this line, and form a pair of vectors called a screw . For comparison, the six parameters that define a spatial displacement can also be given by three Euler angles that define the rotation and the three components of the translation vector.
A screw is a six-dimensional vector constructed from a pair of three-dimensional vectors, such as forces and torques and linear and angular velocity, that arise in the study of spatial rigid body movement. The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line.
A twist is a screw used to represent the velocity of a rigid body as an angular velocity around an axis and a linear velocity along this axis. All points in the body have the same component of the velocity along the axis, however the greater the distance from the axis the greater the velocity in the plane perpendicular to this axis. Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis.
The points in a body undergoing a constant twist motion trace helices in the fixed frame. If this screw motion has zero pitch then the trajectories trace circles, and the movement is a pure rotation. If the screw motion has infinite pitch then the trajectories are all straight lines in the same direction.
The force and torque vectors that arise in applying Newton's laws to a rigid body can be assembled into a screw called a wrench . A force has a point of application and a line of action, therefore it defines the Plücker coordinates of a line in space and has zero pitch. A torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw.
Let a screw be an ordered pair
where S and V are three-dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called dual vectors .
Now, introduce the ordered pair of real numbers â = ( a , b ) , called a dual scalar . Let the addition and subtraction of these numbers be componentwise, and define multiplication as a ^ c ^ = ( a , b ) ( c , d ) = ( a c , a d + b c ) . {\displaystyle {\hat {\mathsf {a}}}{\hat {\mathsf {c}}}=(a,b)(c,d)=(ac,ad+bc).} The multiplication of a screw S = ( S , V ) by the dual scalar â = ( a , b ) is computed componentwise to be, a ^ S = ( a , b ) ( S , V ) = ( a S , a V + b S ) . {\displaystyle {\hat {\mathsf {a}}}{\mathsf {S}}=(a,b)(\mathbf {S} ,\mathbf {V} )=(a\mathbf {S} ,a\mathbf {V} +b\mathbf {S} ).}
Finally, introduce the dot and cross products of screws by the formulas: S ⋅ T = ( S , V ) ⋅ ( T , W ) = ( S ⋅ T , S ⋅ W + V ⋅ T ) , {\displaystyle {\mathsf {S}}\cdot {\mathsf {T}}=(\mathbf {S} ,\mathbf {V} )\cdot (\mathbf {T} ,\mathbf {W} )=(\mathbf {S} \cdot \mathbf {T} ,\,\,\mathbf {S} \cdot \mathbf {W} +\mathbf {V} \cdot \mathbf {T} ),} which is a dual scalar, and S × T = ( S , V ) × ( T , W ) = ( S × T , S × W + V × T ) , {\displaystyle {\mathsf {S}}\times {\mathsf {T}}=(\mathbf {S} ,\mathbf {V} )\times (\mathbf {T} ,\mathbf {W} )=(\mathbf {S} \times \mathbf {T} ,\,\,\mathbf {S} \times \mathbf {W} +\mathbf {V} \times \mathbf {T} ),} which is a screw. The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.
Let the dual scalar ẑ = ( φ , d ) define a dual angle , then the infinite series definitions of sine and cosine yield the relations sin z ^ = ( sin φ , d cos φ ) , cos z ^ = ( cos φ , − d sin φ ) , {\displaystyle \sin {\hat {\mathsf {z}}}=(\sin \varphi ,d\cos \varphi ),\,\,\,\cos {\hat {\mathsf {z}}}=(\cos \varphi ,-d\sin \varphi ),} which are also dual scalars. In general, the function of a dual variable is defined to be f (ẑ) = ( f ( φ ), df ′( φ )) , where df ′( φ ) is the derivative of f ( φ ).
These definitions allow the following results:
A common example of a screw is the wrench associated with a force acting on a rigid body. Let P be the point of application of the force F and let P be the vector locating this point in a fixed frame. The wrench W = ( F , P × F ) is a screw. The resultant force and moment obtained from all the forces F i , i = 1, ..., n , acting on a rigid body is simply the sum of the individual wrenches W i , that is
Notice that the case of two equal but opposite forces F and − F acting at points A and B respectively, yields the resultant
This shows that screws of the form
can be interpreted as pure moments.
In order to define the twist of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, D( t ) = ([A( t )], d ( t )) , where [A] is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body coordinates to trace a curve P (t) in the fixed frame given by
The velocity of P is
where v is velocity of the origin of the moving frame, that is d d /dt. Now substitute p = [ A T ]( P − d ) into this equation to obtain,
where [Ω] = [d A /d t ][ A T ] is the angular velocity matrix and ω is the angular velocity vector.
The screw
is the twist of the moving body. The vector V = v + d × ω is the velocity of the point in the body that corresponds with the origin of the fixed frame.
There are two important special cases: (i) when d is constant, that is v = 0, then the twist is a pure rotation about a line, then the twist is
and (ii) when [Ω] = 0, that is the body does not rotate but only slides in the direction v , then the twist is a pure slide given by
For a revolute joint , let the axis of rotation pass through the point q and be directed along the vector ω , then the twist for the joint is given by,
For a prismatic joint , let the vector v pointing define the direction of the slide, then the twist for the joint is given by,
The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of line, which in turn are obtained from the transformations of the coordinate of points on the line.
Let the displacement of a body be defined by D = ([ A ], d ), where [ A ] is the rotation matrix and d is the translation vector. Consider the line in the body defined by the two points p and q , which has the Plücker coordinates ,
then in the fixed frame we have the transformed point coordinates P = [ A ] p + d and Q = [ A ] q + d , which yield.
Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by
The matrix [ D ] is the skew-symmetric matrix that performs the cross product operation, that is [ D ] y = d × y .
The 6×6 matrix obtained from the spatial displacement D = ([ A ], d ) can be assembled into the dual matrix
which operates on a screw s = ( s . v ) to obtain,
The dual matrix [Â] = ([ A ], [ DA ]) has determinant 1 and is called a dual orthogonal matrix .
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
This notation does not distinguish between P = ( X , Y , Z , 1), and P = ( X , Y , Z ), which is hopefully clear in context.
The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
The dot denotes the derivative with respect to time, and because p is constant its derivative is zero.
Substitute the inverse transform for p into the velocity equation to obtain the velocity of P by operating on its trajectory P ( t ), that is
where
Recall that [Ω] is the angular velocity matrix. The matrix [ S ] is an element of the Lie algebra se(3) of the Lie group SE(3) of homogeneous transforms. The components of [ S ] are the components of the twist screw, and for this reason [ S ] is also often called a twist.
From the definition of the matrix [ S ], we can formulate the ordinary differential equation,
and ask for the movement [ T ( t )] that has a constant twist matrix [ S ]. The solution is the matrix exponential
This formulation can be generalized such that given an initial configuration g (0) in SE( n ), and a twist ξ in se( n ), the homogeneous transformation to a new location and orientation can be computed with the formula,
where θ represents the parameters of the transformation.
In transformation geometry , the elemental concept of transformation is the reflection (mathematics) . In planar transformations a translation is obtained by reflection in parallel lines, and rotation is obtained by reflection in a pair of intersecting lines. To produce a screw transformation from similar concepts one must use planes in space : the parallel planes must be perpendicular to the screw axis , which is the line of intersection of the intersecting planes that generate the rotation of the screw. Thus four reflections in planes effect a screw transformation. The tradition of inversive geometry borrows some of the ideas of projective geometry and provides a language of transformation that does not depend on analytic geometry .
The combination of a translation with a rotation effected by a screw displacement can be illustrated with the exponential mapping .
Since ε 2 = 0 for dual numbers , exp( aε ) = 1 + aε , all other terms of the exponential series vanishing.
Let F = {1 + εr : r ∈ H }, ε 2 = 0.
Note that F is stable under the rotation q → p −1 qp and under the translation (1 + εr )(1 + εs ) = 1 + ε ( r + s ) for any vector quaternions r and s . F is a 3-flat in the eight-dimensional space of dual quaternions . This 3-flat F represents space , and the homography constructed, restricted to F , is a screw displacement of space.
Let a be half the angle of the desired turn about axis r , and br half the displacement on the screw axis . Then form z = exp(( a + bε ) r ) and z * = exp(( a − bε ) r ) . Now the homography is
The inverse for z * is
so, the homography sends q to
Now for any quaternion vector p , p * = − p , let q = 1 + pε ∈ F , where the required rotation and translation are effected.
Evidently the group of units of the ring of dual quaternions is a Lie group . A subgroup has Lie algebra generated by the parameters a r and b s , where a , b ∈ R , and r , s ∈ H . These six parameters generate a subgroup of the units, the unit sphere. Of course it includes F and the 3-sphere of versors .
Consider the set of forces F 1 , F 2 ... F n act on the points X 1 , X 2 ... X n in a rigid body. The trajectories of X i , i = 1,..., n are defined by the movement of the rigid body with rotation [ A ( t )] and the translation d ( t ) of a reference point in the body, given by
where x i are coordinates in the moving body.
The velocity of each point X i is
where ω is the angular velocity vector and v is the derivative of d ( t ).
The work by the forces over the displacement δ r i = v i δt of each point is given by
Define the velocities of each point in terms of the twist of the moving body to obtain
Expand this equation and collect coefficients of ω and v to obtain
Introduce the twist of the moving body and the wrench acting on it given by
then work takes the form
The 6×6 matrix [Π] is used to simplify the calculation of work using screws, so that
where
and [I] is the 3×3 identity matrix.
If the virtual work of a wrench on a twist is zero, then the forces and torque of the wrench are constraint forces relative to the twist. The wrench and twist are said to be reciprocal, that is if
then the screws W and T are reciprocal.
In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6×6 matrix [Π] in the calculation of work. [ 1 ] In this case the twist is defined to be
so the calculation of work takes the form
In this case, if
then the wrench W is reciprocal to the twist T.
The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics). [ 4 ]
Felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program . [ 11 ] He also worked out elliptic geometry, and a fresh view of Euclidean geometry, with the Cayley–Klein metric . The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described by Harvey Lipkin. [ 12 ] Other prominent contributors include Julius Plücker , W. K. Clifford , F. M. Dimentberg , Kenneth H. Hunt , J. R. Phillips. [ 13 ]
The homography idea in transformation geometry was advanced by Sophus Lie more than a century ago. Even earlier, William Rowan Hamilton displayed the versor form of unit quaternions as exp( a r )= cos a + r sin a . The idea is also in Euler's formula parametrizing the unit circle in the complex plane .
William Kingdon Clifford initiated the use of dual quaternions for kinematics , followed by Aleksandr Kotelnikov , Eduard Study ( Geometrie der Dynamen ), and Wilhelm Blaschke . However, the point of view of Sophus Lie has recurred. [ 14 ] In 1940, Julian Coolidge described the use of dual quaternions for screw displacements on page 261 of A History of Geometrical Methods . He notes the 1885 contribution of Arthur Buchheim . [ 15 ] Coolidge based his description simply on the tools Hamilton had used for real quaternions. | https://en.wikipedia.org/wiki/Screw_theory |
Scrim and sarking is a method of interior construction widely used in Australia and New Zealand in the late 19th and early 20th centuries. In this method, wooden panels were nailed over the beams and joists of a house frame, and a heavy, loosely woven cloth, called scrim , was then stapled or tacked over the wood panels. This construction method allowed wallpaper to be applied directly. [ 1 ]
In New Zealand, the sarking was often the native rimu (red pine), and the scrim was usually either jute or hessian . [ 2 ] It is easy to tell whether walls have scrim and sarking as their basis: knocking on the wall produces the sound of the wood, and any wallpaper laid over the top has an uneven finish. In many instances, the scrim will come loose from the sarking, in which case the wallpaper will appear to float loose from the wall. [ 1 ]
Compared with more modern forms of interior wall surfacing, scrim and sarking has poor insulation properties and can encourage damp. It is also more costly to insure homes with scrim and sarking walls, as they pose a fire danger. [ 3 ] For these reasons, home renovation will often see it replaced with gypsum -based wallboards . [ 2 ]
This material -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scrim_and_sarking |
ScripTalk is an audible medication label technology designed to give access to individuals who are blind, visually impaired, or print impaired. [ 1 ] [ 2 ] It consists of a device and a microchip attached to the bottom of a prescription drug bottle. [ 3 ] The label information is encoded on a Radio-frequency identification (RFID) electronic label (microchip) using the ScriptAbility software by a pharmacist and placed on the prescription package. ScripTalk prescription labels were introduced in the early 2000s. [ 3 ] As of 2020, the technology was applied through the United States and Canada.
In 1996, Philip Raistrick and David Raistrick founded En-Vision America, which is now based in Palmetto, FL. In 2000, the father and son invented and patented the Audible Prescription Reading Device and Labeling System for individuals who are visually impaired or print impaired. [ 4 ] [ 5 ] Shortly thereafter, the United States Department of Veterans Affairs (VA) began to test the technology for blinded veterans. [ 6 ] ScripTalk was approved for use by the VA in 2004 and began being integrated in VA hospitals across the US. [ 1 ] [ 7 ]
In 2012, Walmart introduced the ScripTalk service through a pilot program. [ 8 ] and by 2019, the company was rolling out the ScripTalk service throughout all Walmart and Sam's Club locations and via mail orders. [ 9 ] [ 10 ] Among other pharmacy and retail chains that have integrated ScripTalk are CVS, Costco, Albertsons, Kaiser Permanente, Veteran's Administration, Winn Dixie and more. [ 2 ] [ 11 ] [ 12 ] In February 2020, the ScripTalk technology was rolling out in Canada through Empire Company Limited, parent company to Sobeys, at its 420 pharmacy locations throughout the country, including Sobeys , Safeway , IGA , Foodland , Farm Boy , FreshCo , Thrifty Foods and Lawtons Drug. [ 4 ] [ 13 ] [ 14 ]
A number of the states in the US, including Oregon and Nevada introduced laws obliging pharmaceutical companies to supply blind and visually impaired patients with the prescription reading devices such as ScripTalk. [ 15 ] [ 16 ] The RFID ScripTalk label technology was granted a number of patents by the United States Patent and Trademark Office . [ 17 ]
The technology is designed for visually impaired people with the purpose to reduce risks for at-home medication errors, such as confusing medications, swallowing the wrong pill, ingesting expired medications, or missing a refill. ScripTalk consists of three main components: an Audible Prescription Reading Device (APRD) or ScripTalk Station Reader, radio-frequency identification (RFID) microchip label and supporting software. [ 18 ] [ 19 ]
In 2019, the company also released ScripTalk Mobile app that runs on iOS and Android devices with installed Near Field Communication (NFC) technology to read ScripTalk Talking Labels without APRD. [ 20 ] [ 21 ] [ 22 ] [ 23 ] | https://en.wikipedia.org/wiki/ScripTalk |
Scrip Intelligence (Scrip) is an English language international pharmaceutical news, analysis and data service. First published as a weekly print newsletter in March 1972, Scrip included articles on side-effects, regulatory changes and mergers and acquisitions .
Scrip World Pharmaceutical News was initially published by advertising company J Walter Thomson but was bought by Dr Philip J Brown in 1976 for £2000, who founded PJB Publishing. Brown sold Scrip, associated titles, and his company to Informa for £150 million in 2003. [ 1 ] [ 2 ] [ 3 ]
Scrip has now developed into an online global pharma news and analysis service. It provides daily news and analysis on the biotechnology and pharmaceutical industry including areas such as policy & regulation, business news, research & development , generics , drug delivery , and clinical trials .
In 2022, Informa sold its Pharma Intelligence division, including Scrip , to Warburg Pincus . [ 4 ] The business was renamed Citeline. Later in 2022, Citeline was acquired by Norstella. [ 5 ]
The Scrip readership has grown to over 100,000 worldwide, [ 6 ] half of which are in Europe. Scrip is read by managers and executives working across all disciplines in pharmaceutical and biotech companies worldwide. [ 7 ]
Scrip has a large independent editorial team of 21. The journalists are based throughout the world in Washington, D.C. , Tokyo , India , and London to ensure a global emphasis to the latest news stories. | https://en.wikipedia.org/wiki/Scrip_World_Pharmaceutical_News |
A script lichen , or graphid lichen , is a member of a group of lichens which have spore producing structures that look like writing on the lichen body. [ 1 ] The structures are elongated and narrow apothecia called lirellae , which look like short scribbles on the thallus. [ 1 ] "Graphid" is derived from Greek for "writing". An example is Graphis mucronata . [ 1 ]
This article about lichens or lichenology is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Script_lichen |
Scripta Materialia is a peer-reviewed scientific journal . It is the "letters" section of Acta Materialia and covers novel properties, or substantially improved properties of materials. Specific materials discussed are metals , ceramics and semiconductors at all length scales, and published research endeavors explore the functional or mechanical behavior of these materials. Articles tend to focus on the materials science and engineering aspects of discovery, characterization, development (including advances), structure, chemistry, theory, experiment, modeling, simulation, physics processes (thermodynamics, mechanics, etc.), synthesis, processing (production), mechanisms, and control.
The journal also publishes comments on papers published in both Acta Materialia and Scripta Materialia and "Viewpoint Sets", which are sets of short articles invited by guest editors. The editor-in-chief is Gregory S. Rohrer, who also edits Acta Materialia .
The journal was established in 1967 as Scripta Metallurgica . [ 2 ] It was renamed Scripta Metallurgica et Materialia in 1990, finally obtaining its current name in 1996. [ 3 ]
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2020 impact factor of 5.611. [ 4 ] | https://en.wikipedia.org/wiki/Scripta_Materialia |
Scriptella is an open source ETL (Extract-Transform-Load) and script execution tool written in Java. It allows the use of SQL or another scripting language suitable for the data source to perform required transformations. [ 1 ] Scriptella does not offer any graphical user interface .
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scriptella |
Scriptural Reasoning ("SR") is one type of interdisciplinary, interfaith scriptural reading. It is an evolving practice of diverse methodologies in which Christians , Jews , Muslims , Hindus , Buddhists , Sikhs , Baháʼís , and members of other faiths, meet in groups to study their sacred scriptures and oral traditions together, and to explore the ways in which such study can help them understand and respond to particular contemporary issues. Originally developed by theologians and religious philosophers as a means of fostering post-critical and postliberal corrections to patterns of modern reasoning, it has now spread beyond academic circles.
Theologians of different faiths have strongly challenged the claims made by some of Scriptural Reasoning's founder practitioners that they have requisite knowledge of ancient traditions of Islamic, Jewish and Christian exegesis and, on that basis, "not only the capacity, but also the authority to correct" or "repair" modernist binarist or fundamentalist interpretations of the Bible or Quran . Published articles by academics have also criticised some Scriptural Reasoning projects in the United Kingdom for alleged lack of parity between participating religions and instrumentalising of sacred texts for political agendas and money, while other scholars have alleged a history in Scriptural Reasoning from earlier SR conferences in the United States of exclusion and bullying of Christian theologian critics, and in later SR projects in the UK of victimisation of Muslim theologian whistleblowers.
Scriptural Reasoning involves participants from multiple religious traditions [ 1 ] meeting, very often in small groups, to read and discuss passages from their sacred texts and oral traditions (e.g., the Tanakh , Talmud , New Testament , Vedas , Qur'an , Hadith , or Guru Granth Sahib ). [ 2 ] The texts will often relate to a common topic — say, the figure of Abraham , or consideration of legal and moral issues of property-holding. [ 3 ] Participants discuss the content of the texts, and will often explore the variety of ways in which their religious communities have worked with them and continue to work with them, and the ways in which those texts might shape their understanding of and engagement with a range of contemporary issues. [ 4 ]
A participant from any one religious tradition might therefore:
Most forms of SR exhibit the following basic features:
To leave space for the variety of ways in which Scriptural Reasoning may be practiced and developed, SR practitioners often find it more fruitful to characterize SR open-endedly in terms of metaphors, often drawn from the Abrahamic traditions themselves.
Scriptural Reasoning has sometimes been described as a "tent of meeting" — a Biblical mishkan ( Hebrew :ׁ משׁכן Arabic : مسكن) — a reference to the story of Genesis 18. Steven Kepnes, a Jewish philosopher, writes:
Participants in SR practice come to it as both representatives of academic institutions and particular "houses" (churches, mosques, synagogues) of worship. SR meets, however, outside of these institutions and houses in special times and in separate spaces that are likened to Biblical "tents of meeting". Practitioners come together in these tents of meeting to read and reason with scriptures. They then return to their academic and religious institutions and to the world with renewed energy and wisdom for these institutions and the world. [ 11 ]
Scriptural Reasoning has been compared to gathering around the warmth of a hearth, where — Ochs explains — the hearth represents "those dimensions of life that members of a religion turn to in times of crisis, tension, or uncertainty in the hope of drawing nearer to the source of their deepest values and identities." [ 12 ] This metaphor builds on the rabbinic notion of Torah as a "fire," drawn from texts like Jeremiah 23:29--"Is not my word like fire, says the LORD?" and Deuteronomy 33:2, as interpreted midrashically by the rabbis. In Sifre Devarim 343, the editor concludes that "the words of Torah are compared to fire" before developing this comparison in various respects. Most relevant to SR is that, "Just as a person that is too close to a fire is burned and if he is too far coldness [results], so too with the words of the Torah. As long as a person is involved in them, they are life-giving, but when one removes himself from them, they kill him..."
In this vein, James and Rashkover write:
The same sacredness and life that rewards l'shma study can also be the cause of absolutism and violence when a community feels under threat. Scripture is powerful : "Is not my word like fire, says the Lord?" (Jer. 23:29). The same fire that warms and gives life can also kill and destroy. Ochs discerns that the impulse to guard the sacredness of scripture, even violently, is often an index of the community's love of their sacred scriptures as a primal source of divine life. Rather than unleashing the destroying fire of scriptural passion, SR is a practice of offering a measure of scripture's warmth to others. [ 13 ]
More recently, Ochs has generalized his concept of scripture into that of a hearth, "those dimensions of life that members of a religion turn to in times of crisis, tension, or uncertainty in the hope of drawing nearer to the source of their deepest values and identities." [ 14 ] SR, in this view, becomes a prototype of a broader family of "hearth-to-hearth" engagements.
It is impossible to give a definitive or authoritative account of the purpose of SR. Scriptural Reasoning is first and foremost a practice, and individuals and communities may engage in a practice for many and various reasons, while furthermore the purposes or agendas in SR of some practitioners have been contested or rejected by others. Moreover, the actual effects of a practice may outstrip the intentions of its practitioners. Thus Scriptural Reasoners frequently emphasize that doing and experimenting with SR as a practice logically precedes theoretical accounts of its grounds or function. According to Nicholas Adams, 'Scriptural reasoning is a practice which can be theorised, not a theory which can be put into practice. More accurately, it is a variety of practices whose interrelations can be theorised to an extent, but not in any strong sense of fully explanatory theory.' [ 15 ] Peter Ochs makes the same point with reference to a midrash on Exodus 24:7 in b. Shab. 88a:
In the book of Exodus, when Moses tried to deliver the Ten Commandments for the second time, the Israelites respond with the declaration naaseh v'nishmah! Literally, their declaration means "We shall do it and understand it," but, it was more likely an idiomatic expression for "We are on the job!" or "Consider it done!" The later rabbinic sages offered a homiletic rereading: "We shall first act and then understand"...We have nurtured SR in the same fashion, seeking to experiment with many forms of practice before discovering the one that best fits our goals and working over many years to refine it. We proceeded through experimentation first and only later through theoretical reflection. [ 16 ]
Nevertheless, it is possible to distinguish three commonly-cited and not mutually-exclusive purposes.
According to David Ford, one should practice SR because studying scripture is intrinsically valuable. On this view, one practices SR for the same reasons and in the same spirit that most traditional Abrahamic readers have studied their scriptures. David Ford makes this point using the Hebrew term " l'shma ":
This practice of shared reading could be done for its own sake — or, better, for God’s sake. Each of the three traditions has its own ways of valuing the study of its scriptures as something worth doing quite apart from any ulterior motive. Scriptural Reasoning might of course have all sorts of practical implications, but to do it above all for God’s sake — as Jews say, l’shma — encourages purity of intention and discourages the mere instrumentalising of inter-faith engagement. [ 17 ]
The term l'shma, which literally means "for the name," is ambiguous, capable of signifying Torah study "for its own sake" or "for God's sake." [ 18 ]
Under the heading of SR as study l'shma, we might include those who approach SR as a practice that promotes the development of "wisdom," a central theme of David Ford's work on SR. [ 19 ] In the same vein Peter Ochs speaks of SR as "open[ing] unexpected levels of textual and hermeneutical inquiry...for its own sake," an opening made possible by the affective warmth of SR study circles. [ 20 ] Others frame SR as a kind of ritual practice or even something approaching an act of worship. Marianne Moyaert, for example, argues that SR can be characterized as a formative "ritualized practice." [ 21 ]
Study l'shma is motivated by desire, by love for the scriptures and/or for God. For this reason, by inviting participants to share l'shma study together, SR provides what Ochs calls "a venue for members of different traditions or modes of inquiry to share their affection for scripture." This affective aspect of SR, in turn, contributes to SR's capacity to form unexpected interreligious friendships.
The most likely source of these friendships is that the style of Formational Scriptural Reasoning tempts participants (often unawares) to reveal at least a bit of the warmth and ingenuousness they display in intimate settings of scripture study among coreligionists at home. [ 22 ]
As originally conceived, SR was an academic practice involving theologians, religious philosophers, and text scholars, and was said to be aimed at 'repairing' or 'correcting' patterns of modern philosophical and theological reasoning. [ 23 ] These patterns of reasoning persist both in the Western academy and in religious traditions influenced by modernity. Thus according to Peter Ochs, SR was originally intended to repair academic methods of study and the habits of mind that they presuppose.
For the founders of Scriptural Reasoning, the original purpose was to repair what they judged to be inadequate academic methods for teaching scripture and scripturally-based religions, such as the Abrahamic religions...Over time, both Scriptural Reasoning and Textual Reasoning acquired new purposes as participants discovered additional consequences of these practices. [ 24 ]
Nicholas Adams characterizes SR as a practice of "reparative reasoning" capable of advancing "the pragmatic repair of secular universalism." [ 25 ] Building on this description, Ochs frequently emphasizes SR's reparative capacity to accustom practitioners to new ways of reasoning and habits of mind. He says that "the primary purpose of Scriptural reasoning is to correct "binarism in modern Western civilization and in religious groups that have, willy-nilly, adopted this binarism as if it were an engine of indigenous religious discourse and belief." [ 26 ] Binarism is this logical tendency to assume that difference entails opposition. As Ochs says, "All I mean by "binarism" is a strong tendency to overstate and over-generalize the usefulness of either/or distinctions." [ 27 ] SR repairs this tendency, in part, by training practitioners in alternative habits of mind:
[To affirm] that scripture tolerates, say, two meanings of a crucial verse, and not only one, is already to soften the rage that such participants may feel towards those whose readings different from theirs. In place of rage, such participants may adopt, for example, a superior and patronizing--but nonviolent--attitude towards these others as errant, but guilty only of a weaker reading of scripture rather than a reading that defies the very truth of things. [ 28 ]
SR also tends to repair the binarism that is a persistent feature of modern religious traditions.
Scriptural Reasoning is stimulated by the perception, furthermore, that the religious institutions that reside in the modern West have tended to assimilate these binarist tendencies into their theological discourses. One result is that many movements labeled " fundamentalist " display tendencies to a modern Western-style binarism that has been written into the tissue of traditional religious practices and discourses. [ 29 ]
SR thus implies a distinction between fundamentalism and traditionalism: the former tends to apply when the indigenous logic of a religious tradition has been superseded by modern binarism. For this reason, SR can undermine fundamentalism without attacking religious tradition per se, and indeed, purporting to draw its repair from traditional texts and interpretive practices. SR, by contrast, undermines fundamentalism while adopting an optimistic posture towards religious tradition. "Liberal" religion itself tends to operate with the same modern logic; indeed, the opposition between "liberal" and "fundamentalist" forms of religion is plausible, in part, because both operate with similar logics. For this reason, as Kepnes says, SR is "neither Liberal nor Fundamentalist." [ 30 ] This is one reason that SR has often been described as a 'postliberal' or 'postcritical' theological or philosophical movement. [ 31 ]
Its purpose is sometimes described as 'humbling and creative' interfaith encounter [ 32 ] or 'deeper mutual understanding'. [ 33 ]
Scriptural Reasoning is just one type of inter-textual discussion of the sacred scriptures of different religions, something which have been practiced by many scholars over many centuries. In the case of SR, it began as an intra-faith practice of Textual Reasoning by Jewish academics of Jewish texts, before becoming an inter-faith activity of Scriptural Reasoning when the conversation was joined by members of other religions. Over time SR has been developed by different scholars in a variety of diverse and contrasting ways.
Scriptural Reasoning has roots in a variety of classical practices of scriptural interpretation, particularly rabbinic midrash. Its proximate origins, however, lie in a related practice, "Textual Reasoning" ("TR"), [ 34 ] which involved Jewish philosophers reading Talmud in conversation with scholars of rabbinics. [ 35 ] Peter Ochs was one of the leading participants in Textual Reasoning (TR). [ 36 ] As James and Rashkover say,
Textual Reasoning (TR) emerged in the 1980s from conversations among Jewish philosophers disappointed by the failure of modern Western philosophy to provide principles of inquiry capable of addressing the pressing concerns of living Jewish communities. These philosophers developed a novel practice of Jewish text study rooted in the Jewish textual tradition itself which they aspired to activate as a source of communal repair. Textual Reasoning brought text scholars familiar with rabbinic reading practices together with Jewish philosophers skilled in illuminating logics of reading and reasoning. [ 37 ]
In 1990, Ochs and his colleagues founded what they then called the "Postmodern Jewish Philosophy Network" which hosted lively online exchanges, biannual meetings, an online journal. In 1996 they adopted the term "textual reasoning" for this practice, evoking classical Jewish practices of interpretation, and renamed their group the Society for Textual Reasoning. [ 38 ] In 2002, they founded a Journal of Textual Reasoning .
Textual Reasoning already displayed many features of what would become SR. According to Ochs, these include a tendency to pursue text study "for its own sake"; to both seek the plain sense of a text and to go explore various other dimensions of meaning; to value intense individual thought and group dialogue; and a combination of scholarly discipline with humor and laughter. [ 39 ] "Textual Reasoning" is often distinguished, as a Jewish practice of study, from Biblical Reasoning (Christian) and Qur'anic Reasoning (Muslim).
According to James and Rashkover, "Textual Reasoning gave birth to Scriptural Reasoning (SR) as early Textual Reasoners developed friendships with Christian and Muslim scholars and began to experiment with reading scripture together." [ 40 ] Ochs recounts the early history:
Beginning in 1994, a group of scholars of Islam, Judaism, and Christianity joined together to discover a way to conduct dialogue across the borders of these three Abrahamic scriptural traditions … We met for five years of biannual study until we discovered and refined the best method, which we called "Scriptural Reasoning" (SR). [ 41 ]
The term "Scriptural Reasoning" was coined by Peter Ochs [ 42 ] to distinguish the interfaith practice of scripture study from its tradition-specific antecedents. Ochs also argues, however, that SR presupposes parallel formation in practices of study across difference like TR:
In its broadest meaning, SR includes two sub-practices: study-across-difference within a single scriptural tradition and study across the borders of different scriptural traditions … [T]he former, which we label "Textual Reasoning" (or TR), also makes an irreplaceable contribution to the overall practice of SR. [ 43 ]
The international Society for Scriptural Reasoning (SSR) was founded in 1995. [ 44 ] The founders include Ochs himself, David F. Ford , Daniel W. Hardy , and Basit Koshul. [ 45 ] In 2001, the SSR established a Journal of Scriptural Reasoning to publish research into SR and to displays the academic fruits of SR as a practice.
Scriptural Reasoning began as an academic practice and expanded rapidly in academic circles. SR scholars formed an "additional meeting group" at the American Academy of Religion which later became the official Scriptural Reasoning Program Unit. [ 46 ]
They began a Scriptural Reasoning Theory Group at Cambridge University , in partnership with the Cambridge Interfaith Program . It was renamed the Scriptural Reasoning in the University group in 2007 and continued meeting through 2020. [ 47 ] ) This group focused on applying Scriptural Reasoning in academia and producing original scholarship about SR. [ 48 ] Out of this group emerged the Scripture & Violence Project , which has published academic work on the relationship between violence and the Abrahamic scriptures and makes available resources for laypeople to engage with these issues. [ 49 ]
Other academic developments of SR include a Scriptural Reasoning project at the Center of Theological Inquiry in Princeton, which examined SR and the history of medieval scriptural commentaries; [ 50 ] the Scriptures in Dialogue project founded by Leo Baeck College ; and the SR Oxford group of the Scriptural Reasoning Society ("Oxford School") founded by the Interfaith Alliance UK .
Scriptural Reasoning has also become a "civic practice" in the community, examples of which include the Central Virginia Scriptural Reasoning Group sponsored by Eastern Mennonite University , at St Ethelburga's Centre for Reconciliation and Peace at St Ethelburga's Bishopsgate , the SR Camden and SR Westminster groups of the Scriptural Reasoning Society sponsored by Camden Faith Communities Partnership, Liberal Judaism (United Kingdom) and different places of worship in London.
Civic developments from Scriptural Reasoning carrying different names, include the Faith and Citizenship programme of London Metropolitan University , and the Three Faiths Forum , which develops modes of scriptural study for young people in schools and local communities.
One early fruit of Scriptural Reasoning was Dabru Emet , a document on Jewish-Christian relations published in 2000 in The New York Times . [ 51 ] This document, authored by four Jewish scholars — Peter Ochs, David Novak, Tikva Frymer-Kensky, and Michael Singer — and signed by over 200 rabbis and scholars from most strands of Judaism, aimed to lay the groundwork for more sympathetic and productive engagement between Judaism and Christianity. [ 52 ]
In 2007, independent Islamic authorities in London issued a fatwa [ 53 ] advising Muslims about participation in the practice of Scriptural Reasoning. [ 54 ]
The Rose Castle Foundation was founded in 2014 to equip leaders for peace and reconciliation work between the Abrahamic religions, with Scriptural Reasoning being central to its training. The Rose Castle Foundation also maintains a database of SR groups around the world. [ 55 ]
Criticisms of Scriptural Reasoning which have been made by academics from different traditions address some of its founding practitioners' claims to their having requisite knowledge of ancient traditions of Islamic, Jewish and Christian exegesis and, on that basis, the purported authority to "correct" or "repair" binarist or fundamentalist interpretations of Scripture. Scholars also challenge SR's underlying presuppositions, and raise concerns about the dynamics of power, money and control in SR's practical outworking.
Theologian Adrian Thatcher has questioned whether Scriptural Reasoning flattens theological differences in the way the three traditions approach their respective Scriptures, arguing that "Christian people are not the people of a book, even a very holy book. They are people of a Savior, the One who reveals a loving God who, by God’s Spirit, remakes and renews humankind in the image of the Son...Its danger lies in the implication that the relation between believers and their respective sacred texts lies along an axis of similarity". He notes "the paucity of references to Jesus Christ" in the essays in The Promise of Scriptural Reasoning (see, e.g., Ford and Pecknold 2006), and asking whether this "may indicate … the further erosion of Christocentric biblical interpretation." [ 56 ]
Christian theologian, James M. Gustafson , questions the claim implied by Peter Ochs' descriptions of Scriptural Reasoning that it "has not only the capacity, but also the authority to correct 'modernist reason'" — and asking whether Scriptural Reasoning has been sufficiently open to the critical discourses fostered in modernity. He writes, "One is startled to read 'scriptural reasoning' in the singular … the use of 'scriptural reasoning' implies a canon within the canon, the parameters and perimeters which are undisclosed". [ 57 ] His points have been responded to by S. Mark Heim. [ 58 ]
Christina Grenholm and Daniel Patte critique Scriptural Reasoning's presuppositions of Christian self-understanding and context for biblical interpretation. They write:
The so-called "scriptural reasoning" movement presupposes Christianity as a separate nation with clear borders and set markers and Scripture as its authorised map … but without adopting the critical perspective that would reveal that there are different kinds of "scriptural reasonings" [ 59 ]
Catholic theologian, Gavin D'Costa offers a four-fold criticism of David Ford's presentation of Scriptural Reasoning. Firstly, he asserts that Christological and ecclesiological doctrine necessary for Christian biblical reading is marginalized by SR. He comments:
"Ford's tent insinuates (and nothing stronger can be said here) the logic of liberalism: the Bible has no binding authority, nor has the church's reading of it got primary status, nor can Christian scripture/Christ actually narrate the other texts of scriptures: Jewish and Muslim." [ 60 ]
D'Costa moreover argues that "SR seems to eschew any canopy over the project, but the metaphysics of Christian scriptural reading generates precisely such a canopy". Furthermore, he asserts that "there is a vaguely pluralistic agenda present" and that "SR is analogised [by Ford] to shared worship". D'Costa states that SR neglects scope for witness and evangelism. His critique has been responded to by Darren Sarisky. [ 61 ]
Under the title, The Broken Promise of Scriptural Reasoning , Muslim theologian, Muhammad Al-Hussaini, presents a critique of David Ford's Anglican-led Scriptural Reasoning initiatives, which he argues lack parity between participant religions, have been characterised by colonialist politics of control, and which he categorises as ' amalīyya fāsida ( Arabic : عملية فاسدة), "corrupt practice". [ 62 ] He states that Fordian Scriptural Reasoning has "No minhag/minhaj , no timeless established Judaeo-Islamic discipline of dialectical exegesis traditionis , of thickly-reading holy books using instruments of philology, grammar, received oral tradition and sensitive exposition of concentric layers of literal through to allegorical readings of a verse". He contends, "Instead, Ford’s Anglican-led SR becomes merely a poor kind of inter-faith Protestant Bible study fashioned within the competency limitations of its self-appointed leadership". He expresses concern at what he suggests "appeared to be SR’s failure to respect indigenous ways of reading Islamic Scripture, namely alongside hadith and classical commentaries", and further asserts, "Over time I became increasingly offended at the instrumentalising of biblical and Quranic materials for political and funding agendas". [ 63 ]
Muslim theologian, Timothy Winter , argues that the presuppositions and motivations of Scriptural Reasoning are alien to the Islamic context. He states, "Scriptural reasoning is not method, but rather a promiscuous openness to methods of a kind unfamiliar to Islamic conventions of reading". He also asserts that Scriptural Reasoning's claims to correct secular reasonings through a re-engagement with traditional reading have little resonance for Islam that has not experienced such changes in any meaningful sense. He writes, "There cannot be a 'return to Scripture' in Peter Ochs's sense, since the Qur’an has nowhere been abandoned, and Muslim interlocutors in SR are much more likely to feel part of an unbroken tradition than advocates of a latter-day ressourcement". He asserts the closer proximity of Jewish-Islamic traditional exegesis: "The three-way dynamic helps to reduce binary polarisations, but it does carry a bias towards the ‘Semitic’. Muslim-Jewish relations turn out to be privileged for several reasons which may relate to this traditional category". He goes on to state, "The cognate quality of Arabic and Hebrew, which frequently enriches the practice of comparative SR", but states, "If SR tends to exclude the search for precision, and to celebrate an ‘irremediable vagueness’ (Ochs), Muslims may demur". [ 64 ]
Muslim theologian, Mohamed Elsharkawy, positively contrasts practices of Scriptural Reasoning in different contexts but sees SR in the United Kingdom as particularly "heavily contaminated with a Church of England Orientalism and a state counter-extremism agenda". He writes:
The monied UK interfaith agenda exists in part to give credibility to a declining Church of England, and David Ford's Scriptural Reasoning openly admits its Anglican origins and dominant polity. Funding of some Church-led Scriptural Reasoning projects with British government counter-extremism cash betrays the overarching agenda towards Islam, Muslims and our classical hermeneutics, as do proposed grand interfaith projects with the likes of Tony Blair. In place of our ancient tafsir al-qur'an , humbly seeking Allah's multifaceted meanings in every Arabic verse of His Book, Fordian Scriptural Reasoning is at times crude reading with an agenda, and those who have spoken out against this have been hurt. [ 65 ]
He asserts that from the early days of SR there has been exclusion and bullying of some Christian theologians and later Muslim scholars who have raised concerns about alleged malfeasance within Scriptural Reasoning projects, and he proposes a "Reform of Scriptural Reasoning" through repentance, engagement with SR's critics and an end to what he calls "the endless uncritical self-marketing of Scriptural Reasoning by a dominant clique".
Elsharkawy asserts, "There are serious issues of the scholarly integrity of Scriptural Reasoning when so much of the allegedly 'independent academic literature' advocating for it is published in journals like Modern Theology , the Journal of Scriptural Reasoning and other publications on whose editorial boards sit David Ford , Peter Ochs and other SR promoters themselves! How are these independent reliable sources? Have they ever had the honesty to publish work by the opponents of SR? Rather, along with its colonialism, the defining characteristics of Scriptural Reasoning in some contexts have been the gatekeepers and minders of the 'brand', the vigorous and expensive marketing of SR, 'invitation-only' tactics for some events, and the excluding and in some cases harming the academic lives of some of its Christian and Muslim critics. So much for 'better quality disagreement'". | https://en.wikipedia.org/wiki/Scriptural_reasoning |
Scrubber systems (e.g. chemical scrubbers, gas scrubbers) are a diverse group of air pollution control devices that can be used to remove some particulates and/or gases from industrial exhaust streams. An early application of a carbon dioxide scrubber was in the submarine the Ictíneo I , in 1859; a role for which they continue to be used today. Traditionally, the term "scrubber" has referred to pollution control devices that use liquid to wash unwanted pollutants from a gas stream. Recently, the term has also been used to describe systems that inject a dry reagent or slurry into a dirty exhaust stream to "wash out" acid gases . Scrubbers are one of the primary devices that control gaseous emissions, especially acid gases. Scrubbers can also be used for heat recovery from hot gases by flue-gas condensation . [ 1 ] They are also used for the high flows in solar, PV, or LED processes. [ 2 ]
There are several methods to remove toxic or corrosive compounds from exhaust gas and neutralize it.
Combustion is sometimes the cause of harmful exhausts, but, in many cases, combustion may also be used for exhaust gas cleaning if the temperature is high enough and enough oxygen is available. [ 3 ]
The exhaust gases of combustion may contain substances considered harmful to the environment, and the scrubber may remove or neutralize those.
A wet scrubber is used for cleaning air , fuel gas or other gases of various pollutants and dust particles. Wet scrubbing works via the contact of target compounds or particulate matter with the scrubbing solution. Water is the most common solvent used to remove inorganic contaminants, particularly for dust, but solutions of reagents that specifically target certain compounds may also be used. [ 4 ]
Process exhaust gas can also contain water-soluble toxic and/or corrosive gases like hydrochloric acid (HCl) or ammonia (NH 3 ). These can be removed very well by a wet scrubber. [ 5 ]
Removal efficiency of pollutants is improved by increasing residence time in the scrubber or by the increase of surface area of the scrubber solution by the use of a spray nozzle , packed towers or an aspirator . Wet scrubbers may increase the proportion of water in the gas, resulting in a visible stack plume, if the gas is sent to a stack.
Wet scrubbers can also be used for heat recovery from hot gases by flue-gas condensation . [ 1 ] In this mode, termed a condensing scrubber, water from the scrubber drain is circulated through a cooler to the nozzles at the top of the scrubber. The hot gas enters the scrubber at the bottom. If the gas temperature is above the water dew point , it is initially cooled by evaporation of water drops. Further cooling causes water vapors to condense , adding to the amount of circulating water.
The condensation of water releases significant amounts of low temperature heat due to the high value of the specific latent heat of the vaporisation of water (more than 2 gigajoules (560 kWh) per ton of water [ 6 ] ), which can be recovered by the cooler for e.g. district heating purposes.
Excess condensed water must continuously be removed from the circulating water.
A dry or semi-dry scrubbing system, unlike the wet scrubber , does not saturate the flue gas stream that is being treated with moisture. In some cases no moisture is added, while in others only the amount of moisture that can be evaporated in the flue gas without condensing is added. Therefore, dry scrubbers generally do not have a stack steam plume or wastewater handling/disposal requirements. Dry scrubbing systems are used to remove acid gases (such as SO 2 and HCl ) primarily from combustion sources.
There are a number of dry type scrubbing system designs. However, all consist of two main sections or devices: a device to introduce the acid gas sorbent material into the gas stream and a particulate matter control device to remove reaction products, excess sorbent material as well as any particulate matter already in the flue gas .
Dry scrubbing systems can be categorized as dry sorbent injectors (DSIs) or as spray dryer absorbers (SDAs) . Spray dryer absorbers are also called semi-dry scrubbers or spray dryers.
Dry scrubbing systems are often used for the removal of odorous and corrosive gases from wastewater treatment plant operations. The medium used is typically an activated alumina compound impregnated with materials to handle specific gases such as hydrogen sulfide . Media used can be mixed together to offer a wide range of removal for other odorous compounds such as methyl mercaptans , aldehydes , volatile organic compounds , dimethyl sulfide , and dimethyl disulfide .
Dry sorbent injection involves the addition of an alkaline material (usually hydrated lime , soda ash , or sodium bicarbonate ) into the gas stream to react with the acid gases . The sorbent can be injected directly into several different locations: the combustion process, the flue gas duct (ahead of the particulate control device), or an open reaction chamber (if one exists). The acid gases react with the alkaline sorbents to form solid salts which are removed in the particulate control device. These simple systems can achieve only limited acid gas (SO 2 and HCl) removal efficiencies. Higher collection efficiencies can be achieved by increasing the flue gas humidity (i.e., cooling using water spray). These devices have been used on medical waste incinerators and a few municipal waste combustors.
In spray dryer absorbers , the flue gases are introduced into an absorbing tower (dryer) where the gases are contacted with a finely atomized alkaline slurry . Acid gases are absorbed by the slurry mixture and react to form solid salts which are removed by the particulate control device. The heat of the flue gas is used to evaporate all the water droplets, leaving a non-saturated flue gas to exit the absorber tower. Spray dryers are capable of achieving high (80+%) acid gas removal efficiencies. These devices have been used on industrial and utility boilers and municipal waste incinerators .
