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Aspect ratio is another way to measure ram-air parachutes. Aspect ratios of parachutes are measured the same way as aircraft wings, by comparing span with chord. Low aspect ratio parachutes, i.e., span 1.8 times the chord, are now limited to precision landing competitions. Popular precision landing parachutes include Jalbert (now NAA) Para-Foils and John Eiff's series of Challenger Classics. While low aspect ratio parachutes tend to be extremely stable, with gentle stall characteristics, they suffer from steep glide ratios and a small tolerance, or "sweet spot", for timing the landing flare.
Because of their predictable opening characteristics, parachutes with a medium aspect ratio around 2.1 are widely used for reserves, BASE, and canopy formation competition. Most medium aspect ratio parachutes have seven cells.
High aspect ratio parachutes have the flattest glide and the largest tolerance for timing the landing flare, but the least predictable openings. An aspect ratio of 2.7 is about the upper limit for parachutes. High aspect ratio canopies typically have nine or more cells. All reserve ram-air parachutes are of the square variety, because of the greater reliability, and the less-demanding handling characteristics.
Paragliders | Parachute | Wikipedia | 254 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
Paragliders - virtually all of which use ram-air canopies - are more akin to today's sport parachutes than, say, parachutes of the mid-1970s and earlier. Technically, they are ascending parachutes, though that term is not used in the paragliding community, and they have the same basic airfoil design of today's 'square' or 'elliptical' sports parachuting canopy, but generally have more sectioned cells, higher aspect ratio and a lower profile. Cell count varies widely, typically from the high 20s to the 70s, while aspect ratio can be 8 or more, though aspect ratio (projected) for such a canopy might be down at 6 or so - both outrageously higher than a representative skydiver's parachute. The wing span is typically so great that it's far closer to a very elongated rectangle or ellipse than a square and that term is rarely used by paraglider pilots. Similarly, span might be ~15 m with span (projected) at 12 m. Canopies are still attached to the harness by suspension lines and (four or six) risers, but they use lockable carabiners as the final connection to the harness. Modern high-performance paragliders often have the cell openings closer to the bottom of the leading edge and the end cells might appear to be closed, both for aerodynamic streamlining (these apparently closed end cells are vented and inflated from the adjacent cells, which have venting in the cell walls).
The main difference is in paragliders' usage, typically longer flights that can last all day and hundreds of kilometres in some cases. The harness is also quite different from a parachuting harness and can vary dramatically from ones for the beginner (which might be just a bench seat with nylon material and webbing to ensure the pilot is secure, no matter the position), to seatboardless ones for high altitude and cross-country flights (these are usually full-body cocoon- or hammock-like devices to include the outstretched legs - called speedbags, aerocones, etc. - to ensure aerodynamic efficiency and warmth). In many designs, there will be protection for the back and shoulder areas built-in, and support for a reserve canopy, water container, etc. Some even have windshields. | Parachute | Wikipedia | 478 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
Because paragliders are made for foot- or ski-launch, they aren't suitable for terminal velocity openings and there is no slider to slow down an opening (paraglider pilots typically start with an open but uninflated canopy). To launch a paraglider, one typically spreads out the canopy on the ground to closely approximate an open canopy with the suspension lines having little slack and less tangle - see more in Paragliding. Depending on the wind, the pilot has three basic options: 1) a running forward launch (typically in no wind or slight wind), 2) a standing launch (in ideal winds) and 3) a reverse launch (in higher winds). In ideal winds, the pilot pulls on the top risers to have the wind inflate the cells and simply eases the brakes down, much like an aircraft's flaps, and takes off. Or if there is no wind, the pilot runs or skis to make it inflate, typically at the edge of a cliff or hill. Once the canopy is above one's head, it's a gentle pull down on both toggles in ideal winds, a tow (say, behind a vehicle) on flat ground, a continued run down the hill, etc. Ground handling in a variety of winds is important and there are even canopies made strictly for that practice, to save on wear and tear of more expensive canopies designed for say, XC, competition or just recreational flying.
General characteristics
Main parachutes used by skydivers today are designed to open softly. Overly rapid deployment was an early problem with ram-air designs. The primary innovation that slows the deployment of a ram-air canopy is the slider; a small rectangular piece of fabric with a grommet near each