Many chemicals can be removed from exhaust gas also by using adsorber material. The flue gas is passed through a cartridge which is filled with one or several adsorber materials and has been adapted to the chemical properties of the components to be removed. [ 7 ] This type of scrubber is sometimes also called dry scrubber. The adsorber material has to be replaced after its surface is saturated. Note: adsorption is a surface phenomena, absorption involves the entire material. Ex: Activated carbon an adsorbent, used for the adsorption of odorous compounds.
Mercury is a highly toxic element commonly found in coal and municipal waste. Wet scrubbers are only effective for removal of soluble mercury species, such as oxidized mercury, Hg 2+ . Mercury vapor in its elemental form, Hg 0 , is insoluble in the scrubber slurry and not removed. Therefore, an additional process of Hg 0 conversion is required to complete mercury capture. Usually halogens are added to the flue gas for this purpose. The type of coal burned as well as the presence of a selective catalytic reduction unit both affect the ratio of elemental to oxidized mercury in the flue gas and thus the degree to which the mercury is removed.
In July 2015, one study found that some mercury scrubbers installed on coal power plants inadvertently capture PAH (polycyclic aromatic hydrocarbons) emissions as well. [ 8 ] [ 9 ]
One side effect of scrubbing is that the process only moves the unwanted substance from the exhaust gases into a liquid solution, solid paste or powder form. This must be disposed of safely, if it can not be reused.
For example, mercury removal results in a waste product that either needs further processing to extract the raw mercury, or must be buried in a special hazardous wastes landfill that prevents the mercury from seeping out into the environment. There are issues with that, as it is extremely dangerous to the environment, and many factories cannot process them or have it moved to a landfill.
As an example of reuse, limestone-based scrubbers in coal-fired power plants can produce a synthetic gypsum of sufficient quality that can be used to manufacture drywall and other industrial products. [ 10 ]
Poorly maintained scrubbers have the potential to spread disease-causing bacteria. The problem is a result of inadequate cleaning. For example, the cause of a 2005 outbreak of Legionnaires' disease in Norway was just a few infected scrubbers. The outbreak caused 10 deaths and more than 50 cases of infection. [ 11 ]
Scrubbers were first used on board ships for the production of inert gas for oil tanker operations.
Later, in preparation for the global 0.5% sulfur cap in 2020, the International Maritime Organization (IMO) adopted guidelines on the approval, installation and use of exhaust gas scrubbers (exhaust gas cleaning systems) on board ships to ensure compliance with the sulfur regulation of MARPOL Annex VI . [ 12 ] Flag states must approve such systems and port states can (as part of their port state control ) ensure that such systems are functioning correctly. If a scrubber system is not functioning properly (and the IMO procedures for such malfunctions are not adhered to), port states can sanction the ship. The United Nations Convention on the Law of the Sea also bestows port states with a right to regulate (and even ban) the use of open loop scrubber systems within ports and internal waters. [ 13 ] [ 14 ] | https://en.wikipedia.org/wiki/Scrubber |
A scruffy dome is a steel grille that is placed over the inlet of a manhole , and is usually dome shaped. They function as a way for stormwater to enter the pipe network without allowing larger debris in, such as people or animals. [ 1 ] [ 2 ] Scruffy domes are usually placed in parks and wetlands , and are usually made with galvanised steel . [ 2 ]
This article about a civil engineering topic is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scruffy_dome |
A Scuderi cycle is a thermodynamic cycle that is constructed out of the following series of thermodynamic processes : [ 1 ]
The adiabatic processes are impermeable to heat: heat flows rapidly into the loop through the left expanding process, resulting in increasing pressure while volume is increasing; some of it flows back out through the right depressurizing process; the remaining heat does the work. | https://en.wikipedia.org/wiki/Scuderi_cycle |
From the 7th to 3rd Century BC, the Scythian people of the Pontic–Caspian steppe engaged in the widespread practice of metallurgy . Though Scythian society was heavily based around a nomadic, mobile lifestyle, the culture was capable of practicing metallurgy and of producing metal objects. Many works of Scythian metalworking have subsequently been found throughout the range of the people.
The Scythians emerged as a people prior to the 7th Century BC, when they were first mentioned in historical records. [ 1 ] The Scythian civilization consisted of a number of distinct tribal groups scattered across the Pontic Steppes , Caucasus , and Central Asia. Though primarily a nomadic people, the Scythians established a number of settlements across their territory; these establishments in turn allowed for the development of a sedentary society and the accompanying development of trade skills, including metalworking. [ 1 ]
Scythian knowledge of metalworking likely originated with the peoples of Iran and China, with this knowledge spreading along trade routes and arriving in the steppes from the 2nd to 1st Millennium BC. [ 2 ] Early Scythian metallurgy was centered around bronzeworking, as these skills had already been widely adopted by the Scythians' neighbors. The Minusinsk Basin of Siberia has been speculated as the origin point for the raw materials used in Bronze-age Scythian metallurgy, and Scythian access to this region fueled the peoples' later centuries of expansion. [ 1 ] During the 8th Century BC Scythians were often employed by nations in the Near East and these returning soldiers may have brought knowledge of iron-working back to their homeland, and by the start of the 6th-century BC the practice was widespread in the Pontic steppes. [ 3 ] In addition to bronze and iron working, gold and copper-working were also present in Scythian society; in his commentary on the Scythian people, Greek historian Herodotus remarked on their fondness for making things from gold and copper. [ 4 ] Metallurgy held a major place in Scythian society as metalworkers were needed to produce material goods to support the Scythian way of life. As a nomadic society with broad borders, the Scythians often raided neighboring peoples and as such required metal weaponry - particularly iron swords and bronze arrowheads. [ 5 ] It has been speculated that the Scythian's use of stylized metal adornments may have been copied from their opponents during these conflicts. [ 4 ] In addition, jewelry and other adornment was in demand among all classes of society, as can be seen with the discovery of metal adornments in the burial tombs attributed to the Scythians. One notable aspect of Scythian clothing was the widespread use of metal belts. [ 4 ]
Other signs of Scythian metalworking can be found throughout sites attributed to the people. Several notable Scythian archeological sites contain the remnants of metalworking operations; at one settlement along the Dnieper , remnants of blast furnaces and slag have been found, implying the existence of a large metallurgical center. [ 6 ] Studies of other Scythian sites have also led to the remains of metal workshops and tools being found, further supporting the theory that the Scythians were organized craftspeople. [ 7 ] Scythian metalworkers were particularly renowned for the high quality of their copper crafting. [ 7 ] During war, portable molds were brought to forge arrowheads for the Scythian cavalry. [ 5 ] Scythian metallurgy also influenced the metallurgy of the Koban people of the North Caucasus . [ 5 ] | https://en.wikipedia.org/wiki/Scythian_metallurgy |
Scytovirin is a 95- amino acid antiviral protein isolated from the cyanobacteria Scytonema varium . [ 1 ] It has been cultured in E. coli and its structure investigated in detail. [ 2 ] [ 3 ] [ 4 ] Scytovirin is thought to be produced by the bacteria to protect itself from viruses that might otherwise attack it, but as it has broad-spectrum antiviral activity against a range of enveloped viruses, scytovirin has also been found to be useful against a range of major human pathogens, most notably HIV / AIDS but also including SARS coronavirus and filoviruses such as Ebola virus and Marburg virus . [ 5 ] [ 6 ] [ 7 ] While some lectins such as cyanovirin and Urtica dioica agglutinin are thought likely to be too allergenic to be used internally in humans, studies so far on scytovirin and griffithsin have not shown a similar level of immunogenicity. [ 8 ] Scytovirin and griffithsin are currently being investigated as potential microbicides for topical use. [ 9 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Scytovirin |
Selenium monochloride or diselenium dichloride is an inorganic compound with the formula Se 2 Cl 2 . Although a common name for the compound is selenium monochloride, reflecting its empirical formula , IUPAC does not recommend that name, instead preferring the more descriptive diselenium dichloride.
Diselenium dichloride is a reddish-brown, oily liquid that hydrolyses slowly. It exists in chemical equilibrium with SeCl 2 , SeCl 4 , chlorine , and elemental selenium . [ 1 ] Diselenium dichloride is mainly used as a reagent for the synthesis of Se-containing compounds.
Dielenium dichloride has the connectivity Cl−Se−Se−Cl . With a nonplanar structure, it has C 2 molecular symmetry , similar to hydrogen peroxide and disulfur dichloride , which is referred to as gauche . The Se-Se bond length is 223 pm , and the Se-Cl bond lengths are 220 pm. The dihedral angle between the Cl a −Se−Se and Se−Se−Cl b planes is 87°. [ 2 ]
Early routes to diselenium dichloride involved chlorination of elemental selenium . [ 3 ] An improved method involves the reaction of a mixture of selenium, selenium dioxide , and hydrochloric acid : [ 4 ]
A dense layer of diselenium dichloride settles from the reaction mixture, which can be purified by dissolving it in fuming sulfuric acid and reprecipitating it with hydrochloric acid. A second method for the synthesis involves the reaction of selenium with oleum and hydrochloric acid: [ 4 ]
The crude diselenium dichloride is removed via separatory funnel. Diselenium dichloride cannot be distilled without decomposition, even at reduced pressure. [ 4 ]
In acetonitrile solutions, it exists in equilibrium with SeCl 2 and SeCl 4 . [ 5 ] Selenium dichloride degrades to diselenium dichloride after a few minutes at room temperature: [ 6 ]
Diselenium dichloride is an electrophilic selenizing agent, and thus it reacts with simple alkenes to give bis( β -chloroalkyl)selenide and bis(chloroalkyl)selenium dichloride. It converts hydrazones of hindered ketones into the corresponding selenoketones , the structural analogs of ketones whereby the oxygen atom is replaced with a selenium atom. [ 7 ] Finally, the compound has been used to introduce bridging selenium ligands between the metal atoms of some iron and chromium carbonyl complexes. [ 7 ] | https://en.wikipedia.org/wiki/Se2Cl2 |
Selenium hexasulfide is a chemical compound with the chemical formula Se 2 S 6 . Its molecular structure is an 8-membered ring, consisting of two selenium and six sulfur atoms ( diselenacyclooctasulfane ), analogous to the S 8 ring, an allotrope of sulfur ( cyclooctasulfur or cyclooctasulfane), and other 8-membered rings of selenium sulfides with formula Se n S 8− n . [ 2 ]
There are several isomers depending on the relative placement of the selenium atoms in the ring: 1,2-diselenacyclooctasulfane (with the two Se atoms adjacent), 1,3-diselenacyclooctasulfane , 1,4-diselenacyclooctasulfane , and 1,5-diselenacyclooctasulfane (with the Se atoms opposite). [ 3 ] It is an oxidizing agent.
The 1,2 isomer can be prepared by reaction of chlorosulfanes and dichlorodiselane with potassium iodide in carbon disulfide . The reaction produces also cyclooctaselenium Se 8 and all other eight-member cyclic selenium sulfides, except selenacyclooctasulfane SeS 7 , and several six- and seven-membered rings. [ 2 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Se2S6 |
Tetrasulfur tetranitride is an inorganic compound with the formula S 4 N 4 . This vivid orange, opaque, crystalline explosive is the most important binary sulfur nitride , which are compounds that contain only the elements sulfur and nitrogen . It is a precursor to many S-N compounds and has attracted wide interest for its unusual structure and bonding. [ 1 ] [ 2 ]
Nitrogen and sulfur have similar electronegativities . When the properties of atoms are so highly similar , they often form extensive families of covalently bonded structures and compounds. Indeed, a large number of S-N and S-NH compounds are known with S 4 N 4 as their parent.
S 4 N 4 adopts an unusual "extreme cradle" structure, with D 2d point group symmetry . It can be viewed as a derivative of a (hypothetical) eight-membered ring (or more simply a 'deformed' eight-membered ring) of alternating sulfur and nitrogen atoms. The pairs of sulfur atoms across the ring are separated by 2.586 Å , resulting in a cage-like structure as determined by single crystal X-ray diffraction. [ 3 ] The nature of the transannular S–S interactions remains a matter of investigation because it is significantly shorter than the sum of the van der Waals radii [ 4 ] but has been explained in the context of molecular orbital theory . [ 1 ] One pair of the transannular S atoms have valence 4, and the other pair of the transannular S atoms have valence 2. [ citation needed ] The bonding in S 4 N 4 is considered to be delocalized , which is indicated by the fact that the bond distances between neighboring sulfur and nitrogen atoms are nearly identical. S 4 N 4 has been shown to co-crystallize with benzene and the C 60 molecule. [ 5 ]
S 4 N 4 is stable to air . It is, however, unstable in the thermodynamic sense with a positive enthalpy of formation of +460 kJ/mol. This endothermic enthalpy of formation originates in the difference in energy of S 4 N 4 compared to its highly stable decomposition products:
S 4 N 4 is shock and friction sensitive and because one of its decomposition products is a gas, it is considered a primary explosive. [ 1 ] [ 6 ] Purer samples tend to be more sensitive. [ 7 ] Small samples can be detonated by striking with a hammer. S 4 N 4 is thermochromic , changing from pale yellow below −30 °C to orange at room temperature to deep red above 100 °C. [ 1 ]
S 4 N 4 was first prepared in 1835 by M. Gregory by the reaction of disulfur dichloride with ammonia , [ 8 ] a process that has been optimized: [ 9 ]
Coproducts of this reaction include heptasulfur imide ( S 7 NH ) and elemental sulfur, and the latter equilibrates with more S 4 N 4 and ammonium sulfide : [ 10 ]
A related synthesis employs [NH 4 ]Cl instead: [ 1 ]
An alternative synthesis entails the use of (((CH 3 ) 3 Si) 2 N) 2 S as a precursor with pre-formed S–N bonds. (((CH 3 ) 3 Si) 2 N) 2 S is prepared by the reaction of lithium bis(trimethylsilyl)amide and SCl 2 .
The (((CH 3 ) 3 Si) 2 N) 2 S reacts with the combination of SCl 2 and SO 2 Cl 2 to form S 4 N 4 , trimethylsilyl chloride , and sulfur dioxide : [ 11 ]
S 4 N 4 is a Lewis base at nitrogen. It binds to strong Lewis acids , such as SbCl 5 and SO 3 , or H[BF 4 ] :
The cage is distorted in these adducts . [ 1 ]
S 4 N 4 reacts with metal complexes, but the bonding situation may be quite complex. The cage remains intact in some cases but in other cases, it is degraded. [ 2 ] [ 12 ] For example, the soft Lewis acid CuCl forms a coordination polymer : [ 1 ]
Reportedly, [Pt 2 Cl 4 (P(CH 3 ) 2 Ph ) 2 ] initially forms a complex with S 4 N 4 at sulfur. This compound, upon standing, isomerizes to additionally bond through a nitrogen atom. S 4 N 4 oxidatively adds to Vaska's complex ( [Ir(Cl)(CO)(P Ph 3 ) 2 ] to form a hexacoordinate iridium complex where the S 4 N 4 binds through two sulfur atoms and one nitrogen atom. [ 2 ]
Dilute NaOH hydrolyzes S 4 N 4 as follows, yielding thiosulfate and trithionate : [ 1 ]
More concentrated base yields sulfite :
Many S-N compounds are prepared from S 4 N 4 . [ 13 ]
In electrophilic substitution or 1,3-dipolar cycloaddition reactions, S 4 N 4 behaves as a combination of the dithionitronium synthon and the sulfide synthon. Thus it adds to arenes and electron-rich alkynes to give 1,2,5‑ thiadiazoles . [ 14 ] Electron-poor alkynes attack S 4 N 4 to give a different cycloadduct of stoichiometry RC(NS) 2 SCR ′ . [ 15 ] [ 14 ] With electron-rich alkenes , S 4 N 4 behaves as a Diels-Alder diene. [ 14 ]
Passing gaseous S 4 N 4 over silver metal yields the low temperature superconductor polythiazyl or polysulfurnitride (transition temperature (0.26±0.03) K [ 16 ] ), often simply called "(SN) x ". In the conversion, the silver first becomes sulfided, and the resulting Ag 2 S catalyzes the conversion of the S 4 N 4 into the four-membered ring S 2 N 2 , which readily polymerizes . [ 1 ]
Oxidation of S 4 N 4 with elemental chlorine gives thiazyl chloride , [ citation needed ] but milder reagents give S 4 N + 3 :
That cation is relatively non- electrophilic and planar, with a delocalized π system . However, it adds triphenylphosphine to give [S(NPPh 3 ) 3 ] 3+ [Cl − ] 3 , a triimide analogue to sulfur trioxide . Conversely, S 4 N + 3 salts react with aluminum azide to recover S 4 N 4 . [ 14 ]
Treatment with tetramethylammonium azide produces the similar 10-π heterocycle [S 3 N 3 ] − :
In a related reaction, the use of the bis(triphenylphosphine)iminium azide gives a salt containing the blue [NS 4 ] − anion: [ 13 ]
[NS 4 ] − has a chain structure approximated by the resonance [S=S=N−S−S − ] ↔ [ − S−S−N=S=S] .
Reaction with piperidine generates [S 4 N 5 ] − :
A related cation is also known, i.e. [S 4 N 5 ] + .
Triphenylphosphine abstracts a sulfur atom, replacing it with another triphenylphosphine moiety: [ 14 ]
S 4 N 4 is a categorized as a primary explosive that is shock and friction sensitive. While comparable to pentaerythritol tetranitrate (PETN) in terms of impact sensitivity, its friction sensitivity is equal to or even lower than lead azide. [ 17 ] Purer samples are more shock-sensitive than those contaminated with elemental sulfur. [ 9 ] [ 7 ] | https://en.wikipedia.org/wiki/Se4N4 |
Carbonyl selenide is a chemical compound with the chemical formula O = C = Se . It is a linear molecule that is primarily of interest for research purposes.
Carbonyl selenide is a colorless gas with an unpleasant odor. [ 1 ] Although the compound is quite stable, its solutions gradually revert to elemental selenium and carbon monoxide . [ 2 ]
Carbonyl selenide can be produced by treating selenium with carbon monoxide in the presence of amines . [ 3 ]
It is used in organoselenium chemistry as a means of incorporating selenium into organic compounds , e.g. for the preparation of selenocarbamates ( O -selenocarbamates R−O−C(=Se)−NR'R" and Se -selenocarbamates, R−Se−C(=O)−NR'R" , where R is organyl and R' and R" are any group, typically H or organyl). [ 3 ] | https://en.wikipedia.org/wiki/SeCO |
Selenium tetrachloride is the inorganic compound composed with the formula SeCl 4 . This compound exists as yellow to white volatile solid. It is one of two commonly available selenium chlorides , the other example being selenium monochloride , Se 2 Cl 2 . SeCl 4 is used in the synthesis of other selenium compounds.
The compound is prepared by treating selenium with chlorine . [ 3 ] When the reacting selenium is heated, the product sublimes from the reaction flask. The volatility of selenium tetrachloride can be exploited to purification of selenium.
Solid SeCl 4 is actually a tetrameric cubane-type cluster , for which the Se atom of an SeCl 6 octahedron sits on four corners of the cube and the bridging Cl atoms sit on the other four corners. The bridging Se-Cl distances are longer than the terminal Se-Cl distances, but all Cl-Se-Cl angles are approximately 90°. [ 4 ]
SeCl 4 has often been used as an example for teaching VSEPR rules of hypervalent molecules . As such, one would predict four bonds but five electron groups giving rise to a seesaw geometry . This clearly is not the case in the crystal structure . Others have suggested that the crystal structure can be represented as SeCl 3 + and Cl − . This formulation would predict a pyramidal geometry for the SeCl 3 + cation with a Cl-Se-Cl bond angle of approximately 109°. However, this molecule is an excellent example of a situation where maximal bonding cannot be achieved with the simplest molecular formula. The formation of the tetramer (SeCl 4 ) 4 , [ 5 ] with delocalized sigma bonding of the bridging chloride is clearly preferred over a "hypervalent" small molecule.
Gaseous SeCl 4 contains SeCl 2 and chlorine, which recombine upon condensation.
Selenium tetrachloride can be reduced in situ to the dichloride using triphenylstibine :
Selenium tetrachloride reacts with water to give selenous and hydrochloric acids : [ 6 ] [ page needed ]
Upon treatment with selenium dioxide , it gives selenium oxychloride : [ 6 ] [ page needed ] | https://en.wikipedia.org/wiki/SeCl4 |
Selenium tetrafluoride ( Se F 4 ) is an inorganic compound . It is a colourless liquid that reacts readily with water. It can be used as a fluorinating reagent in organic syntheses (fluorination of alcohols, carboxylic acids or carbonyl compounds) and has advantages over sulfur tetrafluoride in that milder conditions can be employed and it is a liquid rather than a gas.
The first reported synthesis of selenium tetrafluoride was by Paul Lebeau in 1907, who treated selenium with fluorine : [ 1 ]
A synthesis involving more easily handled reagents entails the fluorination of selenium dioxide with sulfur tetrafluoride : [ 2 ]
An intermediate in this reaction is seleninyl fluoride (SeOF 2 ).
Other methods of preparation include fluorinating elemental selenium with chlorine trifluoride :
Selenium in SeF 4 has an oxidation state of +4. Its shape in the gaseous phase is similar to that of SF 4 , having a see-saw shape. VSEPR theory predicts a pseudo-trigonal pyramidal disposition of the five electron pairs around the selenium atom. The axial Se-F bonds are 177 pm with an F-Se-F bond angle of 169.2°. The two other fluorine atoms are attached by shorter bonds (168 pm), with an F-Se-F bond angle of 100.6°. In solution at low concentrations this monomeric structure predominates, but at higher concentrations evidence suggests weak association between SeF 4 molecules leading to a distorted octahedral coordination around the selenium atom. In the solid the selenium center also has a distorted octahedral environment.
In HF , SeF 4 behaves as a weak base, weaker than sulfur tetrafluoride , SF 4 (K b = 2 X 10 −2 ):
Ionic adducts containing the SeF 3 + cation are formed with SbF 5 , AsF 5 , NbF 5 , TaF 5 , and BF 3 . [ 3 ] With caesium fluoride , CsF, the SeF 5 − anion is formed, which has a square pyramidal structure similar to the isoelectronic chlorine pentafluoride , ClF 5 and bromine pentafluoride , BrF 5 . [ 4 ] With 1,1,3,3,5,5-hexamethylpiperidinium fluoride or 1,2-dimethylpropyltrimethylammonium fluoride, the SeF 6 2− anion is formed. This has a distorted octahedral shape which contrasts to the regular octahedral shape of the analogous SeCl 6 2− . [ 5 ] | https://en.wikipedia.org/wiki/SeF4 |
Selenium hexafluoride is the inorganic compound with the formula SeF 6 . It is a very toxic colourless gas described as having a "repulsive" odor. [ 5 ] It is not widely encountered and has no commercial applications. [ 6 ]
SeF 6 has octahedral molecular geometry with an Se−F bond length of 168.8 pm . In terms of bonding, it is hypervalent .
SeF 6 can be prepared from the elements. [ 7 ] It also forms by the reaction of bromine trifluoride (BrF 3 ) with selenium dioxide . The crude product can be purified by sublimation.
The relative reactivity of the hexafluorides of S, Se, and Te follows the order TeF 6 > SeF 6 > SF 6 , the latter being completely inert toward hydrolysis until high temperatures. SeF 6 also resists hydrolysis. [ 3 ] The gas can be passed through 10% NaOH or KOH without change, but reacts with gaseous ammonia at 200 °C. [ 8 ]
Although selenium hexafluoride is quite inert and slow to hydrolyze, it is toxic even at low concentrations, [ 9 ] especially by longer exposure. In the U.S., OSHA and ACGIH standards for selenium hexafluoride exposure is an upper limit of 0.05 ppm in air averaged over an eight-hour work shift. Additionally, selenium hexafluoride is designated as IDLH chemical with a maximum allowed exposure limit of 2 ppm. [ 10 ] | https://en.wikipedia.org/wiki/SeF6 |
Selenium dioxide is the chemical compound with the formula SeO 2 . This colorless solid is one of the most frequently encountered compounds of selenium . It is used in making specialized glasses as well as a reagent in organic chemistry. [ 4 ]
Solid SeO 2 is a one-dimensional polymer , the chain consisting of alternating selenium and oxygen atoms. Each Se atom is pyramidal and bears a terminal oxide group. The bridging Se-O bond lengths are 179 pm and the terminal Se-O distance is 162 pm. [ 5 ] The relative stereochemistry at Se alternates along the polymer chain ( syndiotactic ).
In the gas phase selenium dioxide is present as dimers and other oligomeric species, at higher temperatures it is monomeric. [ 6 ] The monomeric form adopts a bent structure very similar to that of sulfur dioxide with a bond length of 161 pm. [ 6 ] The dimeric form has been isolated in a low temperature argon matrix and vibrational spectra indicate that it has a centrosymmetric chair form. [ 5 ] Dissolution of SeO 2 in selenium oxydichloride give the trimer [Se(O)O] 3 . [ 6 ] Monomeric SeO 2 is a polar molecule, with the dipole moment of 2.62 D [ 7 ] pointed from the midpoint of the two oxygen atoms to the selenium atom.
The solid sublimes readily. At very low concentrations the vapour has a revolting odour, resembling decayed horseradishes. At higher concentrations the vapour has an odour resembling horseradish sauce and can burn the nose and throat on inhalation. Whereas SO 2 tends to be molecular and SeO 2 is a one-dimensional chain, TeO 2 is a cross-linked polymer. [ 5 ]
SeO 2 is considered an acidic oxide : it dissolves in water to form selenous acid . [ 6 ] Often the terms selenous acid and selenium dioxide are used interchangeably. It reacts with base to form selenite salts containing the SeO 2− 3 anion. For example, reaction with sodium hydroxide produces sodium selenite :
Selenium dioxide is prepared by oxidation of selenium by burning in air or by reaction with nitric acid or hydrogen peroxide , but perhaps the most convenient preparation is by the dehydration of selenous acid .
The natural form of selenium dioxide, downeyite, is a very rare mineral. It is only found at a small number of burning coal banks , where it forms around vents created from escaping gasses. [ 8 ]
SeO 2 is an important reagent in organic synthesis . Oxidation of paraldehyde (acetaldehyde trimer) with SeO 2 gives glyoxal [ 9 ] and the oxidation of cyclohexanone gives 1,2-cyclohexanedione . [ 10 ] The selenium starting material is reduced to selenium, and precipitates as a red amorphous solid which can easily be filtered off. [ 10 ] This type of reaction is called a Riley oxidation . It is also renowned as a reagent for allylic oxidation , [ 11 ] a reaction that entails the following conversion
This can be described more generally as;
where R, R', R" may be alkyl or aryl substituents.
Selenium dioxide can also be used to synthesize 1,2,3-selenadiazoles from acylated hydrazone derivatives. [ 12 ]
Selenium dioxide imparts a red colour to glass . It is used in small quantities to counteract the colour due to iron impurities and so to create (apparently) colourless glass. In larger quantities, it gives a deep ruby red colour.
Selenium dioxide is the active ingredient in some cold-bluing solutions.
It was also used as a toner in photographic developing .
Selenium is an essential element , but ingestion of more than 5 mg/day leads to nonspecific symptoms . [ 13 ] | https://en.wikipedia.org/wiki/SeO2 |
Selenoyl fluoride , selenoyl difluoride , selenium oxyfluoride , or selenium dioxydifluoride is a chemical compound with the formula SeO 2 F 2 .
The shape of the molecule is a distorted tetrahedron with the O-Se-O angle being 126.2°, the O-Se-F angle being 108.0° and F-Se-F being 94.1°. [ 3 ] The Se-F bond length is 1.685 Å and the selenium to oxygen bond is 1.575 Å long. [ 4 ]
Selenoyl fluoride can be formed by the action of warm fluorosulfonic acid on barium selenate [ 5 ] or selenic acid . SeO 3 + SeF 4 can give this gas along with other oxyfluorides.
Selenoyl fluoride is more reactive than its analogon sulfuryl fluoride . It is easier to hydrolyse and to reduce. It may react violently upon contact with ammonia .
Selenoyl fluoride reacting with xenon difluoride gives FXeOSeF 5 . [ 6 ] | https://en.wikipedia.org/wiki/SeO2F2 |
Selenium trioxide is the inorganic compound with the formula Se O 3 . It is white, hygroscopic solid. It is also an oxidizing agent and a Lewis acid . It is of academic interest as a precursor to Se(VI) compounds. [ 4 ]
Selenium trioxide is difficult to prepare because it is unstable with respect to the dioxide :
It has been generated in a number of ways despite the fact that the dioxide does not combust under normal conditions. [ 4 ] One method entails dehydration of anhydrous selenic acid with phosphorus pentoxide at 150–160 °C. Another method is the reaction of liquid sulfur trioxide with potassium selenate .
In its chemistry SeO 3 generally resembles sulfur trioxide , SO 3 , rather than tellurium trioxide , TeO 3 . [ 4 ] The substance reacts explosively with oxidizable organic compounds . [ 5 ]
At 120 °C SeO 3 reacts with selenium dioxide to form the Se(VI)-Se(IV) compound diselenium pentaoxide: [ 6 ]
It reacts with selenium tetrafluoride to form selenoyl fluoride , the selenium analogue of sulfuryl fluoride
As with SO 3 adducts are formed with Lewis bases such as pyridine , dioxane and ether . [ 4 ]
With lithium oxide and sodium oxide it reacts to form salts of Se VI O 5 4− and Se VI O 6 6− : [ 7 ] With Li 2 O, it gives Li 4 SeO 5 , containing the trigonal pyramidal anion Se VI O 5 4− with equatorial bonds, 170.6–171.9 pm; and longer axial Se−O bonds of 179.5 pm. With Na 2 O it gives Na 4 SeO 5 , containing the square pyramidal Se VI O 5 4− , with Se−O bond lengths ranging from range 172.9 → 181.5 pm, and Na 12 (SeO 4 ) 3 (SeO 6 ), containing octahedral Se VI O 6 6− . Se VI O 6 6− is the conjugate base of the unknown orthoselenic acid (Se(OH) 6 ).
In the solid phase SeO 3 consists of cyclic tetramers, with an 8 membered (Se−O) 4 ring. Selenium atoms are 4-coordinate, bond lengths being Se−O bridging are 175 pm and 181 pm, non-bridging 156 and 154 pm. [ 7 ]
SeO 3 in the gas phase consists of tetramers and monomeric SeO 3 which is trigonal planar with an Se−O bond length of 168.78 pm. [ 8 ] | https://en.wikipedia.org/wiki/SeO3 |
Selenium oxydichloride is the inorganic compound with the formula SeOCl 2 . It is a colorless liquid. With a high dielectric constant (55) and high specific conductance, it is an attractive solvent. Structurally, it is a close chemical relative of thionyl chloride SOCl 2 , being a pyramidal molecule.
Selenium oxydichloride can be prepared by several methods, and a common one involves the conversion of selenium dioxide to dichloroselenious acid followed by dehydration: [ 3 ]
The original synthesis involved the redistribution reaction of selenium dioxide and selenium tetrachloride .
Pure selenium oxydichloride autoionizes to a dimer: [ 4 ]
The SeOCl 2 is generally a labile Lewis acid and solutions of sulfur trioxide in SeOCl 2 likely form [SeOCl] + [SO 3 Cl] − the same way. [ 5 ]
The compound hydrolyzes readily to form hydrogen chloride and selenium dioxide , [ citation needed ] and very few organic compounds dissolve in it without reaction. At elevated temperatures, it is a strong oxidizer, yielding a chloride, selenium dioxide , and diselenium dichloride . [ 6 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/SeOCl2 |
Selenium disulfide , also known as selenium sulfide , is a chemical compound and medication used to treat seborrheic dermatitis , dandruff , and pityriasis versicolor . [ 1 ] [ 2 ] It is applied to the affected area as a lotion or shampoo. [ 3 ] Symptoms frequently return if treatment is stopped. [ 4 ]
Side effects may include hair discoloration, skin irritation, and risk of systemic absorption and toxicity , among others. [ 1 ] Use is not recommended in children less than 2–5 years old. [ 1 ] [ 4 ] Use in pregnancy or breastfeeding has not been studied. [ 5 ] It consists of a mixture of inorganic covalent compounds having an approximate empirical formulas of SeS 2 . [ 6 ] Selenium disulfide acts as a keratolytic and antifungal agent. [ 7 ] [ 8 ] [ 9 ]
Selenium disulfide was approved for medical use in the United States at least as early as 1951. [ 4 ] It is on the World Health Organization's List of Essential Medicines . [ 10 ] Selenium disulfide is available as a generic medication and over the counter . [ 3 ]
Selenium disulfide is sold as an antifungal agent in shampoos (such as Selsun Blue ) for the treatment of dandruff and seborrheic dermatitis associated in the scalp with fungi of genus Malassezia . [ 11 ] [ 12 ] [ 13 ] It is also used on the body to treat tinea versicolor (pityriasis versicolor), a type of fungal skin infection caused by a different species of Malassezia . [ 2 ] [ 14 ]
A 2015 systematic review of topical treatments for seborrheic dermatitis of the scalp identified only a single randomized controlled trial evaluating selenium disulfide for the condition. [ 15 ] It was a three-arm trial of 246 people with moderate to severe dandruff and compared treatment with 2% ketoconazole shampoo (n=97), 2.5% selenium disulfide shampoo (n=100), and placebo (shampoo base with no antiseborrheic agent) (n=49) for 29 days. [ 15 ] [ 16 ] The study found a 73% reduction in dandruff score with ketoconazole, a 67% reduction with selenium disulfide, and a 45% reduction with placebo. [ 15 ] [ 16 ] Based on the study, the systematic review concluded that selenium disulfide may be effective in the treatment of dandruff but that the available evidence is limited and overall evidence quality is low. [ 15 ] It also found that while selenium disulfide has infrequent side effects, it seems to have more side effects than ketoconazole shampoo. [ 15 ] Consequently, the review concluded that selenium disulfide should not be considered as a first-line therapy but instead should be used as an alternative treatment after other therapies like ketoconazole shampoo have proven not effective. [ 15 ]
A 2015 review recommended topical antifungal agents, topical corticosteroids, and topical calcineurin inhibitors like tacrolimus as the main treatments for seborrheic dermatitis based on good-quality evidence, rather than selenium disulfide for which evidence is much more limited. [ 17 ] However, the review did suggest use of over-the-counter selenium disulfide shampoos as an inexpensive option for managing mild symptoms of seborrheic dermatitis. [ 17 ]
Selenium disulfide is available in the form of a prescription drug as a 2.25% medical shampoo . [ 18 ] [ 19 ] In the United States , a 1% strength is available over-the-counter , and a 2.5% strength is also available with a prescription . [ 19 ] In Canada, the 2.5% strength is available over-the-counter. [ 19 ] Selsun Blue is an over-the-counter shampoo for dandruff with 1% selenium disulfide as its active ingredient. [ 20 ] [ 19 ]
Side effects of selenium disulfide shampoo for dandruff appear to be infrequent. [ 15 ] [ 16 ] A randomized controlled trial of 100 people who received selenium disulfide reported side effects of itching or burning sensation of the scalp (3 people), eruption near the hairline (1 person), psoriasis (1 person), lightening or bleaching of hair color (2 people), orange staining of the scalp (1 person), and a chemical taste while shampooing (1 person). [ 15 ] [ 16 ]
Selenium disulfide can cause discoloration of nails and light hair [ 7 ] and can alter the color of hair dyes . Several scattered case reports of orange to red–brown scalp discoloration with selenium sulfide shampoo exist. [ 7 ] [ 21 ] The discoloration resolved shortly following discontinuation of selenium disulfide shampoo and its removal could be facilitated by lightly swabbing with isopropyl alcohol . [ 21 ] Selenium disulfide may also discolor metallic jewellery . Case reports of temporary diffuse hair loss with selenium disulfide shampoo exist as well. [ 22 ] [ 23 ] Excessive environmental or occupational exposure to selenium has also been associated with hair loss and other adverse effects. [ 22 ] However, hair loss has not been reported with topical selenium disulfide in several large studies. [ 21 ]
Selenium disulfide should not be applied to damaged skin as there is a risk of systemic absorption and associated toxicity . [ 1 ] Systemic symptoms may include tremors , weakness , lethargy , lower abdominal pain , and occasional vomiting . [ 1 ] These symptoms usually resolve within 10 days following exposure. [ 1 ]
Selenium disulfide acts as an antifungal and keratolytic agent to treat seborrheic dermatitis and dandruff. [ 7 ] [ 8 ] [ 9 ]
The systemic absorption and toxicity of orally administered selenium sulfide has been studied in animals. [ 24 ] [ 7 ] Topical use of selenium sulfide in the form of a shampoo or lotion in humans does not appear to normally increase circulating or urinary selenium levels. [ 25 ] [ 26 ] [ 27 ] [ 28 ] [ 29 ] However, application of selenium disulfide to damaged skin can result in systemic absorption and has been associated with cases of toxicity. [ 1 ] Selenium disulfide appears to be much less toxic than other selenium salts , which may be attributed to its low aqueous solubility and very poor systemic absorption. [ 7 ]
Selenium disulfide has a composition that approximates to SeS 2 and is sometimes called selenium sulfide. However, as used in proprietary formulations, it is not a pure chemical compound but a mixture of eight-membered-ring compounds where the overall Se:S ratio is 1:2. The specific chemicals contain a variable number of S and Se atoms, Se n S 8−n . [ 30 ]
Many selenium sulfides are known, as indicated by 77 Se- NMR spectroscopy. [ 31 ]
Selenium sulfide was introduced for medical use in the United States in 1951. [ 4 ] [ 19 ]
Selenium monosulfide, along with elemental selenium and sulfur, has been used in medicinal preparations in the past, [ 32 ] causing confusion and contradiction [ 33 ] as to exactly what form selenium is in any given topical preparation.
In the film Evolution selenium was mentioned as an active ingredient of Head & Shoulders . A group of academics, therefore, tried to use this brand of shampoo to stop an alien invasion after discovering that the alien life form was sensitive to selenium. [ 34 ]
Selenium disulfide has been suggested to be effective as a treatment for hyperkeratosis based on a small case series of three treated patients. [ 35 ] It has also been reported to be effective in the treatment of scalp psoriasis based on clinical observation of over 100 treated patients and two case reports of dramatic response. [ 36 ]
Selenium sulfide is under development for the treatment of meibomianitis (meibomian gland dysfunction) and dry eyes in topical and ophthalmic formulations. [ 37 ] [ 38 ] [ 19 ] As of March 2021, it is in phase 2 / 3 clinical trials for meibomianitis and phase 2 trials for dry eyes. [ 37 ] The developmental code name of selenium sulfide for these uses is AZR-MD-001 and it is being developed by Azura Ophthalmics. [ 37 ] | https://en.wikipedia.org/wiki/SeS2 |
The SeaWiFS Bio-optical Archive and Storage System (SeaBASS) is a data archive of in situ oceanographic data used to support satellite remote sensing research of ocean color . SeaBASS is used for developing algorithms for satellite-derived variables (such as chlorophyll-a concentration ) and for validating or “ground-truthing” satellite-derived data products. [ 1 ] [ 2 ] The acronym begins with “S” for SeaWiFS, because the data repository began in the 1990s around the time of the launch of the SeaWiFS satellite sensor, and the same data archive has been used ever since. Oceanography projects funded by the NASA Earth Science program are required to upload data collected on research campaigns to the SeaBASS data repository to increase the volume of open-access data available to the public. [ 3 ] [ 4 ] As of 2021 the data archive contained information from thousands of field campaigns uploaded by over 100 principal investigators. [ 5 ]
This database -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/SeaBASS_(data_archive) |
SeaSeep is a combination of 2D seismic data (a group of seismic lines acquired individually, as opposed to multiple closely space lines 1 ), high resolution multibeam sonar which is an evolutionary advanced form of side-scan sonar , navigated piston coring (one of the more common sea floor sampling methods 2 ), heat flow sampling (which serve a critical purpose in oil exploration and production 3 ) and possibly gravity and magnetic data (refer to Dick Gibson's Primer on Gravity and Magnetics 4 ).
The term SeaSeep originally belonged to Black Gold Energy LLC 5 and refers to a dataset that combines all of the available data into one integrated package that can be used in hydrocarbon exploration . With the acquisition of Black Gold Energy LLC by Niko Resources Ltd. 6 in December 2009 the term now belongs to Niko Resources
The concept of a SeaSeep dataset is the modern day offshore derivative of how many oil fields were found in the late 19th and early 20th century; by finding a large anticline structure with an associated oil seep. In the United States , many of the first commercial fields in California were found using this method including the Newhall Field discovered in 1876 and the Kern River Field discovered in 1899 7 . Seeps have also been used to find offshore fields including the Cantarell Field in Mexico in 1976; the largest oil field in Mexico and one of the largest in the world. The field is named after a fisherman, Rudesindo Cantarell, who complained to PEMEX about his fishing nets being stained by oil seeps in the Bay of Campeche .
The biological and geochemical manifestations of seepage leads to distinct bathymetrical features including positive relief mounds, pinnacles, mud volcanoes and negative relief pockmarks . These features can be detected by multibeam sonar and then sampled by navigated piston coring. Spec and proprietary multibeam seep mapping and core geochemistry by Texas A&M University 's Geochemical & Environmental Research Group 8 and later TDI Brooks 9 demonstrated thermogenic charge in deepwater Angola and deepwater Nigeria leading to an aggressive exploration program by a number of oil companies and subsequent discoveries. The emphasis on, and marketplace acceptance of, multibeam mapping combined with navigated coring as an improvement over grid-based approaches to geochemical exploration is attributed to AOA Geophysics Inc 10 .
1. Schlumberger Oilfield Glossary ( [1] )
2. Piston Coring ( [2] )
3. TDI Brooks ( [3] )
4. Primer on Gravity and Magnetics ( [4] )
5. Black Gold Energy ( [5] )
6. Niko Resources Ltd. ( [6] )
7. Natural Oil and Gas Seeps in California: ( [7] )
8. Geochemical & Environmental Research Group ( [8] )
9. TDI Brooks ( [9] )
10. AOA Geophysics ( [10] ) | https://en.wikipedia.org/wiki/SeaSeep |
The Sea or the Water is an area of the sky in which many water-related, and few land-related, constellations occur. This may be because the Sun passed through this part of the sky during the rainy season . [ 1 ]
Most of these constellations are named by Ptolemy :
Sometimes included are the ship Argo and Crater the Water Cup.
Some water-themed constellations are newer, so are not in this region. They include Hydrus , the lesser water snake; Volans , the flying fish; and Dorado , the swordfish.
This constellation -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sea_(astronomy) |
Sea air has traditionally been thought to offer health benefits associated with its unique odor, which is caused by dimethyl sulfide , released by microbes. [ 1 ]
Salts generally do not dissolve in air, but can be carried by sea spray in the form of particulate matter .
In the early 19th century, a lower prevalence of disease in coastal regions or islands was attributed to the sea air. [ 2 ] Such medical beliefs were translated into the literature of Jane Austen and other authors. [ 3 ] Victorians mistakenly attributed the odor of sea air to ozone .
Later that century, such beliefs led to the establishment of seaside resorts for the treatment of tuberculosis, [ 4 ] with medical beliefs of its efficacy continuing into the 20th century. [ 5 ] However, the quality of sea air was often degraded by pollution from wood- and coal-burning ships. Today those fuels are gone, replaced by high sulphur oil in diesel engines, which generate sulphate aerosols . [ 6 ]
This oceanography article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sea_air |
Sea foam , ocean foam , beach foam , or spume is a type of foam created by the agitation of seawater , particularly when it contains higher concentrations of dissolved organic matter (including proteins , lignins , and lipids ) derived from sources such as the offshore breakdown of algal blooms . [ 1 ] These compounds can act as surfactants or foaming agents . As the seawater is churned by breaking waves in the surf zone adjacent to the shore, the surfactants under these turbulent conditions trap air, forming persistent bubbles that stick to each other through surface tension .
Sea foam is a global phenomenon, [ 1 ] and it varies depending on location and the potential influence of the surrounding marine, freshwater, and/or terrestrial environments. [ 2 ] Due to its low density and persistence, foam can be blown by strong on-shore winds inland, towards the beach. Human activities, such as production, transport or spillage of petroleum products or detergents, can also contribute to the formation of sea foam.
Sea foam is formed under conditions that are similar to the formation of sea spray . One of the main distinctions from sea spray formation is the presence of higher concentrations of dissolved organic matter from macrophytes and phytoplankton . The dissolved organic matter in the surface water, which can be derived from the natural environment or human-made sources, provides stability to the resulting sea foam. [ 3 ]
The physical processes that contribute to sea foam formation are breaking surface waves , bubble entrainment, a process of bubbles being incorporated or captured within a liquid such as sea water and whitecap formation. [ 4 ] Breaking of surface waves injects air from the atmosphere into the water column, leading to bubble creation. These bubbles get transported around the top few meters of the surface ocean due to their buoyancy . The smallest bubbles entrained in the water column dissolve entirely, leading to higher ratios of dissolved gases in the surface ocean. The bubbles that do not dissolve eventually make it back to the surface. As they rise, these bubbles accumulate hydrophobic substances. Presence of dissolved organic matter stabilizes the bubbles, aggregating together as sea foam. [ 1 ] Some studies on sea foam report that breaking of algal cells in times of heavy swells makes sea foam production more likely. [ 3 ]
Falling rain drops on the sea surface can also contribute to sea foam formation and destruction. [ 5 ] There have been some non-mechanistic studies demonstrating increased sea foam formation due to high rainfall events. [ 2 ] Turbulence in the surface mixed layer can affect the concentration of dissolved organic matter and aids in the formation of nutrient-dense foam. [ 6 ]
The composition of sea foam is generally a mixture of decomposed organic materials, including zooplankton , phytoplankton , algae (including diatoms [ 7 ] ), bacteria , fungi , protozoans , and vascular plant detritus , [ 6 ] though each occurrence of sea foam varies in its specific contents. In some areas, sea foam is found to be made up of primarily protein , dominant in both fresh and old foam, as well as lipids and carbohydrates . The high protein and low carbohydrate concentration suggest that sugars originally present in the surrounding mucilage created by algae or plant matter has been quickly consumed by bacteria. [ 3 ] Additional research has shown that a small fraction of the dry weight in sea foam is organic carbon , which contains phenolics , sugars, amino sugars , and amino acids . In the Bay of Fundy , high mortality rates of an abundant tube-dwelling amphipod ( Corophium volutator ) by natural die-offs as well as predation by migrating seabirds contributed to amino sugars released in the surrounding environment and thus, in sea foam. [ 6 ]
The organic matter in sea foam has been found to increase dramatically during phytoplankton blooms in the area. [ 8 ] Some research has shown very high concentrations of microplankton in sea foam, with significantly higher numbers of autotrophic phytoplankton than heterotrophs [ 7 ] Some foams are particularly rich in their diatom population which can make up the majority of the microalgal biomass in some cases. [ 7 ] A diversity of bacteria is also present in sea foam; old foam tends to have a higher density of bacteria. One study found that 95% of sea foam bacteria were rod-shaped, while the surrounding surface water contained mostly coccoid-form bacteria and only 5% - 10% rod-shaped bacteria. [ 3 ] There is also seasonal variability of sea foam composition; [ 6 ] in some regions there is a seasonal occurrence of pollen in sea foam which can alter its chemistry. [ 2 ] Though foam is not inherently toxic, it may contain high concentrations of contaminants. [ 1 ] Foam bubbles can be coated with or contain these materials which can include petroleum compounds, pesticides , and herbicides . [ 1 ]
Structurally, sea foam is thermodynamically unstable, though some sea foam can persist in the environment for several days at most. There are two types of sea foam categorized based on their stability: 1) Unstable or transient foams have very short lifetimes of only seconds. The bubbles formed in sea foam may burst releasing aerosols into the air, contributing to sea spray . 2) Metastable foams can have a lifetime of several hours to several days; their duration is sometimes attributed to small particles of silica , calcium , or iron which contribute to foam stability and longevity. [ 1 ] Additionally, seawater that contains released dissolved organic material from phytoplankton and macrophytic algae that is then agitated in its environment is most likely to produce stable, longer-lasting foam when compared with seawater lacking one of those components. For example, filtered seawater when added to the fronds of the kelp , Ecklonia maxima, produced foam but it lacked the stability that unfiltered seawater provided. Additionally, kelp fronds that were maintained in flowing water therefore reducing their mucus coating, were unable to help foam form. [ 3 ] Different types of salt are also found to have varying effects on bubble proximity within sea foam, therefore contributing to its stability. [ 1 ]
The presence of sea foam in the marine environment plays a number of ecological roles including providing sources of food and creating habitat. As a food source, sea foam with a stable composition is more important ecologically, as it is able to persist longer and can transport nutrients within the marine environment. [ 3 ] Longer decay times result in a higher chance that energy contained in sea foam will move up the food web into higher trophic levels . [ 3 ] In the Bay of Fundy for example, a tube-dwelling amphipod , Corophium volutator, can potentially attain 70% of its nutritional requirements from the sugars and amino acids derived from sea foam in its environment. At times however, the sea foam was found to be toxic to this species. It is thought that high concentrations of phenolics and/or the occasional presence of heavy metals or pesticides incorporated into the sea foam from the sea surface contributed to its toxicity. [ 6 ] On the west coast of Cape Peninsula , South Africa , sea foam often occurs in nearshore marine areas with large kelp beds during periods of strong westerly winds. It is thought that the foam generated in these conditions is an important food source for local organisms due to the presence of organic detritus in the sea foam. [ 3 ]
Sea foam also acts as a mode of transport for both organisms and nutrients within the marine environment and, at times, into the intertidal or terrestrial environments. Wave action can deposit foam into intertidal areas where it can remain when the tide recedes, bringing nutrients to the intertidal zone. [ 6 ] Additionally, sea foam can become airborne in windy conditions, transporting materials between marine and terrestrial environments. [ 2 ] The ability of sea foam to transport materials is also thought to benefit macroalgal organisms, as macroalgae propagules can be carried to different microenvironments, thus influencing the tidal landscape and contributing to new possible ecological interactions. [ 9 ] As sea foam is a wet environment, it is conducive habitat to algal spores where propagules can attach to the substrate and avoid risk of dissemination . [ 9 ] When sea foam contains fungi, it can also aid in the decomposition of plant and animal remains in coastal ecosystems. [ 2 ]
Additionally, sea foam is a habitat for a number of marine microorganisms . Some research has shown the presence of various microphytoplanktonic, nanophytoplanktonic, and diatom groups in seafoam; the phytoplankton groups appeared in significantly higher abundance than in sea surface film and the top pelagic zone [ 7 ]
Naturally occurring sea foam is not inherently toxic ; however, it can be exposed to high concentrations of contaminants in the surface microlayer derived from the breakdown of algal blooms, fossil fuel production and transport, and stormwater runoff. [ 1 ] These contaminants contribute to the formation of noxious sea foam through adsorption onto bubbles. Bubbles may burst and release toxins into the atmosphere in the form of sea spray or aerosol , or they may persist in foams. Toxins released through aerosols and breaking bubbles can be inhaled by humans. The microorganisms that occupy sea foams as habitat have increased susceptibility for contaminant exposure. [ 10 ] Consequently, these toxic substances can be integrated into the trophic food web . [ 1 ]
Foams can form following the degradation of harmful algal blooms (HABs). These are primarily composed of algal species, but can also consist of dinoflagellates and cyanobacteria . [ 11 ] Biomass from algae in the bloom is integrated into sea foam in the sea surface microlayer. [ 9 ] When the impacted sea foam breaks down, toxins from the algae are released into the air causing respiratory issues and occasionally initiating asthma attacks. [ 12 ] Phaeocystis globosa is one algal species that is considered problematic, as observed in a study in the Netherlands. [ 11 ] Its high biomass accumulation allows it to create large quantities of toxic foam that often wash onto beaches. P. globosa blooms are initiated in areas of high nutrient availability, often affiliated with coastal locations with a lot of stormwater runoff and eutrophication . Studies suggest that the development of foam is directly correlated to blooms caused by P. globosa , despite that foam formation typically occurs approximately two weeks after the appearance of an algal bloom offshore. Organic material from P. globosa was observed decomposing while suspended at the sea surface, but was not observed lower in the water column. P. globosa is also considered a nuisance species because its large foam formations impair the public's ability to enjoy the beach. [ 11 ]
While sea foam is a common result of the agitation of seawater mixing with organic material in the surface ocean, human activities can contribute to the production of excess and often toxic foam. [ 1 ] In addition to the organic oils, acids, and proteins that amass in the sea surface microlayer , compounds derived from petroleum production and transport, synthetic surfactants, and pesticide use can enter the sea surface and be incorporated into foam. The pollutants present can also affect the persistence of the foam produced. Crude oil discharged from tankers, motor oil, sewage, and detergents from polluted runoff can create longer-lasting foams. [ 1 ] In one study, polychlorinated biphenyls (PCBs), a persistent organic pollutant , were found to amass in sea foams. [ 10 ] Some experts and health authorities recommend avoiding contact with sea foam in lakes and rivers and seas that are contaminated with PFAS , since these substances were found to accumulate in sea foam in high concentrations. [ 13 ] [ 14 ] [ 15 ] Man-made microplastic pollution can accumulate in breaking waves and increase sea foam stability. [ 16 ]
Natural gas terminals have been cited as contributors to the production of modified foams due to the process of using seawater to convert natural gas to liquified natural gas . [ 17 ] One study showed a much greater abundance of heterotrophic prokaryotes (archaea and bacteria) and cyanobacteria in foam that was generated near a liquified natural gas terminal. These prokaryotes were able to recycle chemical materials discharged from the terminal, which enhanced microbial growth. Additionally, higher levels of total organic carbon (TOC) and plankton biomass were recorded in foam generated in close proximity to the terminal. Organic carbon was transferred readily into the pelagic food web after uptake by prokaryotes and ingestion by grazers. [ 17 ] | https://en.wikipedia.org/wiki/Sea_foam |
Sea ice refreezing refers to various climate engineering techniques aimed at directly facilitating the formation or restoration of ice in polar regions , particularly in the Arctic Ocean . These approaches are being investigated as potential interventions to counter the accelerating loss of sea ice due to climate change , especially to avert a potential blue ocean event and its potential runaway climate impacts. [ 1 ] [ 2 ]
Sea ice , especially in the Arctic region, has declined in recent decades in area and volume due to climate change . It has been melting more in summer than it refreezes in winter. Global warming , caused by greenhouse gas forcing is responsible for the decline in Arctic sea ice. The decline of sea ice has been accelerating during the early twenty-first century, with a decline rate of 4.7% per decade (it has declined over 50% since the first satellite records). [ 3 ] [ 4 ] [ 5 ] Summertime sea ice will likely cease to exist sometime during the 21st century. [ 6 ]
Sea ice loss is one of the main drivers of Arctic amplification , the phenomenon that the Arctic warms faster than the rest of the world under climate change. It is plausible that sea ice decline also makes the jet stream weaker, which would cause more persistent and extreme weather in mid-latitudes . [ 7 ] [ 8 ]
In 2025, researchers at Purdue University led by Tian Li developed a modified wood material, termed "ice-wood," designed to facilitate ice formation in marine environments . The team selected wood as their base material due to its natural properties and lower environmental impact compared to synthetic alternatives. [ 1 ]
The ice-wood material is fabricated from American basswood ( Tilia americana ). To modify the material, a section is removed from a 10×10×1.5 centimeter piece of basswood, which undergoes partial carbonization on one side. The larger remaining piece is treated with hydrogen peroxide and heat to extract lignin , after which the carbonized section is reinserted into the larger piece. The removal of lignin, which gives wood its characteristic color, renders the treated portion significantly whiter and more reflective . The resulting modifications create a dual-surface material, with each side having distinct thermal properties. [ 1 ]
The ice-wood helps to seed the formation of ice through a combination of physical processes. When deployed, water rises through the ice-wood's natural microchannels via capillary action . Solar radiation heats the carbonized dark section, causing water evaporation. From there, water vapor re- condenses on the colder, lignin-depleted white surface. The elevated position of the structure combined with its reflective properties allows the surface to maintain sub-freezing temperatures even when ambient air temperatures rise several degrees above freezing, facilitating ice formation. In controlled tests conducted in 2.0 °C (35.6 °F) water, the ice-wood's surface remained below freezing despite ambient air temperatures reaching 7.0–8.0 °C (44.6–46.4 °F). Ice formation was observed beginning at the edges of the material and extending outward. [ 1 ]
Purdue researchers determined that while covering the entire Arctic with ice-wood would impractical, that it was feasible to deploy larger units in coastal areas, particularly for Indigenous communities who depend on sea ice for fishing and hunting activities. The technology would intend to accelerate winter ice formation and reduce summer melt rates. [ 1 ]
Computer modeling examining hypothetical widespread deployment of ice-wood throughout the Arctic from 2005 to 2022 indicated that by the 2022 melting season, the technology could have increased ice growth rates by approximately 0.3 centimeters per day. The models also predicted that the deployment would reduce sea surface temperatures by approximately 3.0 °C (37.4 °F) compared to actual 2022 measurements. [ 1 ]
Proponents of the technology note several factors favoring potential large-scale implementation. These include that wood is relatively inexpensive and abundant in nature, the lignin removal process is already performed at industrial scale in paper manufacturing , and that the production methods employ other established technologies. [ 1 ]
Some climate researchers such as Cecilia Bitz at the University of Washington , expressed skepticism about the technology's effectiveness during Arctic summers, when air temperatures typically reach around 10.0 °C (50.0 °F)—potentially too warm for the ice-wood to maintain freezing conditions at its surface. Other climate researchers including Julienne Stroeve at University College London questioned the allocation of resources toward such interventions rather than focusing on reducing carbon dioxide emissions . [ 1 ]
One approach Arctic refreezing involves drilling through the existing winter ice layer to access the ocean water beneath, then pumping this water onto the snow cover atop the ice. The approach was initially proposed by Steven Desch and colleagues at Arizona State University in 2016. The process is based upon how when seawater saturates the snow layer, it fills air pockets within the snow structure. The water-saturated snow freezes, effectively converting snow to solid ice and increases the thermal conductivity of the ice sheet. According to the research team, the resulting enhanced thermal conductivity allows cold Arctic air temperatures to penetrate more efficiently through the ice, accelerating natural ice formation on existing ice sheet's underside. The research team's early modeling suggested that implementing this approach across just 10% of the Arctic could potentially reverse recent ice loss in the polar region. [ 2 ] [ 9 ] [ 10 ]
In 2024, a United Kingdom -based organization called Real Ice , working in collaboration with the University of Cambridge 's Centre for Climate Repair, conducted field trials of seawater pumping technology in Cambridge Bay on Victoria Island , Canada . In the study, test sites with a single borehole demonstrated ice thickening of approximately 50 centimetres (20 in) compared to control sites between January and May, while stimulating an additional 25 centimetres (9.8 in) of natural ice growth on the ice sheet's underside. The saline water created during the freezing process successfully percolated back through the ice into the ocean rather than forming a detrimental surface layer that could weaken the ice structure. The resulting thickened ice maintained the structural integrity necessary to extend its lifespan through seasonal warming periods, according to Real Ice. [ 2 ] [ 9 ] [ 10 ]
Following the promising results of initial testing, Real Ice has begun developing more advanced implementation systems. The company has established a partnership with the BioRobotics Institute at the Sant'Anna School of Advanced Studies in Pisa , Italy , to design autonomous underwater drones capable of navigating beneath Arctic ice sheets, creating boreholes at strategic locations, and pumping seawater onto the ice surface. Real Ice faculty believed that it could cover approximately 2 square kilometres (0.77 sq mi) per drone per winter season. [ 2 ]
According to estimates by Real Ice, meaningful climate intervention would require the treatment of approximately 1,000,000 square kilometres (390,000 sq mi) of Arctic sea ice, and the deployment of roughly 500,000 autonomous drones resulting in the production of an additional 500 cubic kilometers of sea ice each winter. The company estimated annual operational costs of approximately $6 billion. According to development timelines announced in 2024, prototype drones were scheduled for completion in 2025, with Arctic field testing planned for the winter of 2026-2027. [ 2 ] [ 11 ]
Critics stated that operation of hundreds of thousands of seawater pumps would require significant energy resources , potentially creating carbon emissions that could partially offset its climate benefits unless powered by renewable sources . Some climate researchers, such as Woodwell Climate Research Center senior scientist Jennifer Francis , questioned whether such interventions could be deployed at sufficient scale to meaningfully impact Arctic ice conditions, particularly given the accelerating pace of climate change. Furthermore, polar researchers such University of Bristol associate professor Liz Bagshaw, expressed concern regarding the potential disruption to Arctic wildlife , especially regarding the existing vulnerability of the Arctic ecosystem. This included the possible impact of a reduction in snow cover that some species require for denning and breeding activities. [ 2 ] [ 9 ] | https://en.wikipedia.org/wiki/Sea_ice_refreezing |
The Sea of Faith Network is an organisation with the stated aim to explore and promote religious faith as a human creation.
The Sea of Faith movement started in 1984 as a response to Don Cupitt 's book and television series , both titled Sea of Faith . [ 1 ] Cupitt was educated in both science and theology at the University of Cambridge in the 1950s, and is a philosopher, theologian, Anglican priest, and former Dean of Emmanuel College, Cambridge . [ 2 ] In the book and TV series, he surveyed western thinking about religion and charted a transition from traditional realist religion to the view that religion is simply a human creation. [ 3 ]
The name Sea of Faith is taken from Matthew Arnold 's nostalgic mid-19th century poem " Dover Beach ," in which the poet expresses regret that belief in a supernatural world is slowly slipping away; the "sea of faith" is withdrawing like the ebbing tide. [ 4 ]
Following the television series, a small group of radical Christian clergy and laity began meeting to explore how they might promote this new understanding of religious faith. Starting with a mailing list of 143 sympathisers, they organised the first UK conference in 1988. [ 5 ] A second conference was held in the following year shortly after which the Sea of Faith Network was officially launched. Annual national conferences have been a key event of the network ever since. [ 6 ]
The Sea of Faith Network is a loose network rather than a formal religious organisation. It holds national and regional conferences and promotional events each year. There is an active network of local groups who meet regularly for discussion and exploration. [ 7 ]
The group's magazine Sofia is published quarterly in the United Kingdom. The group also maintains a web site and an on-line discussion group. [ 7 ]
Currently there are national networks in the UK, New Zealand and Australia with scattered membership in the USA, Northern Ireland, South Africa, France and The Netherlands. The world-wide membership, as of 2004, stood at about 2,000. Each national network is run by a steering committee elected from its members. [ 7 ]
The organisation has no official creed or statement of belief to which members are required to assent. Its stated aim is to "explore and promote religious faith as a human creation," [ 8 ] In this it spans a broad spectrum of faith positions from uncompromising non-realism at one end to critical realism at the other. [ 8 ] Some members describe themselves as on the liberal or radical wing of conventional belief (see liberal Christianity ) while others choose to call themselves religious or Christian humanists (see humanism ). Some even refer to themselves as agnostic, atheist, or simply nontheist (see Christian atheism ).
Sea of Faith possesses no religious writings or ceremonies of its own; many members remain active in their own religion (mainly but not exclusively Christian) while others have no religious affiliation at all. [ 7 ]
A number of commentators have identified Sea of Faith as closely associated with the non-realist approach to religion. [ 9 ] [ 10 ] This refers to the belief that God has no "real," objective, or empirical existence, independent of human language and culture; God is "real" in the sense of being a potent symbol, metaphor or projection, but having no objective existence outside and beyond the practice of religion. [ 11 ] Non-realism therefore entails a rejection of all supernaturalism, including concepts such as miracles, the afterlife, and the agency of spirits. [ 12 ]
Cupitt wrote, "God is the sum of our values, representing to us their ideal unity, their claims upon us and their creative power," [ 13 ] Cupitt calls this "a voluntarist interpretation of faith: a fully demythologized version of Christianity," [ 12 ] It entails the claim that even after we have given up the idea that religious beliefs can be grounded in anything beyond the human realm, religion can still be believed and practised in new ways. [ 14 ]
Since he began writing in 1971, Cupitt has produced 36 books. During this time his views have continued to evolve and change. [ 15 ] In his early books such as Taking Leave of God and The Sea of Faith Cupitt talks of God alone as non-real, [ 16 ] but by the end of the 1980s he moved into postmodernism , describing his position as empty radical humanism: [ 17 ] that is, there is nothing but our language, our world, and the meanings, truths and interpretations that we have generated. Everything is non-real, including God. [ 18 ]
While Cupitt was the founding influence of Sea of Faith and is much respected for his work for the network, it would not be true to say that he is regarded as a guru or leader of Sea of Faith. Members are free to dissent from his views and Cupitt himself has argued strongly that Sea of Faith should never be a fan club. [ 12 ] Both Cupitt and the network emphasise the importance of autonomous critical thought and reject authoritarianism in all forms. [ 12 ]
Alvin Plantinga called the movement "an amiable sort of dottiness," [ 19 ] Anthony Campbell also pointed to the contradictions in Cupitt's intellectual project. At once destroying the tenets of Christianity and then claiming to be a "non-realist" Christian seemed to Campbell to be the same as being an atheist. [ 20 ] | https://en.wikipedia.org/wiki/Sea_of_Faith |
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky . In the narrowest sense of the word, it consists of a flat plate (the dial ) and a gnomon , which casts a shadow onto the dial. As the Sun appears to move through the sky, the shadow aligns with different hour-lines, which are marked on the dial to indicate the time of day. The style is the time-telling edge of the gnomon, though a single point or nodus may be used. The gnomon casts a broad shadow; the shadow of the style shows the time. The gnomon may be a rod, wire, or elaborately decorated metal casting. The style must be parallel to the axis of the Earth's rotation for the sundial to be accurate throughout the year. The style's angle from horizontal is equal to the sundial's geographical latitude .
The term sundial can refer to any device that uses the Sun's altitude or azimuth (or both) to show the time. Sundials are valued as decorative objects, metaphors , and objects of intrigue and mathematical study.
The passing of time can be observed by placing a stick in the sand or a nail in a board and placing markers at the edge of a shadow or outlining a shadow at intervals. It is common for inexpensive, mass-produced decorative sundials to have incorrectly aligned gnomons, shadow lengths, and hour-lines, which cannot be adjusted to tell correct time. [ 2 ]
There are several different types of sundials. Some sundials use a shadow or the edge of a shadow while others use a line or spot of light to indicate the time.
The shadow-casting object, known as a gnomon , may be a long thin rod or other object with a sharp tip or a straight edge. Sundials employ many types of gnomon. The gnomon may be fixed or moved according to the season. It may be oriented vertically, horizontally, aligned with the Earth's axis, or oriented in an altogether different direction determined by mathematics.
Given that sundials use light to indicate time, a line of light may be formed by allowing the Sun's rays through a thin slit or focusing them through a cylindrical lens . A spot of light may be formed by allowing the Sun's rays to pass through a small hole, window, oculus , or by reflecting them from a small circular mirror. A spot of light can be as small as a pinhole in a solargraph or as large as the oculus in the Pantheon.
Sundials also may use many types of surfaces to receive the light or shadow. Planes are the most common surface, but partial spheres , cylinders , cones and other shapes have been used for greater accuracy or beauty.
Sundials differ in their portability and their need for orientation. The installation of many dials requires knowing the local latitude , the precise vertical direction (e.g., by a level or plumb-bob), and the direction to true north . Portable dials are self-aligning: for example, it may have two dials that operate on different principles, such as a horizontal and analemmatic dial, mounted together on one plate. In these designs, their times agree only when the plate is aligned properly.
Sundials may indicate the local solar time only. To obtain the national clock time, three corrections are required:
The principles of sundials are understood most easily from the Sun 's apparent motion. [ 3 ] The Earth rotates on its axis, and revolves in an elliptical orbit around the Sun. An excellent approximation assumes that the Sun revolves around a stationary Earth on the celestial sphere , which rotates every 24 hours about its celestial axis. The celestial axis is the line connecting the celestial poles . Since the celestial axis is aligned with the axis about which the Earth rotates, the angle of the axis with the local horizontal is the local geographical latitude .
Unlike the fixed stars , the Sun changes its position on the celestial sphere, being (in the northern hemisphere) at a positive declination in spring and summer, and at a negative declination in autumn and winter, and having exactly zero declination (i.e., being on the celestial equator ) at the equinoxes . The Sun's celestial longitude also varies, changing by one complete revolution per year. The path of the Sun on the celestial sphere is called the ecliptic . The ecliptic passes through the twelve constellations of the zodiac in the course of a year.
This model of the Sun's motion helps to understand sundials. If the shadow-casting gnomon is aligned with the celestial poles , its shadow will revolve at a constant rate, and this rotation will not change with the seasons. This is the most common design. In such cases, the same hour lines may be used throughout the year. The hour-lines will be spaced uniformly if the surface receiving the shadow is either perpendicular (as in the equatorial sundial) or circular about the gnomon (as in the armillary sphere ).
In other cases, the hour-lines are not spaced evenly, even though the shadow rotates uniformly. If the gnomon is not aligned with the celestial poles, even its shadow will not rotate uniformly, and the hour lines must be corrected accordingly. The rays of light that graze the tip of a gnomon, or which pass through a small hole, or reflect from a small mirror, trace out a cone aligned with the celestial poles. The corresponding light-spot or shadow-tip, if it falls onto a flat surface, will trace out a conic section , such as a hyperbola , ellipse or (at the North or South Poles) a circle .
This conic section is the intersection of the cone of light rays with the flat surface. This cone and its conic section change with the seasons, as the Sun's declination changes; hence, sundials that follow the motion of such light-spots or shadow-tips often have different hour-lines for different times of the year. This is seen in shepherd's dials, sundial rings, and vertical gnomons such as obelisks. Alternatively, sundials may change the angle or position (or both) of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial.
The earliest sundials known from the archaeological record are shadow clocks (1500 BC or BCE ) from ancient Egyptian astronomy and Babylonian astronomy . Presumably, humans were telling time from shadow-lengths at an even earlier date, but this is hard to verify. In roughly 700 BC, the Old Testament describes a sundial—the “dial of Ahaz ” in Isaiah 38:8 and 2 Kings 20:11 . By 240 BC, Eratosthenes had estimated the circumference of the world using an obelisk and a water well and a few centuries later, Ptolemy had charted the latitude of cities using the angle of the sun. The people of Kush created sun dials through geometry. [ 4 ] [ 5 ] The Roman writer Vitruvius lists dials and shadow clocks known at that time in his De architectura . The Tower of the Winds in Athens included both a sundial and a water clock for telling time. A canonical sundial is one that indicates the canonical hours of liturgical acts, and these were used from the 7th to the 14th centuries by religious orders . The Italian astronomer Giovanni Padovani published a treatise on the sundial in 1570, in which he included instructions for the manufacture and laying out of mural (vertical) and horizontal sundials. Giuseppe Biancani 's Constructio instrumenti ad horologia solaria (c. 1620) discusses how to make a perfect sundial. They have been in common use since the 16th century.
In general, sundials indicate the time by casting a shadow or throwing light onto a surface known as a dial face or dial plate . Although usually a flat plane, the dial face may also be the inner or outer surface of a sphere, cylinder, cone, helix, and various other shapes.
The time is indicated where a shadow or light falls on the dial face, which is usually inscribed with hour lines. Although usually straight, these hour lines may also be curved, depending on the design of the sundial (see below). In some designs, it is possible to determine the date of the year, or it may be required to know the date to find the correct time. In such cases, there may be multiple sets of hour lines for different months, or there may be mechanisms for setting/calculating the month. In addition to the hour lines, the dial face may offer other data—such as the horizon, the equator and the tropics—which are referred to collectively as the dial furniture.
The entire object that casts a shadow or light onto the dial face is known as the sundial's gnomon . [ 6 ] However, it is usually only an edge of the gnomon (or another linear feature) that casts the shadow used to determine the time; this linear feature is known as the sundial's style . The style is usually aligned parallel to the axis of the celestial sphere, and therefore is aligned with the local geographical meridian. In some sundial designs, only a point-like feature, such as the tip of the style, is used to determine the time and date; this point-like feature is known as the sundial's nodus . [ 6 ] [ a ] Some sundials use both a style and a nodus to determine the time and date.
The gnomon is usually fixed relative to the dial face, but not always; in some designs such as the analemmatic sundial, the style is moved according to the month. If the style is fixed, the line on the dial plate perpendicularly beneath the style is called the substyle , [ 6 ] meaning "below the style". The angle the style makes with the plane of the dial plate is called the substyle height, an unusual use of the word height to mean an angle . On many wall dials, the substyle is not the same as the noon line (see below). The angle on the dial plate between the noon line and the substyle is called the substyle distance , an unusual use of the word distance to mean an angle .
By tradition, many sundials have a motto . The motto is usually in the form of an epigram : sometimes sombre reflections on the passing of time and the brevity of life, but equally often humorous witticisms of the dial maker. One such quip is, I am a sundial, and I make a botch, Of what is done much better by a watch. [ 7 ]
A dial is said to be equiangular if its hour-lines are straight and spaced equally. Most equiangular sundials have a fixed gnomon style aligned with the Earth's rotational axis, as well as a shadow-receiving surface that is symmetrical about that axis; examples include the equatorial dial, the equatorial bow, the armillary sphere, the cylindrical dial and the conical dial. However, other designs are equiangular, such as the Lambert dial, a version of the analemmatic sundial with a moveable style.
A sundial at a particular latitude in one hemisphere must be reversed for use at the opposite latitude in the other hemisphere. [ 8 ] A vertical direct south sundial in the Northern Hemisphere becomes a vertical direct north sundial in the Southern Hemisphere . To position a horizontal sundial correctly, one has to find true north or south . The same process can be used to do both. [ 9 ] The gnomon, set to the correct latitude, has to point to the true south in the Southern Hemisphere as in the Northern Hemisphere it has to point to the true north. [ 10 ] The hour numbers also run in opposite directions, so on a horizontal dial they run anticlockwise (US: counterclockwise) rather than clockwise. [ 11 ]
Sundials which are designed to be used with their plates horizontal in one hemisphere can be used with their plates vertical at the complementary latitude in the other hemisphere. For example, the illustrated sundial in Perth , Australia , which is at latitude 32° South, would function properly if it were mounted on a south-facing vertical wall at latitude 58° (i.e. 90° − 32°) North, which is slightly further north than Perth, Scotland . The surface of the wall in Scotland would be parallel with the horizontal ground in Australia (ignoring the difference of longitude), so the sundial would work identically on both surfaces. Correspondingly, the hour marks, which run counterclockwise on a horizontal sundial in the southern hemisphere, also do so on a vertical sundial in the northern hemisphere. (See the first two illustrations at the top of this article.) On horizontal northern-hemisphere sundials, and on vertical southern-hemisphere ones, the hour marks run clockwise.
The most common reason for a sundial to differ greatly from clock time is that the sundial has not been oriented correctly or its hour lines have not been drawn correctly. For example, most commercial sundials are designed as horizontal sundials as described above. To be accurate, such a sundial must have been designed for the local geographical latitude and its style must be parallel to the Earth's rotational axis; the style must be aligned with true north and its height (its angle with the horizontal) must equal the local latitude. To adjust the style height, the sundial can often be tilted slightly "up" or "down" while maintaining the style's north-south alignment. [ 12 ]
Some areas of the world practice daylight saving time , which changes the official time, usually by one hour. This shift must be added to the sundial's time to make it agree with the official time.
A standard time zone covers roughly 15° of longitude, so any point within that zone which is not on the reference longitude (generally a multiple of 15°) will experience a difference from standard time that is equal to 4 minutes of time per degree. For illustration, sunsets and sunrises are at a much later "official" time at the western edge of a time-zone, compared to sunrise and sunset times at the eastern edge. If a sundial is located at, say, a longitude 5° west of the reference longitude, then its time will read 20 minutes slow, since the Sun appears to revolve around the Earth at 15° per hour. This is a constant correction throughout the year. For equiangular dials such as equatorial, spherical or Lambert dials, this correction can be made by rotating the dial surface by an angle equaling the difference in longitude, without changing the gnomon position or orientation. However, this method does not work for other dials, such as a horizontal dial; the correction must be applied by the viewer.
However, for political and practical reasons, time-zone boundaries have been skewed. At their most extreme, time zones can cause official noon, including daylight savings, to occur up to three hours early (in which case the Sun is actually on the meridian at official clock time of 3 PM ). This occurs in the far west of Alaska , China , and Spain . For more details and examples, see time zones .
Although the Sun appears to rotate uniformly about the Earth, in reality this motion is not perfectly uniform. This is due to the eccentricity of the Earth's orbit (the fact that the Earth's orbit about the Sun is not perfectly circular, but slightly elliptical ) and the tilt (obliquity) of the Earth's rotational axis relative to the plane of its orbit. Therefore, sundial time varies from standard clock time . On four days of the year, the correction is effectively zero. However, on others, it can be as much as a quarter-hour early or late. The amount of correction is described by the equation of time . This correction is equal worldwide: it does not depend on the local latitude or longitude of the observer's position. It does, however, change over long periods of time, (centuries or more, [ 13 ] )
because of slow variations in the Earth's orbital and rotational motions. Therefore, tables and graphs of the equation of time that were made centuries ago are now significantly incorrect. The reading of an old sundial should be corrected by applying the present-day equation of time, not one from the period when the dial was made.
In some sundials, the equation of time correction is provided as an informational plaque affixed to the sundial, for the observer to calculate. In more sophisticated sundials the equation can be incorporated automatically. For example, some equatorial bow sundials are supplied with a small wheel that sets the time of year; this wheel in turn rotates the equatorial bow, offsetting its time measurement. In other cases, the hour lines may be curved, or the equatorial bow may be shaped like a vase, which exploits the changing altitude of the sun over the year to effect the proper offset in time. [ 14 ]
A heliochronometer is a precision sundial first devised in about 1763 by Philipp Hahn and improved by Abbé Guyoux in about 1827. [ 15 ] It corrects apparent solar time to mean solar time or another standard time . Heliochronometers usually indicate the minutes to within 1 minute of Universal Time .
The Sunquest sundial , designed by Richard L. Schmoyer in the 1950s, uses an analemmic-inspired gnomon to cast a shaft of light onto an equatorial time-scale crescent. Sunquest is adjustable for latitude and longitude, automatically correcting for the equation of time, rendering it "as accurate as most pocket watches". [ 16 ] [ 17 ] [ 18 ] [ 19 ]
Similarly, in place of the shadow of a gnomon the sundial at Miguel Hernández University uses the solar projection of a graph of the equation of time intersecting a time scale to display clock time directly.
An analemma may be added to many types of sundials to correct apparent solar time to mean solar time or another standard time . These usually have hour lines shaped like "figure eights" ( analemmas ) according to the equation of time . This compensates for the slight eccentricity in the Earth's orbit and the tilt of the Earth's axis that causes up to a 15 minute variation from mean solar time. This is a type of dial furniture seen on more complicated horizontal and vertical dials.
Prior to the invention of accurate clocks, in the mid 17th century, sundials were the only timepieces in common use, and were considered to tell the "right" time. The equation of time was not used. After the invention of good clocks, sundials were still considered to be correct, and clocks usually incorrect. The equation of time was used in the opposite direction from today, to apply a correction to the time shown by a clock to make it agree with sundial time. Some elaborate " equation clocks ", such as one made by Joseph Williamson in 1720, incorporated mechanisms to do this correction automatically. (Williamson's clock may have been the first-ever device to use a differential gear.) Only after about 1800 was uncorrected clock time considered to be "right", and sundial time usually "wrong", so the equation of time became used as it is today. [ 20 ]
The most commonly observed sundials are those in which the shadow-casting style is fixed in position and aligned with the Earth's rotational axis, being oriented with true north and south, and making an angle with the horizontal equal to the geographical latitude. This axis is aligned with the celestial poles , which is closely, but not perfectly, aligned with the pole star Polaris . For illustration, the celestial axis points vertically at the true North Pole , whereas it points horizontally on the equator . The world's largest axial gnomon sundial is the mast of the Sundial Bridge at Turtle Bay in Redding, California . A formerly world's largest gnomon is at Jaipur , raised 26°55′ above horizontal, reflecting the local latitude. [ 21 ]
On any given day, the Sun appears to rotate uniformly about this axis, at about 15° per hour, making a full circuit (360°) in 24 hours. A linear gnomon aligned with this axis will cast a sheet of shadow (a half-plane) that, falling opposite to the Sun, likewise rotates about the celestial axis at 15° per hour. The shadow is seen by falling on a receiving surface that is usually flat, but which may be spherical, cylindrical, conical or of other shapes. If the shadow falls on a surface that is symmetrical about the celestial axis (as in an armillary sphere, or an equatorial dial), the surface-shadow likewise moves uniformly; the hour-lines on the sundial are equally spaced. However, if the receiving surface is not symmetrical (as in most horizontal sundials), the surface shadow generally moves non-uniformly and the hour-lines are not equally spaced; one exception is the Lambert dial described below.
Some types of sundials are designed with a fixed gnomon that is not aligned with the celestial poles like a vertical obelisk. Such sundials are covered below under the section, "Nodus-based sundials".
The formulas shown in the paragraphs below allow the positions of the hour-lines to be calculated for various types of sundial. In some cases, the calculations are simple; in others they are extremely complicated. There is an alternative, simple method of finding the positions of the hour-lines which can be used for many types of sundial, and saves a lot of work in cases where the calculations are complex. [ 22 ] This is an empirical procedure in which the position of the shadow of the gnomon of a real sundial is marked at hourly intervals. The equation of time must be taken into account to ensure that the positions of the hour-lines are independent of the time of year when they are marked. An easy way to do this is to set a clock or watch so it shows "sundial time" [ b ] which is standard time , [ c ] plus the equation of time on the day in question. [ d ] The hour-lines on the sundial are marked to show the positions of the shadow of the style when this clock shows whole numbers of hours, and are labelled with these numbers of hours. For example, when the clock reads 5:00, the shadow of the style is marked, and labelled "5" (or "V" in Roman numerals ). If the hour-lines are not all marked in a single day, the clock must be adjusted every day or two to take account of the variation of the equation of time.
The distinguishing characteristic of the equatorial dial (also called the equinoctial dial ) is the planar surface that receives the shadow, which is exactly perpendicular to the gnomon's style. [ 25 ] This plane is called equatorial, because it is parallel to the equator of the Earth and of the celestial sphere. If the gnomon is fixed and aligned with the Earth's rotational axis, the sun's apparent rotation about the Earth casts a uniformly rotating sheet of shadow from the gnomon; this produces a uniformly rotating line of shadow on the equatorial plane. Since the Earth rotates 360° in 24 hours, the hour-lines on an equatorial dial are all spaced 15° apart (360/24).
The uniformity of their spacing makes this type of sundial easy to construct. If the dial plate material is opaque, both sides of the equatorial dial must be marked, since the shadow will be cast from below in winter and from above in summer. With translucent dial plates (e.g. glass) the hour angles need only be marked on the sun-facing side, although the hour numberings (if used) need be made on both sides of the dial, owing to the differing hour schema on the sun-facing and sun-backing sides.
Another major advantage of this dial is that equation of time (EoT) and daylight saving time (DST) corrections can be made by simply rotating the dial plate by the appropriate angle each day. This is because the hour angles are equally spaced around the dial. For this reason, an equatorial dial is often a useful choice when the dial is for public display and it is desirable to have it show the true local time to reasonable accuracy. The EoT correction is made via the relation
Near the equinoxes in spring and autumn, the sun moves on a circle that is nearly the same as the equatorial plane; hence, no clear shadow is produced on the equatorial dial at those times of year, a drawback of the design.
A nodus is sometimes added to equatorial sundials, which allows the sundial to tell the time of year. On any given day, the shadow of the nodus moves on a circle on the equatorial plane, and the radius of the circle measures the declination of the sun. The ends of the gnomon bar may be used as the nodus, or some feature along its length. An ancient variant of the equatorial sundial has only a nodus (no style) and the concentric circular hour-lines are arranged to resemble a spider-web. [ 26 ]
In the horizontal sundial (also called a garden sundial ), the plane that receives the shadow is aligned horizontally, rather than being perpendicular to the style as in the equatorial dial. [ 27 ] Hence, the line of shadow does not rotate uniformly on the dial face; rather, the hour lines are spaced according to the rule. [ 28 ]
Or in other terms:
where L is the sundial's geographical latitude (and the angle the gnomon makes with the dial plate), H H {\displaystyle \ H_{H}\ } is the angle between a given hour-line and the noon hour-line (which always points towards true north ) on the plane, and t is the number of hours before or after noon. For example, the angle H H {\displaystyle \ H_{H}\ } of the 3 PM hour-line would equal the arctangent of sin L , since tan 45° = 1. When L = 90 ∘ {\displaystyle \ L=90^{\circ }\ } (at the North Pole ), the horizontal sundial becomes an equatorial sundial; the style points straight up (vertically), and the horizontal plane is aligned with the equatorial plane; the hour-line formula becomes H H = 15 ∘ × t , {\displaystyle \ H_{H}=15^{\circ }\times t\ ,} as for an equatorial dial. A horizontal sundial at the Earth's equator , where L = 0 ∘ , {\displaystyle \ L=0^{\circ }\ ,} would require a (raised) horizontal style and would be an example of a polar sundial (see below).
The chief advantages of the horizontal sundial are that it is easy to read, and the sunlight lights the face throughout the year. All the hour-lines intersect at the point where the gnomon's style crosses the horizontal plane. Since the style is aligned with the Earth's rotational axis, the style points true north and its angle with the horizontal equals the sundial's geographical latitude L . A sundial designed for one latitude can be adjusted for use at another latitude by tilting its base upwards or downwards by an angle equal to the difference in latitude. For example, a sundial designed for a latitude of 40° can be used at a latitude of 45°, if the sundial plane is tilted upwards by 5°, thus aligning the style with the Earth's rotational axis. [ citation needed ]
Many ornamental sundials are designed to be used at 45 degrees north. Some mass-produced garden sundials fail to correctly calculate the hourlines and so can never be corrected. A local standard time zone is nominally 15 degrees wide, but may be modified to follow geographic or political boundaries. A sundial can be rotated around its style (which must remain pointed at the celestial pole) to adjust to the local time zone. In most cases, a rotation in the range of 7.5° east to 23° west suffices. This will introduce error in sundials that do not have equal hour angles. To correct for daylight saving time , a face needs two sets of numerals or a correction table. An informal standard is to have numerals in hot colors for summer, and in cool colors for winter. [ citation needed ] Since the hour angles are not evenly spaced, the equation of time corrections cannot be made via rotating the dial plate about the gnomon axis. These types of dials usually have an equation of time correction tabulation engraved on their pedestals or close by. Horizontal dials are commonly seen in gardens, churchyards and in public areas.
In the common vertical dial , the shadow-receiving plane is aligned vertically; as usual, the gnomon's style is aligned with the Earth's axis of rotation. [ 29 ] As in the horizontal dial, the line of shadow does not move uniformly on the face; the sundial is not equiangular . If the face of the vertical dial points directly south, the angle of the hour-lines is instead described by the formula: [ 30 ]
where L is the sundial's geographical latitude , H V {\displaystyle \ H_{V}\ } is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t is the number of hours before or after noon. For example, the angle H V {\displaystyle \ H_{V}\ } of the 3 P.M. hour-line would equal the arctangent of cos L , since tan 45° = 1 . The shadow moves counter-clockwise on a south-facing vertical dial, whereas it runs clockwise on horizontal and equatorial north-facing dials.
Dials with faces perpendicular to the ground and which face directly south, north, east, or west are called vertical direct dials . [ 31 ] It is widely believed, and stated in respectable publications, that a vertical dial cannot receive more than twelve hours of sunlight a day, no matter how many hours of daylight there are. [ 32 ] However, there is an exception. Vertical sundials in the tropics which face the nearer pole (e.g. north facing in the zone between the Equator and the Tropic of Cancer) can actually receive sunlight for more than 12 hours from sunrise to sunset for a short period around the time of the summer solstice. For example, at latitude 20° North, on June 21, the sun shines on a north-facing vertical wall for 13 hours, 21 minutes. [ 33 ] Vertical sundials which do not face directly south (in the northern hemisphere) may receive significantly less than twelve hours of sunlight per day, depending on the direction they do face, and on the time of year. For example, a vertical dial that faces due East can tell time only in the morning hours; in the afternoon, the sun does not shine on its face. Vertical dials that face due East or West are polar dials , which will be described below. Vertical dials that face north are uncommon, because they tell time only during the spring and summer, and do not show the midday hours except in tropical latitudes (and even there, only around midsummer). For non-direct vertical dials – those that face in non-cardinal directions – the mathematics of arranging the style and the hour-lines becomes more complicated; it may be easier to mark the hour lines by observation, but the placement of the style, at least, must be calculated first; such dials are said to be declining dials . [ 34 ]
Vertical dials are commonly mounted on the walls of buildings, such as town-halls, cupolas and church-towers, where they are easy to see from far away. In some cases, vertical dials are placed on all four sides of a rectangular tower, providing the time throughout the day. The face may be painted on the wall, or displayed in inlaid stone; the gnomon is often a single metal bar, or a tripod of metal bars for rigidity. If the wall of the building faces toward the south, but does not face due south, the gnomon will not lie along the noon line, and the hour lines must be corrected. Since the gnomon's style must be parallel to the Earth's axis, it always "points" true north and its angle with the horizontal will equal the sundial's geographical latitude; on a direct south dial, its angle with the vertical face of the dial will equal the colatitude , or 90° minus the latitude. [ 35 ]
In polar dials , the shadow-receiving plane is aligned parallel to the gnomon-style. [ 36 ] Thus, the shadow slides sideways over the surface, moving perpendicularly to itself as the Sun rotates about the style. As with the gnomon, the hour-lines are all aligned with the Earth's rotational axis. When the Sun's rays are nearly parallel to the plane, the shadow moves very quickly and the hour lines are spaced far apart. The direct East- and West-facing dials are examples of a polar dial. However, the face of a polar dial need not be vertical; it need only be parallel to the gnomon. Thus, a plane inclined at the angle of latitude (relative to horizontal) under the similarly inclined gnomon will be a polar dial. The perpendicular spacing X of the hour-lines in the plane is described by the formula
where H is the height of the style above the plane, and t is the time (in hours) before or after the center-time for the polar dial. The center time is the time when the style's shadow falls directly down on the plane; for an East-facing dial, the center time will be 6 A.M. , for a West-facing dial, this will be 6 P.M. , and for the inclined dial described above, it will be noon. When t approaches ±6 hours away from the center time, the spacing X diverges to +∞ ; this occurs when the Sun's rays become parallel to the plane.
A declining dial is any non-horizontal, planar dial that does not face in a cardinal direction, such as (true) north , south , east or west . [ 37 ] As usual, the gnomon's style is aligned with the Earth's rotational axis, but the hour-lines are not symmetrical about the noon hour-line. For a vertical dial, the angle H VD {\displaystyle \ H_{\text{VD}}\ } between the noon hour-line and another hour-line is given by the formula below. Note that H VD {\displaystyle \ H_{\text{VD}}\ } is defined positive in the clockwise sense w.r.t. the upper vertical hour angle; and that its conversion to the equivalent solar hour requires careful consideration of which quadrant of the sundial that it belongs in. [ 38 ]
where L {\displaystyle \ L\ } is the sundial's geographical latitude ; t is the time before or after noon; D {\displaystyle \ D\ } is the angle of declination from true south , defined as positive when east of south; and s o {\displaystyle \ s_{o}\ } is a switch integer for the dial orientation. A partly south-facing dial has an s o {\displaystyle \ s_{o}\ } value of +1 ; those partly north-facing, a value of −1 . When such a dial faces south ( D = 0 ∘ {\displaystyle \ D=0^{\circ }\ } ), this formula reduces to the formula given above for vertical south-facing dials, i.e.
When a sundial is not aligned with a cardinal direction, the substyle of its gnomon is not aligned with the noon hour-line. The angle B {\displaystyle \ B\ } between the substyle and the noon hour-line is given by the formula [ 38 ]
If a vertical sundial faces trUe south Or north ( D = 0 ∘ {\displaystyle \ D=0^{\circ }\ } or D = 180 ∘ , {\displaystyle \ D=180^{\circ }\ ,} respectively), the angle B = 0 ∘ {\displaystyle \ B=0^{\circ }\ } and the substyle is aligned with the noon hour-line.
The height of the gnomon, that is the angle the style makes to the plate, G , {\displaystyle \ G\ ,} is given by :
The sundials described above have gnomons that are aligned with the Earth's rotational axis and cast their shadow onto a plane. If the plane is neither vertical nor horizontal nor equatorial, the sundial is said to be reclining or inclining . [ 40 ] Such a sundial might be located on a south-facing roof, for example. The hour-lines for such a sundial can be calculated by slightly correcting the horizontal formula above [ 41 ] [ 42 ]
where R {\displaystyle \ R\ } is the desired angle of reclining relative to the local vertical, L is the sundial's geographical latitude, H R V {\displaystyle \ H_{RV}\ } is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t is the number of hours before or after noon. For example, the angle H R V {\displaystyle \ H_{RV}\ } of the 3pm hour-line would equal the arctangent of cos ( L + R ) , since tan 45° = 1 . When R = 0° (in other words, a south-facing vertical dial), we obtain the vertical dial formula above.
Some authors use a more specific nomenclature to describe the orientation of the shadow-receiving plane. If the plane's face points downwards towards the ground, it is said to be proclining or inclining , whereas a dial is said to be reclining when the dial face is pointing away from the ground. Many authors also often refer to reclined, proclined and inclined sundials in general as inclined sundials. It is also common in the latter case to measure the angle of inclination relative to the horizontal plane on the sun side of the dial.
In such texts, since I = 90 ∘ + R , {\displaystyle \ I=90^{\circ }+R\ ,} the hour angle formula will often be seen written as :
The angle between the gnomon style and the dial plate, B, in this type of sundial is :
or :
Some sundials both decline and recline, in that their shadow-receiving plane is not oriented with a cardinal direction (such as true north or true south) and is neither horizontal nor vertical nor equatorial. For example, such a sundial might be found on a roof that was not oriented in a cardinal direction.
The formulae describing the spacing of the hour-lines on such dials are rather more complicated than those for simpler dials.
There are various solution approaches, including some using the methods of rotation matrices, and some making a 3D model of the reclined-declined plane and its vertical declined counterpart plane, extracting the geometrical relationships between the hour angle components on both these planes and then reducing the trigonometric algebra. [ 43 ]
One system of formulas for Reclining-Declining sundials: (as stated by Fennewick) [ 44 ]
The angle H RD {\displaystyle \ H_{\text{RD}}\ } between the noon hour-line and another hour-line is given by the formula below. Note that H RD {\displaystyle \ H_{\text{RD}}\ } advances counterclockwise with respect to the zero hour angle for those dials that are partly south-facing and clockwise for those that are north-facing.
within the parameter ranges : D < D c {\displaystyle \ D<D_{c}\ } and − 90 ∘ < R < ( 90 ∘ − L ) . {\displaystyle -90^{\circ }<R<(90^{\circ }-L)~.}
Or, if preferring to use inclination angle, I , {\displaystyle \ I\ ,} rather than the reclination, R , {\displaystyle \ R\ ,} where I = ( 90 ∘ + R ) {\displaystyle \ I=(90^{\circ }+R)\ } :
within the parameter ranges : D < D c {\displaystyle \ D<D_{c}~~} and 0 ∘ < I < ( 180 ∘ − L ) . {\displaystyle ~~0^{\circ }<I<(180^{\circ }-L)~.}
Here L {\displaystyle \ L\ } is the sundial's geographical latitude; s o {\displaystyle \ s_{o}\ } is the orientation switch integer; t is the time in hours before or after noon; and R {\displaystyle \ R\ } and D {\displaystyle \ D\ } are the angles of reclination and declination, respectively.
Note that R {\displaystyle \ R\ } is measured with reference to the vertical. It is positive when the dial leans back towards the horizon behind the dial and negative when the dial leans forward to the horizon on the Sun's side. Declination angle D {\displaystyle \ D\ } is defined as positive when moving east of true south.
Dials facing fully or partly south have s o = + 1 , {\displaystyle \ s_{o}=+1\ ,} while those partly or fully north-facing have an s o = − 1 . {\displaystyle \ s_{o}=-1~.} Since the above expression gives the hour angle as an arctangent function, due consideration must be given to which quadrant of the sundial each hour belongs to before assigning the correct hour angle.
Unlike the simpler vertical declining sundial, this type of dial does not always show hour angles on its sunside face for all declinations between east and west. When a northern hemisphere partly south-facing dial reclines back (i.e. away from the Sun) from the vertical, the gnomon will become co-planar with the dial plate at declinations less than due east or due west. Likewise for southern hemisphere dials that are partly north-facing.
Were these dials reclining forward, the range of declination would actually exceed due east and due west.
In a similar way, northern hemisphere dials that are partly north-facing and southern hemisphere dials that are south-facing, and which lean forward toward their upward pointing gnomons, will have a similar restriction on the range of declination that is possible for a given reclination value.
The critical declination D c {\displaystyle \ D_{c}\ } is a geometrical constraint which depends on the value of both the dial's reclination and its latitude :
As with the vertical declined dial, the gnomon's substyle is not aligned with the noon hour-line. The general formula for the angle B , {\displaystyle \ B\ ,} between the substyle and the noon-line is given by :
The angle G , {\displaystyle \ G\ ,} between the style and the plate is given by :
Note that for G = 0 ∘ , {\displaystyle \ G=0^{\circ }\ ,} i.e. when the gnomon is coplanar with the dial plate, we have :
i.e. when D = D c , {\displaystyle \ D=D_{c}\ ,} the critical declination value. [ 44 ]
Because of the complexity of the above calculations, using them for the practical purpose of designing a dial of this type is difficult and prone to error. It has been suggested that it is better to locate the hour lines empirically, marking the positions of the shadow of a style on a real sundial at hourly intervals as shown by a clock and adding/deducting that day's equation of time adjustment. [ 22 ] See Empirical hour-line marking , above.
The surface receiving the shadow need not be a plane, but can have any shape, provided that the sundial maker is willing to mark the hour-lines. If the style is aligned with the Earth's rotational axis, a spherical shape is convenient since the hour-lines are equally spaced, as they are on the equatorial dial shown here; the sundial is equiangular . This is the principle behind the armillary sphere and the equatorial bow sundial. [ 45 ] However, some equiangular sundials – such as the Lambert dial described below – are based on other principles.
In the equatorial bow sundial , the gnomon is a bar, slot or stretched wire parallel to the celestial axis. The face is a semicircle, corresponding to the equator of the sphere, with markings on the inner surface. This pattern, built a couple of meters wide out of temperature-invariant steel invar , was used to keep the trains running on time in France before World War I. [ 46 ]
Among the most precise sundials ever made are two equatorial bows constructed of marble found in Yantra mandir . [ 47 ] This collection of sundials and other astronomical instruments was built by Maharaja Jai Singh II at his then-new capital of Jaipur , India between 1727 and 1733. The larger equatorial bow is called the Samrat Yantra (The Supreme Instrument); standing at 27 meters, its shadow moves visibly at 1 mm per second, or roughly a hand's breadth (6 cm) every minute.
Other non-planar surfaces may be used to receive the shadow of the gnomon.
As an elegant alternative, the style (which could be created by a hole or slit in the circumference) may be located on the circumference of a cylinder or sphere, rather than at its central axis of symmetry.
In that case, the hour lines are again spaced equally, but at twice the usual angle, due to the geometrical inscribed angle theorem. This is the basis of some modern sundials, but it was also used in ancient times; [ e ]
In another variation of the polar-axis-aligned cylindrical, a cylindrical dial could be rendered as a helical ribbon-like surface, with a thin gnomon located either along its center or at its periphery.
Sundials can be designed with a gnomon that is placed in a different position each day throughout the year. In other words, the position of the gnomon relative to the centre of the hour lines varies. The gnomon need not be aligned with the celestial poles and may even be perfectly vertical (the analemmatic dial). These dials, when combined with fixed-gnomon sundials, allow the user to determine true north with no other aid; the two sundials are correctly aligned if and only if they both show the same time. [ citation needed ]
A universal equinoctial ring dial (sometimes called a ring dial for brevity, although the term is ambiguous), is a portable version of an armillary sundial, [ 49 ] or was inspired by the mariner's astrolabe . [ 50 ] It was likely invented by William Oughtred around 1600 and became common throughout Europe. [ 51 ]
In its simplest form, the style is a thin slit that allows the Sun's rays to fall on the hour-lines of an equatorial ring. As usual, the style is aligned with the Earth's axis; to do this, the user may orient the dial towards true north and suspend the ring dial vertically from the appropriate point on the meridian ring. Such dials may be made self-aligning with the addition of a more complicated central bar, instead of a simple slit-style. These bars are sometimes an addition to a set of Gemma's rings . This bar could pivot about its end points and held a perforated slider that was positioned to the month and day according to a scale scribed on the bar. The time was determined by rotating the bar towards the Sun so that the light shining through the hole fell on the equatorial ring. This forced the user to rotate the instrument, which had the effect of aligning the instrument's vertical ring with the meridian.
When not in use, the equatorial and meridian rings can be folded together into a small disk.
In 1610, Edward Wright created the sea ring , which mounted a universal ring dial over a magnetic compass. This permitted mariners to determine the time and magnetic variation in a single step. [ 52 ]
Analemmatic sundials are a type of horizontal sundial that has a vertical gnomon and hour markers positioned in an elliptical pattern. There are no hour lines on the dial and the time of day is read on the ellipse. The gnomon is not fixed and must change position daily to accurately indicate time of day.
Analemmatic sundials are sometimes designed with a human as the gnomon. Human gnomon analemmatic sundials are not practical at lower latitudes where a human shadow is quite short during the summer months. A 66 inch tall person casts a 4 inch shadow at 27° latitude on the summer solstice. [ 53 ]
The Foster-Lambert dial is another movable-gnomon sundial. [ 54 ] In contrast to the elliptical analemmatic dial, the Lambert dial is circular with evenly spaced hour lines, making it an equiangular sundial , similar to the equatorial, spherical, cylindrical and conical dials described above. The gnomon of a Foster-Lambert dial is neither vertical nor aligned with the Earth's rotational axis; rather, it is tilted northwards by an angle α = 45° - (Φ/2), where Φ is the geographical latitude . Thus, a Foster-Lambert dial located at latitude 40° would have a gnomon tilted away from vertical by 25° in a northerly direction. To read the correct time, the gnomon must also be moved northwards by a distance
where R is the radius of the Foster-Lambert dial and δ again indicates the Sun's declination for that time of year.
Altitude dials measure the height of the Sun in the sky, rather than directly measuring its hour-angle about the Earth's axis. They are not oriented towards true north , but rather towards the Sun and generally held vertically. The Sun's elevation is indicated by the position of a nodus, either the shadow-tip of a gnomon, or a spot of light.
In altitude dials, the time is read from where the nodus falls on a set of hour-curves that vary with the time of year. Many such altitude-dials' construction is calculation-intensive, as also the case with many azimuth dials. But the capuchin dials (described below) are constructed and used graphically.
Altitude dials' disadvantages:
Since the Sun's altitude is the same at times equally spaced about noon (e.g., 9am and 3pm), the user had to know whether it was morning or afternoon. At, say, 3:00 pm, that is not a problem. But when the dial indicates a time 15 minutes from noon, the user likely will not have a way of distinguishing 11:45 from 12:15.
Additionally, altitude dials are less accurate near noon, because the sun's altitude is not changing rapidly then.
Many of these dials are portable and simple to use. As is often the case with other sundials, many altitude dials are designed for only one latitude. But the capuchin dial (described below) has a version that's adjustable for latitude. [ 55 ]
Mayall & Mayall (1994) , p. 169 describe the Universal Capuchin sundial.
The length of a human shadow (or of any vertical object) can be used to measure the sun's elevation and, thence, the time. [ 56 ] The Venerable Bede gave a table for estimating the time from the length of one's shadow in feet, on the assumption that a monk's height is six times the length of his foot. Such shadow lengths will vary with the geographical latitude and with the time of year. For example, the shadow length at noon is short in summer months, and long in winter months.
Chaucer evokes this method a few times in his Canterbury Tales , as in his Parson's Tale . [ f ]
An equivalent type of sundial using a vertical rod of fixed length is known as a backstaff dial .
A shepherd's dial – also known as a shepherd's column dial , [ 57 ] [ 58 ] pillar dial , cylinder dial or chilindre – is a portable cylindrical sundial with a knife-like gnomon that juts out perpendicularly. [ 59 ] It is normally dangled from a rope or string so the cylinder is vertical. The gnomon can be twisted to be above a month or day indication on the face of the cylinder. This corrects the sundial for the equation of time. The entire sundial is then twisted on its string so that the gnomon aims toward the Sun, while the cylinder remains vertical. The tip of the shadow indicates the time on the cylinder. The hour curves inscribed on the cylinder permit one to read the time. Shepherd's dials are sometimes hollow, so that the gnomon can fold within when not in use.
The shepherd's dial is evoked in Henry VI, Part 3 , [ g ] among other works of literature. [ h ]
The cylindrical shepherd's dial can be unrolled into a flat plate. In one simple version, [ 62 ] the front and back of the plate each have three columns, corresponding to pairs of months with roughly the same solar declination (June:July, May:August, April:September, March:October, February:November, and January:December). The top of each column has a hole for inserting the shadow-casting gnomon, a peg. Often only two times are marked on the column below, one for noon and the other for mid-morning / mid-afternoon.
Timesticks, clock spear , [ 57 ] or shepherds' time stick , [ 57 ] are based on the same principles as dials. [ 57 ] [ 58 ] The time stick is carved with eight vertical time scales for a different period of the year, each bearing a time scale calculated according to the relative amount of daylight during the different months of the year. Any reading depends not only on the time of day but also on the latitude and time of year. [ 58 ] A peg gnomon is inserted at the top in the appropriate hole or face for the season of the year, and turned to the Sun so that the shadow falls directly down the scale. Its end displays the time. [ 57 ]
In a ring dial (also known as an Aquitaine or a perforated ring dial ), the ring is hung vertically and oriented sideways towards the sun. [ 63 ] A beam of light passes through a small hole in the ring and falls on hour-curves that are inscribed on the inside of the ring. To adjust for the equation of time, the hole is usually on a loose ring within the ring so that the hole can be adjusted to reflect the current month.
Card dials are another form of altitude dial. [ 64 ] A card is aligned edge-on with the sun and tilted so that a ray of light passes through an aperture onto a specified spot, thus determining the sun's altitude. A weighted string hangs vertically downwards from a hole in the card, and carries a bead or knot. The position of the bead on the hour-lines of the card gives the time. In more sophisticated versions such as the Capuchin dial, there is only one set of hour-lines, i.e., the hour lines do not vary with the seasons. Instead, the position of the hole from which the weighted string hangs is varied according to the season.
The Capuchin sundials are constructed and used graphically, as opposed the direct hour-angle measurements of horizontal or equatorial dials; or the calculated hour angle lines of some altitude and azimuth dials.
In addition to the ordinary Capuchin dial, there is a universal Capuchin dial, adjustable for latitude.
A navicula de Venetiis or "little ship of Venice" was an altitude dial used to tell time and which was shaped like a little ship. The cursor (with a plumb line attached) was slid up / down the mast to the correct latitude. The user then sighted the Sun through the pair of sighting holes at either end of the "ship's deck". The plumb line then marked what hour of the day it was. [ citation needed ]
Another type of sundial follows the motion of a single point of light or shadow, which may be called the nodus . For example, the sundial may follow the sharp tip of a gnomon's shadow, e.g., the shadow-tip of a vertical obelisk (e.g., the Solarium Augusti ) or the tip of the horizontal marker in a shepherd's dial. Alternatively, sunlight may be allowed to pass through a small hole or reflected from a small (e.g., coin-sized) circular mirror, forming a small spot of light whose position may be followed. In such cases, the rays of light trace out a cone over the course of a day; when the rays fall on a surface, the path followed is the intersection of the cone with that surface. Most commonly, the receiving surface is a geometrical plane , so that the path of the shadow-tip or light-spot (called declination line ) traces out a conic section such as a hyperbola or an ellipse . The collection of hyperbolae was called a pelekonon (axe) by the Greeks, because it resembles a double-bladed ax, narrow in the center (near the noonline) and flaring out at the ends (early morning and late evening hours).
There is a simple verification of hyperbolic declination lines on a sundial: the distance from the origin to the equinox line should be equal to harmonic mean of distances from the origin to summer and winter solstice lines. [ 65 ]
Nodus-based sundials may use a small hole or mirror to isolate a single ray of light; the former are sometimes called aperture dials . The oldest example is perhaps the antiborean sundial ( antiboreum ), a spherical nodus-based sundial that faces true north ; a ray of sunlight enters from the south through a small hole located at the sphere's pole and falls on the hour and date lines inscribed within the sphere, which resemble lines of longitude and latitude, respectively, on a globe. [ 66 ]
Isaac Newton developed a convenient and inexpensive sundial, in which a small mirror is placed on the sill of a south-facing window. [ 67 ] The mirror acts like a nodus, casting a single spot of light on the ceiling. Depending on the geographical latitude and time of year, the light-spot follows a conic section, such as the hyperbolae of the pelikonon. If the mirror is parallel to the Earth's equator, and the ceiling is horizontal, then the resulting angles are those of a conventional horizontal sundial. Using the ceiling as a sundial surface exploits unused space, and the dial may be large enough to be very accurate.
Sundials are sometimes combined into multiple dials. If two or more dials that operate on different principles — such as an analemmatic dial and a horizontal or vertical dial — are combined, the resulting multiple dial becomes self-aligning, most of the time. Both dials need to output both time and declination. In other words, the direction of true north need not be determined; the dials are oriented correctly when they read the same time and declination. However, the most common forms combine dials are based on the same principle and the analemmatic does not normally output the declination of the sun, thus are not self-aligning. [ 68 ]
The diptych consisted of two small flat faces, joined by a hinge. [ 69 ] Diptychs usually folded into little flat boxes suitable for a pocket. The gnomon was a string between the two faces. When the string was tight, the two faces formed both a vertical and horizontal sundial. These were made of white ivory, inlaid with black lacquer markings. The gnomons were black braided silk, linen or hemp string. With a knot or bead on the string as a nodus, and the correct markings, a diptych (really any sundial large enough) can keep a calendar well-enough to plant crops. A common error describes the diptych dial as self-aligning. This is not correct for diptych dials consisting of a horizontal and vertical dial using a string gnomon between faces, no matter the orientation of the dial faces. Since the string gnomon is continuous, the shadows must meet at the hinge; hence, any orientation of the dial will show the same time on both dials. [ 70 ]
A common type of multiple dial has sundials on every face of a Platonic solid (regular polyhedron), usually a cube . [ 71 ]
Extremely ornate sundials can be composed in this way, by applying a sundial to every surface of a solid object.
In some cases, the sundials are formed as hollows in a solid object, e.g., a cylindrical hollow aligned with the Earth's rotational axis (in which the edges play the role of styles) or a spherical hollow in the ancient tradition of the hemisphaerium or the antiboreum . (See the History section above.) In some cases, these multiface dials are small enough to sit on a desk, whereas in others, they are large stone monuments.
A Polyhedral's dial faces can be designed to give the time for different time-zones simultaneously. Examples include the Scottish sundial of the 17th and 18th centuries, which was often an extremely complex shape of polyhedral, and even convex faces.
Prismatic dials are a special case of polar dials, in which the sharp edges of a prism of a concave polygon serve as the styles and the sides of the prism receive the shadow. [ 72 ] Examples include a three-dimensional cross or star of David on gravestones.
The Benoy dial was invented by Walter Gordon Benoy of Collingham, Nottinghamshire , England. Whereas a gnomon casts a sheet of shadow, his invention creates an equivalent sheet of light by allowing the Sun's rays through a thin slit, reflecting them from a long, slim mirror (usually half-cylindrical), or focusing them through a cylindrical lens . Examples of Benoy dials can be found in the United Kingdom at: [ 73 ]
Invented by the German mathematician Hugo Michnik in 1922, the bifilar sundial has two non-intersecting threads parallel to the dial. Usually the second thread is orthogonal to the first. [ 75 ] The intersection of the two threads' shadows gives the local solar time.
A digital sundial indicates the current time with numerals formed by the sunlight striking it. Sundials of this type are installed in the Deutsches Museum in Munich and in the Sundial Park in Genk (Belgium), and a small version is available commercially. There is a patent for this type of sundial. [ 76 ]
The globe dial is a sphere aligned with the Earth's rotational axis, and equipped with a spherical vane. [ 77 ] Similar to sundials with a fixed axial style, a globe dial determines the time from the Sun's azimuthal angle in its apparent rotation about the earth. This angle can be determined by rotating the vane to give the smallest shadow.
The simplest sundials do not give the hours, but rather note the exact moment of 12:00 noon. [ 78 ] In centuries past, such dials were used to set mechanical clocks, which were sometimes so inaccurate as to lose or gain significant time in a single day. The simplest noon-marks have a shadow that passes a mark. Then, an almanac can translate from local solar time and date to civil time. The civil time is used to set the clock. Some noon-marks include a figure-eight that embodies the equation of time , so that no almanac is needed.
In some U.S. colonial-era houses, a noon-mark might be carved into a floor or windowsill. [ 79 ] Such marks indicate local noon, and provide a simple and accurate time reference for households to set their clocks. Some Asian countries had post offices set their clocks from a precision noon-mark. These in turn provided the times for the rest of the society. The typical noon-mark sundial was a lens set above an analemmatic plate. The plate has an engraved figure-eight shape, which corresponds to the equation of time (described above) versus the solar declination. When the edge of the Sun's image touches the part of the shape for the current month, this indicates that it is 12:00 noon.
A sundial cannon , sometimes called a 'meridian cannon', is a specialized sundial that is designed to create an 'audible noonmark', by automatically igniting a quantity of gunpowder at noon. These were novelties rather than precision sundials, sometimes installed in parks in Europe mainly in the late 18th or early 19th centuries. They typically consist of a horizontal sundial, which has in addition to a gnomon a suitably mounted lens , set to focus the rays of the sun at exactly noon on the firing pan of a miniature cannon loaded with gunpowder (but no ball ). To function properly the position and angle of the lens must be adjusted seasonally. [ citation needed ]
A horizontal line aligned on a meridian with a gnomon facing the noon-sun is termed a meridian line and does not indicate the time, but instead the day of the year. Historically they were used to accurately determine the length of the solar year . Examples are the Bianchini meridian line in Santa Maria degli Angeli e dei Martiri in Rome , and the Cassini line in San Petronio Basilica at Bologna . [ 80 ]
The association of sundials with time has inspired their designers over the centuries to display mottoes as part of the design. Often these cast the device in the role of memento mori , inviting the observer to reflect on the transience of the world and the inevitability of death. "Do not kill time, for it will surely kill thee." Other mottoes are more whimsical: "I count only the sunny hours," and "I am a sundial and I make a botch / of what is done far better by a watch." Collections of sundial mottoes have often been published through the centuries. [ citation needed ]
If a horizontal-plate sundial is made for the latitude in which it is being used, and if it is mounted with its plate horizontal and its gnomon pointing to the celestial pole that is above the horizon, then it shows the correct time in apparent solar time . Conversely, if the directions of the cardinal points are initially unknown, but the sundial is aligned so it shows the correct apparent solar time as calculated from the reading of a clock , its gnomon shows the direction of True north or south, allowing the sundial to be used as a compass. The sundial can be placed on a horizontal surface, and rotated about a vertical axis until it shows the correct time. The gnomon will then be pointing to the north, in the northern hemisphere , or to the south in the southern hemisphere. This method is much more accurate than using a watch as a compass and can be used in places where the magnetic declination is large, making a magnetic compass unreliable. An alternative method uses two sundials of different designs. (See #Multiple dials , above.) The dials are attached to and aligned with each other, and are oriented so they show the same time. This allows the directions of the cardinal points and the apparent solar time to be determined simultaneously, without requiring a clock. [ citation needed ] | https://en.wikipedia.org/wiki/Sea_ring |
Sea salt is salt that is produced by the evaporation of seawater . It is used as a seasoning in foods, cooking , cosmetics and for preserving food. It is also called bay salt , [ 1 ] solar salt , [ 2 ] or simply salt . Like mined rock salt , production of sea salt has been dated to prehistoric times .
Commercially available sea salts on the market today vary widely in their chemical composition. Although the principal component is sodium chloride , the remaining portion can range from less than 0.2 to 22% of other salts. These are mostly calcium, potassium, and magnesium salts of chloride and sulfate with substantially lesser amounts of many trace elements found in natural seawater. Though the composition of commercially available salt may vary, the ionic composition of natural saltwater is relatively constant. [ 3 ]
Sea salt is mentioned in the Vinaya Pitaka , a Buddhist scripture compiled in the mid-5th century BC. [ 4 ] The principle of production is evaporation of the water from the sea brine . In warm and dry climates this may be accomplished entirely by using solar energy, but in other climates fuel sources have been used. Modern sea salt production is almost entirely found in Mediterranean and other warm, dry climates. [ 5 ]
Such places are today called salt works, instead of the older English word saltern . An ancient or medieval saltern was established where there was:
In this way, salt marsh , pasture (salting), and salt works (saltern) enhanced each other economically. This was the pattern during the Roman and medieval periods around The Wash , in eastern England. [ 6 ] There, the tide brought the brine, the extensive saltings provided the pasture, the fens and moors provided the peat fuel, and the sun sometimes shone.
The dilute brine of the sea was largely evaporated by the sun. In Roman areas, this was done using ceramic containers known as briquetage . [ 6 ] Workers scraped up the concentrated salt and mud slurry and washed it with clean sea water to settle impurities out of the now concentrated brine. They poured the brine into shallow pans (lightly baked from local marine clay ) and set them on fist-sized clay pillars over a peat fire for final evaporation. Then they scraped out the dried salt and sold it.
In traditional salt production in the Visayas Islands of the Philippines , salt are made from coconut husks , driftwood , or other plant matter soaked in seawater for at least several months. These are burned into ash then seawater is run through the ashes on a filter. The resulting brine is then evaporated in containers. Coconut milk is sometimes added to the brine before evaporation. The practice is endangered due to competition with cheap industrially-produced commercial salt. Only two traditions survive to the present day: asín tibuok and túltul (or dúkdok). [ 7 ] [ 8 ]
In the colonial New World , slaves were brought from Africa to rake salt on various islands in the West Indies , Bahamas and particularly Turks and Caicos Islands .
Today, salt labelled "sea salt" in the US might not have actually come from the sea, as long as it meets the FDA's purity requirements. [ 9 ] All mined salts were originally sea salts since they originated from a marine source at some point in the distant past, usually from an evaporating shallow sea. [ 10 ]
Some gourmets believe sea salt tastes better and has a better texture than ordinary table salt. [ 11 ] In applications that retain sea salt's coarser texture, it can provide a different mouthfeel , and may change flavor due to its different rate of dissolution . The mineral content also affects the taste. The colors and variety of flavors are due to local clays and algae found in the waters the salt is harvested from. For example, some boutique salts from Korea and France are pinkish gray and some from India are black. Black and red salts from Hawaii may even have powdered black lava and baked red clay added in. [ 12 ] Some sea salt contains sulfates . [ 13 ] It may be difficult to distinguish sea salt from other salts, such as pink Himalayan salt , Maras salt from the ancient Inca hot springs, or rock salt ( halite ) [ citation needed ] .
Black lava salt is a marketing term for sea salt harvested from various places around the world that has been blended and colored with activated charcoal . The salt is used as a decorative condiment to be shown at the table. [ 14 ]
The nutritional value of sea salt and table salt are about the same as they are both primarily sodium chloride . [ 15 ] [ 16 ] Table salt is more processed than sea salt to eliminate minerals and usually contains an additive such as silicon dioxide to prevent clumping. [ 15 ]
Iodine , an element essential for human health , [ 17 ] is present only in small amounts in sea salt. [ 18 ] Iodised salt is table salt mixed with a minute amount of various salts of the element iodine.
Studies have found some microplastic contamination in sea salt from the US, Europe and China. [ 19 ] Sea salt has also been shown to be contaminated by fungi that can cause food spoilage as well as some that may be mycotoxigenic . [ 20 ]
In traditional Korean cuisine, jugyeom ( 죽염 , 竹鹽), which means "bamboo salt", is prepared by roasting salt at temperatures between 800 and 2000 °C [ 21 ] in a bamboo container plugged with mud at both ends. This product absorbs minerals from the bamboo and the mud, and is claimed to increase the anticlastogenic and antimutagenic properties of the fermented soybean paste known in Korea as doenjang . [ 22 ] However, these claims are not substantiated by high-quality studies. | https://en.wikipedia.org/wiki/Sea_salt |
Sea salt aerosol , which originally comes from sea spray , is one of the most widely distributed natural aerosols . Sea salt aerosols are characterized as non-light-absorbing, highly hygroscopic , and having coarse particle size . Some sea salt dominated aerosols could have a single scattering albedo as large as ~0.97. [ 1 ] Due to the hygroscopy, a sea salt particle can serve as a very efficient cloud condensation nuclei (CCN), altering cloud reflectivity , lifetime, and precipitation process. According to the IPCC report, the total sea salt flux from ocean to atmosphere is ~3300 teragrams (Tg) per year. [ 2 ] [ obsolete source ]
Many physical processes over ocean surface can generate sea salt aerosols. One common cause is the bursting of air bubbles , which are entrained by the wind stress during the whitecap formation. Another is tearing of drops from wave tops. [ 3 ] Wind speed is the key factor to determine the production rate in both mechanisms. Sea salt particle number concentration can reach 50 cm −3 or more with high winds (>10 m s −1 ), compared to ~10 cm −3 or less under moderate wind regimes. [ 3 ] Due to the dependence on wind speed, it could be expected that sea-salt particle production and its impacts on climate may vary with climate change . [ needs update ]
Sea salt aerosols are mainly constituted of sodium chloride (NaCl), but other chemical ions which are common in sea water, such as K + , Mg 2+ , Ca 2+ , SO 4 2− and so on, can also be found. A recent study revealed that sea salt aerosols also contain a substantial amount of organic matter . [ 4 ] [ 5 ] Mostly, organic materials are internally mixed due to the drying of air bubbles at the organic-rich sea surface. [ 3 ] The fraction of organic components increases with the decreasing particle size. The contained organic materials change the optical properties of sea salt as well as the hygroscopicity , especially when some insoluble organic matter is induced.
Size of sea salt aerosols ranges widely from ~0.05 to 10 μm in diameter, with most of masses concentrated in super-micron range (coarse mode), and highest number concentration in sub-micron range. Correspondingly, sea salt aerosols have a wide range of atmospheric lifetimes . As the sea salt aerosols are hygroscopic , their particle sizes may vary with humidity by up to a factor of 2. Sea salt aerosols influence the sulfate aerosol formation in different ways due to the different sizes. Very small sea salt aerosols, which are below the critical diameter for droplet activation at low supersaturations , can serve as nuclei for the growth of sulfate particles, while larger sea salt particles serve as a sink for gaseous hydrogen sulfate (H 2 SO 4 ) molecules, reducing the amount of sulfate available for the formation of accumulation mode particles. [ 3 ]
Sea salt aerosols can alter the Earth radiation budget through directly scattering solar radiation (direct effect), and indirectly changing the cloud albedo by serving as CCN (indirect effect). Different models give different predictions of annual mean radiative forcing induced by sea salt direct effect, but most of the previous studies give a number around 0.6-1.0 W m −2 . [ 6 ] [ 7 ] Radiative forcing caused by indirect effects show even greater variations in model prediction because of the parameterization of aerosol indirect effect. However, model results [ 6 ] [ 7 ] present a stronger indirect effect on the Southern Hemisphere . The aerosols might be used for marine cloud brightening .
Like all other soluble aerosols, increasing normal-sized sea salts suppresses the precipitation process in warm clouds by increasing cloud droplet number concentration and reducing the cloud droplet size. Also, they invigorate precipitation in mix-phase clouds because once the suppressed smaller cloud droplets are lifted above freezing level, more latent heat content would be released due to the freezing of cloud drops. [ 8 ] Besides that, adding giant sea salt aerosols to polluted clouds can accelerate the precipitation process because giant CCNs could be nucleated into large particles which collect other smaller cloud drops and grow into rain droplets. [ 9 ] Cloud drops formed on giant sea salt aerosols may grow much more rapidly by condensation that cloud drops formed on small soluble aerosol particles, as giant sea salt cloud drops may remain concentrated solution drops for long times after they are carried into cloud. Such drops may have condensational growth rates more than two times faster than drops formed on small aerosol particles, and unlike normal cloud drops, drops formed on the largest of the giant sea salt aerosols may even grow by condensation in otherwise subsaturated cloudy downdrafts. [ 10 ] | https://en.wikipedia.org/wiki/Sea_salt_aerosol |
The sea surface microlayer ( SML ) is the boundary interface between the atmosphere and ocean , covering about 70% of Earth 's surface. With an operationally defined thickness between 1 and 1,000 μm (1.0 mm ), the SML has physicochemical and biological properties that are measurably distinct from underlying waters. Recent studies now indicate that the SML covers the ocean to a significant extent, and evidence shows that it is an aggregate-enriched biofilm environment with distinct microbial communities . Because of its unique position at the air-sea interface, the SML is central to a range of global marine biogeochemical and climate-related processes. [ 1 ]
The sea surface microlayer is the boundary layer where all exchange occurs between the atmosphere and the ocean. [ 2 ] The chemical, physical, and biological properties of the SML differ greatly from the sub-surface water just a few centimeters beneath. [ 3 ]
Despite the huge extent of the ocean's surface, until now relatively little attention has been paid to the sea surface microlayer (SML) as the ultimate interface where heat , momentum and mass exchange between the ocean and the atmosphere takes place. Via the SML, large-scale environmental changes in the ocean such as warming , acidification , deoxygenation , and eutrophication potentially influence cloud formation , precipitation , and the global radiation balance . Due to the deep connectivity between biological, chemical, and physical processes, studies of the SML may reveal multiple sensitivities to global and regional changes. [ 4 ]
Understanding the processes at the ocean's surface, in particular involving the SML as an important and determinant interface, could provide an essential contribution to the reduction of uncertainties regarding ocean-climate feedbacks. As of 2017, processes occurring within the SML, as well as the associated rates of material exchange through the SML, remained poorly understood and were rarely represented in marine and atmospheric numerical models. [ 4 ]
The sea surface microlayer (SML) is the boundary interface between the atmosphere and ocean, covering about 70% of the Earth's surface. The SML has physicochemical and biological properties that are measurably distinct from underlying waters. Because of its unique position at the air-sea interface, the SML is central to a range of global biogeochemical and climate-related processes. Although known for the last six decades, the SML often has remained in a distinct research niche, primarily as it was not thought to exist under typical oceanic conditions. Recent studies now indicate that the SML covers the ocean to a significant extent, [ 5 ] highlighting its global relevance as the boundary layer linking two major components of the Earth system – the ocean and the atmosphere. [ 1 ]
In 1983, Sieburth hypothesised that the SML was a hydrated gel-like layer formed by a complex mixture of carbohydrates , proteins , and lipids . [ 6 ] In recent years, his hypothesis has been confirmed, and scientific evidence indicates that the SML is an aggregate-enriched biofilm environment with distinct microbial communities. [ 7 ] In 1999 Ellison et al. estimated that 200 Tg C yr −1 (200 million tonnes of carbon per year) accumulates in the SML, similar to sedimentation rates of carbon to the ocean's seabed, though the accumulated carbon in the SML probably has a very short residence time . [ 8 ] Although the total volume of the microlayer is very small compared to the ocean's volume, Carlson suggested in his seminal 1993 paper that unique interfacial reactions may occur in the SML that may not occur in the underlying water or at a much slower rate there. [ 9 ] He therefore hypothesised that the SML plays an important role in the diagenesis of carbon in the upper ocean. [ 9 ] Biofilm-like properties and highest possible exposure to solar radiation leads to an intuitive assumption that the SML is a biochemical microreactor. [ 10 ] [ 1 ]
Historically, the SML has been summarized as being a microhabitat composed of several layers distinguished by their ecological, chemical and physical properties with an operational total thickness of between 1 and 1000 μm. In 2005 Hunter defined the SML as a "microscopic portion of the surface ocean which is in contact with the atmosphere and which may have physical, chemical or biological properties that are measurably different from those of adjacent sub-surface waters". [ 12 ] He avoids a definite range of thickness as it depends strongly on the feature of interest. A thickness of 60 μm has been measured based on sudden changes of the pH, [ 13 ] and could be meaningfully used for studying the physicochemical properties of the SML. At such thickness, the SML represents a laminar layer, free of turbulence, and greatly affecting the exchange of gases between the ocean and atmosphere. As a habitat for neuston (surface-dwelling organisms ranging from bacteria to larger siphonophores), the thickness of the SML in some ways depends on the organism or ecological feature of interest. In 2005, Zaitsev described the SML and associated near- surface layer (down to 5 cm) as an incubator or nursery for eggs and larvae for a wide range of aquatic organisms. [ 14 ] [ 1 ]
Hunter's definition includes all interlinked layers from the laminar layer to the nursery without explicit reference to defined depths. [ 15 ] In 2017, Wurl et al. proposed Hunter's definition be validated with a redeveloped SML paradigm that includes its global presence, biofilm-like properties and role as a nursery. The new paradigm pushes the SML into a new and wider context relevant to many ocean and climate sciences. [ 1 ]
According to Wurl et al. , the SML can never be devoid of organics due to the abundance of surface-active substances (e.g., surfactants) in the upper ocean [ 5 ] and the phenomenon of surface tension at air-liquid interfaces. [ 16 ] The SML is analogous to the thermal boundary layer, and remote sensing of the sea surface temperature shows ubiquitous anomalies between the sea surface skin and bulk temperature. [ 17 ] Even so, the differences in both are driven by different processes. Enrichment, defined as concentration ratios of an analyte in the SML to the underlying bulk water, has been used for decades as evidence for the existence of the SML. Consequently, depletions of organics in the SML are debatable; however, the question of enrichment or depletion is likely to be a function of the thickness of the SML (which varies with sea state; [ 18 ] including losses via sea spray, the concentrations of organics in the bulk water, [ 5 ] and the limitations of sampling techniques to collect thin layers . [ 19 ] Enrichment of surfactants, and changes in the sea surface temperature and salinity, serve as universal indicators for the presence of the SML. Organisms are perhaps less suitable as indicators of the SML because they can actively avoid the SML and/or the harsh conditions in the SML may reduce their populations. However, the thickness of the SML remains "operational" in field experiments because the thickness of the collected layer is governed by the sampling method. Advances in SML sampling technology are needed to improve our understanding of how the SML influences air-sea interactions. [ 1 ]
Marine surface habitats sit at the interface between the atmosphere and the ocean. The biofilm-like habitat at the surface of the ocean harbours surface-dwelling microorganisms, commonly referred to as neuston . [ 20 ] The sea surface microlayer (SML) constitutes the uppermost layer of the ocean, only 1–1000 μm thick, with unique chemical and biological properties that distinguish it from the underlying water (ULW). [ 21 ] [ 2 ] Due to the location at the air-sea interface, the SML can influence exchange processes across this boundary layer, such as air-sea gas exchange and the formation of sea spray aerosols. [ 2 ] [ 22 ] [ 23 ] [ 4 ] [ 24 ]
Due to its exclusive position between the atmosphere and the hydrosphere and by spanning about 70% of the Earth's surface, the sea-surface microlayer (sea-SML) is regarded as a fundamental component in air–sea exchange processes and in biogeochemical cycling. [ 7 ] Although having a minor thickness of <1000 μm, [ 2 ] the elusive SML is long known for its distinct physicochemical characteristics compared to the underlying water, [ 25 ] e.g., by featuring the accumulation of dissolved and particulate organic matter, [ 25 ] [ 26 ] transparent exopolymer particles (TEP), and surface-active molecules. [ 27 ] [ 22 ] Therefore, the SML is a gelatinous biofilm, [ 28 ] maintaining physical stability through surface tension forces. [ 29 ] It also forms a vast habitat for different organisms, collectively termed as neuston [ 29 ] with a recent global estimate of 2 × 1023 microbial cells for the sea-SML. [ 30 ] [ 20 ]
Life at air–water interfaces has never been considered easy, mainly because of the harsh environmental conditions that influence the SML. [ 31 ] However, high abundances of microorganisms, especially of bacteria and picophytoplankton, accumulating in the SML compared to the underlying water were frequently reported, [ 26 ] [ 32 ] [ 33 ] accompanied by a predominant heterotrophic activity. [ 34 ] [ 35 ] [ 36 ] This is because primary production at the immediate air–water interface is often hindered by photoinhibition. [ 37 ] [ 38 ] However, some exceptions of photosynthetic organisms, e.g., Trichodesmium, Synechococcus, or Sargassum, show more tolerance towards high light intensities and, hence, can become enriched in the SML. [ 26 ] [ 39 ] [ 40 ] Previous research has provided evidence that neustonic organisms can cope with wind and wave energy, [ 32 ] [ 41 ] [ 42 ] solar and ultraviolet (UV) radiation, [ 43 ] [ 44 ] [ 45 ] fluctuations in temperature and salinity, [ 46 ] [ 47 ] and a higher potential predation risk by the zooneuston. [ 48 ] Furthermore, wind action promoting sea spray formation and bubbles rising from deeper water and bursting at the surface release SML-associated microbes into the atmosphere. [ 49 ] In addition to being more concentrated compared to planktonic counterparts, the bacterioneuston, algae, and protists display distinctive community compositions compared to the underlying water, in both marine [ 29 ] [ 39 ] [ 40 ] [ 41 ] [ 50 ] [ 51 ] and freshwater habitats. [ 52 ] [ 53 ] Furthermore, the bacterial community composition was often dependent on the SML sampling device being used. [ 54 ] [ 55 ] [ 56 ] While being well defined with respect to bacterial community composition, little is known about viruses in the SML, i.e., the virioneuston. This review has its focus on virus–bacterium dynamics at air–water interfaces, even if viruses likely interact with other SML microbes, including archaea and the phytoneuston, as can be deduced from viral interference with their planktonic counterparts. [ 57 ] [ 58 ] Although viruses were briefly mentioned as pivotal SML components in a recent review on this unique habitat, [ 4 ] a synopsis of the emerging knowledge and the major research gaps regarding bacteriophages at air–water interfaces is still missing in the literature. [ 20 ]
Organic compounds such as amino acids , carbohydrates , fatty acids , and phenols are highly enriched in the SML interface. Most of these come from biota in the sub-surface waters, which decay and become transported to the surface, [ 59 ] [ 60 ] though other sources exist also such as atmospheric deposition , coastal runoff , and anthropogenic nutrification. [ 2 ] The relative concentration of these compounds is dependent on the nutrient sources as well as climate conditions such as wind speed and precipitation . [ 60 ] These organic compounds on the surface create a "film," referred to as a "slick" when visible, [ 3 ] which affects the physical and optical properties of the interface. These films occur because of the hydrophobic tendencies of many organic compounds, which causes them to protrude into the air-interface. [ 2 ] [ 61 ] The existence of organic surfactants on the ocean surface impedes wave formation for low wind speeds. For increasing concentrations of surfactant there is an increasing critical wind speed necessary to create ocean waves. [ 2 ] [ 3 ] Increased levels of organic compounds at the surface also hinders air-sea gas exchange at low wind speeds. [ 62 ] One way in which particulates and organic compounds on the surface are transported into the atmosphere is the process called "bubble bursting". [ 2 ] [ 63 ] Bubbles generate the major portion of marine aerosols . [ 62 ] [ 64 ] [ 65 ] They can be dispersed to heights of several meters, picking up whatever particles latch on to their surface. However, the major supplier of materials comes from the SML. [ 59 ]
Surfaces and interfaces are critical zones where major physical, chemical, and biological exchanges occur. As the ocean covers 362 million km 2 , about 71% of the Earth's surface, the ocean-atmosphere interface is plausibly one of the largest and most important interfaces on the planet. Every substance entering or leaving the ocean from or to the atmosphere passes through this interface, which on the water-side -and to a lesser extent on the air-side- shows distinct physical, chemical, and biological properties. On the water side the uppermost 1 to 1000 μm of this interface are referred to as the sea surface microlayer (SML). [ 66 ] Like a skin, the SML is expected to control the rates of exchange of energy and matter between air and sea, thereby potentially exerting both short-term and long-term impacts on various Earth system processes, including biogeochemical cycling, production and uptake of radiately active gases like CO 2 or DMS, [ 67 ] thus ultimately climate regulation. [ 68 ] As of 2017, processes occurring within the SML, as well as the associated rates of material exchange through the SML, remained poorly understood and were rarely represented in marine and atmospheric numerical models. [ 4 ]
An improved understanding of the biological, chemical, and physical processes at the ocean's upper surface could provide an essential contribution to the reduction of uncertainties regarding ocean-climate feedbacks. Due to its positioning between atmosphere and ocean, the SML is the first to be exposed to climate changes including temperature, climate relevant trace gases, wind speed, and precipitation as well as to pollution by human waste, including nutrients, toxins, nanomaterials, and plastic debris. [ 4 ]
The term neuston describes the organisms in the SML and was first suggested by Naumann in 1917. [ 69 ] As in other marine ecosystems, bacterioneuston communities have important roles in SML functioning. [ 70 ] Bacterioneuston community composition of the SML has been analysed and compared to the underlying water in different habitats with varying results, and has primarily focused on coastal waters and shelf seas, with limited study of the open ocean . [ 71 ] [ 29 ] [ 50 ] In the North Sea, a distinct bacterial community was found in the SML with Vibrio spp. and Pseudoalteromonas spp. dominating the bacterioneuston. [ 29 ] During an artificially induced phytoplankton bloom in a fjord mesocosm experiment, the most dominant denaturing gradient gel electrophoresis (DGGE) bands of the bacterioneuston consisted of two bacterial families: Flavobacteriaceae and Alteromonadaceae . [ 50 ] Other studies have however, found little or no differences in the bacterial community composition of the SML and the ULW. [ 71 ] [ 72 ] Difficulties in direct comparisons between studies can arise because of the different methods used to sample the SML, which result in varied sampling depths. [ 73 ] [ 56 ] [ 70 ] [ 24 ]
Even less is known about the community control mechanisms in the SML and how the bacterial community assembles at the air-sea interface. The bacterioneuston community could be altered by differing wind conditions and radiation levels, [ 44 ] [ 74 ] [ 41 ] [ 42 ] with high wind speeds inhibiting the formation of a distinct bacterioneuston community. [ 41 ] [ 42 ] Wind speed and radiation levels refer to external controls, however, bacterioneuston community composition might also be influenced by internal factors such as nutrient availability and organic matter (OM) produced either in the SML or in the ULW. [ 75 ] [ 76 ] [ 77 ] [ 24 ]
One of the principal OM components consistently enriched in the SML are transparent exopolymer particles (TEP), [ 78 ] [ 79 ] [ 80 ] which are rich in carbohydrates and form by the aggregation of dissolved precursors excreted by phytoplankton in the euphotic zone . [ 81 ] [ 82 ] [ 83 ] [ 84 ] Higher TEP formation rates in the SML, facilitated through wind shear and dilation of the surface water, have been proposed as one explanation for the observed enrichment in TEP. [ 79 ] [ 85 ] Also, due to their natural positive buoyancy, when not ballasted by other particles sticking to them, TEP ascend through the water column and ultimately end up at the SML . [ 86 ] A second possible pathway of TEP from the water column to the SML is by bubble scavenging. [ 87 ] [ 24 ]
Next to rising bubbles, another potential transport mechanism for bacteria from the ULW to the SML could be ascending particles [ 71 ] [ 74 ] or more specifically TEP. [ 86 ] Bacteria readily attach to TEP in the water column . [ 88 ] [ 89 ] [ 90 ] TEP can serve as microbial hotspots and can be used directly as a substrate for bacterial degradation, [ 91 ] [ 92 ] [ 93 ] and as grazing protection for attached bacteria, e.g., by acting as an alternate food source for zooplankton. [ 94 ] [ 95 ] [ 96 ] TEP have also been suggested to serve as light protection for microorganisms in environments with high irradiation. [ 97 ] [ 24 ]
Viruses in the sea surface microlayer, the so-called virioneuston , have recently become of interest to researchers as enigmatic biological entities in the boundary surface layers with potentially important ecological impacts. Given this vast air–water interface sits at the intersection of major air–water exchange processes spanning more than 70% of the global surface area, it is likely to have profound implications for marine biogeochemical cycles , on the microbial loop and gas exchange, as well as the marine food web structure, the global dispersal of airborne viruses originating from the sea surface microlayer, and human health. [ 20 ]
Viruses are the most abundant biological entities in the water column of the world's oceans. [ 98 ] In the free water column, the virioplankton typically outnumbers the bacterioplankton by one order of magnitude reaching typical bulk water concentrations of 10 7 viruses mL −1 . [ 99 ] Moreover, they are known as integral parts of global biogeochemical cycles [ 99 ] to shape and drive microbial diversity [ 100 ] and to structure trophic networks. [ 101 ] Like other neuston members, the virioneuston likely originates from the bulk seawater. For instance, in 1977 Baylor et al. postulated adsorption of viruses onto air bubbles as they rise to the surface, [ 102 ] or viruses can stick to organic particles [ 103 ] also being transported to the SML via bubble scavenging. [ 104 ] [ 20 ]
Within the SML, viruses interacting with the bacterioneuston will probably induce the viral shunt, a phenomenon that is well known for marine pelagic systems. The term viral shunt describes the release of organic carbon and other nutritious compounds from the virus-mediated lysis of host cells, and its addition to the local dissolved organic matter (DOM) pool. [ 105 ] The enriched and densely packed bacterioneuston forms an excellent target for viruses compared to the bacterioplankton populating the subsurface. This is because high host-cell numbers will increase the probability of host–virus encounters. The viral shunt might effectively contribute to the SML's already high DOM content enhancing bacterial production as previously suggested for pelagic ecosystems [ 101 ] and in turn replenishing host cells for viral infections. By affecting the DOM pool, viruses in the SML might directly interfere with the microbial loop being initiated when DOM is microbially recycled, converted into biomass, and passed along the food web. In addition, the release of DOM from lysed host cells by viruses contributes to organic particle generation. [ 106 ] However, the role of the virioneuston for the microbial loop has never been investigated. [ 20 ]
Devices used to sample the concentrations of particulates and compounds of the SML include a glass fabric, metal mesh screens, and other hydrophobic surfaces. These are placed on a rotating cylinder which collects surface samples as it rotates on top of the ocean surface. [ 107 ]
The glass plate sampler is commonly used. [ 108 ] It was first described in 1972 by Harvey and Burzell as a simple but effective method of collecting small sea surface microlayer samples. [ 109 ] A clean glass plate is immersed vertically into the water and then withdrawn in a controlled manner. Harvey and Burzell used a plate which was 20 cm square and 4 mm thick. They withdrew it from the sea at the rate of 20 cm per second. [ 109 ] Typically the uppermost 20–150 μm of the surface microlayer adheres to the plate as it is withdrawn. [ 68 ] The sample is then wiped from both sides of the plate into a sampling vial. [ 110 ]
For a plate of the size used by Harvey and Burzel, the resulting sample volumes are between about 3 and 12 cubic centimetres. The sampled SML thickness h in micrometres is given by:
where V is the sample volume in cm 3 , A is the total immersed plate area of both sides in cm 2 , and N is the number of times the sample was dipped. [ 110 ]
Ocean surface habitats sit at the interface between the ocean and the atmosphere. The biofilm-like habitat at the surface of the ocean harbours surface-dwelling microorganisms, commonly referred to as neuston . This vast air–water interface sits at the intersection of major air–water exchange processes spanning more than 70% of the global surface area . Bacteria in the surface microlayer of the ocean, called bacterioneuston , are of interest due to practical applications such as air-sea gas exchange of greenhouse gases, production of climate-active marine aerosols, and remote sensing of the ocean. [ 111 ] Of specific interest is the production and degradation of surfactants (surface active materials) via microbial biochemical processes. Major sources of surfactants in the open ocean include phytoplankton, [ 112 ] terrestrial runoff, and deposition from the atmosphere. [ 111 ]
Unlike coloured algal blooms, surfactant-associated bacteria may not be visible in ocean colour imagery. Having the ability to detect these "invisible" surfactant-associated bacteria using synthetic aperture radar has immense benefits in all-weather conditions, regardless of cloud, fog, or daylight. [ 111 ] This is particularly important in very high winds, because these are the conditions when the most intense air-sea gas exchanges and marine aerosol production take place. Therefore, in addition to colour satellite imagery, SAR satellite imagery may provide additional insights into a global picture of biophysical processes at the boundary between the ocean and atmosphere, air-sea greenhouse gas exchanges and production of climate-active marine aerosols. [ 111 ]
A stream of airborne microorganisms, including marine viruses , bacteria and protists , circles the planet above weather systems but below commercial air lanes. [ 113 ] Some peripatetic microorganisms are swept up from terrestrial dust storms, but most originate from marine microorganisms in sea spray . In 2018, scientists reported that hundreds of millions of these viruses and tens of millions of bacteria are deposited daily on every square meter around the planet. [ 114 ] [ 115 ]
Compared to the sub-surface waters, the sea surface microlayer contains elevated concentration of bacteria and viruses , as well as toxic metals and organic pollutants. [ 2 ] [ 116 ] [ 117 ] [ 118 ] [ 119 ] These materials can be transferred from the sea-surface to the atmosphere in the form of wind-generated aqueous aerosols due to their high vapor tension and a process known as volatilisation . [ 63 ] When airborne, these microbes can be transported long distances to coastal regions. If they hit land they can have detrimental effects on animals, vegetation and human health. [ 120 ] Marine aerosols that contain viruses can travel hundreds of kilometers from their source and remain in liquid form as long as the humidity is high enough (over 70%). [ 121 ] [ 122 ] [ 123 ] These aerosols are able to remain suspended in the atmosphere for about 31 days. [ 59 ] Evidence suggests that bacteria can remain viable after being transported inland through aerosols. Some reached as far as 200 meters at 30 meters above sea level. [ 124 ] It was also noted that the process which transfers this material to the atmosphere causes further enrichment in both bacteria and viruses in comparison to either the SML or sub-surface waters (up to three orders of magnitude in some locations). [ 125 ] [ 124 ]
The stagnant film model is a mathematical model used to simulate the sea surface microlayer. It is a kinematic model which can be used to describe how gas exchange from the ocean's surface and the atmosphere reaches equilibrium . [ 126 ] [ 127 ] The model assumes both the ocean and atmosphere are composed mostly of well-mixed, constantly moving fluid layers with the sea surface microlayer present as a permanent thin-film layer in the middle. Gas exchange occurs by molecular diffusion between the two fluid layers through the sea surface microlayer. [ 126 ] [ 128 ] | https://en.wikipedia.org/wiki/Sea_surface_microlayer |
Sealcoating , or pavement sealing , is the process of applying a protective coating to asphalt -based pavements to provide a layer of protection from the elements: water, oils, and U.V. damage.
The effects of asphalt sealers have been debated. Asphalt sealing is marketed as increasing the life of the asphalt, but there is no independent research that proves these claims. Asphalt sealing can also make the asphalt more slippery and impact the environment .
Pavement sealcoat products come in a variety of standards. For example, refined tar-based sealer offers the best protecting against water penetration and chemical resistance. Asphalt-based sealer typically offers poor protection against environmental, chemical and harsher climates (salt water). Petroleum-based sealer offer protection against water and chemicals somewhere between the other two sealers. Another difference between coatings is in terms of wear. Again, refined tar-based sealer offers the best wear characteristics (typically 3–5 years) while asphalt-based sealer may last 1–3 years. Petroleum-based sealer falls between refined tar and asphalt.
There are concerns about pavement sealer polluting the environment after it is abraded from the surface of the pavement. Some states in North America have banned the use of coal tar–based sealants primarily based on United States Geological Survey studies. [ 1 ]
There are primarily five types of pavement sealers.
These are commonly known as:
All five have their advantages and disadvantages, but are typically chosen by the preference unless otherwise specified.
Prior to application the surface must be completely clean and dry using sweeping methods and/or blowers. If the surface is not clean and dry, then poor adhesion will result. Pavement sealers are applied with either pressurized spray equipment, or self-propelled squeegee machines or by hand with a brush. Equipment must have continuous agitation to maintain consistency of the sealcoat mix. The process is typically a two-coat application which requires 24 to 48 hours of curing before vehicles can be allowed back on the surface. Once the surface is properly prepared, then properly mixed sealer will be applied at about 1.5 square meters per liter (60 sq ft per U.S. gal; 72 sq ft per Imp gal) per coat.
Some studies suggest that refined coal tar sealants are a significant contributor to polycyclic aromatic hydrocarbon levels in streams and creek beds and that the continual application of sealcoats may be a significant factor. As a result, more than 100 units of government in the United States have banned this material. [ 2 ] The same studies also suggest that it can be harmful if ingested before curing and ingesting soil or dust contaminated by eroded coal tar sealant. [ 3 ] It is also known to have effects on fish and other animals that live in water. | https://en.wikipedia.org/wiki/Sealcoat |
Seamount microbial communities refer to microorganisms living within the surrounding environment of seamounts . Seamounts are often called the hotspot of marine life [ 1 ] [ 2 ] serving as a barrier that disrupts the current and flow in the ocean, which is referred to as the seamount effect. [ 3 ] [ 4 ]
Around 25 million seamounts are known to exist, [ 2 ] [ 4 ] [ 5 ] however, the research on microbial communities are focused on volcanically active seamounts. [ 3 ] [ 4 ] The microbial interactions with marine life, such as sponges and corals, demonstrates the potential importance of microbes in the foundational success of seamount communities.
Seamounts are submarine mountains which are widely prevalent across the globe, often with an active hydrothermal system. [ 5 ] [ 6 ] Seamounts also contribute to the global circulation of oceanic primary productivity as biodiversity hotspots, enhancing biomass and species diversity within and beyond marine ecosystems. Their distinctive topographies alter the regional oceanic circulation, resulting in upwelling that transport nutrient-rich water from deep ocean to the surface. [ 5 ] This results in unique habitats that sustain the lives of marine organisms, including a diverse range of microbial communities. These unique features of seamounts overall contribute significantly to biogeochemical cycles , particularly due to the involvement of microbial communities in nutrient recycling. [ 4 ] [ 5 ] [ 7 ] Given that microbes are most abundant in aquatic environments, seamounts serve as an excellent focus for research. [ 5 ] [ 2 ]
Seamounts are primarily formed due to underwater volcanic and tectonic activities beneath the ocean floor at the plate boundaries of lithosphere . [ 6 ] Additionally, mid-ocean ridges at divergent plate boundaries are common locations for high-temperature hydrothermal venting , which are distinctive habitats, especially to many chemoautotrophic microbes. However, these systems tend to be more uniform in their physical and chemical characteristics compared to vent systems found near volcanic arcs and back-arc regions. [ 6 ] [ 4 ] Seamounts features a variety of geological characteristics that affect hydrodynamic conditions in a local and regional scale, thus impacting the entire marine ecosystem globally. [ 6 ]
The diverse landscapes of seamounts, from cone-shaped to chain-like formations with multiple summits, influence the hydrodynamic patterns which define the distribution and composition of microbial communities within the regions, as observed in the environments of Seine and Sedlo seamounts. [ 7 ] [ 9 ] Research on various seamounts, such as Kocebu Guyot, the seamounts in the South China Sea, and those like Sedlo and Seine, shows a notable influence on the diversity of habitats and seasonal changes in microbial populations. The variety of microbes found on seamounts is often correlated with levels of Total Organic Carbon (TOC) , where higher TOC concentrations are associated with a richer microbial diversity. Specific microbial groups, like Zetaproteobacteria and Epsilonproteobacteria, are predominant in certain seamount areas, their presence shaped by the unique geochemical conditions resulting from the geological activities of the seamounts. [ 3 ] [ 7 ] [ 2 ] [ 10 ] This phenomenon, known as the 'seamount effect', alters the patterns of ocean currents, leading to enhanced biodiversity and increased ecological productivity. This makes seamounts critical study areas for understanding the patterns and structures of microbial communities. [ 4 ] [ 7 ]
Diversity and connectivity of bacteria , protists , and fungi was investigated through studies around the Kocebu Guyot in the Magellan Seamount chain in western Pacific Ocean. The studies showed that bacterial communities had higher interlayer connectivity than protists and fungi. Fungi appeared to have the lowest connectivity of the three groups. The seamount notably decreased the protist connectivity in the horizontal scale, while increasing vertical connectivity for both bacteria and protists, creating ecological opportunities for species' diversity . [ 10 ] The differences observed between bacteria and protists may be due to bacteria's smaller size and free dispersal. [ 9 ] This study in the Magellan Seamount observes an unusual finding regarding protists, where protists show the highest diversity in deeper water layers. [ 10 ] Typically, eukaryotic diversity tends to be lower in deep water layers due to food limitation. [ 11 ]
Microbial community dominance around the South China Sea (SCS) seamounts was investigated through examining the similarities and differences of seamount and non-seamount sites. Low microbial diversity is associated with low Total Organic Carbon (TOC) concentration in deep waters, while high microbial diversity correlates with high TOC in shallow waters. [ 4 ] In both seamount and non-seamount sites, Thaumarchaeota , Proteobacteria , Chloroflexi, Actinobacteriota , Planctomycetota , Bacteroidota , and Acidobacteriota were identified. [ 4 ] [ 12 ] The dominant microbial communities in SCS seamounts included N itrosopumilales , Alteromonadales , Anaerolineales, Rhodobacterales , Flavobacteriales , Steroidobacterales, and “ Candidatus Actinomarinales.” These microbes were not exclusive to SCS seamounts as they were also found in non-seamount sediments. In seamount seafloor ecosystems, Oceanospirillales , Corynebacteriales, Rhodobacterales, S085, and Kordiimonadales are the dominant microbial communities. [ 4 ]
In volcanically active seamounts, within the surface hydrothermal vents , Zetaproteobacteria and Epsilonproteobacteria are the dominant microbial communities. [ 3 ] Archaea are not abundant in this habitat, except in the Mariana forearc, where they dominate. [ 13 ] Epsilonproteobacteria and Zetaproteobacteria are rare in the seawater surrounding seamounts. In general, microbial communities are influenced by the geochemistry shaped by the seamount geological processes. [ 3 ]
Zetaproteobacteria/FeOB
Zetaproteobacteria/FeOB (detected)
Gammaproteobacteria/SOB (sulfur oxidizing bacteria)
Epsilonproteobacteria/HSOB
Gammaproteobacteria/SOB
Deltaproteobacteria/SRB (sulfur reducing bacteria)
Zetaproteobacteria/FeOB (detected)
Euryarchaeota ; Methanobacteria & Methanosarcinales /AMO (anaerobic methane oxidation) (implicated)
During all seasons, heterotrophic microbial communities dominate for both Seine and Sedlo seamounts. The autotrophic community of both seamounts are primarily composed of small cells. In winter, Prochlorococcus-2 existed on both seamounts. [ 7 ]
The Seine region is in close proximity to the African coast. This area exhibits regional strong upwelling that provides ideal conditions for larger phytoplanktons . [ 32 ] During springtime, the Seine seamount has more contributions from nanoplankton and microphytoplankton, however, less from picoplankton. The integrated abundance of Picoeukaryotes and Synechococcus is highest in spring as well. Comparably, the Prochlorococcus dominates in summer and winter. [ 7 ]
In winter, the Sedlo seamount experiences more contributions from picophytoplankton near the thermocline . During summer, there is a threefold increase in total heterotrophic activity. The integrated abundance of Synechococcus is high in the southern region of Sedlo, suggesting surface water nutrient enrichment. [ 7 ]
The seamount effect is the principle that the topography of seamounts disrupts the oceanic flow in their area. [ 4 ] Due to increased turbulence in the water, this phenomenon results in a greater amount of particulate organic matter (POC) coming down from the surface ocean, which fuels higher biodiversity and production rates than surrounding areas. Seamounts have a higher biodiversity than both coastal and open oceans, but this decreases with distance. [ 1 ] [ 4 ] Increased production rates lead to increased community growth, especially for unrelated organisms. A large community means that an abundance of niches are occupied, some of which include consuming the waste that is produced when organic matter is consumed. [ 4 ] [ 33 ]
The microbial communities that thrive within the sediments of seamounts differ from those which occupy the sediments in non-seamount areas throughout the world's oceans. [ 4 ] Many sediments have ferromanganese crusts, which include ammonia, that can be used as an electron donor for growth. [ 34 ] The sediments at the surface of seamounts house more heterotrophs than autotrophs as a result of the large amounts of POC that become trapped within the sediments following turbulence created by the seamount effect. [ 4 ]
The high production levels present in seamounts contribute to a substantial food source for larger organisms, providing sustenance to higher trophic levels. [ 4 ] [ 35 ] Food sources also come from a disruption in the diel vertical migration (DVM) cycle, as zooplankton which descend back into the deep ocean in at daybreak following their daily cycle can become trapped by high points on the seamount, where they become a prey item for larger consumers. [ 4 ] Both pelagic and benthic organisms are found in seamounts as part of their diverse biological communities. [ 1 ] [ 35 ] When the water flow is changed, it moves microorganisms horizontally between regions of the seamount, further increasing community biodiversity, but also increasing ecological interactions between different trophic levels. [ 35 ]
The topographical effects on physical oceanography can boost diversity, enhance biomass , and alter production levels. [ 36 ] [ 37 ] In biologically abundant seamounts, microbes are no strangers to interacting with other marine life forms.
Sponges and corals are two of many organisms that find water movements in seamounts favorable. As they grow abundant, their encounter with seamount microbes gets more prevalent. Some sponges can be categorized as high-microbial abundance species, [ 38 ] harboring up to 100 million microbes g −1 of sponge tissue. [ 39 ] This relationship provides mutually beneficial interactions, [ 40 ] such as those related to metabolism, nutrient availability,and microbe protection. [ 38 ] Although coral-microbe interaction studies specifically in seamount environments are limited, corals in other deep sea environments also exhibit such important relationships. These mutually beneficial interactions include those related to nutrient availability, anti-pathogen protection, and microbe protection. [ 41 ]
Although other seamount microbe-marine life interactions are currently very limited, the microbial life supports higher levels of biodiversity through sponges and corals. These two are foundational organisms that provide substrate stability, allowing other life forms to exist in that environment. [ 42 ] [ 43 ] [ 44 ] With the right conditions, these can flourish into a web of complex ecology indirectly supported by microbes. | https://en.wikipedia.org/wiki/Seamount_microbial_communities |
Sean Roberts Eddy is Professor of Molecular & Cellular Biology and of Applied Mathematics at Harvard University . Previously he was based at the Janelia Research Campus from 2006 to 2015 [ 1 ] [ 7 ] [ 8 ] in Virginia . His research interests are in bioinformatics , computational biology and biological sequence analysis . [ 9 ] [ 10 ] [ 11 ] [ 12 ] As of 2016 [update] projects include the use of Hidden Markov models [ 13 ] [ 14 ] in HMMER , Infernal [ 15 ] Pfam and Rfam . [ 16 ] [ 17 ] [ 18 ] [ 19 ]
Eddy graduated June, 1982 from Marion Center Area High School, Marion Center, Pennsylvania . He then completed a Bachelor of Science in Biology at California Institute of Technology in 1986, [ 20 ] followed by a Doctor of Philosophy in molecular biology at the University of Colorado under the supervision of Larry Gold in 1991 studying the T4 phage . [ 2 ] [ 21 ] [ 22 ]
From 1992 to 1995 he was a postdoctoral research fellow at the Medical Research Council (MRC) Laboratory of Molecular Biology (LMB) in Cambridge UK working with John Sulston and Richard Durbin . From 1995 to 2007 he worked at Washington University School of Medicine and has been working for the Howard Hughes Medical Institute since 2000.
In 2007, Sean was the winner of the Benjamin Franklin Award in Bioinformatics for contributions to Open Access in the Life Sciences. [ 23 ]
In 2022, Eddy was elected as a Fellow of the International Society for Computational Biology . [ 24 ] | https://en.wikipedia.org/wiki/Sean_Eddy |
The search for extraterrestrial intelligence (usually shortened as SETI ) is an expression that refers to the diverse efforts and scientific projects intended to detect extraterrestrial signals, or any evidence of intelligent life beyond Earth.
Researchers use methods such as monitoring electromagnetic radiation, searching for optical signals, and investigating potential extraterrestrial artifacts for any signs of transmission from civilizations present on other planets. [ 1 ] [ 2 ] [ 3 ] Some initiatives have also attempted to send messages to hypothetical alien civilizations, such as NASA 's Golden Record . [ 4 ]
Modern SETI research began in the early 20th century after the advent of radio , expanding with projects like Project Ozma, the Wow! signal detection, and the Breakthrough Listen initiative; a $100 million, 10-year attempt to detect signals from nearby stars, announced in 2015 by Stephen Hawking , and Yuri Milner . Since the 1980s, international efforts have been ongoing, with community led projects such as SETI@home and Project Argus, engaging in analyzing data. [ 5 ] While SETI remains a respected scientific field, it often gets compared to conspiracy theory, UFO research, bringing unawarrented skepticism from the public, despite its reliance on rigorous scientific methods and verifiable data and research. Similar studies on Unidentified Aerial Phenomena (UAP) such as the Avi Loeb's Galileo Project have brought further attention to SETI research.
Despite decades of searching, no confirmed evidence of alien intelligence has been found, bringing criticism onto SETI for being 'overly hopeful'. Critics argue that SETI is speculative and unfalsifiable, while supporters see it as a crucial step in addressing the Fermi Paradox and understanding extraterrestrial technosignature . [ 6 ]
There have been many earlier searches for extraterrestrial intelligence within the Solar System . In 1896, Nikola Tesla suggested that an extreme version of his wireless electrical transmission system could be used to contact beings on Mars . [ 7 ] In 1899, while conducting experiments at his Colorado Springs experimental station , he thought he had detected a signal from Mars since an odd repetitive static signal seemed to cut off when Mars set in the night sky. Analysis of Tesla's research has led to a range of explanations including:
In the early 1900s, Guglielmo Marconi , Lord Kelvin and David Peck Todd also stated their belief that radio could be used to contact Martians , with Marconi stating that his stations had also picked up potential Martian signals. [ 10 ] [ 11 ]
On August 21–23, 1924, Mars entered an opposition closer to Earth than at any time in the century before or the next 80 years. [ 12 ] In the United States, a "National Radio Silence Day" was promoted during a 36-hour period from August 21–23, with all radios quiet for five minutes on the hour, every hour. At the United States Naval Observatory , a radio receiver was lifted 3 kilometres (1.9 miles) above the ground in a dirigible tuned to a wavelength between 8 and 9 km, using a "radio-camera" developed by Amherst College and Charles Francis Jenkins . The program was led by David Peck Todd with the military assistance of Admiral Edward W. Eberle ( Chief of Naval Operations ), with William F. Friedman (chief cryptographer of the United States Army), assigned to translate any potential Martian messages. [ 13 ] [ 14 ]
A 1959 paper by Philip Morrison and Giuseppe Cocconi first pointed out the possibility of searching the microwave spectrum. It proposed frequencies and a set of initial targets. [ 15 ] [ 16 ]
In 1960, Cornell University astronomer Frank Drake performed the first modern SETI experiment, named " Project Ozma " after the Queen of Oz in L. Frank Baum 's fantasy books. [ 17 ] Drake used a radio telescope 26 metres (85 ft) in diameter at Green Bank, West Virginia , to examine the stars Tau Ceti and Epsilon Eridani near the 1.420 gigahertz marker frequency, a region of the radio spectrum dubbed the " water hole " due to its proximity to the hydrogen and hydroxyl radical spectral lines. A 400 kilohertz band around the marker frequency was scanned using a single-channel receiver with a bandwidth of 100 hertz. He found nothing of interest.
Soviet scientists took a strong interest in SETI during the 1960s and performed a number of searches with omnidirectional antennas in the hope of picking up powerful radio signals. Soviet astronomer Iosif Shklovsky wrote the pioneering book in the field, Universe, Life, Intelligence (1962), which was expanded upon by American astronomer Carl Sagan as the best-selling book Intelligent Life in the Universe (1966). [ 18 ]
In the March 1955 issue of Scientific American , John D. Kraus described an idea to scan the cosmos for natural radio signals using a flat-plane radio telescope equipped with a parabolic reflector . Within two years, his concept was approved for construction by Ohio State University . With a total of US$ 71,000 (equivalent to $794,880 in 2024)
in grants from the National Science Foundation , construction began on an 8-hectare (20-acre) plot in Delaware, Ohio . This Ohio State University Radio Observatory telescope was called "Big Ear". Later, it began the world's first continuous SETI program, called the Ohio State University SETI program.
In 1971, NASA funded a SETI study that involved Drake, Barney Oliver of Hewlett-Packard Laboratories , and others. The resulting report proposed the construction of an Earth-based radio telescope array with 1,500 dishes known as " Project Cyclops ". The price tag for the Cyclops array was US$10 billion. Cyclops was not built, but the report [ 19 ] formed the basis of much SETI work that followed.
The Ohio State SETI program gained fame on August 15, 1977, when Jerry Ehman , a project volunteer, witnessed a startlingly strong signal received by the telescope. He quickly circled the indication on a printout and scribbled the exclamation "Wow!" in the margin. Dubbed the Wow! signal , it is considered by some to be the best candidate for a radio signal from an artificial, extraterrestrial source ever discovered, but it has not been detected again in several additional searches. [ 20 ]
On 24 May 2023, a test extraterrestrial signal, in the form of a "coded radio signal from Mars", was transmitted to radio telescopes on Earth, according to a report in The New York Times . [ 21 ]
In 1980, Carl Sagan , Bruce Murray , and Louis Friedman founded the U.S. Planetary Society , partly as a vehicle for SETI studies. [ 3 ]
In the early 1980s, Harvard University physicist Paul Horowitz took the next step and proposed the design of a spectrum analyzer specifically intended to search for SETI transmissions. Traditional desktop spectrum analyzers were of little use for this job, as they sampled frequencies using banks of analog filters and so were restricted in the number of channels they could acquire. However, modern integrated-circuit digital signal processing (DSP) technology could be used to build autocorrelation receivers to check far more channels. This work led in 1981 to a portable spectrum analyzer named "Suitcase SETI" that had a capacity of 131,000 narrow band channels. After field tests that lasted into 1982, Suitcase SETI was put into use in 1983 with the 26-meter (85 ft) Harvard/Smithsonian radio telescope at Oak Ridge Observatory in Harvard, Massachusetts . This project was named "Sentinel" and continued into 1985.
Even 131,000 channels were not enough to search the sky in detail at a fast rate, so Suitcase SETI was followed in 1985 by Project "META", for "Megachannel Extra-Terrestrial Assay". The META spectrum analyzer had a capacity of 8.4 million channels and a channel resolution of 0.05 hertz. An important feature of META was its use of frequency Doppler shift to distinguish between signals of terrestrial and extraterrestrial origin. The project was led by Horowitz with the help of the Planetary Society, and was partly funded by movie maker Steven Spielberg . A second such effort, META II, was begun in Argentina in 1990, to search the southern sky, receiving an equipment upgrade in 1996–1997. [ 22 ] [ 23 ]
The follow-on to META was named "BETA", for "Billion-channel Extraterrestrial Assay", and it commenced observation on October 30, 1995. The heart of BETA's processing capability consisted of 63 dedicated fast Fourier transform (FFT) engines, each capable of performing a 2 22 -point complex FFTs in two seconds, and 21 general-purpose personal computers equipped with custom digital signal processing boards. This allowed BETA to receive 250 million simultaneous channels with a resolution of 0.5 hertz per channel. It scanned through the microwave spectrum from 1.400 to 1.720 gigahertz in eight hops, with two seconds of observation per hop. An important capability of the BETA search was rapid and automatic re-observation of candidate signals, achieved by observing the sky with two adjacent beams, one slightly to the east and the other slightly to the west. A successful candidate signal would first transit the east beam, and then the west beam and do so with a speed consistent with Earth 's sidereal rotation rate. A third receiver observed the horizon to veto signals of obvious terrestrial origin. On March 23, 1999, the 26-meter radio telescope on which Sentinel, META and BETA were based was blown over by strong winds and seriously damaged. [ 24 ] This forced the BETA project to cease operation.
In 1978, the NASA SETI program had been heavily criticized by Senator William Proxmire , and funding for SETI research was removed from the NASA budget by Congress in 1981; [ 26 ] however, funding was restored in 1982, after Carl Sagan talked with Proxmire and convinced him of the program's value. [ 26 ] In 1992, the U.S. government funded an operational SETI program, in the form of the NASA Microwave Observing Program (MOP). MOP was planned as a long-term effort to conduct a general survey of the sky and also carry out targeted searches of 800 specific nearby stars. MOP was to be performed by radio antennas associated with the NASA Deep Space Network , as well as the 140-foot (43 m) radio telescope of the National Radio Astronomy Observatory at Green Bank, West Virginia and the 1,000-foot (300 m) radio telescope at the Arecibo Observatory in Puerto Rico. The signals were to be analyzed by spectrum analyzers, each with a capacity of 15 million channels. These spectrum analyzers could be grouped together to obtain greater capacity. Those used in the targeted search had a bandwidth of 1 hertz per channel, while those used in the sky survey had a bandwidth of 30 hertz per channel.
MOP drew the attention of the United States Congress , where the program met opposition [ 27 ] and canceled one year after its start. [ 26 ] SETI advocates continued without government funding, and in 1995 the nonprofit SETI Institute of Mountain View, California resurrected the MOP program under the name of Project "Phoenix", backed by private sources of funding. In 2012 it cost around $2 million per year to maintain SETI research at the SETI Institute and around 10 times that to support different SETI activities globally. [ 28 ] Project Phoenix , under the direction of Jill Tarter , was a continuation of the targeted search program from MOP and studied roughly 1,000 nearby Sun -like stars until approximately 2015. [ 29 ] From 1995 through March 2004, Phoenix conducted observations at the 64-meter (210 ft) Parkes radio telescope in Australia , the 140-foot (43 m) radio telescope of the National Radio Astronomy Observatory in Green Bank, West Virginia, and the 1,000-foot (300 m) radio telescope at the Arecibo Observatory in Puerto Rico. The project observed the equivalent of 800 stars over the available channels in the frequency range from 1200 to 3000 MHz. The search was sensitive enough to pick up transmitters with 1 GW EIRP to a distance of about 200 light-years .
Many radio frequencies penetrate Earth's atmosphere quite well, and this led to radio telescopes that investigate the cosmos using large radio antennas. Furthermore, human endeavors emit considerable electromagnetic radiation as a byproduct of communications such as television and radio. These signals would be easy to recognize as artificial due to their repetitive nature and narrow bandwidths . Earth has been sending radio waves from broadcasts into space for over 100 years. [ 30 ] These signals have reached over 1,000 stars, most notably Vega , Aldebaran , Barnard's Star , Sirius , and Proxima Centauri . If intelligent alien life exists on any planet orbiting these nearby stars, these signals could be heard and deciphered, even though some of the signal is garbled by the Earth's ionosphere .
Many international radio telescopes are currently [ when? ] being used for radio SETI searches, including the Low Frequency Array (LOFAR) in Europe, the Murchison Widefield Array (MWA) in Australia, and the Lovell Telescope in the United Kingdom. [ 31 ]
The SETI Institute collaborated with the Radio Astronomy Laboratory at the Berkeley SETI Research Center to develop a specialized radio telescope array for SETI studies, similar to a mini-cyclops array. Formerly known as the One Hectare Telescope (1HT), the concept was renamed the "Allen Telescope Array" (ATA) after the project's benefactor, Paul Allen . Its sensitivity is designed to be equivalent to a single large dish more than 100 meters in diameter, if fully completed. Presently [ when? ] , the array has 42 operational dishes at the Hat Creek Radio Observatory in rural northern California. [ 32 ] [ 33 ]
The full array (ATA-350) is planned to consist of 350 or more offset- Gregorian radio dishes, each 6.1 meters (20 feet) in diameter. These dishes are the largest producible with commercially available satellite television dish technology. The ATA was planned for a 2007 completion date, at a cost of US$25 million. The SETI Institute provided money for building the ATA while University of California, Berkeley designed the telescope and provided operational funding. The first portion of the array (ATA-42) became operational in October 2007 with 42 antennas. The DSP system planned for ATA-350 is extremely ambitious. Completion of the full 350 element array will depend on funding and the technical results from ATA-42.
ATA-42 (ATA) is designed to allow multiple observers simultaneous access to the interferometer output at the same time. Typically, the ATA snapshot imager (used for astronomical surveys and SETI) is run in parallel with a beamforming system (used primarily for SETI). [ 34 ] ATA also supports observations in multiple synthesized pencil beams at once, through a technique known as "multibeaming". Multibeaming provides an effective filter for identifying false positives in SETI, since a very distant transmitter must appear at only one point on the sky. [ 35 ] [ 36 ] [ 37 ]
SETI Institute's Center for SETI Research (CSR) uses ATA in the search for extraterrestrial intelligence, observing 12 hours a day, 7 days a week. From 2007 to 2015, ATA identified hundreds of millions of technological signals. So far, all these signals have been assigned the status of noise or radio frequency interference because a) they appear to be generated by satellites or Earth-based transmitters, or b) they disappeared before the threshold time limit of ~1 hour. [ 38 ] [ 39 ] Researchers in CSR are working on ways to reduce the threshold time limit, and to expand ATA's capabilities for detection of signals that may have embedded messages. [ 40 ]
Berkeley astronomers used the ATA to pursue several science topics, some of which might have transient SETI signals, [ 41 ] [ 42 ] [ 43 ] until 2011, when the collaboration between the University of California, Berkeley and the SETI Institute was terminated.
CNET published an article and pictures about the Allen Telescope Array (ATA) on December 12, 2008. [ 44 ] [ 45 ]
In April 2011, the ATA entered an 8-month "hibernation" due to funding shortfalls. Regular operation of the ATA resumed on December 5, 2011. [ 46 ] [ 47 ]
In 2012, the ATA was revitalized with a $3.6 million donation by Franklin Antonio , co-founder and Chief Scientist of QUALCOMM Incorporated. [ 48 ] This gift supported upgrades of all the receivers on the ATA dishes to have (2× to 10× over the range 1–8 GHz) greater sensitivity than before and supporting observations over a wider frequency range from 1–18 GHz, though initially the radio frequency electronics only go to 12 GHz. As of July 2013, the first of these receivers was installed and proven, with full installation on all 42 antennas being expected for June 2017. ATA is well suited to the search for extraterrestrial intelligence (SETI) and to discovery of astronomical radio sources , such as heretofore unexplained non-repeating, possibly extragalactic, pulses known as fast radio bursts or FRBs. [ 49 ] [ 50 ]
SERENDIP (Search for Extraterrestrial Radio Emissions from Nearby Developed Intelligent Populations) is a SETI program launched in 1979 by the Berkeley SETI Research Center . [ 51 ] [ 52 ] SERENDIP takes advantage of ongoing "mainstream" radio telescope observations as a " piggy-back " or " commensal " program, using large radio telescopes including the NRAO 90m telescope at Green Bank and, formerly, the Arecibo 305m telescope . Rather than having its own observation program, SERENDIP analyzes deep space radio telescope data that it obtains while other astronomers are using the telescopes. The most recently deployed SERENDIP spectrometer, SERENDIP VI, was installed at both the Arecibo Telescope and the Green Bank Telescope in 2014–2015. [ 53 ]
Breakthrough Listen is a ten-year initiative with $100 million funding begun in July 2015 to actively search for intelligent extraterrestrial communications in the universe , in a substantially expanded way, using resources that had not previously been extensively used for the purpose. [ 54 ] [ 55 ] [ 56 ] [ 3 ] It has been described as the most comprehensive search for alien communications to date. [ 55 ] The science program for Breakthrough Listen is based at Berkeley SETI Research Center , [ 57 ] [ 58 ] located in the Astronomy Department [ 59 ] at the University of California, Berkeley .
Announced in July 2015, the project is observing for thousands of hours every year on two major radio telescopes, the Green Bank Observatory in West Virginia, and the Parkes Observatory in Australia . [ 60 ] Previously, only about 24 to 36 hours of telescope time per year were used in the search for alien life. [ 55 ] Furthermore, the Automated Planet Finder at Lick Observatory is searching for optical signals coming from laser transmissions. The massive data rates from the radio telescopes (24 GB/s at Green Bank) necessitated the construction of dedicated hardware at the telescopes to perform the bulk of the analysis. [ 61 ] Some of the data are also analyzed by volunteers in the SETI@home volunteer computing network. [ 60 ] Founder of modern SETI Frank Drake was one of the scientists on the project's advisory committee. [ 62 ] [ 54 ] [ 55 ]
In October 2019, Breakthrough Listen started a collaboration with scientists from the TESS team ( Transiting Exoplanet Survey Satellite ) to look for signs of advanced extraterrestrial life. Thousands of new planets found by TESS will be scanned for technosignatures by Breakthrough Listen partner facilities across the globe. Data from TESS monitoring of stars will also be searched for anomalies. [ 63 ]
China's 500 meter Aperture Spherical Telescope (FAST) lists detecting interstellar communication signals as part of its science mission. It is funded by the National Development and Reform Commission (NDRC) and managed by the National Astronomical observatories (NAOC) of the Chinese Academy of Sciences (CAS). FAST is the first radio observatory built with SETI as a core scientific goal. [ 64 ] FAST consists of a fixed 500 m (1,600 ft) diameter spherical dish constructed in a natural depression sinkhole caused by karst processes in the region. It is the world's largest filled-aperture radio telescope. [ 65 ] According to its website, FAST can search to 28 light-years, and is able to reach 1,400 stars. If the transmitter's radiated power were to be increased to 1,000,000 MW, FAST would be able to reach one million stars. This is compared to the former Arecibo 305 meter telescope detection distance of 18 light-years. [ 66 ]
On 14 June 2022, astronomers, working with China's FAST telescope , reported the possibility of having detected artificial (presumably alien) signals, but cautioned that further studies were required to determine if a natural radio interference may be the source. [ 67 ] More recently, on 18 June 2022, Dan Werthimer , chief scientist for several SETI-related projects, reportedly noted, "These signals are from radio interference; they are due to radio pollution from earthlings, not from E.T.". [ 68 ]
Since 2016, University of California Los Angeles (UCLA) undergraduate and graduate students have been participating in radio searches for technosignatures with the Green Bank Telescope. Targets include the Kepler field, TRAPPIST-1 , and solar-type stars. [ 69 ] The search is sensitive to Arecibo-class transmitters located within 420 light years of Earth and to transmitters that are 1,000 times more powerful than Arecibo located within 13,000 light years of Earth. [ 70 ]
The SETI@home project used volunteer computing to analyze signals acquired by the SERENDIP project.
SETI@home was conceived by David Gedye along with Craig Kasnoff and is a popular volunteer computing project that was launched by the Berkeley SETI Research Center at the University of California, Berkeley , in May 1999. It was originally funded by The Planetary Society and Paramount Pictures , and later by the state of California . The project is run by director David P. Anderson and chief scientist Dan Werthimer . Any individual could become involved with SETI research by downloading the Berkeley Open Infrastructure for Network Computing (BOINC) software program, attaching to the SETI@home project, and allowing the program to run as a background process that uses idle computer power. The SETI@home program itself ran signal analysis on a "work unit" of data recorded from the central 2.5 MHz wide band of the SERENDIP IV instrument. After computation on the work unit was complete, the results were then automatically reported back to SETI@home servers at University of California, Berkeley. By June 28, 2009, the SETI@home project had over 180,000 active participants volunteering a total of over 290,000 computers. These computers gave SETI@home an average computational power of 617 teraFLOPS . [ 71 ] In 2004 radio source SHGb02+14a set off speculation in the media that a signal had been detected but researchers noted the frequency drifted rapidly and the detection on three SETI@home computers fell within random chance . [ 72 ] [ 73 ]
By 2010, after 10 years of data collection, SETI@home had listened to that one frequency at every point of over 67 percent of the sky observable from Arecibo with at least three scans (out of the goal of nine scans), which covers about 20 percent of the full celestial sphere. [ 74 ] On March 31, 2020, with 91,454 active users, the project stopped sending out new work to SETI@home users, bringing this particular SETI effort to an indefinite hiatus. [ 75 ]
SETI Network was the only fully operational private search system. [ 76 ] The SETI Net station consisted of off-the-shelf, consumer-grade electronics to minimize cost and to allow this design to be replicated as simply as possible. It had a 3-meter parabolic antenna that could be directed in azimuth and elevation, an LNA that covered 100 MHz of the 1420 MHz spectrum, a receiver to reproduce the wideband audio, and a standard personal computer as the control device and for deploying the detection algorithms. The antenna could be pointed and locked to one sky location in Ra and DEC which enabling the system to integrate on it for long periods. The Wow! signal area was monitored for many long periods. All search data was collected and is available on the Internet archive.
SETI Net started operation in the early 1980s as a way to learn about the science of the search, and developed several software packages for the amateur SETI community. It provided an astronomical clock, a file manager to keep track of SETI data files, a spectrum analyzer optimized for amateur SETI, remote control of the station from the Internet, and other packages.
SETI Net went dark and was decommissioned on 2021-12-04. The collected data is available on their website.
Founded in 1994 in response to the United States Congress cancellation of the NASA SETI program, The SETI League, Incorporated is a membership-supported nonprofit organization with 1,500 members in 62 countries. This grass-roots alliance of amateur and professional radio astronomers is headed by executive director emeritus H. Paul Shuch , the engineer credited with developing the world's first commercial home satellite TV receiver. Many SETI League members are licensed radio amateurs and microwave experimenters. Others are digital signal processing experts and computer enthusiasts.
The SETI League pioneered the conversion of backyard satellite TV dishes 3 to 5 m (10–16 ft) in diameter into research-grade radio telescopes of modest sensitivity. [ 77 ] The organization concentrates on coordinating a global network of small, amateur-built radio telescopes under Project Argus, an all-sky survey seeking to achieve real-time coverage of the entire sky. [ 78 ] Project Argus was conceived as a continuation of the all-sky survey component of the late NASA SETI program (the targeted search having been continued by the SETI Institute's Project Phoenix). There are currently 143 Project Argus radio telescopes operating in 27 countries. Project Argus instruments typically exhibit sensitivity on the order of 10 −23 Watts/square metre, or roughly equivalent to that achieved by the Ohio State University Big Ear radio telescope in 1977, when it detected the landmark "Wow!" candidate signal. [ 79 ]
The name "Argus" derives from the mythical Greek guard-beast who had 100 eyes, and could see in all directions at once. In the SETI context, the name has been used for radio telescopes in fiction (Arthur C. Clarke, " Imperial Earth " ; Carl Sagan, " Contact " ), was the name initially used for the NASA study ultimately known as "Cyclops," and is the name given to an omnidirectional radio telescope design being developed at the Ohio State University. [ 80 ]
While most SETI sky searches have studied the radio spectrum, some SETI researchers have considered the possibility that alien civilizations might be using powerful lasers for interstellar communications at optical wavelengths. [ 81 ] [ 82 ] [ 83 ] The idea was first suggested by R. N. Schwartz and Charles Hard Townes in a 1961 paper published in the journal Nature titled "Interstellar and Interplanetary Communication by Optical Masers". However, the 1971 Cyclops study discounted the possibility of optical SETI, reasoning that construction of a laser system that could outshine the bright central star of a remote star system would be too difficult. In 1983, Townes published a detailed study of the idea in the United States journal Proceedings of the National Academy of Sciences , [ 84 ] which was met with interest by the SETI community.
There are two problems with optical SETI. The first problem is that lasers are highly "monochromatic", that is, they emit light only on one frequency, making it troublesome to figure out what frequency to look for. [ 85 ] However, emitting light in narrow pulses results in a broad spectrum of emission; the spread in frequency becomes higher as the pulse width becomes narrower, making it easier to detect an emission.
The other problem is that while radio transmissions can be broadcast in all directions, lasers are highly directional. Interstellar gas and dust is almost transparent to near infrared, so these signals can be seen from greater distances, but the extraterrestrial laser signals would need to be transmitted in the direction of Earth in order to be detected. [ 86 ] [ 87 ]
Optical SETI supporters have conducted paper studies [ 88 ] of the effectiveness of using contemporary high-energy lasers and a ten-meter diameter mirror as an interstellar beacon. The analysis shows that an infrared pulse from a laser, focused into a narrow beam by such a mirror, would appear thousands of times brighter than the Sun to a distant civilization in the beam's line of fire. The Cyclops study proved incorrect in suggesting a laser beam would be inherently hard to see.
Such a system could be made to automatically steer itself through a target list, sending a pulse to each target at a constant rate. This would allow targeting of all Sun-like stars within a distance of 100 light-years. The studies have also described an automatic laser pulse detector system with a low-cost, two-meter mirror made of carbon composite materials, focusing on an array of light detectors. This automatic detector system could perform sky surveys to detect laser flashes from civilizations attempting contact.
Several optical SETI experiments are now in progress. A Harvard-Smithsonian group that includes Paul Horowitz designed a laser detector and mounted it on Harvard's 155-centimeter (61-inch) optical telescope. This telescope is currently being used for a more conventional star survey, and the optical SETI survey is " piggybacking " on that effort. Between October 1998 and November 1999, the survey inspected about 2,500 stars. Nothing that resembled an intentional laser signal was detected, but efforts continue. The Harvard-Smithsonian group is now working with Princeton University to mount a similar detector system on Princeton's 91-centimeter (36-inch) telescope. The Harvard and Princeton telescopes will be "ganged" to track the same targets at the same time, with the intent being to detect the same signal in both locations as a means of reducing errors from detector noise.
The Harvard-Smithsonian SETI group led by Professor Paul Horowitz built a dedicated all-sky optical survey system along the lines of that described above, featuring a 1.8-meter (72-inch) telescope. The new optical SETI survey telescope is being set up at the Oak Ridge Observatory in Harvard, Massachusetts .
The University of California, Berkeley, home of SERENDIP and SETI@home , is also conducting optical SETI searches and collaborates with the NIROSETI program. The optical SETI program at Breakthrough Listen was initially directed by Geoffrey Marcy , an extrasolar planet hunter, and it involves examination of records of spectra taken during extrasolar planet hunts for a continuous, rather than pulsed, laser signal. This survey uses the Automated Planet Finder 2.4-m telescope at the Lick Observatory , situated on the summit of Mount Hamilton, east of San Jose, California. [ 89 ] The other Berkeley optical SETI effort is being pursued by the Harvard-Smithsonian group and is being directed by Dan Werthimer of Berkeley, who built the laser detector for the Harvard-Smithsonian group. This survey uses a 76-centimeter (30-inch) automated telescope at Leuschner Observatory and an older laser detector built by Werthimer.
The SETI Institute also runs a program called ' Laser SETI ' with an instrument composed of several cameras that continuously survey the entire night sky searching for millisecond singleton laser pulses of extraterrestrial origin. [ 90 ] [ 91 ]
In January 2020, two Pulsed All-sky Near-infrared Optical SETI (PANOSETI) project telescopes were installed in the Lick Observatory Astrograph Dome. The project aims to commence a wide-field optical SETI search and continue prototyping designs for a full observatory. The installation can offer an "all-observable-sky" optical and wide-field near-infrared pulsed technosignature and astrophysical transient search for the northern hemisphere. [ 92 ] [ 83 ]
In May 2017, astronomers reported studies related to laser light emissions from stars, as a way of detecting technology-related signals from an alien civilization . The reported studies included Tabby's Star (designated KIC 8462852 in the Kepler Input Catalog ), an oddly dimming star in which its unusual starlight fluctuations may be the result of interference by an artificial megastructure, such as a Dyson swarm , made by such a civilization. No evidence was found for technology-related signals from KIC 8462852 in the studies. [ 93 ] [ 94 ] [ 95 ]
In a 2020 paper, Berera examined sources of decoherence in the interstellar medium and made the observation that quantum coherence
of photons in certain frequency bands could be sustained to interstellar distances.
It was suggested this would allow for quantum communication at these distances. [ 96 ]
In a 2021 preprint , astronomer Michael Hipke described for the first time how one could search for quantum communication transmissions sent by ETI using existing telescope and receiver technology. He also provides arguments for why future searches of ETI should also target interstellar quantum communication networks. [ 97 ] [ 98 ]
A 2022 paper by Arjun Berera and Jaime Calderón-Figueroa noted that interstellar quantum communication by other civilizations could be possible and may be advantageous, identifying some potential challenges and factors for detecting technosignatures . They may, for example, use X-ray photons for remotely established quantum communication and quantum teleportation as the communication mode. [ 99 ] [ 100 ]
The possibility of using interstellar messenger probes in the search for extraterrestrial intelligence was first suggested by Ronald N. Bracewell in 1960 (see Bracewell probe ), and the technical feasibility of this approach was demonstrated by the British Interplanetary Society's starship study Project Daedalus in 1978. Starting in 1979, Robert Freitas advanced arguments [ 101 ] [ 102 ] [ 103 ] for the proposition that physical space-probes are a superior mode of interstellar communication to radio signals (see Voyager Golden Record ).
In recognition that any sufficiently advanced interstellar probe in the vicinity of Earth could easily monitor the terrestrial Internet , 'Invitation to ETI' was established by Allen Tough in 1996, as a Web-based SETI experiment inviting such spacefaring probes to establish contact with humanity. The project's 100 signatories includes prominent physical, biological, and social scientists, as well as artists, educators, entertainers, philosophers and futurists. H. Paul Shuch , executive director emeritus of The SETI League, serves as the project's Principal Investigator.
Inscribing a message in matter and transporting it to an interstellar destination can be enormously more energy efficient than communication using electromagnetic waves if delays larger than light transit time can be tolerated. [ 104 ] That said, for simple messages such as "hello," radio SETI could be far more efficient. [ 105 ] If energy requirement is used as a proxy for technical difficulty, then a solarcentric Search for Extraterrestrial Artifacts (SETA) [ 106 ] may be a useful supplement to traditional radio or optical searches. [ 107 ] [ 108 ]
Much like the "preferred frequency" concept in SETI radio beacon theory, the Earth-Moon or Sun-Earth libration orbits [ 109 ] might therefore constitute the most universally convenient parking places for automated extraterrestrial spacecraft exploring arbitrary stellar systems. A viable long-term SETI program may be founded upon a search for these objects.
In 1979, Freitas and Valdes conducted a photographic search of the vicinity of the Earth-Moon triangular libration points L 4 and L 5 , and of the solar-synchronized positions in the associated halo orbits, seeking possible orbiting extraterrestrial interstellar probes, but found nothing to a detection limit of about 14th magnitude. [ 109 ] The authors conducted a second, more comprehensive photographic search for probes in 1982 [ 110 ] that examined the five Earth-Moon Lagrangian positions and included the solar-synchronized positions in the stable L4/L5 libration orbits, the potentially stable nonplanar orbits near L1/L2, Earth-Moon L 3 , and also L 2 in the Sun-Earth system. Again no extraterrestrial probes were found to limiting magnitudes of 17–19th magnitude near L3/L4/L5, 10–18th magnitude for L 1 / L 2 , and 14–16th magnitude for Sun-Earth L 2 .
In June 1983, Valdes and Freitas used the 26 m radiotelescope at Hat Creek Radio Observatory to search for the tritium hyperfine line at 1516 MHz from 108 assorted astronomical objects, with emphasis on 53 nearby stars including all visible stars within a 20 light-year radius. The tritium frequency was deemed highly attractive for SETI work because (1) the isotope is cosmically rare, (2) the tritium hyperfine line is centered in the SETI water hole region of the terrestrial microwave window, and (3) in addition to beacon signals, tritium hyperfine emission may occur as a byproduct of extensive nuclear fusion energy production by extraterrestrial civilizations. The wideband- and narrowband-channel observations achieved sensitivities of 5–14 × 10 −21 W/m 2 /channel and 0.7–2 × 10 −24 W/m 2 /channel, respectively, but no detections were made. [ 111 ]
Others have speculated, that we might find traces of past civilizations in our very own Solar System, on planets like Venus or Mars , although the traces would be found most likely underground. [ 112 ] [ 113 ]
Technosignatures, including all signs of technology, are a recent avenue in the search for extraterrestrial intelligence. [ 114 ] [ 3 ] Technosignatures may originate from various sources, from megastructures such as Dyson spheres [ 115 ] [ 116 ] and space mirrors or space shaders [ 117 ] to the atmospheric contamination created by an industrial civilization, [ 118 ] or city lights on extrasolar planets, and may be detectable in the future with large hypertelescopes . [ 119 ]
Technosignatures can be divided into three broad categories: astroengineering projects, signals of planetary origin, and spacecraft within and outside the Solar System .
An astroengineering installation such as a Dyson sphere , designed to convert all of the incident radiation of its host star into energy, could be detected through the observation of an infrared excess from a solar analog star, [ 120 ] or by the star's apparent disappearance in the visible spectrum over several years. [ 121 ] After examining some 100,000 nearby large galaxies, a team of researchers has concluded that none of them display any obvious signs of highly advanced technological civilizations. [ 122 ] [ 123 ]
Another hypothetical form of astroengineering, the Shkadov thruster , moves its host star by reflecting some of the star's light back on itself, and would be detected by observing if its transits across the star abruptly end with the thruster in front. [ 124 ] Asteroid mining within the Solar System is also a detectable technosignature of the first kind. [ 125 ]
Individual extrasolar planets can be analyzed for signs of technology. Avi Loeb of the Center for Astrophysics | Harvard & Smithsonian has proposed that persistent light signals on the night side of an exoplanet can be an indication of the presence of cities and an advanced civilization. [ 126 ] [ 127 ] In addition, the excess infrared radiation [ 119 ] [ 128 ] and chemicals [ 129 ] [ 130 ] produced by various industrial processes or terraforming efforts [ 131 ] may point to intelligence.
Light and heat detected from planets need to be distinguished from natural sources to conclusively prove the existence of civilization on a planet. However, as argued by the Colossus team, [ 132 ] a civilization heat signature should be within a "comfortable" temperature range, like terrestrial urban heat islands , i.e., only a few degrees warmer than the planet itself. In contrast, such natural sources as wild fires, volcanoes, etc. are significantly hotter, so they will be well distinguished by their maximum flux at a different wavelength.
Other than astroengineering, technosignatures such as artificial satellites around exoplanets , particularly such in geostationary orbit , might be detectable even with today's technology and data, and would allow, similar to fossils on Earth, to find traces of extrasolar life from long ago. [ 133 ]
Extraterrestrial craft are another target in the search for technosignatures. Magnetic sail interstellar spacecraft should be detectable over thousands of light-years of distance through the synchrotron radiation they would produce through interaction with the interstellar medium ; other interstellar spacecraft designs may be detectable at more modest distances. [ 134 ] In addition, robotic probes within the Solar System are also being sought with optical and radio searches. [ 135 ] [ 136 ]
For a sufficiently advanced civilization, hyper energetic neutrinos from Planck scale accelerators should be detectable at a distance of many Mpc. [ 137 ]
A notable advancement in technosignature detection is the development of an algorithm for signal reconstruction in zero-knowledge one-way communication channels. [ 138 ] This algorithm decodes signals from unknown sources without prior knowledge of the encoding scheme, using principles from Algorithmic Information Theory to identify the geometric and topological dimensions of the encoding space. It successfully reconstructed the Arecibo message despite significant noise. The work establishes a connection between syntax and semantics in SETI and technosignature detection, enhancing fields like cryptography and Information Theory . [ 139 ]
Based on fractal theory and the Weierstrass function , a known fractal, another method authored by the same group called fractal messaging offers a framework for space-time scale-free communication. This method leverages properties of self-similarity and scale invariance, enabling spatio-temporal scale-independent and parallel infinite-frequency communication. It also embodies the concept of sending a self-encoding/self-decoding signal as a mathematical formula, equivalent to self-executable computer code that unfolds to read a message at all possible time scales and in all possible channels simultaneously. [ 140 ]
Italian physicist Enrico Fermi suggested in the 1950s that if technologically advanced civilizations are common in the universe, then they should be detectable in one way or another. According to those who were there, Fermi either asked "Where are they?" or "Where is everybody?" [ 141 ]
The Fermi paradox is commonly understood as asking why extraterrestrials have not visited Earth, [ 142 ] but the same reasoning applies to the question of why signals from extraterrestrials have not been heard. The SETI version of the question is sometimes referred to as "the Great Silence".
The Fermi paradox can be stated more completely as follows:
The size and age of the universe incline us to believe that many technologically advanced civilizations must exist. However, this belief seems logically inconsistent with our lack of observational evidence to support it. Either (1) the initial assumption is incorrect and technologically advanced intelligent life is much rarer than we believe, or (2) our current observations are incomplete, and we simply have not detected them yet, or (3) our search methodologies are flawed and we are not searching for the correct indicators, or (4) it is the nature of intelligent life to destroy itself.
There are multiple explanations proposed for the Fermi paradox, [ 143 ] ranging from analyses suggesting that intelligent life is rare (the " Rare Earth hypothesis "), to analyses suggesting that although extraterrestrial civilizations may be common, they would not communicate with us, would communicate in a way we have not discovered yet, could not travel across interstellar distances, or destroy themselves before they master the technology of either interstellar travel or communication.
The German astrophysicist and radio astronomer Sebastian von Hoerner suggested [ 144 ] that the average duration of civilization was 6,500 years. After this time, according to him, it disappears for external reasons (the destruction of life on the planet, the destruction of only rational beings) or internal causes (mental or physical degeneration). According to his calculations, on a habitable planet (one in three million stars) there is a sequence of technological species over a time distance of hundreds of millions of years, and each of them "produces" an average of four technological species. With these assumptions, the average distance between civilizations in the Milky Way is 1,000 light years. [ 145 ] [ 146 ] [ 147 ]
Science writer Timothy Ferris has posited that since galactic societies are most likely only transitory, an obvious solution is an interstellar communications network, or a type of library consisting mostly of automated systems. They would store the cumulative knowledge of vanished civilizations and communicate that knowledge through the galaxy. Ferris calls this the "Interstellar Internet", with the various automated systems acting as network "servers". If such an Interstellar Internet exists, the hypothesis states, communications between servers are mostly through narrow-band, highly directional radio or laser links. Intercepting such signals is, as discussed earlier, very difficult. However, the network could maintain some broadcast nodes in hopes of making contact with new civilizations.
Although somewhat dated in terms of "information culture" arguments, not to mention the obvious technological problems of a system that could work effectively for billions of years and requires multiple lifeforms agreeing on certain basics of communications technologies, this hypothesis is actually testable (see below).
A significant problem is the vastness of space. Despite piggybacking on the world's most sensitive radio telescope, astronomer and initiator of SERENDIP Charles Stuart Bowyer noted the then world's largest instrument could not detect random radio noise emanating from a civilization like ours, [ citation needed ] which has been leaking radio and TV signals for less than 100 years. [ 148 ] For SERENDIP and most other SETI projects to detect a signal from an extraterrestrial civilization, the civilization would have to be beaming a powerful signal directly at us. It also means that Earth civilization will only be detectable within a distance of 100 light-years. [ 149 ]
The International Academy of Astronautics (IAA) has a long-standing SETI Permanent Study Group (SPSG, formerly called the IAA SETI Committee), which addresses matters of SETI science , technology , and international policy . The SPSG meets in conjunction with the International Astronautical Congress (IAC) , held annually at different locations around the world, and sponsors two SETI Symposia at each IAC. In 2005, the IAA established the SETI: Post-Detection Science and Technology Taskgroup (chairman, Professor Paul Davies ) "to act as a Standing Committee to be available to be called on at any time to advise and consult on questions stemming from the discovery of a putative signal of extraterrestrial intelligent (ETI) origin." [ 150 ]
However, the protocols mentioned apply only to radio SETI rather than for METI ( Active SETI ). [ 151 ] The intention for METI is covered under the SETI charter "Declaration of Principles Concerning Sending Communications with Extraterrestrial Intelligence".
In October 2000 astronomers Iván Almár and Jill Tarter presented a paper to The SETI Permanent Study Group in Rio de Janeiro, Brazil which proposed a scale (modelled after the Torino scale ) which is an ordinal scale between zero and ten that quantifies the impact of any public announcement regarding evidence of extraterrestrial intelligence; [ 152 ] the Rio scale has since inspired the 2005 San Marino Scale (in regard to the risks of transmissions from Earth) and the 2010 London Scale (in regard to the detection of extraterrestrial life). [ 153 ] The Rio scale itself was revised in 2018. [ 154 ]
The SETI Institute does not officially recognize the Wow! signal as of extraterrestrial origin as it was unable to be verified, although in a 2020 Twitter post the organization stated that ''an astronomer might have pinpointed the host star''. [ 155 ] The SETI Institute has also publicly denied that the candidate signal Radio source SHGb02+14a is of extraterrestrial origin. [ 156 ] [ 157 ] Although other volunteering projects such as Zooniverse credit users for discoveries, there is currently no crediting or early notification by SETI@Home following the discovery of a signal.
Some people, including Steven M. Greer , [ 158 ] have expressed cynicism that the general public might not be informed in the event of a genuine discovery of extraterrestrial intelligence due to significant vested interests. Some, such as Bruce Jakosky [ 159 ] have also argued that the official disclosure of extraterrestrial life may have far reaching and as yet undetermined implications for society, particularly for the world's religions .
Active SETI , also known as messaging to extraterrestrial intelligence (METI), consists of sending signals into space in the hope that they will be detected by an alien intelligence.
In November 1974, a largely symbolic attempt was made at the Arecibo Observatory to send a message to other worlds. Known as the Arecibo Message , it was sent towards the globular cluster M13 , which is 25,000 light-years from Earth. Further IRMs Cosmic Call , Teen Age Message , Cosmic Call 2 , and A Message From Earth were transmitted in 1999, 2001, 2003 and 2008 from the Evpatoria Planetary Radar.
Whether or not to attempt to contact extraterrestrials has attracted significant academic debate in the fields of space ethics and space policy . [ 160 ] [ 161 ] [ 162 ] Physicist Stephen Hawking , in his book A Brief History of Time , suggests that "alerting" extraterrestrial intelligences to our existence is foolhardy, citing humankind's history of treating its own kind harshly in meetings of civilizations with a significant technology gap, e.g., the extermination of Tasmanian aborigines. He suggests, in view of this history, that we "lay low". [ 163 ] In one response to Hawking, in September 2016, astronomer Seth Shostak sought to allay such concerns. [ 164 ] Astronomer Jill Tarter also disagrees with Hawking, arguing that aliens developed and long-lived enough to communicate and travel across interstellar distances would have evolved a cooperative and less violent intelligence. She however thinks it is too soon for humans to attempt active SETI and that humans should be more advanced technologically first but keep listening in the meantime. [ 165 ]
As various SETI projects have progressed, some have criticized early claims by researchers as being too "euphoric". For example, Peter Schenkel, while remaining a supporter of SETI projects, wrote in 2006 that:
[i]n light of new findings and insights, it seems appropriate to put excessive euphoria to rest and to take a more down-to-earth view [...] We should quietly admit that the early estimates—that there may be a million, a hundred thousand, or ten thousand advanced extraterrestrial civilizations in our galaxy—may no longer be tenable. [ 1 ]
Critics claim that the existence of extraterrestrial intelligence has no good Popperian criteria for falsifiability , as explained in a 2009 editorial in Nature , which said:
Seti... has always sat at the edge of mainstream astronomy. This is partly because, no matter how scientifically rigorous its practitioners try to be, SETI can't escape an association with UFO believers and other such crackpots. But it is also because SETI is arguably not a falsifiable experiment. Regardless of how exhaustively the Galaxy is searched, the null result of radio silence doesn't rule out the existence of alien civilizations. It means only that those civilizations might not be using radio to communicate. [ 6 ]
Nature added that SETI was "marked by a hope, bordering on faith" that aliens were aiming signals at us, that a hypothetical alien SETI project looking at Earth with "similar faith" would be "sorely disappointed", despite our many untargeted radar and TV signals, and our few targeted Active SETI radio signals denounced by those fearing aliens, and that it had difficulties attracting even sympathetic working scientists and government funding because it was "an effort so likely to turn up nothing". [ 6 ]
However, Nature also added, "Nonetheless, a small SETI effort is well worth supporting, especially given the enormous implications if it did succeed" and that "happily, a handful of wealthy technologists and other private donors have proved willing to provide that support". [ 6 ]
Supporters of the Rare Earth Hypothesis argue that advanced lifeforms are likely to be very rare, and that, if that is so, then SETI efforts will be futile. [ 166 ] [ 167 ] [ 168 ] However, the Rare Earth Hypothesis itself faces many criticisms . [ 168 ]
In 1993, Roy Mash stated that "Arguments favoring the existence of extraterrestrial intelligence nearly always contain an overt appeal to big numbers, often combined with a covert reliance on generalization from a single instance" and concluded that "the dispute between believers and skeptics is seen to boil down to a conflict of intuitions which can barely be engaged, let alone resolved, given our present state of knowledge". [ 169 ] In response, in 2012, Milan M. Ćirković , then research professor at the Astronomical Observatory of Belgrade and a research associate of the Future of Humanity Institute at the University of Oxford , [ 170 ] said that Mash was unrealistically over-reliant on excessive abstraction that ignored the empirical information available to modern SETI researchers. [ 171 ]
George Basalla , Emeritus Professor of History at the University of Delaware , [ 172 ] is a critic of SETI who argued in 2006 that "extraterrestrials discussed by scientists are as imaginary as the spirits and gods of religion or myth", [ 173 ] [ 174 ] and was in turn criticized by Milan M. Ćirković [ 170 ] for, among other things, being unable to distinguish between "SETI believers" and "scientists engaged in SETI", who are often sceptical (especially about quick detection), such as Freeman Dyson and, at least in their later years, Iosif Shklovsky and Sebastian von Hoerner, and for ignoring the difference between the knowledge underlying the arguments of modern scientists and those of ancient Greek thinkers. [ 174 ]
Massimo Pigliucci , Professor of Philosophy at CUNY – City College , [ 175 ] asked in 2010 whether SETI is "uncomfortably close to the status of pseudoscience " due to the lack of any clear point at which negative results cause the hypothesis of Extraterrestrial Intelligence to be abandoned, [ 176 ] before eventually concluding that SETI is "almost-science", which is described by Milan M. Ćirković [ 170 ] as Pigliucci putting SETI in "the illustrious company of string theory , interpretations of quantum mechanics , evolutionary psychology and history (of the 'synthetic' kind done recently by Jared Diamond )", while adding that his justification for doing so with SETI "is weak, outdated, and reflecting particular philosophical prejudices similar to the ones described above in Mash [ 169 ] and Basalla [ 173 ] ". [ 177 ]
Richard Carrigan , a particle physicist at the Fermi National Accelerator Laboratory near Chicago, Illinois , suggested that passive SETI could also be dangerous and that a signal released onto the Internet could act as a computer virus . [ 178 ] Computer security expert Bruce Schneier dismissed this possibility as a "bizarre movie-plot threat". [ 179 ]
Ufologist Stanton Friedman has often criticized SETI researchers for, among other reasons, what he sees as their unscientific criticisms of Ufology, [ 180 ] [ 181 ] but, unlike SETI, Ufology has generally not been embraced by academia as a scientific field of study, [ 182 ] [ 183 ] and it is usually characterized as a partial [ 184 ] or total [ 185 ] [ 186 ] pseudoscience . In a 2016 interview, Jill Tarter pointed out that it is still a misconception that SETI and UFOs are related. [ 187 ] She states, "SETI uses the tools of the astronomer to attempt to find evidence of somebody else's technology coming from a great distance. If we ever claim detection of a signal, we will provide evidence and data that can be independently confirmed. UFOs—none of the above." [ 187 ] The Galileo Project headed by Harvard astronomer Avi Loeb is one of the few scientific efforts to study UFOs or UAPs. [ 188 ] Loeb criticized that the study of UAP is often dismissed and not sufficiently studied by scientists and should shift from "occupying the talking points of national security administrators and politicians" to the realm of science. [ 189 ] The Galileo Project's position after the publication of the 2021 UFO Report by the U.S. Intelligence community is that the scientific community needs to "systematically, scientifically and transparently look for potential evidence of extraterrestrial technological equipment". [ 190 ] | https://en.wikipedia.org/wiki/Search_for_extraterrestrial_intelligence |
In molecular modelling , docking is a method which predicts the preferred orientation of one molecule to another when bound together in a stable complex . In the case of protein docking , the search space consists of all possible orientations of the protein with respect to the ligand . Flexible docking in addition considers all possible conformations of the protein paired with all possible conformations of the ligand. [ 1 ]
With present computing resources , it is impossible to exhaustively explore these search spaces; instead, there are many strategies which attempt to sample the search space with optimal efficiency. Most docking programs in use account for a flexible ligand, and several attempt to model a flexible protein receptor. Each "snapshot" of the pair is referred to as a pose.
In this approach, proteins are typically held rigid, and the ligand is allowed to freely explore their conformational space. The generated conformations are then docked successively into the protein, and an MD simulation consisting of a simulated annealing protocol is performed. This is usually supplemented with short MD energy minimization steps, and the energies determined from the MD runs are used for ranking the overall scoring. Although this is a computer-expensive method (involving potentially hundreds of MD runs), it has some advantages: for example, no specialized energy/scoring functions are required. MD force-fields can typically be used to find poses that are reasonable and can be compared with experimental structures.
The Distance Constrained Essential Dynamics method (DCED) has been used to generate multiple structures for docking, called eigenstructures. This approach, although avoiding most of the costly MD calculations, can capture the essential motions involved in a flexible receptor, representing a form of coarse-grained dynamics. [ 2 ]
The most common technique used in many docking programs, shape-complementarity methods focus on the match between the receptor and the ligand in order to find an optimal pose. Programs include DOCK , [ 3 ] FRED, [ 4 ] GLIDE, [ 5 ] SURFLEX, [ 6 ] eHiTS [ 7 ] and many more. Most methods describe the molecules in terms of a finite number of descriptors that include structural complementarity and binding complementarity. Structural complementarity is mostly a geometric description of the molecules, including solvent-accessible surface area , overall shape and geometric constraints between atoms in the protein and ligand. Binding complementarity takes into account features like hydrogen bonding interactions, hydrophobic contacts and van der Waals interactions to describe how well a particular ligand will bind to the protein. Both kinds of descriptors are conveniently represented in the form of structural templates which are then used to quickly match potential compounds (either from a database or from the user-given inputs) that will bind well at the active site of the protein. Compared to the all-atom molecular dynamics approaches, these methods are very efficient in finding optimal binding poses for the protein and ligand.
Two of the most used docking programs belong to this class: GOLD [ 8 ] and AutoDock . [ 9 ] Genetic algorithms allow the exploration of a large conformational space – which is basically spanned by the protein and ligand jointly in this case – by representing each spatial arrangement of the pair as a “gene” with a particular energy. The entire genome thus represents the complete energy landscape which is to be explored. The simulation of the evolution of the genome is carried out by cross-over techniques similar to biological evolution , where random pairs of individuals (conformations) are “mated” with the possibility for a random mutation in the offspring. These methods have proven very useful in sampling the vast state-space while maintaining closeness to the actual process involved.
Although genetic algorithms are quite successful in sampling the large conformational space, many docking programs require the protein to remain fixed, while allowing only the ligand to flex and adjust to the active site of the protein. Genetic algorithms also require multiple runs to obtain reliable answers regarding ligands that may bind to the protein. The time it takes to typically run a genetic algorithm in order to allow a proper pose may be longer, hence these methods may not be as efficient as shape complementarity-based approaches in screening large databases of compounds. Recent improvements in using grid-based evaluation of energies, limiting the exploration of the conformational changes at only local areas (active sites) of interest, and improved tabling methods have significantly enhanced the performance of genetic algorithms and made them suitable for virtual screening applications. | https://en.wikipedia.org/wiki/Searching_the_conformational_space_for_docking |
Seascape ecology is a scientific discipline that deals with the causes and ecological consequences of spatial pattern in the marine environment , drawing heavily on conceptual and analytical frameworks developed in terrestrial landscape ecology . [ 1 ]
Seascape ecology, the application of landscape ecology concepts to the marine environment [ 2 ] has been slowly emerging since the 1970s, [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] yielding new ecological insights and showing growing potential to support the development of ecologically meaningful science-based management practices. [ 8 ] [ 9 ] [ 10 ] [ 11 ] For marine systems, the application of landscape ecology came about through a recognition that many of the concepts developed in the theory of island biogeography [ 12 ] [ 13 ] and the study of patch dynamics (precursors to modern landscape ecology) could be applicable to a range of marine environments from plankton patches [ 14 ] to patch reefs , [ 15 ] inter-tidal mussel beds [ 16 ] and seagrass meadows . [ 17 ] [ 18 ]
Progress in the ecological understanding of spatial patterning was not confined to shallow seafloor environments. For the open ocean , advances in ocean observing systems since the 1970s have allowed scientists to map, classify, quantify and track dynamic spatial structure in the form of eddies , surface roughness , currents , runoff plumes, ice , temperature fronts and plankton patches using oceanographic technologies – a theme increasingly referred to as pelagic seascape ecology. [ 19 ] [ 20 ] [ 21 ] Subsurface structures too, such as internal waves , thermoclines , haloclines , boundary layers and stratification resulting in distinct layering of organisms, is increasingly being mapped and modelled in multiple dimensions.
Like landscape ecologists , seascape ecologists are interested in the spatially explicit geometry of patterns and the relationships between pattern, ecological processes and environmental change. A central tenet in landscape ecology is that patch context matters, where local conditions are influenced by attributes of the surroundings. For instance, the physical arrangement of objects in space, and their location relative to other things, influences how they function. [ 22 ]
A landscape ecologist will ask different questions focused at different scales than other scientists, such as: What are the ecological consequences of different shaped patches, patch size, quality, edge geometry, spatial arrangement and diversity of patches across the landscape? At what scale(s) is structure most influential? How do landscape patterns influence the way that animals find food, evade predators and interact with competitors? How does human activity alter the structure and function of landscapes?
Several guiding principles that exist at the core of landscape ecology have made major contributions to terrestrial landscape planning and conservation, but in marine systems our understanding is still in its infancy. The first book on seascape ecology was published in December 2018. [ 23 ]
Seascapes are defined broadly as spatially heterogeneous and dynamic spaces that can be delineated at a wide range of scales in time and space. With regard to seascapes defined by sampling units, a 1 m 2 quadrant can be a valid seascape sample unit (SSU), just as can a 1 km 2 analytical window in a geographical information system. The wide diversity of possible focal scales in marine ecology means that the term seascape cannot be used as an indication of scale, or a level of organization.
The sea exhibits complex spatial patterning that can be mapped and quantified, such as gradients in plant communities across tidal saltmarshes or the intricate mosaics of patches typical of coral reefs . [ 24 ] In the open ocean too, dynamic spatial structure in the form of water currents, eddies, temperature fronts and plankton patches can be measured readily. [ 25 ] [ 26 ] Physical processes such as storms dramatically influence spatial patterning in the environment and human activity can also directly create patch structure, modify mosaic composition and even completely remove elements of the seascape. Furthermore, climate-change induced shifts in species related to water temperature change and sea level rise are driving a gradual reconfiguration of the geography of species and habitats.
The patterns revealed by remote sensing devices are most often mapped and represented using two types of model: (1) collections of discrete patches forming mosaics e.g. as represented in two-dimensional benthic habitat map, or (2) continuously varying gradients in three-dimensional terrain models e.g. in remotely sensed Bathymetric data. [ 27 ] [ 28 ] In landscape ecology, patches can be classified into a binary patch-matrix model based on island biogeography theory where a focal habitat patch type (e.g. seagrasses) is surrounded by an inhospitable matrix (e.g. sand), or a patch-mosaic of interconnected patches, where the interactions of the parts influence the ecological function of the whole mosaic. Both patch and gradient models have provided important insights into the spatial ecology of marine species and biodiversity.
Scale, the spatial or temporal dimensions of a phenomenon, is central to seascape ecology and the topic permeates all applications of a seascape ecology approach from conceptual models through to design of sampling, analyses and interpretation of results. [ 29 ] Species and life-stage responses to patchiness and gradients in environmental structure are likely to be scale dependent, therefore, scale selection is an important task in any ecological study. Seascape ecology acknowledges that decisions made for scaling ecological studies influence our perspective and ultimately our understanding of ecological patterns and processes. [ 30 ] Historically, marine scientists have played a significant role in communicating the importance of scale in ecology [ 31 ]
In 1963, a physical oceanographer, Henry Stommel , published a conceptual diagram that was to have a profound effect on all of the environmental sciences. [ 32 ] The diagram [ 33 ] depicted variation in sea level height at spatial scales from centimeters to that of the planet and at time scales from seconds to tens of millennia. [ 34 ] The oceanographer John Steele (1978) adapted the Stommel diagram to depict the spatial and temporal scales of patchiness in phytoplankton , zooplankton and fish .
Measuring habitat structure at multiple scales is typical in seascape ecology, particularly where a single meaningful scale is not known or not meaningful to the ecological process of interest. Multi-scale measurements have been used to discover the scale at which populations are associated with key habitat features. [ 35 ] [ 36 ] With regard to scaling seascapes, one approach is to select spatial and temporal scales to be ecologically meaningful to the organism’s movements or other processes of interest. [ 37 ]
The absence of information, or the lack of continuous observation on the way animals use space through time, can all too often result in insufficient consideration of seascape context potentially resulting in misleading conclusions on the primary drivers of ecological patterns and processes. Pittman and McAlpine (2003) [ 38 ] offer a multi-scale framework for scaling ecological studies that integrates hierarchy theory with movement ecology and the concept of ecological neighborhoods. [ 39 ] Here the focal scale is guided by the spatial and temporal scales relevant to an ecological process of interest. The focal scale is nested within a spatial hierarchy that incorporates patterns and processes at both finer and broader scales [ 40 ]
Understanding seascape ecology is important in the study of juvenile fish development, particularly when examining estuaries . Prior investigations into nursery function have predominantly focused on individual habitats or specific species, lacking a comprehensive understanding of the intricate relationships within and among diverse habitat patches. [ 41 ] [ 42 ] [ 43 ] This is of concern due to the frequent oncogenic shifts [ clarification needed ] observed in many juvenile species, which causes them to use multiple habitats throughout their developmental phases. Various habitats, including mangrove roots, macroalgae, seagrasses, and others, offer distinct services and resources crucial for the development of juvenile fish. The failure to consider the entirety of habitats occupied during development may impede ecologists and resource managers in effectively evaluating the conservation value and priority of estuarine regions. | https://en.wikipedia.org/wiki/Seascape_ecology |
A seashell or sea shell , also known simply as a shell , is a hard, protective outer layer usually created by an animal or organism that lives in the sea. Most seashells are made by mollusks , such as snails , clams , and oysters to protect their soft insides. [ 1 ] Empty seashells are often found washed up on beaches by beachcombers . The shells are empty because the animal has died and the soft parts have decomposed or been eaten by another organism.
A seashell is usually the exoskeleton of an invertebrate (an animal without a backbone), and is typically composed of calcium carbonate [ 1 ] or chitin. Most shells that are found on beaches are the shells of marine mollusks, partly because these shells are usually made of calcium carbonate, and endure better than shells made of chitin.
Apart from mollusk shells , other shells that can be found on beaches are those of barnacles , horseshoe crabs and brachiopods . Marine annelid worms in the family Serpulidae create shells which are tubes made of calcium carbonate cemented onto other surfaces. The shells of sea urchins are called " tests ", and the moulted shells of crabs and lobsters are exuviae . While most seashells are external, some cephalopods have internal shells.
Seashells have been used by humans for many different purposes throughout history and prehistory. However, seashells are not the only kind of shells; in various habitats, there are shells from freshwater animals such as freshwater mussels and freshwater snails , and shells of land snails .
When the word "seashells" refers only to the shells of marine mollusks, then studying seashells is part of conchology . Conchologists or serious collectors who have a scientific bias are in general careful not to disturb living populations and habitats: even though they may collect a few live animals, most responsible collectors do not often over-collect or otherwise disturb ecosystems.
The study of mollusks (including their shells) is known as malacology ; a person who studies mollusks is known as a malacologist .
Seashells are commonly found in beach drift, which is natural detritus deposited along strandlines on beaches by the waves and the tides . Shells are very often washed up onto a beach empty and clean, the animal having already died.
Empty seashells are often picked up by beachcombers. However, the majority of seashells which are offered for sale commercially have been collected alive (often in bulk) and then killed and cleaned, specifically for the commercial trade. [ 2 ] This type of large-scale exploitation can sometimes have a strong negative impact on local ecosystems , and sometimes can significantly reduce the distribution of rare species .
Seashells are created by the molluscs that use them for protection. [ 3 ] Molluscs have an outside layer of tissues on their bodies – the mantle – which creates the shell material and which connects the shell to the mollusc. The specialized cells in the mantle form the shell using different minerals and proteins . [ 3 ] The proteins are then used to create the framework that supports the growing shell. Calcium carbonate is the main compound of shell structure, aiding in adhesion . [ 3 ]
The word seashell is often used to mean only the shell of a marine mollusk . Marine mollusk shells that are familiar to beachcombers and thus most likely to be called "seashells" are the shells of marine species of bivalves (or clams ), gastropods (or snails ), scaphopods (or tusk shells ), polyplacophorans (or chitons ), and cephalopods (such as nautilus and spirula ). These shells are very often the most commonly encountered, both in the wild, and for sale as decorative objects.
Marine species of gastropods and bivalves are more numerous than land and freshwater species, and the shells are often larger and more robust. The shells of marine species also often have more sculpture and more color, although this is by no means always the case.
In the tropical and sub-tropical areas of the planet, there are far more species of colorful, large, shallow water shelled marine mollusks than there are in the temperate zones and the regions closer to the poles.
Although there are a number of species of shelled mollusks that are quite large, there are vast numbers of extremely small species too, see micromollusks .
Not all mollusks are marine. There are numerous land and freshwater mollusks, see for example snail and freshwater bivalves . In addition, not all mollusks have an external shell: some mollusks such as some cephalopods (squid and octopuses) have an internal shell, and many mollusks have no shell, see for example slug and nudibranch .
Bivalves are often the most common seashells that wash up on large sandy beaches or in sheltered lagoons . They can sometimes be extremely numerous. Very often the two valves become separated.
There are more than 15,000 species of bivalves that live in both marine and freshwater. Examples of bivalves are clams, scallops, mussels, and oysters. The majority of bivalves consist of two identical shells that are held together by a flexible hinge. The animal's body is held protectively inside these two shells. Bivalves that do not have two shells either have one shell or they lack a shell altogether. The shells are made of calcium carbonate and are formed in layers by secretions from the mantle. Bivalves, also known as pelecypods, are mostly filter feeders; through their gills, they draw in water, in which is trapped tiny food particles. Some bivalves have eyes and an open circulatory system. Bivalves are used all over the world as food and as a source of pearls. The larvae of some freshwater mussels can be dangerous to fish and can bore through wood.
Shell Beach, Western Australia , is a beach which is entirely made up of the shells of the cockle Fragum erugatum .
Certain species of gastropod seashells (the shells of sea snails ) can sometimes be common, washed up on sandy beaches, and also on beaches that are surrounded by rocky marine habitat.
Chiton plates or valves often wash up on beaches in rocky areas where chitons are common. Chiton shells, which are composed of eight separate plates and a girdle, usually come apart not long after death, so they are almost always found as disarticulated plates. Plates from larger species of chitons are sometimes known as "butterfly shells" because of their shape.
Only a few species of cephalopods have shells (either internal or external) that are sometimes found washed up on beaches.
Some cephalopods such as Sepia , the cuttlefish, have a large internal shell, the cuttlefish bone , and this often washes up on beaches in parts of the world where cuttlefish are common.
Spirula spirula is a deep water squid-like cephalopod. It has an internal shell which is small (about 1 in or 24 mm) but very light and buoyant. This chambered shell floats very well and therefore washes up easily and is familiar to beachcombers in the tropics.
Nautilus is the only genus of cephalopod that has a well-developed external shell. Females of the cephalopod genus Argonauta create a papery egg case which sometimes washes up on tropical beaches and is referred to as a "paper nautilus".
The largest group of shelled cephalopods, the ammonites , are extinct, but their shells are very common in certain areas as fossils .
Empty molluscan seashells are a sturdy, and usually readily available, "free" resource which is often easily found on beaches, in the intertidal zone , and in the shallow subtidal zone. As such they are sometimes used second-hand by animals other than humans for various purposes, including for protection (as in hermit crabs ) and for construction.
There are numerous popular books and field guides on the subject of shell-collecting. Although there are a number of books about land and freshwater mollusks, the majority of popular books emphasize, or focus exclusively on, the shells of marine mollusks. Both the science of studying mollusk shells and the hobby of collecting and classifying them are known as conchology . The line between professionals and amateur enthusiasts is often not well defined in this subject, because many amateurs have contributed to, and continue to contribute to, conchology and the larger science of malacology . Many shell collectors belong to "shell clubs" where they can meet others who share their interests. A large number of amateurs collect the shells of marine mollusks, and this is partly because many shells wash up empty on beaches, or live in the intertidal or sub-tidal zones, and are therefore easily found and preserved without much in the way of specialized equipment or expensive supplies. Some shell collectors find their own material and keep careful records, or buy only "specimen shells", which means shells which have full collecting data : information including how, when, where, in what habitat, and by whom, the shells were collected. On the other hand, some collectors buy the more widely available commercially imported exotic shells, the majority of which have very little data, or none at all. To museum scientists, having full collecting data (when, where, and by whom it was collected) with a specimen is far more important than having the shell correctly identified. Some owners of shell collections hope to be able to donate their collection to a major natural history or zoology museum at some point, however, shells with little or no collecting data are usually of no value to science, and are likely not to be accepted by a major museum. Apart from any damage to the shell that may have happened before it was collected, shells can also suffer damage when they are stored or displayed. For an example of one rather serious kind of damage see Byne's disease .
There are a number of clubs or societies which consist of people who are united by a shared interest in shells. In the US, these clubs are more common in southerly coastal areas, such as Florida and California , where the marine fauna is rich in species.
Seashells are usually identified by consulting general or regional shell-collecting field guides , and specific scientific books on different taxa of shell-bearing mollusks ( monographs ) or "iconographies" (limited text – mainly photographs or other illustrations). (For a few titles on this subject in the US, see the list of books at the foot of this article.) Identifications to the species level are generally achieved by examining illustrations and written descriptions, rather than by the use of Identification keys , as is often the case in identifying plants and other phyla of invertebrates. The construction of functional keys for the identification of the shells of marine mollusks to the species level can be very difficult, because of the great variability within many species and families. The identification of certain individual species is often very difficult, even for a specialist in that particular family. Some species cannot be differentiated on the basis of shell character alone.
Numerous smaller and more obscure mollusk species (see micromollusk ) are yet to be discovered and named. In other words, they have not yet been differentiated from similar species and assigned scientific (binomial) names in articles in journals recognized by the International Commission on Zoological Nomenclature (ICZN). Large numbers of new species are published in the scientific literature each year. There are currently an estimated 100,000 species of mollusks worldwide.
The term seashell is also applied loosely to mollusk shells that are not of marine origin, for example by people walking the shores of lakes and rivers using the term for the freshwater mollusk shells they encounter. Seashells purchased from tourist shops or dealers may include various freshwater and terrestrial shells as well. Non-marine items offered may include large and colorful tropical land snail shells, freshwater apple snail shells, and pearly freshwater unionid mussel shells. This can be confusing to collectors, as non-marine shells are often not included in their reference books.
Seashells have been used as a medium of exchange in various places, including many Indian Ocean and Pacific Ocean islands, also in North America, Africa and the Caribbean.
Seashells have often been used as tools , because of their strength and the variety of their shapes.
Because seashells are in some areas a readily available bulk source of calcium carbonate, shells such as oyster shells are sometimes used as soil conditioners in horticulture . The shells are broken or ground into small pieces in order to have the desired effect of raising the pH and increasing the calcium content in the soil.
Seashells have played a part in religion and spirituality, sometimes even as ritual objects.
Seashells have been used as musical instruments, wind instruments for many hundreds if not thousands of years. Most often the shells of large sea snails are used, as trumpets, by cutting a hole in the spire of the shell or cutting off the tip of the spire altogether. Various different kinds of large marine gastropod shells can be turned into "blowing shells"; however, the most commonly encountered species used as " conch " trumpets are:
Children in some cultures are often told the myth that you can hear the sound of the ocean by holding a seashell to ones ear. This is due to the effect of seashell resonance .
Whole seashells or parts of sea shells have been used as jewelry or in other forms of adornment since prehistoric times. Mother of pearl was historically primarily a seashell product, although more recently some mother of pearl comes from freshwater mussels. Also see pearl .
" Sailor's Valentines " were late 19th-century decorative keepsakes which were made from the Caribbean, and which were often purchased by sailors to give to their loved ones back home for example in England. These valentines consisted of elaborate arrangements of small seashells glued into attractive symmetrical designs, which were encased on a wooden (usually octagonal) hinged box-frame. The patterns used often featured heart-shaped designs, or included a sentimental expression of love spelled out in small shells.
The making of shell work artifacts is a practice of Aboriginal women from La Perouse in Sydney , dating back to the 19th century. Shell work objects include baby shoes, jewelry boxes and replicas of famous landmarks, including the Sydney Harbour Bridge and the Sydney Opera House . The shellwork tradition began as an Aboriginal women's craft which was adapted and tailored to suit the tourist souvenir market, and which is now considered high art. [ 12 ]
Small pieces of colored and iridescent shell have been used to create mosaics and inlays , which have been used to decorate walls, furniture and boxes. Large numbers of whole seashells, arranged to form patterns, have been used to decorate mirror frames, furniture and human-made shell grottos .
A very large outdoor sculpture at Akkulam of a gastropod seashell is a reference to the sacred chank shell Turbinella pyrum of India. In 2003, Maggi Hambling designed a striking 13 ft (4 m) high sculpture of a scallop shell which stands on the beach at Aldeburgh , in England. The goddess of love, Venus or Aphrodite , is often traditionally depicted rising from the sea on a seashell. In The Birth of Venus , Botticelli depicted the goddess Venus rising from the ocean on a scallop shell.
Sea shells found in the creek and backwater of the coast of west India are used as an additive to poultry feed. They are crushed and mixed with jowar maize and dry fish. [ citation needed ]
Seashells, namely from bivalves [ 13 ] and gastropods, are fundamentally composed of calcium carbonate. In this sense, they have potential to be used as raw material in the production of lime .
Along the Gulf Coast of the United States , oyster shells were mixed into cement to make "shellcrete" which could form bricks, blocks and platforms. It could also be applied over logs. [ 14 ] A notable example is the 19th-century Sabine Pass Lighthouse in Louisiana, near Texas. [ 15 ]
Many arthropods have sclerites , or hardened body parts, which form a stiff exoskeleton made up mostly of chitin . In crustaceans , especially those of the class Malacostraca (crabs, shrimps and lobsters, for instance), the plates of the exoskeleton may be fused to form a more or less rigid carapace . Moulted carapaces of a variety of marine malacostraceans often wash up on beaches. The horseshoe crab is an arthropod of the family Limulidae . The shells or exuviae of these arachnid relatives are common in beach drift in certain areas of the world.
Some echinoderms such as sea urchins , including heart urchins and sand dollars , have a hard "test" or shell. After the animal dies, the flesh rots out and the spines fall off, and then fairly often the empty test washes up whole onto a beach, where it can be found by a beachcomber. These tests are fragile and easily broken into pieces.
The brachiopods , or lamp shells, superficially resemble clams, but the phylum is not closely related to mollusks. Most lines of brachiopods ended during the Permian-Triassic extinction event , and their ecological niche was filled by bivalves. A few of the remaining species of brachiopods occur in the low intertidal zone and thus can be found live by beachcombers.
Some polychaetes , marine annelid worms in the family Serpulidae , secrete a hard tube made of calcium carbonate, adhering to stones or other shells. This tube resembles, and can be confused with, the shell of marine gastropod mollusks in the family Vermetidae , the worm snails.
A few other categories of marine animals leave remains which might be considered "seashells" in the widest possible sense of the word.
Sea turtles have a carapace and plastron of bone and cartilage which is developed from their ribs . Infrequently a turtle "shell" will wash up on a beach.
Pieces of the hard skeleton of corals commonly wash up on beaches in areas where corals grow.
The construction of the shell-like structures of corals are aided by a symbiotic relationship with a class of algae , zooxanthellae . Typically a coral polyp will harbor particular species of algae, which will photosynthesise and thereby provide energy for the coral and aid in calcification, [ 16 ] while living in a safe environment and using the carbon dioxide and nitrogenous waste produced by the polyp. Coral bleaching is a disruption of the balance between polyps and algae, and can lead to the breakdown and death of coral reefs.
The skeletons of soft corals such as gorgonians , also known as sea fans and sea whips , commonly wash ashore in the tropics after storms.
Plant-like diatoms and animal-like radiolarians are two forms of plankton which form hard silicate shells. Foraminifera and coccolithophores create shells known as " tests " which are made of calcium carbonate. These shells and tests are usually microscopic in size, though in the case of foraminifera, they are sometimes visible to the naked eye, often resembling miniature mollusk shells. | https://en.wikipedia.org/wiki/Seashell |
Season cracking is a form of stress-corrosion cracking of brass cartridge cases originally reported from British forces in India. During the monsoon season, military activity was temporarily reduced, and ammunition was stored in stables until the dry weather returned. Many brass cartridges were subsequently found to be cracked, especially where the case was crimped to the bullet. It was not until 1921 that the phenomenon was explained by Moor, Beckinsale and Mallinson: ammonia from horse urine, combined with the residual stress in the cold-drawn metal of the cartridges, was responsible for the cracking. [ 1 ]
Season cracking is characterised by deep brittle cracks which penetrate into affected components. If the cracks reach a critical size, the component can suddenly fracture, sometimes with disastrous results. However, if the concentration of ammonia is very high, then attack is much more severe, and attack over all exposed surfaces occurs. The problem was solved by annealing the brass cases after forming so as to relieve the residual stresses.
The attack takes the form of a reaction between ammonia and copper to form the cuprammonium ion, formula [Cu(NH 3 ) 4 ] 2+ , a chemical complex which is water-soluble, and hence washed from the growing cracks. The problem of cracking can therefore also occur in copper and any other copper alloy, such as bronze . The tendency of copper to react with ammonia was exploited in making rayon , and the deep blue colour of the aqueous solution of copper(II) oxide in ammonia is known as Schweizer's reagent . [ clarification needed ]
Although the problem was first found in brass, any alloy containing copper will be susceptible to the problem. It includes copper itself (as used in pipe for example), bronzes and other alloys with a significant copper content. Like all problems with hairline cracks , detection in the early stages of attack is difficult, but the characteristic blue coloration may give a clue to attack. Microscopic inspection will often reveal the cracks, and x-ray analysis using the EDX facility on the scanning electron microscope or SEM should reveal the presence of elemental nitrogen from ammoniacal traces. | https://en.wikipedia.org/wiki/Season_cracking |
Seasonal thermal energy storage ( STES ), also known as inter-seasonal thermal energy storage , [ 1 ] is the storage of heat or cold for periods of up to several months. The thermal energy can be collected whenever it is available and be used whenever needed, such as in the opposing season. For example, heat from solar collectors or waste heat from air conditioning equipment can be gathered in hot months for space heating use when needed, including during winter months. Waste heat from industrial process can similarly be stored and be used much later [ 2 ] or the natural cold of winter air can be stored for summertime air conditioning. [ 3 ] [ 4 ]
STES stores can serve district heating systems, as well as single buildings or complexes. Among seasonal storages used for heating, the design peak annual temperatures generally are in the range of 27 to 80 °C (81 to 180 °F), and the temperature difference occurring in the storage over the course of a year can be several tens of degrees. Some systems use a heat pump to help charge and discharge the storage during part or all of the cycle. For cooling applications, often only circulation pumps are used.
Sorption and thermochemical heat storage are considered the most suitable for seasonal storage due to the theoretical absence of heat loss between charging and discharging. [ 5 ] However, studies have shown that actual heat losses currently are usually significant. [ 6 ]
Examples for district heating include Drake Landing Solar Community where ground storage provides 97% of yearly consumption without heat pumps , [ 7 ] and Danish pond storage with boosting. [ 8 ]
There are several types of STES technology, covering a range of applications from single small buildings to community district heating networks. Generally, efficiency increases and the specific construction cost decreases with size.
UTES (underground thermal energy storage), in which the storage medium may be geological strata ranging from earth or sand to solid bedrock, or aquifers. UTES technologies include:
The International Energy Agency's Energy Conservation through Energy Storage (ECES) Programme [ 30 ] [ 31 ] has held triennial global energy conferences since 1981. The conferences originally focused exclusively on STES, but now that those technologies are mature other topics such as phase change materials (PCM) and electrical energy storage are also being covered. Since 1985 each conference has had "stock" (for storage) at the end of its name; e.g. EcoStock, ThermaStock. [ 32 ] They are held at various locations around the world. Most recent were InnoStock 2012 (the 12th International Conference on Thermal Energy Storage) in Lleida, Spain [ 33 ] and GreenStock 2015 in Beijing. [ 34 ] EnerStock 2018 will be held in Adana, Turkey in April 2018. [ 35 ]
The IEA-ECES programme continues the work of the earlier International Council for Thermal Energy Storage which from 1978 to 1990 had a quarterly newsletter and was initially sponsored by the U.S. Department of Energy. The newsletter was initially called ATES Newsletter, and after BTES became a feasible technology it was changed to STES Newsletter. [ 36 ] [ 37 ]
Small passively heated buildings typically use the soil adjoining the building as a low-temperature seasonal heat store that in the annual cycle reaches a maximum temperature similar to average annual air temperature, with the temperature drawn down for heating in colder months. Such systems are a feature of building design, as some simple but significant differences from 'traditional' buildings are necessary. At a depth of about 20 feet (6 m) in the soil, the temperature is naturally stable within a year-round range, [ 38 ] if the drawdown does not exceed the natural capacity for solar restoration of heat. Such storage systems operate within a narrow range of storage temperatures over the course of a year, as opposed to the other STES systems described above for which large annual temperature differences are intended.
Two basic passive solar building technologies were developed in the US during the 1970s and 1980s. They use direct heat conduction to and from thermally isolated, moisture-protected soil as a seasonal storage method for space heating, with direct conduction as the heat return mechanism. In one method, "passive annual heat storage" (PAHS), [ 39 ] the building's windows and other exterior surfaces capture solar heat which is transferred by conduction through the floors, walls, and sometimes the roof, into adjoining thermally buffered soil. When the interior spaces are cooler than the storage medium, heat is conducted back to the living space. [ 40 ] [ 41 ]
The other method, “annualized geothermal solar” (AGS) uses a separate solar collector to capture heat. The collected heat is delivered to a storage device (soil, gravel bed or water tank) either passively by the convection of the heat transfer medium (e.g. air or water) or actively by pumping it. This method is usually implemented with a capacity designed for six months of heating.
A number of examples of the use of solar thermal storage from across the world include: Suffolk One a college in East Anglia, England, that uses a thermal collector of pipe buried in the bus turning area to collect solar energy that is then stored in 18 boreholes each 100 metres (330 ft) deep for use in winter heating. Drake Landing Solar Community in Canada uses solar thermal collectors on the garage roofs of 52 homes, which is then stored in an array of 35 metres (115 ft) deep boreholes. The ground can reach temperatures in excess of 70 °C which is then used to heat the houses passively. The scheme has been running successfully since 2007. In Brædstrup , Denmark, some 8,000 square metres (86,000 sq ft) of solar thermal collectors are used to collect some 4,000,000 kWh/year similarly stored in an array of 50 metres (160 ft) deep boreholes.
Architect Matyas Gutai [ 42 ] obtained an EU grant to construct a house in Hungary [ 43 ] which uses extensive water filled wall panels as heat collectors and reservoirs with underground heat storage water tanks. The design uses microprocessor control.
A number of homes and small apartment buildings have demonstrated combining a large internal water tank for heat storage with roof-mounted solar-thermal collectors. Storage temperatures of 90 °C (194 °F) are sufficient to supply both domestic hot water and space heating. The first such house was MIT Solar House #1, in 1939. An eight-unit apartment building in Oberburg , Switzerland was built in 1989, with three tanks storing a total of 118 m 3 (4,167 cubic feet) that store more heat than the building requires. Since 2011, that design is now being replicated in new buildings. [ 44 ]
In Berlin , the “Zero Heating Energy House”, was built in 1997 in as part of the IEA Task 13 low energy housing demonstration project. It stores water at temperatures up to 90 °C (194 °F) inside a 20 m 3 (706 cubic feet) tank in the basement . [ 45 ]
A similar example was built in Ireland in 2009, as a prototype. The solar seasonal store [ 46 ] consists of a 23 m 3 (812 cu ft) tank, filled with water, [ 47 ] which was installed in the ground, heavily insulated all around, to store heat from evacuated solar tubes during the year. The system was installed as an experiment to heat the world's first standardized pre-fabricated passive house [ 48 ] in Galway, Ireland . The aim was to find out if this heat would be sufficient to eliminate the need for any electricity in the already highly efficient home during the winter months.
Based on improvements in glazing the Zero heating buildings are now possible without seasonal energy storage.
STES is also used extensively for the heating of greenhouses. [ 49 ] [ 50 ] [ 51 ] ATES is the kind of storage commonly in use for this application. In summer, the greenhouse is cooled with ground water, pumped from the “cold well” in the aquifer. The water is heated in the process, and is returned to the “warm well” in the aquifer. When the greenhouse needs heat, such as to extend the growing season, water is withdrawn from the warm well, becomes chilled while serving its heating function, and is returned to the cold well. This is a very efficient system of free cooling , which uses only circulation pumps and no heat pumps.
Annualized geo-solar (AGS) enables passive solar heating in even cold, foggy north temperate areas. It uses the ground under or around a building as thermal mass to heat and cool the building. After a designed, conductive thermal lag of 6 months the heat is returned to, or removed from, the inhabited spaces of the building. In hot climates, exposing the collector to the frigid night sky in winter can cool the building in summer.
The six-month thermal lag is provided by about three meters (ten feet) of dirt. A six-meter-wide (20 ft) buried skirt of insulation around the building keeps rain and snow melt out of the dirt, which is usually under the building. The dirt does radiant heating and cooling through the floor or walls. A thermal siphon moves the heat between the dirt and the solar collector. The solar collector may be a sheet-metal compartment in the roof , or a wide flat box on the side of a building or hill. The siphons may be made from plastic pipe and carry air. Using air prevents water leaks and water-caused corrosion. Plastic pipe doesn't corrode in damp earth, as metal ducts can.
AGS heating systems typically consist of:
Usually it requires several years for the storage earth-mass to fully preheat from the local at-depth soil temperature (which varies widely by region and site-orientation) to an optimum Fall level at which it can provide up to 100% of the heating requirements of the living space through the winter. This technology continues to evolve, with a range of variations (including active-return devices) being explored. The listserve where this innovation is most often discussed is "Organic Architecture" at Yahoo.
This system is almost exclusively deployed in northern Europe. One system has been built at Drake Landing in North America. A more recent system is a Do-it-yourself energy-neutral home in progress in Collinsville, IL that will rely solely on Annualized Solar for conditioning. | https://en.wikipedia.org/wiki/Seasonal_thermal_energy_storage |
Seasoning is the process of coating the surface of cookware with fat which is heated in order to produce a corrosion resistant layer of polymerized fat. [ 1 ] [ 2 ] It is required for raw cast-iron cookware [ 3 ] and carbon steel , which otherwise rust rapidly in use, but is also used for many other types of cookware. An advantage of seasoning is that it helps prevent food sticking.
Some cast-iron and carbon steel cookware is pre-seasoned by manufacturers to protect the pan from oxidation (rust), but will need to be further seasoned by the end-users for the cookware to become ready for best nonstick cooking results. [ 4 ] To form a strong seasoning, the raw iron item is thoroughly cleaned, coated in a very thin layer of unsaturated fat or oil, and then heated until the bioplastic layer forms, and left to completely cool. Multiple layers are required for the best long-term results.
Stainless steel and aluminium cookware do not require protection from corrosion, but seasoning reduces sticking, and can help with browning as the seasoning coating has high thermal emissivity . [ 5 ] [ 6 ] Other cookware surfaces are generally not seasoned.
A seasoned surface is hydrophobic and highly attractive to oils and fats used for cooking. These form a layer that prevents foods, which typically contain water, from touching and cooking onto the hydrophilic metallic cooking surface underneath. [ 7 ] These properties are useful when frying, roasting and baking.
Food sticks easily to a bare metal cooking surface; it must either be oiled or seasoned before use. [ 8 ] The coating known as seasoning is formed by a process of repeatedly layering extremely thin coats of oil on the cookware and oxidizing each layer with medium-high heat for a time. This process is known as "seasoning"; the color of the coating is commonly known as its "patina" - the base coat will darken with use. [ 9 ]
To season cookware (e.g., to season a new pan, or to replace damaged seasoning on an old pan), the following is a typical process:
First the cookware is thoroughly cleaned to remove old seasoning, manufacturing residues or a possible manufacturer-applied anti corrosion coating and to expose the bare metal. If it is not pre-seasoned, a new cast-iron skillet or dutch oven typically comes from the manufacturer with a protective coating of wax or shellac ; otherwise it would rust. [ 10 ] This needs to be removed before the cookware is used. [ 11 ] An initial scouring with hot soapy water will usually remove the protective coating. Alternatively, for woks , it is common to burn off the coating over high heat (outside or under a vent hood) to expose the bare metal surface. For already-used cookware that are to be re-seasoned, the cleaning process can be more complex, involving rust removal and deep cleaning (with strong soap or lye, [ 12 ] or by burning in a campfire or self-cleaning oven [ 13 ] [ better source needed ] ) to remove existing seasoning and build-up.
Then several times the following is performed:
The precise details of the seasoning process differ from one source to another, and there is much disagreement regarding the correct oil to use. There is also no clear consensus about the best temperature and duration. Lodge Manufacturing uses a proprietary soybean blend in their base coats as stated on their website, but states that all oils and fats can be used. [ 18 ] The temperature recommended for seasoning varies from high temperatures above 260 °C (500 °F) to temperatures below 150 °C (302 °F).
Seasoning a cast-iron or carbon steel wok is a common process in Asia and Asian-American culture. While the vegetable oil method of seasoning is also used in Asia, a traditional process for seasoning also includes the use of Chinese chives or scallions as part of the process. [ 19 ]
In conventional seasoning, the oil or fat is converted into a hard surface at or above the high temperatures used for cooking, analogous to the reaction of drying oils . When oils or fats are heated, multiple degradation reactions occur, including decomposition, autoxidation, thermal oxidation , polymerization , and cyclization . Among other processes at high temperatures triglycerides in the oil break down into glycerol and fatty acids. The soluble glycerol then burns off creating (toxic) smoke while the fatty acids in the oil polymerise and harden. [ 20 ] [ 21 ] [ page range too broad ]
Often cookware's seasoning is uneven, and over time it will spread to the whole item. Heating the cookware (such as in a hot oven or on a stovetop ) facilitates the oxidation of the iron; the fats and/or oils protect the metal from contact with the air during the reaction, which would otherwise cause rust to form. Some cast iron users advocate heating the cookware slightly before applying the fat or oil to ensure it is completely dry. [ 22 ] [ 23 ]
The seasoned surface is hydrophobic and highly attractive to oils and fats used for cooking ( oleophilic ). These form a layer that prevents foods, which typically contain water, from touching and cooking on to the hydrophilic metallic cooking surface underneath.
The seasoned surface will deteriorate at the temperature where the coating breaks down. This is typically higher than the smoke point of the original oils and fats used to season the cookware. Thus old seasoning can be removed at a sufficiently high temperature (~500 °C), as found in oven self-cleaning cycles .
Some Chinese cookware is seasoned at a much higher temperature than conventional seasoning at 450 °C. More akin to bluing , this type of seasoning mainly involves a chemical change of the iron pan itself and not the oil. When beef tallow is heated at this temperature, it evaporates on the iron surface and increases the partial pressure of O 2 (oxygen gas) on the pot surface. This transport of oxygen encourages the formation of Fe 3 O 4 nanoballs. The surface formed is broadly speaking hydrophobic and oleophilic, but is more versatile in that it temporarily turns hydrophilic on contact with high-water ingredients. [ 24 ]
Some food writers advise against using seasoned pans and Dutch ovens to cook foods containing tomatoes, vinegar, or other acidic ingredients because these foods would eventually remove the protective layer created during the seasoning process. [ 25 ] [ 26 ] [ 27 ] Tests conducted by America's Test Kitchen found that, while cooking a highly acidic tomato sauce for over 30 minutes produced a metallic taste, cooking acidic food in a well-seasoned pan for a short time is unlikely to have negative consequences. [ 28 ] [ 29 ] [ 30 ] [ 31 ] [ 32 ]
Cast iron pots are best suited to cook food high in oil or fat, such as chicken, bacon, or sausage, or used for deep frying . Cleaning (except prior to seasoning) is often carried out without the use of detergent. Some cookbook authors recommend only wiping seasoned cookware clean after each use or using other cleaning methods such as a salt scrub or boiling water. [ 33 ] The protective layer itself is not very susceptible to soaps, and many users do briefly use detergents and soaps. [ 28 ] However, cast iron is very prone to rust, and the protective layer may have pinholes, so soaking for long periods is contraindicated as the layer may start to flake off.
Unlike commercial non-stick coatings such as Teflon, with which metal cooking utensils are not used because they damage the surface, seasoned surfaces tend to be self-reforming, so they allow the use of such utensils. These are of course much more effective in scraping off food than the softer utensils used with non-stick pans. [ 34 ] [ page needed ] [ 32 ]
In the process of bluing , an oxidizing chemical reaction on an iron surface selectively forms magnetite (Fe 3 O 4 ), the black oxide of iron (as opposed to rust , the red oxide of iron (Fe 2 O 3 )). Black oxide provides some protection against corrosion if also treated with a water-displacing oil to reduce wetting and galvanic action. Bluing is often used with carbon steel and cast iron pans in conjunction with seasoning. | https://en.wikipedia.org/wiki/Seasoning_(cookware) |
Seasoning , or the Seasoning , was the period of adjustment that slave traders and slaveholders subjected African slaves to following their arrival in the Americas . While modern scholarship has occasionally applied this term to the brief period of acclimatization undergone by European immigrants to the Americas, [ 1 ] [ 2 ] [ 3 ] it most frequently and formally referred to the process undergone by enslaved people. [ 4 ] Slave traders used the term "seasoning" to refer to the process of adjusting the enslaved Africans to the new climate, diet, geography, and ecology of the Americas. [ 5 ] The term applied to both the physical acclimatization of the enslaved person to the environment, as well as that person's adjustment to a new social environment , labor regimen, and language. [ 6 ] Slave traders and owners believed that if slaves survived this critical period of environmental seasoning, they were less likely to die and the psychological element would make them more easily controlled. This process took place immediately after the arrival of enslaved people during which their mortality rates were particularly high. These "new" or "saltwater" slaves were described as "outlandish" on arrival. Those who survived this process became "seasoned", and typically commanded a higher price in the market. [ 3 ] [ 7 ] [ 5 ] For example, in eighteenth century Brazil, the price differential between "new" and "seasoned" slaves was about fifteen percent. [ 8 ] [ clarification needed ] [ failed verification ]
Atlantic Creoles made up the first generations of enslaved people. Atlantic creoles were often mixed-race, integrated into and familiar with European society and gained freedom at higher rates prior to the eighteenth century. The first half of the 18th century saw a shift in Atlantic slavery. While tobacco, sugar, and rice took root in the Caribbean and North American colonies, the enslaved population of the New World shifted from a "society with slaves" to a "slave society" with the predominance of "saltwater slavery" -- enslavement through the Atlantic slave trade . [ 9 ] With the expansion of the slave trade in the mid-eighteenth century, the nature of slavery changed. Operating on a larger scale, slave traders transported enslaved Africans to various European colonies throughout the Americas (both before and after the decolonization of the Americas ), systematizing both the voyage and the process of seasoning, though it varied locationally and temporally. While slave traders and owners practiced seasoning in both North and South America, it was not practiced consistently in the Southern Colonies where planters often forced "new" slaves to work immediately upon their arrival to the colonies. [ 10 ]
Slave traders and slaveowners adopted the term "seasoning" during the transatlantic slave trade when newly arrived slaves died at high rates in the years following disembarkation. Death rates differed among regions in the Americas, though both the Middle Passage and the seasoning period were exceptionally deadly across the Americas. A "Dr. Collins" writing in 1803 attributed the high mortality rates to disease, change in climate, diet, labor, "severity", and suicide. [ 11 ] In the Thirteen Colonies , death rates during seasoning were at an estimated 25 to 50 percent. [ 12 ] In Cuba , deaths in a single year were between 7 and 12 percent while the mortality rate reached as high as 33 percent in Jamaica . [ 13 ] In Brazil, an estimated 25 percent of enslaved people died during the seasoning process, where the law also required that slaves be baptized during their first year in the country. [ 14 ]
A contemporary observer noted that seasoning was a "training not only to hard work, but to scanty diet." [ 12 ] Slaveowners drastically limited the slaves' diets, both in breadth and depth, to the diet of the plantations, which was chiefly composed of maize, rice, or flour. [ 15 ] Battered by this inadequate diet, enslaved people often suffered "dropsies" ( edema ) and "fluxes" ( diarrhea ), compounding their severe and widespread malnutrition . [ 16 ]
Newly arrived slaves experienced high rates of disease and death during the seasoning process. During the Middle Passage , slave traders forced enslaved Africans to live in tight quarters without ventilation, sufficient food, or water, and with no opportunity for hygiene. In such conditions, enslaved people often contracted scurvy or amoebic dysentery ; of which amoebic dysentery, or the "bloody flux", claimed the most lives. [ 17 ] [ 18 ] [ 19 ] Once ashore, enslaved people lived in appalling conditions similar to those of the Middle Passage . Underfed and exposed to a new ecology, enslaved people then had to battle the new climate and forced hard labor. Weakened by the voyage and immediate brutality of slavery, many enslaved people died of smallpox , measles , influenza , and unidentified diseases at high rates in the first several years after arrival. [ 20 ]
Though it took many different forms, seasoning universally involved the further commodification of human beings and their preparation by enslavers for the marketplace and labor. Enslavers accomplished this preparation by treating their slaves harshly, subjecting them to a brutal regimen of training and violence. [ 21 ] Lasting between one and three years, this process of adjustment was physically and psychologically taxing, and marked by brutality and coercion. Slaveholders resorted to force and violence in order to subdue the "saltwater" slaves and extract their labor. [ 22 ] Enslavers regularly beat slaves, maimed them, and placed them in stocks or solitary confinement. In one particularly cruel practice, the slaveholder would whip a naked woman, often pregnant, and pour salt, pepper, or wax into her open wounds. [ 23 ] In addition to violence, enslaved people had to adjust to hard labor over the seasoning period. In the Caribbean, newly arrived slaves were given baskets for fertilizing the sugar fields the week they arrived. This was the first step in the essential process of training the new arrivals in the technologies of sugar cultivation . Elsewhere, too, enslaved people were taught how to cultivate and process crops, often including the ones meant to sustain the enslaved population during the seasoning. [ 15 ] [ 8 ] Over the seasoning period, slaveowners wanted their slaves to acquire both knowledge of the labor and to become accustomed to the extreme workload. Training did not only take the form of labor. Enslaved people were also taught the language of the colony either by other slaves who had already undergone the seasoning process or by the white overseers of the plantation. [ 8 ] [ 22 ]
Though constantly threatened with beatings and further ill-treatment, enslaved people resisted their enslavement in the seasoning in several visible ways. Scholars have considered widespread suicide among newly enslaved people an act of resistance. Indeed, enslavers feared suicide alongside disease, and contemporary manuals for the seasoning included recommendations for improving an enslaved person's "disposition" to best avoid suicide. [ 24 ] Hunger plagued enslaved people during and after the seasoning and reports of food theft at any opportunity -- and the beatings from enslavers that followed such thefts -- were common. [ 25 ] Still others refused to eat entirely and were similarly punished. Runaway attempts were common, though these recently enslaved arrivals rarely escaped successfully, as they had little familiarity with their surroundings and were isolated on the major plantations of the Americas. [ 26 ] | https://en.wikipedia.org/wiki/Seasoning_(slavery) |
A seawater greenhouse is a greenhouse structure that enables the growth of crops and the production of fresh water in arid regions. Arid regions constitute about one third of the Earth's land area. Seawater greenhouse technology aims to mitigate issues such as global water scarcity , peak water and soil becoming salted . [ 1 ] The system uses seawater and solar energy , and has a similar structure to the pad-and-fan greenhouse, but with additional evaporators and condensers. [ 1 ] The seawater is pumped into the greenhouse to create a cool and humid environment, the optimal conditions for the cultivation of temperate crops. [ 1 ] The freshwater is produced in a condensed state created by the solar desalination principle, which removes salt and impurities. [ 2 ] Finally, the remaining humidified air is expelled from the greenhouse and used to improve growing conditions for outdoor plants.
The seawater greenhouse concept was first researched and developed in 1991 by Charlie Paton's company Light Works Ltd, which is now known as the Seawater Greenhouse Ltd. Charlie Paton and Philip Davies worked on the first pilot project commenced in 1992, on the Canary Island of Tenerife . A prototype seawater greenhouse was assembled in the UK and constructed on the site in Tenerife covering an area of 360 m 2 . [ 1 ] The temperate crops successfully cultivated included tomatoes, spinach, dwarf peas, peppers, artichokes, French beans, and lettuce.
The second pilot design was installed in 2000 on the coast of Al-Aryam Island , Abu Dhabi, United Arab Emirates. The design is a light steel structure, similar to a multi-span polytunnel, which relies purely on solar energy. A pipe array is installed to improve the design of the greenhouse by decreasing the temperature and increasing the freshwater production. [ 3 ] The greenhouse has an area of 864 m 2 and has a daily water production of 1 m 3 , which nearly meets the crop's irrigation demand. [ 1 ]
The third pilot seawater greenhouse, which is 864 m 2 , is near Muscat in Oman which produces 0.3 to 0.6 m 3 of freshwater per day. This project was created as a collaboration between Sultan Qaboos University. It provides an opportunity to develop a sustainable horticultural sector on the Batinah coast. These projects have enabled the validation of a thermodynamic simulation model which, given appropriate meteorological data, accurately predicts and quantifies how the seawater greenhouse will perform in other parts of the world. [ 4 ]
The fourth project is the commercial installation in Port Augusta , Australia, installed in 2010. It is currently a 20 hectare seawater greenhouse owned and run by Sundrop Farms which has developed it further. [ 3 ] [ 5 ]
The fifth design was constructed in 2017 in Berbera , Somaliland. [ 6 ] The design was researched to be simplified and inexpensive with advanced greenhouse modeling techniques. This design includes a shading system which retains core evaporative cooling elements. [ 6 ]
The Sahara Forest Project (SFP) combines the seawater greenhouse technology and concentrated solar power and constructed pilot projects in Jordan and Qatar. The seawater greenhouse evaporates 50 m 3 of seawater and harvests 5 m 3 of fresh water per hectare per day. [ 7 ] The solar power production capacity through PV panels produces 39 kW on the 3 hectares area with 1350 m 2 growing area. [ 8 ] The greenhouses are 15 degrees cooler than the outside temperatures which enables the production up to 130,000 of kg of vegetables per year and up to 20,000 liters of fresh water per day. [ 8 ] Additionally, the project includes revegetation by soil reclamation of nitrogen-fixing and salt-removing desert plants by repurposed waste products from agriculture and saltwater evaporation. [ 8 ]
A seawater greenhouse uses the surrounding environment to grow temperate crops and produce freshwater. A conventional greenhouse uses solar heat to create a warmer environment to allow adequate growing temperature, whereas the seawater greenhouse does the opposite by creating a cooler environment. The roof traps infrared heat, while allowing visible light through to promote photosynthesis .
The design for cooling the microclimate primarily consists of humidification and dehumidification (HD) desalination process or multiple-effect humidification . [ 9 ] A simple seawater greenhouse consists of two evaporative coolers (evaporators), a condenser, fans, seawater and distilled water pipes and crops in between the two evaporators. [ 10 ] This is shown in schematic figures 1 and 2.
The process recreates the natural hydrological cycle within a controlled environment of the greenhouse by evaporating water from saline water source and regains it as freshwater by condensation. [ 1 ] The first part of the system uses seawater, an evaporator, and a condenser. The front wall of the greenhouse consists of a seawater-wetted evaporator which faces the prevailing wind. These are mostly constituted of corrugated cardboard shown in Figure 3. If the wind is not prevalent enough, fans blow the outside air through the evaporator into the greenhouse. The ambient warm air exchanges the heat with the seawater which cools it down and gets it humidified. [ 10 ] [ 1 ] The cool and humid air creates an adequate growing environment for the crops. The remaining evaporatively-cooled seawater is collected and pumped to the condenser as a coolant. [ 1 ]
The second part of the system has another evaporator. The seawater flows from the first evaporator which preheats it and thereafter flows through the solar thermal collector on the roof to heat it up sufficiently before it flows to the second evaporator. [ 10 ] The seawater, or coolant, flows through a circuit consisting of the evaporators, solar heating pipe, and condenser with an intake of seawater and an output of fresh water. The fresh water is produced by hot and relatively high humidity air which can produce sufficient distilled water for irrigation. [ 10 ] The volume of fresh water is determined by air temperature, relative humidity, solar radiation and the airflow rate. These conditions can be modeled with appropriate meteorological data, enabling the design and process to be optimized for any suitable location.
The technique is applicable to sites in arid regions near the sea. The distance and elevation from the sea must be evaluated considering the energy required to pump water to the site. There are numerous suitable locations on the coasts; others are below sea level, such as the Dead Sea and the Qattara Depression , where hydro schemes have been proposed to exploit the hydraulic pressure to generate power, e.g., Red Sea–Dead Sea Canal . [ 16 ] [ 17 ]
In 1996, Paton and Davies used the Simulink toolkit under MATLAB to model forced ventilation of the greenhouse in Tenerife, Cape Verde, Namibia, and Oman. [ 18 ] The greenhouse is assisted by the prevailing wind, evaporative cooling, transpiration, solar heating, heat transfer through the walls and roof, and condensation which is analyzed in the study. [ 18 ] They found that the amount of water required by the plants is reduced by 80% and 2.6-6.4 kWh electrical energy is needed for m3 of fresh water produced. [ 18 ]
In 2005, Paton and Davis Evaluated design options with thermal modeling using the United Arab Emirates model as a baseline. [ 19 ] They studied three options:perforated screen, C-shaped air path, and pipe array, to find a better seawater circuit to cool the environment and produce the most freshwater. The study found that a pipe array gave the best results: an air temperature decrease of 1 °C, a mean radiant temperature decrease of 7.5 °C, and a freshwater production increase of 63%. This can be implemented to improve seawater greenhouses in hot arid regions such as the second pilot design in the United Arab Emirates. [ 19 ]
In 2018, Paton and Davis researched brine utilization for cooling and salt production in wind-driven seawater greenhouses to design and model it. The brine disposed by the seawater desalination may disturb the ecosystem as the same amount of brine is produced as freshwater. [ 5 ] By using the brine valoristation method of wind-driven air flow by cooling the greenhouse with seawater evaporation, salt can be produced as shown in Figure 4. [ 5 ] This brine is the by-product of the freshwater production, but can also be the ingredient to make salt, making it into a product that can be merchandised.
An additional finding of this research was the importance of the shade-net which is modelled by a thin film in the study shown in Figure 5. [ 5 ] It not only provides cooling, but also elongates the cooling plume by containing the cold air plume from the evaporative cooling pad. [ 5 ] | https://en.wikipedia.org/wiki/Seawater_greenhouse |
Algae fuel , algal biofuel , or algal oil is an alternative to liquid fossil fuels that use algae as the source of energy-rich oils. Also, algae fuels are an alternative to commonly known biofuel sources, such as corn and sugarcane. [ 1 ] [ 2 ] When made from seaweed (macroalgae) it can be known as seaweed fuel or seaweed oil . These fuels have no practical significance but remain an aspirational target in the biofuels research area.
In 1942 Harder and Von Witsch were the first to propose that microalgae be grown as a source of lipids for food or fuel. [ 3 ] [ 4 ] Following World War II, research began in the US, [ 5 ] [ 6 ] [ 7 ] Germany, [ 8 ] Japan, [ 9 ] England, [ 10 ] and Israel [ 11 ] on culturing techniques and engineering systems for growing microalgae on larger scales, particularly species in the genus Chlorella . Meanwhile, H. G. Aach showed that Chlorella pyrenoidosa could be induced via nitrogen starvation to accumulate as much as 70% of its dry weight as lipids. [ 12 ] Since the need for alternative transportation fuel had subsided after World War II, research at this time focused on culturing algae as a food source or, in some cases, for wastewater treatment. [ 13 ]
Interest in the application of algae for biofuels was rekindled during the oil embargo and oil price surges of the 1970s, leading the US Department of Energy to initiate the Aquatic Species Program in 1978. [ 14 ] The Aquatic Species Program spent $25 million over 18 years to develop liquid transportation fuel from algae that would be price-competitive with petroleum-derived fuels. [ 15 ] The research program focused on microalgae cultivation in open outdoor ponds, systems that are low in cost but vulnerable to environmental disturbances like temperature swings and biological invasions. 3,000 algal strains were collected from around the country and screened for desirable properties such as high productivity, lipid content, and thermal tolerance, and the most promising strains were included in the SERI microalgae collection at the Solar Energy Research Institute (SERI) in Golden, Colorado and used for further research. [ 15 ] Among the program's most significant findings were that rapid growth and high lipid production were "mutually exclusive" since the former required high nutrients and the latter required low nutrients. [ 15 ] The final report suggested that genetic engineering may be necessary to be able to overcome this and other natural limitations of algal strains and that the ideal species might vary with place and season. [ 15 ] Although it was successfully demonstrated that large-scale production of algae for fuel in outdoor ponds was feasible, the program failed to do so at a cost that would be competitive with petroleum, especially as oil prices sank in the 1990s. Even in the best-case scenario, it was estimated that unextracted algal oil would cost $59–186 per barrel, [ 15 ] while petroleum cost less than $20 per barrel in 1995. [ 14 ] Therefore, under budget pressure in 1996, the Aquatic Species Program was abandoned. [ 15 ]
Other contributions to algal biofuels research have come indirectly from projects focusing on different applications of algal cultures. For example, in the 1990s Japan's Research Institute of Innovative Technology for the Earth (RITE) implemented a research program with the goal of developing systems to fix CO 2 using microalgae. [ 16 ] Although the goal was not energy production, several studies produced by RITE demonstrated that algae could be grown using flue gas from power plants as a CO 2 source, [ 17 ] [ 18 ] an important development for algal biofuel research. Other work focusing on harvesting hydrogen gas, methane, or ethanol from algae, as well as nutritional supplements and pharmaceutical compounds, has also helped inform research on biofuel production from algae. [ 13 ]
Following the disbanding of the Aquatic Species Program in 1996, there was a relative lull in algal biofuel research. Still, various projects were funded in the US by the Department of Energy , Department of Defense , National Science Foundation , Department of Agriculture , National Laboratories , state funding, and private funding, as well as in other countries. [ 14 ] More recently, rising oil prices in the 2000s spurred a revival of interest in algal biofuels and US federal funding has increased, [ 14 ] numerous research projects are being funded in Australia, New Zealand, Europe, the Middle East, and other parts of the world. [ 19 ]
In December 2022, ExxonMobil , the last large oil company to invest in algae biofuels, ended its research funding. [ 20 ]
In March 2023, researchers said that the commercialization of biofuels would require several billion dollars of funding, plus a long-term dedication to overcoming what appear to be fundamental biological limitations of wild organisms. Most researchers believe that large scale production of biofuels is either "a decade, and more likely two decades, away." [ 20 ]
Algal oil is used as a source of fatty acid supplementation in food products, as it contains mono- and polyunsaturated fats , in particular EPA and DHA . [ 21 ] Its DHA content is roughly equivalent to that of salmon based fish oil. [ 22 ] [ 23 ]
Algae can be converted into various types of fuels, depending on the production technologies and the part of the cells used. The lipid , or oily part of the algae biomass can be extracted and converted into biodiesel through a process similar to that used for any other vegetable oil, or converted in a refinery into "drop-in" replacements for petroleum-based fuels. Alternatively or following lipid extraction, the carbohydrate content of algae can be fermented into bioethanol or butanol fuel . [ 24 ]
Biodiesel is a diesel fuel derived from animal or plant lipids (oils and fats). Studies have shown that some species of algae can produce 60% or more of their dry weight in the form of oil. [ 12 ] [ 15 ] [ 25 ] [ 26 ] [ 27 ] Because the cells grow in aqueous suspension, where they have more efficient access to water, CO 2 and dissolved nutrients, microalgae are capable of producing large amounts of biomass and usable oil in either high rate algal ponds [ 28 ] or photobioreactors . This oil can then be turned into biodiesel which could be sold for use in automobiles. Regional production of microalgae and processing into biofuels will provide economic benefits to rural communities. [ 29 ]
As they do not have to produce structural compounds such as cellulose for leaves, stems, or roots, and because they can be grown floating in a rich nutritional medium, microalgae can have faster growth rates than terrestrial crops. Also, they can convert a much higher fraction of their biomass to oil than conventional crops, e.g. 60% versus 2-3% for soybeans. [ 25 ] The per unit area yield of oil from algae is estimated to be from 58,700 to 136,900 L/ha/year, depending on lipid content, which is 10 to 23 times as high as the next highest yielding crop, oil palm, at 5 950 L/ha/year. [ 30 ]
The U.S. Department of Energy's Aquatic Species Program , 1978–1996, focused on biodiesel from microalgae. The final report suggested that biodiesel could be the only viable method by which to produce enough fuel to replace current world diesel usage. [ 31 ] If algae-derived biodiesel were to replace the annual global production of 1.1bn tons of conventional diesel then a land mass of 57.3 million hectares would be required, which would be highly favorable compared to other biofuels. [ 32 ]
Algae can be used to produce ' green diesel ' (also known as renewable diesel, hydrotreating vegetable oil [ 33 ] or hydrogen-derived renewable diesel) [ 34 ] through a hydrotreating refinery process that breaks molecules down into shorter hydrocarbon chains used in diesel engines. [ 33 ] [ 35 ] It has the same chemical properties as petroleum-based diesel [ 33 ] meaning that it does not require new engines, pipelines or infrastructure to distribute and use. It has yet to be produced at a cost that is competitive with petroleum . [ 34 ] While hydrotreating is currently the most common pathway to produce fuel-like hydrocarbons via decarboxylation/decarbonylation, there is an alternative process offering a number of important advantages over hydrotreating. In this regard, the work of Crocker et al. [ 36 ] and Lercher et al. [ 37 ] is particularly noteworthy. For oil refining, research is underway for catalytic conversion of renewable fuels by decarboxylation . [ 38 ] As the oxygen is present in crude oil at rather low levels, of the order of 0.5%, deoxygenation in petroleum refining is not of much concern, and no catalysts are specifically formulated for oxygenates hydrotreating. Hence, one of the critical technical challenges to make the hydrodeoxygenation of algae oil process economically feasible is related to the research and development of effective catalysts. [ 39 ] [ 40 ]
Butanol can be made from algae or diatoms using only a solar powered biorefinery . This fuel has an energy density 10% less than gasoline, and greater than that of either ethanol or methanol . In most gasoline engines, butanol can be used in place of gasoline with no modifications. In several tests, butanol consumption is similar to that of gasoline, and when blended with gasoline, provides better performance and corrosion resistance than that of ethanol or E85 . [ 41 ]
The green waste left over from the algae oil extraction can be used to produce butanol. In addition, it has been shown that macroalgae (seaweeds) can be fermented by bacteria of genus Clostridia to butanol and other solvents. [ 42 ] Transesterification of seaweed oil (into biodiesel) is also possible with species such as Chaetomorpha linum , Ulva lactuca , and Enteromorpha compressa ( Ulva ). [ 43 ]
The following species are being investigated as suitable species from which to produce ethanol and/or butanol : [ 44 ]
Biogasoline is gasoline produced from biomass . Like traditionally produced gasoline, it contains between 6 ( hexane ) and 12 ( dodecane ) carbon atoms per molecule and can be used in internal-combustion engines . [ 46 ]
Biogas is composed mainly of methane ( CH 4 ) and carbon dioxide (CO 2 ), with some traces of hydrogen sulphide , oxygen, nitrogen, and hydrogen . Macroalgae has high methane production rate compared to plant biomass. Biogas production from macroalgae is more technically viable compared to other fuels, but it is not economically viable due to the high cost of macroalgae feedstock. [ 47 ] Carbohydrate and protein in microalgae can be converted into biogas through anaerobic digestion, which includes hydrolysis, fermentation, and methanogenesis steps. The conversion of algal biomass into methane can potentially recover as much energy as it obtains, but it is more profitable when the algal lipid content is lower than 40%. [ 48 ] Biogas production from microalgae is relatively low because of the high ratio of protein in microalgae, but microalgae can be co-digested with high C/N ratio products such as wastepaper. [ 49 ] Another method to produce biogas is through gasification, where hydrocarbon is converted to syngas through a partial oxidation reaction at high temperature (typically 800 °C to 1000 °C). Gasification is usually performed with catalysts. Uncatalyzed gasification requires temperature to be about 1300 °C. Syngas can be burnt directly to produce energy or used a fuel in turbine engines. It can also be used as feedstock for other chemical productions. [ 50 ]
Methane , [ 51 ] the main constituent of natural gas , can be produced from algae by various methods, namely gasification , pyrolysis and anaerobic digestion . In gasification and pyrolysis methods methane is extracted under high temperature and pressure. Anaerobic digestion [ 52 ] is a straightforward method involved in decomposition of algae into simple components then transforming it into fatty acids using microbes like acidogenic bacteria followed by removing any solid particles and finally adding methanogenic archaea to release a gas mixture containing methane. A number of studies have successfully shown that biomass from microalgae can be converted into biogas via anaerobic digestion. [ 53 ] [ 54 ] [ 55 ] [ 56 ] [ 57 ] Therefore, in order to improve the overall energy balance of microalgae cultivation operations, it has been proposed to recover the energy contained in waste biomass via anaerobic digestion to methane for generating electricity. [ 58 ]
The Algenol system which is being commercialized by BioFields in Puerto Libertad , Sonora , Mexico utilizes seawater and industrial exhaust to produce ethanol. Porphyridium cruentum also have shown to be potentially suitable for ethanol production due to its capacity for accumulating large amount of carbohydrates. [ 59 ]
Trials of using algae as biofuel were carried out by Lufthansa and Virgin Atlantic as early as 2008, although there is little evidence that using algae is a reasonable source for jet biofuels. [ 60 ] By 2015, cultivation of fatty acid methyl esters and alkenones from the algae, Isochrysis , was under research as a possible jet biofuel feedstock . [ 61 ]
In May 2022, scientists at University of Cambridge announced they created an algae energy harvester, that uses natural sunlight to power a small microprocessor , initially powering the processor for six months, and then kept going for a full year. The device, which is about the size of AA battery , is a small container with water and blue green algae. The device does not generate a huge amount of power, but it can be used for Internet of Things devices, eliminating the need for traditional batteries such as lithium-ion batteries. The goal is to have more a environmentally friendly power source that can be used in remote areas. [ 62 ]
Research into algae for the mass-production of oil focuses mainly on microalgae (organisms capable of photosynthesis that are less than 0.4 mm in diameter, including the diatoms and cyanobacteria ) as opposed to macroalgae, such as seaweed . The preference for microalgae has come about due largely to their less complex structure, fast growth rates, and high oil-content (for some species). However, some research is being done into using seaweeds for biofuels, probably due to the high availability of this resource. [ 63 ] [ 64 ]
As of 2012 [update] researchers across various locations worldwide have started investigating the following species for their suitability as a mass oil-producers: [ 65 ] [ 66 ] [ 67 ]
The amount of oil each strain of algae produces varies widely. Note the following microalgae and their various oil yields:
In addition, due to its high growth-rate, Ulva [ 71 ] has been investigated as a fuel for use in the SOFT cycle , (SOFT stands for Solar Oxygen Fuel Turbine), a closed-cycle power-generation system suitable for use in arid, subtropical regions. [ 72 ]
Other species used include Clostridium saccharoperbutylacetonicum , [ 73 ] Sargassum , Gracilaria , Prymnesium parvum , and Euglena gracilis . [ 74 ]
Light is what algae primarily need for growth as it is the most limiting factor. Many companies are investing for developing systems and technologies for providing artificial light. One of them is OriginOil that has developed a Helix BioReactorTM that features a rotating vertical shaft with low-energy lights arranged in a helix pattern. [ 75 ] Water temperature also influences the metabolic and reproductive rates of algae. Although most algae grow at low rate when the water temperature gets lower, the biomass of algal communities can get large due to the absence of grazing organisms. [ 75 ] The modest increases in water current velocity may also affect rates of algae growth since the rate of nutrient uptake and boundary layer diffusion increases with current velocity. [ 75 ]
Other than light and water, phosphorus, nitrogen, and certain micronutrients are also useful and essential in growing algae. Nitrogen and phosphorus are the two most significant nutrients required for algal productivity, but other nutrients such as carbon and silica are additionally required. [ 76 ] Of the nutrients required, phosphorus is one of the most essential ones as it is used in numerous metabolic processes. The microalgae D. tertiolecta was analyzed to see which nutrient affects its growth the most. [ 77 ] The concentrations of phosphorus (P), iron (Fe), cobalt (Co), zinc (Zn), manganese (Mn) and molybdenum (Mo), magnesium (Mg), calcium (Ca), silicon (Si) and sulfur (S) concentrations were measured daily using inductively coupled plasma (ICP) analysis. Among all these elements being measured, phosphorus resulted in the most dramatic decrease, with a reduction of 84% over the course of the culture. [ 77 ] This result indicates that phosphorus, in the form of phosphate, is required in high amounts by all organisms for metabolism.
There are two enrichment media that have been extensively used to grow most species of algae: Walne medium and the Guillard's F/ 2 medium. [ 78 ] These commercially available nutrient solutions may reduce time for preparing all the nutrients required to grow algae. However, due to their complexity in the process of generation and high cost, they are not used for large-scale culture operations. [ 78 ] Therefore, enrichment media used for mass production of algae contain only the most important nutrients with agriculture-grade fertilizers rather than laboratory-grade fertilizers. [ 78 ]
Algae grow much faster than food crops, and can produce hundreds of times more oil per unit area than conventional crops such as rapeseed, palms, soybeans, or jatropha . [ 30 ] As algae have a harvesting cycle of 1–10 days, their cultivation permits several harvests in a very short time-frame, a strategy differing from that associated with annual crops. [ 26 ] In addition, algae can be grown on land unsuitable for terrestrial crops, including arid land and land with excessively saline soil, minimizing competition with agriculture. [ 79 ] Most research on algae cultivation has focused on growing algae in clean but expensive photobioreactors , or in open ponds, which are cheap to maintain but prone to contamination. [ 80 ]
The lack of equipment and structures needed to begin growing algae in large quantities has inhibited widespread mass-production of algae for biofuel production. Maximum use of existing agriculture processes and hardware is the goal. [ 81 ]
Closed systems (not exposed to open air) avoid the problem of contamination by other
organisms blown in by the air. The problem of a closed system is finding a cheap source of sterile CO 2 .
Several experimenters have found the CO 2 from a smokestack works well for growing algae. [ 82 ] [ 83 ] For reasons of economy, some experts think that algae farming for biofuels will have to be done as part of cogeneration , where it can make use of waste heat and help soak up pollution. [ 84 ]
To produce micro-algae at large-scale under controlled environment using PBR system, strategies such as light guides, sparger, and PBR construction materials required should be well considered. [ 85 ]
Most companies pursuing algae as a source of biofuels pump nutrient -rich water through plastic or borosilicate glass tubes (called " bioreactors " ) that are exposed to sunlight (and so-called photobioreactors or PBR). [ 86 ]
Running a PBR is more difficult than using an open pond, and costlier, but may provide a higher level of control and productivity. [ 26 ] In addition, a photobioreactor can be integrated into a closed loop cogeneration system much more easily than ponds or other methods.
Open pond systems consist of simple in ground ponds, which are often mixed by a paddle wheel. These systems have low power requirements, operating costs, and capital costs when compared to closed loop photobioreactor systems. [ 87 ] [ 86 ] Nearly all commercial algae producers for high value algal products utilize open pond systems. [ 88 ]
The Algae scrubber is a system designed primarily for cleaning nutrients and pollutants out of water using algal turfs. An algal turf scrubber (ATS) mimics the algal turfs of a natural coral reef by taking in nutrient rich water from waste streams or natural water sources, and pulsing it over a sloped surface. [ 89 ] This surface is coated with a rough plastic membrane or a screen, which allows naturally occurring algal spores to settle and colonize the surface. Once the algae has been established, it can be harvested every 5–15 days, [ 90 ] and can produce 18 metric tons of algal biomass per hectare per year. [ 91 ] In contrast to other methods, which focus primarily on a single high yielding species of algae, this method focuses on naturally occurring polycultures of algae. As such, the lipid content of the algae in an ATS system is usually lower, which makes it more suitable for a fermented fuel product, such as ethanol, methane, or butanol. [ 91 ] Conversely, the harvested algae could be treated with a hydrothermal liquefaction process, which would make possible biodiesel, gasoline, and jet fuel production. [ 92 ]
There are three major advantages of ATS over other systems. The first advantage is documented higher productivity over open pond systems. [ 93 ] The second is lower operating and fuel production costs. The third is the elimination of contamination issues due to the reliance on naturally occurring algae species. The projected costs for energy production in an ATS system are $0.75/kg, compared to a photobioreactor which would cost $3.50/kg. [ 91 ] Furthermore, due to the fact that the primary purpose of ATS is removing nutrients and pollutants out of water, and these costs have been shown to be lower than other methods of nutrient removal, this may incentivize the use of this technology for nutrient removal as the primary function, with biofuel production as an added benefit. [ 94 ]
After harvesting the algae, the biomass is typically processed in a series of steps, which can differ based on the species and desired product; this is an active area of research [ 26 ] and also is the bottleneck of this technology: the cost of extraction is higher than those obtained. One of the solutions is to use filter feeders to "eat" them. Improved animals can provide both foods and fuels. An alternative method to extract the algae is to grow the algae with specific types of fungi. This causes bio-flocculation of the algae which allows for easier extraction. [ 95 ]
Often, the algae is dehydrated, and then a solvent such as hexane is used to extract energy-rich compounds like triglycerides from the dried material. [ 1 ] [ 96 ] Then, the extracted compounds can be processed into fuel using standard industrial procedures. For example, the extracted triglycerides are reacted with methanol to create biodiesel via transesterification . [ 1 ] The unique composition of fatty acids of each species influences the quality of the resulting biodiesel and thus must be taken into account when selecting algal species for feedstock. [ 26 ]
An alternative approach called Hydrothermal liquefaction employs a continuous process that subjects harvested wet algae to high temperatures and pressures—350 °C (662 °F) and 3,000 pounds per square inch (21,000 kPa). [ 97 ] [ 98 ] [ 99 ]
Products include crude oil, which can be further refined into aviation fuel, gasoline, or diesel fuel using one or many upgrading processes. [ 100 ] The test process converted between 50 and 70 percent of the algae's carbon into fuel. Other outputs include clean water, fuel gas and nutrients such as nitrogen, phosphorus, and potassium. [ 97 ]
Nutrients like nitrogen (N), phosphorus (P), and potassium (K), are important for plant growth and are essential parts of fertilizer. Silica and iron, as well as several trace elements, may also be considered important marine nutrients as the lack of one can limit the growth of, or productivity in, an area. [ 101 ]
Bubbling CO 2 through algal cultivation systems can greatly increase productivity and yield (up to a saturation point). Typically, about 1.8 tonnes of CO 2 will be utilised per tonne of algal biomass (dry) produced, though this varies with algae species. [ 102 ] The Glenturret Distillery in Perthshire percolate CO 2 made during the whisky distillation through a microalgae bioreactor. Each tonne of microalgae absorbs two tonnes of CO 2 . Scottish Bioenergy, who run the project, sell the microalgae as high value, protein-rich food for fisheries . In the future, [ when? ] they will use the algae residues to produce renewable energy through anaerobic digestion . [ 103 ]
Nitrogen is a valuable substrate that can be utilized in algal growth. Various sources of nitrogen can be used as a nutrient for algae, with varying capacities. Nitrate was found to be the preferred source of nitrogen, in regards to amount of biomass grown. Urea is a readily available source that shows comparable results, making it an economical substitute for nitrogen source in large scale culturing of algae. [ 104 ] Despite the clear increase in growth in comparison to a nitrogen-less medium, it has been shown that alterations in nitrogen levels affect lipid content within the algal cells. In one study [ 105 ] nitrogen deprivation for 72 hours caused the total fatty acid content (on a per cell basis) to increase by 2.4-fold. 65% of the total fatty acids were esterified to triacylglycerides in oil bodies, when compared to the initial culture, indicating that the algal cells utilized de novo synthesis of fatty acids. It is vital for the lipid content in algal cells to be of high enough quantity, while maintaining adequate cell division times, so parameters that can maximize both are under investigation.
A possible nutrient source is wastewater from the treatment of sewage, agricultural, or flood plain run-off, all currently major pollutants and health risks. However, this waste water cannot feed algae directly and must first be processed by bacteria, through anaerobic digestion . If waste water is not processed before it reaches the algae, it will contaminate the algae in the reactor, and at the very least, kill much of the desired algae strain. In biogas facilities, organic waste is often converted to a mixture of carbon dioxide, methane , and organic fertilizer. Organic fertilizer that comes out of the digester is liquid, and nearly suitable for algae growth, but it must first be cleaned and sterilized. [ 106 ]
The utilization of wastewater and ocean water instead of freshwater is strongly advocated due to the continuing depletion of freshwater resources. However, heavy metals, trace metals, and other contaminants in wastewater can decrease the ability of cells to produce lipids biosynthetically and also impact various other workings in the machinery of cells. The same is true for ocean water, but the contaminants are found in different concentrations. Thus, agricultural-grade fertilizer is the preferred source of nutrients, but heavy metals are again a problem, especially for strains of algae that are susceptible to these metals. In open pond systems the use of strains of algae that can deal with high concentrations of heavy metals could prevent other organisms from infesting these systems. [ 79 ] In some instances it has even been shown that strains of algae can remove over 90% of nickel and zinc from industrial wastewater in relatively short periods of time. [ 107 ]
In comparison with terrestrial-based biofuel crops such as corn or soybeans, microalgal production results in a much less significant land footprint due to the higher oil productivity from the microalgae than all other oil crops. [ 108 ] Algae can also be grown on marginal lands useless for ordinary crops and with low conservation value, and can use water from salt aquifers that is not useful for agriculture or drinking. [ 84 ] [ 109 ] Algae can also grow on the surface of the ocean in bags or floating screens. [ 110 ] Thus microalgae could provide a source of clean energy with little impact on the provisioning of adequate food and water or the conservation of biodiversity. [ 111 ] Algae cultivation also requires no external subsidies of insecticides or herbicides, removing any risk of generating associated pesticide waste streams. In addition, algal biofuels are much less toxic, and degrade far more readily than petroleum-based fuels. [ 112 ] [ 113 ] [ 114 ] However, due to the flammable nature of any combustible fuel, there is potential for some environmental hazards if ignited or spilled, as may occur in a train derailment or a pipeline leak. [ 115 ] This hazard is reduced compared to fossil fuels , due to the ability for algal biofuels to be produced in a much more localized manner, and due to the lower toxicity overall, but the hazard is still there nonetheless. Therefore, algal biofuels should be treated in a similar manner to petroleum fuels in transportation and use, with sufficient safety measures in place at all times.
Studies have determined that replacing fossil fuels with renewable energy sources, such as biofuels, have the capability of reducing CO 2 emissions by up to 80%. [ 116 ] An algae-based system could capture approximately 80% of the CO 2 emitted from a power plant when sunlight is available. Although this CO 2 will later be released into the atmosphere when the fuel is burned, this CO 2 would have entered the atmosphere regardless. [ 109 ] The possibility of reducing total CO 2 emissions therefore lies in the prevention of the release of CO 2 from fossil fuels. Furthermore, compared to fuels like diesel and petroleum, and even compared to other sources of biofuels, the production and combustion of algal biofuel does not produce any sulfur oxides or nitrous oxides, and produces a reduced amount of carbon monoxide, unburned hydrocarbons, and reduced emission of other harmful pollutants. [ 117 ] Since terrestrial plant sources of biofuel production simply do not have the production capacity to meet current energy requirements, microalgae may be one of the only options to approach complete replacement of fossil fuels.
Microalgae production also includes the ability to use saline waste or waste CO 2 streams as an energy source. This opens a new strategy to produce biofuel in conjunction with waste water treatment, while being able to produce clean water as a byproduct. [ 117 ] When used in a microalgal bioreactor, harvested microalgae will capture significant quantities of organic compounds as well as heavy metal contaminants absorbed from wastewater streams that would otherwise be directly discharged into surface and ground-water. [ 108 ] Moreover, this process also allows the recovery of phosphorus from waste, which is an essential but scarce element in nature – the reserves of which are estimated to have depleted in the last 50 years. [ 118 ] Another possibility is the use of algae production systems to clean up non-point source pollution, in a system known as an algal turf scrubber (ATS). This has been demonstrated to reduce nitrogen and phosphorus levels in rivers and other large bodies of water affected by eutrophication, and systems are being built that will be capable of processing up to 110 million liters of water per day. ATS can also be used for treating point source pollution, such as the waste water mentioned above, or in treating livestock effluent. [ 91 ] [ 119 ] [ 120 ]
Nearly all research in algal biofuels has focused on culturing single species, or monocultures, of microalgae. However, ecological theory and empirical studies have demonstrated that plant and algae polycultures, i.e. groups of multiple species, tend to produce larger yields than monocultures. [ 121 ] [ 122 ] [ 123 ] [ 124 ] Experiments have also shown that more diverse aquatic microbial communities tend to be more stable through time than less diverse communities. [ 125 ] [ 126 ] [ 127 ] [ 128 ] Recent studies found that polycultures of microalgae produced significantly higher lipid yields than monocultures. [ 129 ] [ 130 ] Polycultures also tend to be more resistant to pest and disease outbreaks, as well as invasion by other plants or algae. [ 131 ] Thus culturing microalgae in polyculture may not only increase yields and stability of yields of biofuel, but also reduce the environmental impact of an algal biofuel industry. [ 111 ]
There is clearly a demand for sustainable biofuel production, but whether a particular biofuel will be used ultimately depends not on sustainability but cost efficiency. Therefore, research is focusing on cutting the cost of algal biofuel production to the point where it can compete with conventional petroleum. [ 26 ] [ 132 ] The production of several products from algae has been mentioned [ weasel words ] as the most important factor for making algae production economically viable. Other factors are the improving of the solar energy to biomass conversion efficiency (currently 3%, but 5 to 7% is theoretically attainable [ 133 ] ) and making the oil extraction from the algae easier. [ 134 ]
In a 2007 report [ 26 ] a formula was derived estimating the cost of algal oil in order for it to be a viable substitute to petroleum diesel:
where: C (algal oil) is the price of microalgal oil in dollars per gallon and C (petroleum) is the price of crude oil in dollars per barrel. This equation assumes that algal oil has roughly 80% of the caloric energy value of crude petroleum. [ 135 ]
The IEA estimated in 2017 that algal biomass can be produced for a little as $0.54/kg in open pond in a warm climate to $10.20/kg in photobioreactors in cooler climates. [ 136 ] Assuming that the biomass contains 30% oil by weight, the cost of biomass for providing a liter of oil would be approximately $1.40 ($5.30/gal) and $1.81 ($6.85/gal) for photobioreactors and raceways, respectively. Oil recovered from the lower cost biomass produced in photobioreactors is estimated to cost $2.80/L, assuming the recovery process contributes 50% to the cost of the final recovered oil. [ 26 ] If existing algae projects can achieve biodiesel production price targets of less than $1 per gallon, the United States may realize its goal of replacing up to 20% of transport fuels by 2020 by using environmentally and economically sustainable fuels from algae production. [ 137 ]
Whereas technical problems, such as harvesting, are being addressed successfully by the industry, the high up-front investment of algae-to-biofuels facilities is seen by many as a major obstacle to the success of this technology. As of 2007, only few studies on the economic viability were publicly available, and must often rely on the little data (often only engineering estimates) available in the public domain. Dmitrov [ 138 ] examined the GreenFuel's photobioreactor and estimated that algae oil would only be competitive at an oil price of $800 per barrel. A study by Alabi et al. [ 139 ] examined raceways, photobioreactors and anaerobic fermenters to make biofuels from algae and found that photobioreactors are too expensive to make biofuels. Raceways might be cost-effective in warm climates with very low labor costs, and fermenters may become cost-effective subsequent to significant process improvements. The group found that capital cost, labor cost and operational costs (fertilizer, electricity, etc.) by themselves are too high for algae biofuels to be cost-competitive with conventional fuels. Similar results were found by others, [ 140 ] [ 141 ] [ 142 ] suggesting that unless new, cheaper ways of harnessing algae for biofuels production are found, their great technical potential may never become economically accessible. In 2012, Rodrigo E. Teixeira [ 143 ] demonstrated a new reaction and proposed a process for harvesting and extracting raw materials for biofuel and chemical production that requires a fraction of the energy of current methods, while extracting all cell constituents.
A 2022 study stated that selling fuel from commercially refining biofuel was not feasible due to technological limitations and high costs. The study found that byproducts would require a selling price of $899/ton of residual solids in order to support a competitive price of $2.50 per gallon gasoline equivalent, using the method of converting proteins and carbohydrates into mixed alcohols. Comparatively, the study found converting proteins and carbohydrates into hydrocarbons to require a higher selling price $1033/ton of residual solids in order to be economically feasible. [ 144 ]
Many of the byproducts produced in the processing of microalgae can be used in various applications, many of which have a longer history of production than algal biofuel. Some of the products not used in the production of biofuel include natural dyes and pigments, antioxidants, and other high-value bio-active compounds. [ 80 ] [ 145 ] [ 146 ] These chemicals and excess biomass have found numerous use in other industries. For example, the dyes and oils have found a place in cosmetics, commonly as thickening and water-binding agents. [ 147 ] Discoveries within the pharmaceutical industry include antibiotics and antifungals derived from microalgae, as well as natural health products, which have been growing in popularity over the past few decades. For instance Spirulina contains numerous polyunsaturated fats (Omega 3 and 6), amino acids, and vitamins, [ 148 ] as well as pigments that may be beneficial, such as beta-carotene and chlorophyll. [ 149 ]
One of the main advantages that using microalgae as the feedstock when compared to more traditional crops is that it can be grown much more easily. [ 150 ] Algae can be grown in land that would not be considered suitable for the growth of the regularly used crops. [ 80 ] In addition to this, wastewater that would normally hinder plant growth has been shown to be very effective in growing algae. [ 150 ] Because of this, algae can be grown without taking up arable land that would otherwise be used for producing food crops, and the better resources can be reserved for normal crop production. Microalgae also require fewer resources to grow and little attention is needed, allowing the growth and cultivation of algae to be a very passive process. [ 80 ]
Many traditional feedstocks for biodiesel, such as corn and palm, are also used as feed for livestock on farms, as well as a valuable source of food for humans. Because of this, using them as biofuel reduces the amount of food available for both, resulting in an increased cost for both the food and the fuel produced. Using algae as a source of biodiesel can alleviate this problem in a number of ways. First, algae is not used as a primary food source for humans, meaning that it can be used solely for fuel and there would be little impact in the food industry. [ 151 ] Second, many of the waste-product extracts produced during the processing of algae for biofuel can be used as a sufficient animal feed. This is an effective way to minimize waste and a much cheaper alternative to the more traditional corn- or grain-based feeds. [ 152 ]
Growing algae as a source of biofuel has also been shown to have numerous environmental benefits, and has presented itself as a much more environmentally friendly alternative to current biofuels. For one, it is able to utilize run-off, water contaminated with fertilizers and other nutrients that are a by-product of farming, as its primary source of water and nutrients. [ 150 ] Because of this, it prevents this contaminated water from mixing with the lakes and rivers that currently supply our drinking water. In addition to this, the ammonia, nitrates, and phosphates that would normally render the water unsafe actually serve as excellent nutrients for the algae, meaning that fewer resources are needed to grow the algae. [ 80 ] Many algae species used in biodiesel production are excellent bio-fixers, meaning they are able to remove carbon dioxide from the atmosphere to use as a form of energy for themselves. Because of this, they have found use in industry as a way to treat flue gases and reduce GHG emissions. [ 80 ]
The process of microalgae cultivation is highly water-intensive. Life cycle studies estimated that the production of 1 liter of microalgae based biodiesel requires between 607 and 1944 liters of water. [ 153 ] That said, abundant wastewater and/or seawater , which also contain various nutrients, can theoretically be used for this purpose instead of freshwater.
Algae biodiesel is still a fairly new technology. Despite the fact that research began over 30 years ago, it was put on hold during the mid-1990s, mainly due to a lack of funding and a relatively low petroleum cost. [ 19 ] For the next few years algae biofuels saw little attention; it was not until the gas peak of the early 2000s that it eventually had a revitalization in the search for alternative fuel sources. [ 19 ]
Increasing interest in seaweed farming for carbon sequestration, eutrophication reduction and production of food has resulted in the creation of commercial seaweed cultivation since 2017. [ 154 ] Reductions in the cost of cultivation and harvesting as well as the development of commercial industry will improve the economics of macroalgae biofuels. Climate change has created a proliferation of brown macroalgae mats, which wash up on the shores of the Caribbean. Currently these mats are disposed of but there is interest in developing them into a feedstock for biofuel production. [ 155 ]
The biodiesel produced from the processing of microalgae differs from other forms of biodiesel in the content of polyunsaturated fats. [ 150 ] Polyunsaturated fats are known for their ability to retain fluidity at lower temperatures. While this may seem like an advantage in production during the colder temperatures of the winter, the polyunsaturated fats result in lower stability during regular seasonal temperatures. [ 151 ]
Numerous policies have been put in place since the 1975 oil crisis in order to promote the use of Renewable Fuels in the United States, Canada and Europe. In Canada, these included the implementation of excise taxes exempting propane and natural gas which was extended to ethanol made from biomass and methanol in 1992. The federal government also announced their renewable fuels strategy in 2006 which proposed four components: increasing availability of renewable fuels through regulation, supporting the expansion of Canadian production of renewable fuels, assisting farmers to seize new opportunities in this sector and accelerating the commercialization of new technologies. These mandates were quickly followed by the Canadian provinces:
Policies in the United States have included a decrease in the subsidies provided by the federal and state governments to the oil industry which have usually included $2.84 billion. This is more than what is actually set aside for the biofuel industry. The measure was discussed at the G20 in Pittsburgh where leaders agreed that "inefficient fossil fuel subsidies encourage wasteful consumption, reduce our energy security, impede investment in clean sources and undermine efforts to deal with the threat of climate change". If this commitment is followed through and subsidies are removed, a fairer market in which algae biofuels can compete will be created. In 2010, the U.S. House of Representatives passed a legislation seeking to give algae-based biofuels parity with cellulose biofuels in federal tax credit programs. The algae-based renewable fuel promotion act (HR 4168) was implemented to give biofuel projects access to a $1.01 per gal production tax credit and 50% bonus depreciation for biofuel plant property. The U.S Government also introduced the domestic Fuel for Enhancing National Security Act implemented in 2011. This policy constitutes an amendment to the Federal property and administrative services act of 1949 and federal defense provisions in order to extend to 15 the number of years that the Department of Defense (DOD) multiyear contract may be entered into the case of the purchase of advanced biofuel. Federal and DOD programs are usually limited to a 5-year period [ 156 ]
The European Union (EU) has also responded by quadrupling the credits for second-generation algae biofuels which was established as an amendment to the Biofuels and Fuel Quality Directives. [ 157 ] | https://en.wikipedia.org/wiki/Seaweed_fuel |
Seble-Hiwot Wagaw is an American organic chemist who is a senior leader at AbbVie pharmaceuticals outside Chicago, IL. [ 1 ]
Wagaw was born in Addis Ababa , Ethiopia and emigrated to the United States in 1974. [ 2 ] Her father, Teshome Gebremichael Wagaw , was a faculty member at the University of Michigan for 28 years, and her mother is Tsehai Wolde-Tsadik. [ 3 ] She is one of three children, with an older brother and sister. [ 4 ] She received her Bachelor's and MS degrees in Chemistry from the University of Michigan in 1994, and a Ph.D. in organic chemistry with Stephen L. Buchwald at the Massachusetts Institute of Technology in 1999. [ 5 ] Her research in the Buchwald lab utilized chiral complexes of Palladium to forge new carbon-nitrogen bonds on Aryl rings. [ 6 ]
As a Senior Director for process research and R&D at Abbott Laboratories (later AbbVie ) for her entire career, Wagaw has published research on enantiomerically enriched lead molecules using Pybox ligands. [ 7 ] She has led exploration into new technologies for her process group, including explorations of flow chemistry and commercial-scale electrochemistry. [ 8 ]
She is on the advisory board for Asymchem and co-founded the Cross-industry Women's chemical process group. [ 5 ]
This biographical article about a chemist is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Seble_Wagaw |
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/Sec_(trigonometry) |
Secalin is a prolamin glycoprotein found in the grain rye , Secale cereale . [ 1 ] [ 2 ]
Secalin is one of the forms of gluten proteins that people with coeliac disease cannot tolerate, and thus rye should be avoided by people with this disease. It is generally recommended that such people follow a gluten free diet .
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Secalin |
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/Secans |
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/Secans_complementi |
The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
It was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. [ 3 ] It was adopted by surveyors and civil engineers in the United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. [ 4 ] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,} the exsecant of the complementary angle , [ 5 ] [ 6 ] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. [ 7 ]
As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle.
The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to a circle with one endpoint on the circumference a secans exterior . [ 8 ]
The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant . [ 9 ]
In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . [ 10 ] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. [ 11 ]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} [ 12 ] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track [ 13 ] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. [ 3 ] The same idea was adopted by other authors, such as Searles (1880). [ 14 ] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". [ 15 ]
In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. [ 15 ] [ 16 ]
Solving the same types of problems is required when surveying circular sections of canals [ 17 ] and roads, and the exsecant was still used in mid-20th century books about road surveying. [ 18 ]
The exsecant has sometimes been used for other applications, such as beam theory [ 19 ] and depth sounding with a wire. [ 20 ]
In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. [ 21 ] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), [ 22 ] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing the difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
For example, the secant of 1° is approximately 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1° is approximately −3.817 220 , [ 23 ] all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . [ 24 ] For even smaller angles loss of precision is worse.
If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. [ 25 ] [ 26 ]
For a sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. [ 27 ]
The inverse of the exsecant function, which might be symbolized arcexsec , [ 6 ] is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):
arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
the arctangent expression is well behaved for small angles. [ 28 ]
While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: [ 29 ]
d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}
where ln is the natural logarithm . See also Integral of the secant function .
The exsecant of twice an angle is: [ 6 ]
exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}
Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in the US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55– 70.
Review: Poor, Henry Varnum , ed. (1856-03-22). " Practical Book of Reference, and Engineer's Field Book . By Charles Haslett" . American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX.
Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions" . In Abramowitz, Milton ; Stegun, Irene A. (eds.). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036 .
van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout.
Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation". Synthese . 134 ( 1– 2, Logic and Mathematical Reasoning): 217– 244. doi : 10.1023/A:1022143816001 . JSTOR 20117331 .
Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons .
Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. exsec function, arith.scm lines 61–63 . Retrieved 2024-04-01 .
Review: " Field Manual for Railroad Engineers . By J. C. Nagle" . The Engineer (Review). 84 : 540. 1897-12-03.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . | https://en.wikipedia.org/wiki/Secans_exterior |
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/Secans_interior |
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/Secant_(trigonometry) |
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/Secant_complement |
A secant plane is a plane containing a nontrivial section of a sphere or an ellipsoid , or such a plane that a sphere is projected onto. Secant planes are similar to tangent planes, which contact the sphere's surface at a point, while secant planes contact the surface along curves.
The two-dimensional representations of secant planes are secant lines , the lines that join two distinct points on a curve.
Secant planes are used in map projections . The secant plane intersects a globe along a small circle with no distortion, forming a standard parallel which has true scale. [ 1 ] | https://en.wikipedia.org/wiki/Secant_plane |
seccomp (short for secure computing [ 1 ] ) is a computer security facility in the Linux kernel . seccomp allows a process to make a one-way transition into a "secure" state where it cannot make any system calls except exit() , sigreturn() , read() and write() to already-open file descriptors . Should it attempt any other system calls, the kernel will either just log the event or terminate the process with SIGKILL or SIGSYS . [ 2 ] [ 3 ] In this sense, it does not virtualize the system's resources but isolates the process from them entirely.
seccomp mode is enabled via the prctl(2) system call using the PR_SET_SECCOMP argument, or (since Linux kernel 3.17 [ 4 ] ) via the seccomp(2) system call. [ 5 ] seccomp mode used to be enabled by writing to a file, /proc/self/seccomp , but this method was removed in favor of prctl() . [ 6 ] In some kernel versions, seccomp disables the RDTSC x86 instruction, which returns the number of elapsed processor cycles since power-on, used for high-precision timing. [ 7 ]
seccomp-bpf is an extension to seccomp [ 8 ] that allows filtering of system calls using a configurable policy implemented using Berkeley Packet Filter rules. It is used by OpenSSH [ 9 ] and vsftpd as well as the Google Chrome/Chromium web browsers on ChromeOS and Linux. [ 10 ] (In this regard seccomp-bpf achieves similar functionality, but with more flexibility and higher performance, to the older systrace —which seems to be no longer supported for Linux .)
Some consider seccomp comparable to OpenBSD pledge(2) and FreeBSD capsicum (4) [ citation needed ] .
seccomp was first devised by Andrea Arcangeli in January 2005 for use in public grid computing and was originally intended as a means of safely running untrusted compute-bound programs. It was merged into the Linux kernel mainline in kernel version 2.6.12, which was released on March 8, 2005. [ 11 ] | https://en.wikipedia.org/wiki/Seccomp |
Second-harmonic imaging microscopy ( SHIM ) is based on a nonlinear optical effect known as second-harmonic generation (SHG). SHIM has been established as a viable microscope imaging contrast mechanism for visualization of cell and tissue structure and function. [ 1 ] A second-harmonic microscope obtains contrasts from variations in a specimen's ability to generate second-harmonic light from the incident light while a conventional optical microscope obtains its contrast by detecting variations in optical density , path length, or refractive index of the specimen. SHG requires intense laser light passing through a material with a noncentrosymmetric molecular structure, either inherent or induced externally, for example by an electric field. [ 2 ]
Second-harmonic light emerging from an SHG material is exactly half the wavelength (frequency doubled) of the light entering the material. While two-photon-excited fluorescence (TPEF) is also a two photon process, TPEF loses some energy during the relaxation of the excited state, while SHG is energy conserving. Typically, an inorganic crystal is used to produce SHG light such as lithium niobate (LiNbO 3 ), potassium titanyl phosphate (KTP = KTiOPO 4 ), or lithium triborate (LBO = LiB 3 O 5 ). Though SHG requires a material to have specific molecular orientation in order for the incident light to be frequency doubled, some biological materials can be highly polarizable, and assemble into fairly ordered, large noncentrosymmetric structures. While some biological materials such as collagen, microtubules , and muscle myosin [ 3 ] can produce SHG signals, even water can become ordered and produce second-harmonic signal under certain conditions, which allows SH microscopy to image surface potentials without any labeling molecules. [ 2 ] The SHG pattern is mainly determined by the phase matching condition. A common setup for an SHG imaging system will have a laser scanning microscope with a titanium sapphire mode-locked laser as the excitation source. The SHG signal is propagated in the forward direction. However, some experiments have shown that objects on the order of about a tenth of the wavelength of the SHG produced signal will produce nearly equal forward and backward signals.
SHIM offers several advantages for live cell and tissue imaging. SHG does not involve the excitation of molecules like other techniques such as fluorescence microscopy therefore, the molecules shouldn't suffer the effects of phototoxicity or photobleaching . Also, since many biological structures produce strong SHG signals, the labeling of molecules with exogenous probes is not required which can also alter the way a biological system functions. By using near infrared wavelengths for the incident light, SHIM has the ability to construct three-dimensional images of specimens by imaging deeper into thick tissues.
Two-photons fluorescence ( 2PEF ) is a very different process from SHG : it involves excitation of electrons to higher energy levels, and subsequent de-excitation by photon emission (unlike SHG, although it is also a 2-photon process). Thus, 2PEF is a non coherent process, spatially (emitted isotropically) and temporally (broad, sample-dependent spectrum). It is also not specific to certain structure, unlike SHG. [ 4 ]
It can therefore be coupled to SHG in multiphoton imaging to reveal some molecules that do produce autofluorescence , like elastin in tissues (while SHG reveals collagen or myosin for instance). [ 4 ]
Before SHG was used for imaging, the first demonstration of SHG was performed in 1961 by P. A. Franken, G. Weinreich , C. W. Peters, and A. E. Hill at the University of Michigan, Ann Arbor using a quartz sample. [ 5 ] In 1968, SHG from interfaces was discovered by Bloembergen [ 6 ] and has since been used as a tool for characterizing surfaces and probing interface dynamics. In 1971, Fine and Hansen reported the first observation of SHG from biological tissue samples. [ 7 ] In 1974, Hellwarth and Christensen first reported the integration of SHG and microscopy by imaging SHG signals from polycrystalline ZnSe . [ 8 ] In 1977, Colin Sheppard imaged various SHG crystals with a scanning optical microscope. The first biological imaging experiments were done by Freund and Deutsch in 1986 to study the orientation of collagen fibers in rat tail tendon . [ 9 ] In 1993, Lewis examined the second-harmonic response of styryl dyes in electric fields . He also showed work on imaging live cells. In 2006, Goro Mizutani group developed a non-scanning SHG microscope that significantly shortens the time required for observation of large samples, even if the two-photons wide-field microscope was published in 1996 [ 10 ] and could have been used to detect SHG. The non-scanning SHG microscope was used for observation of plant starch , [ 11 ] [ 12 ] megamolecule, [ 13 ] spider silk [ 14 ] [ 15 ] and so on. In 2010 SHG was extended to whole-animal in vivo imaging. [ 16 ] [ 17 ] In 2019, SHG applications widened when it was applied to the use of selectively imaging agrochemicals directly on leaf surfaces to provide a way to evaluate the effectiveness of pesticides. [ 18 ]
SHG polarization anisotropy can be used to determine the orientation and degree of organization of proteins in tissues since SHG signals have well-defined polarizations. By using the anisotropy equation: [ 19 ]
I p a r − I p e r p I p a r + 2 I p e r p = r {\displaystyle {\frac {I_{par}-I_{perp}}{I_{par}+2I_{perp}}}=r}
and acquiring the intensities of the polarizations in the parallel and perpendicular directions. A high r {\displaystyle r} value indicates an anisotropic orientation whereas a low r {\displaystyle r} value indicates an isotropic structure. In work done by Campagnola and Loew, [ 19 ] it was found that collagen fibers formed well-aligned structures with an r = 0.7 {\displaystyle r=0.7} value.
SHG being a coherent process ( spatially and temporally ), it keeps information on the direction of the excitation and is not emitted isotropically. It is mainly emitted in forward direction (same as excitation), but can also be emitted in backward direction depending on the phase-matching condition . Indeed, the coherence length beyond which the conversion of the signal decreases is:
l c = 2 / Δ k {\displaystyle l_{c}=2/\Delta k}
with Δ k ∝ 1 / ( n 2 ω − n ω ) {\displaystyle \Delta k\propto 1/(n_{2\omega }-n_{\omega })} for forward, but Δ k b w d ∝ 1 / ( n 2 ω + n ω ) {\displaystyle \Delta k_{bwd}\propto 1/(n_{2\omega }+n_{\omega })} for backward such that l c {\displaystyle l_{c}} >> l c , b w d {\displaystyle l_{c,bwd}} . Therefore, thicker structures will appear preferentially in forward, and thinner ones in backward: since the SHG conversion depends at first approximation on the square of the number of nonlinear converters, the signal will be higher if emitted by thick structures, thus the signal in forward direction will be higher than in backward. However, the tissue can scatter the generated light, and a part of the SHG in forward can be retro-reflected in the backward direction. [ 20 ] Then, the forward-over-backward ratio F/B can be calculated, [ 20 ] and is a metric of the global size and arrangement of the SHG converters (usually collagen fibrils). It can also be shown that the higher the out-of-plane angle of the scatterer, the higher its F/B ratio (see fig. 2.14 of [ 21 ] ).
The advantages of polarimetry were coupled to SHG in 2002 by Stoller et al. [ 22 ] Polarimetry can measure the orientation and order at molecular level, and coupled to SHG it can do so with the specificity to certain structures like collagen: polarization-resolved SHG microscopy (p-SHG) is thus an expansion of SHG microscopy. [ 23 ] p-SHG defines another anisotropy parameter, as: [ 24 ]
ρ = I p a r I p e r p {\displaystyle \rho ={\sqrt {\frac {I_{par}}{I_{perp}}}}}
which is, like r , a measure of the principal orientation and disorder of the structure being imaged. Since it is often performed in long cylindrical filaments (like collagen), this anisotropy is often equal to ρ = χ X X X ( 2 ) χ X Y Y ( 2 ) {\displaystyle \rho ={\frac {\chi _{XXX}^{(2)}}{\chi _{XYY}^{(2)}}}} , [ 25 ] where χ ( 2 ) {\displaystyle \chi ^{(2)}} is the nonlinear susceptibility tensor and X the direction of the filament (or main direction of the structure), Y orthogonal to X and Z the propagation of the excitation light.
The orientation ϕ of the filaments in the plane XY of the image can also be extracted from p-SHG by FFT analysis , and put in a map. [ 25 ] [ 26 ]
Collagen (particular case, but widely studied in SHG microscopy), can exist in various forms : 28 different types, of which 5 are fibrillar. One of the challenge is to determine and quantify the amount of fibrillar collagen in a tissue, to be able to see its evolution and relationship with other non-collagenous materials. [ 27 ]
To that end, a SHG microscopy image has to be corrected to remove the small amount of residual fluorescence or noise that exist at the SHG wavelength. After that, a mask can be applied to quantify the collagen inside the image. [ 27 ] Among other quantization techniques, it is probably the one with the highest specificity, reproductibility and applicability despite being quite complex. [ 27 ]
It has also been used to prove that backpropagating action potentials invade dendritic spines without voltage attenuation, establishing a sound basis for future work on Long-term potentiation . Its use here was that it provided a way to accurately measure the voltage in the tiny dendritic spines with an accuracy unattainable with standard two-photon microscopy. [ 28 ] Meanwhile, SHG can efficiently convert near-infrared light to visible light to enable imaging-guided photodynamic therapy, overcoming the penetration depth limitations. [ 29 ]
SHG microscopy and its expansions can be used to study various tissues: some example images are reported in the figure below: collagen inside the extracellular matrix remains the main application. It can be found in tendon, skin, bone, cornea, aorta, fascia, cartilage, meniscus, intervertebral disks...
Myosin can also be imaged in skeletal muscle or cardiac muscle.
Third-Harmonic Generation (THG) microscopy can be complementary to SHG microscopy, as it is sensitive to the transverse interfaces, and to the 3rd order nonlinear susceptibility χ ( 3 ) {\displaystyle \chi ^{(3)}} [ 35 ] [ 36 ]
The mammographic density is correlated with the collagen density, thus SHG can be used for identifying breast cancer . [ 37 ] SHG is usually coupled to other nonlinear techniques such as Coherent anti-Stokes Raman Scattering or Two-photon excitation microscopy , as part of a routine called multiphoton microscopy (or tomography) that provides a non-invasive and rapid in vivo histology of biopsies that may be cancerous. [ 38 ]
The comparison of forward and backward SHG images gives insight about the microstructure of collagen, itself related to the grade and stage of a tumor , and its progression in breast . [ 39 ] Comparison of SHG and 2PEF can also show the change of collagen orientation in tumors . [ 40 ] Even if SHG microscopy has contributed a lot to breast cancer research, it is not yet established as a reliable technique in hospitals , or for diagnostic of this pathology in general. [ 39 ]
Healthy ovaries present in SHG a uniform epithelial layer and well-organized collagen in their stroma , whereas abnormal ones show an epithelium with large cells and a changed collagen structure. [ 39 ] The r ratio (see #Orientational anisotropy ) is also used [ 41 ] to show that the alignment of fibrils is slightly higher for cancerous than for normal tissues.
SHG is, again, combined to 2PEF is used to calculate the ratio:
M F S I = ( shg − tpef ) / ( shg + tpef ) {\displaystyle MFSI=({\text{shg}}-{\text{tpef}})/({\text{shg}}+{\text{tpef}})}
where shg (resp. tpef) is the number of thresholded pixels in the SHG (resp. 2PEF) image, [ 42 ] a high MFSI meaning a pure SHG image (with no fluorescence). The highest MFSI is found in cancerous tissues, [ 39 ] which provides a contrast mode to differentiate from normal tissues.
SHG was also combined to Third-Harmonic Generation (THG) to show that backward (see #Forward over backward SHG ) THG is higher in tumors. [ 43 ]
Changes in collagen ultrastructure in pancreatic cancer can be investigated by multiphoton fluorescence and polarization-resolved SHIM. [ 44 ]
SHG microscopy was reported for the study of lung , colonic , esophageal stroma and cervical cancers. [ 39 ]
Alterations in the organization or polarity of the collagen fibrils can be signs of pathology,. [ 45 ] [ 46 ]
In particular, the anisotropic alignment of collagen fibers allowed the discrimination of healthy dermis from pathological scars in skin . [ 47 ] Also, pathologies in cartilage such as osteoarthritis can be probed by polarization-resolved SHG microscopy,. [ 48 ] [ 49 ] SHIM was later extended to fibro-cartilage ( meniscus ). [ 50 ]
The ability of SHG to image specific molecules can reveal the structure of a certain tissue one material at a time, and at various scales (from macro to micro) using microscopy. For instance, the collagen (type I) is specifically imaged from the extracellular matrix (ECM) of cells, or when it serves as a scaffold or conjonctive material in tissues. [ 51 ] SHG also reveals fibroin in silk , myosin in muscles and biosynthetized cellulose .
All of this imaging capability can be used to design artificials tissues, by targeting specific points of the tissue : SHG can indeed quantitatively measure some orientations, and material quantity and arrangement. [ 51 ] Also, SHG coupled to other multiphoton techniques can serve to monitor the development of engineered tissues, when the sample is relatively thin however. [ 52 ] Of course, they can finally be used as a quality control of the fabricated tissues. [ 52 ]
Cornea , at the surface of the eye , is considered to be made of plywood-like structure of collagen , due to the self-organization properties of sufficiently dense collagen . [ 53 ] Yet, the collagenous orientation in lamellae is still under debate in this tissue . [ 54 ] Keratoconus cornea can also be imaged by SHG to reveal morphological alterations of the collagen . [ 55 ] Third-Harmonic Generation (THG) microscopy is moreover used to image the cornea , which is complementary to SHG signal as THG and SHG maxima in this tissue are often at different places. [ 56 ] | https://en.wikipedia.org/wiki/Second-harmonic_imaging_microscopy |
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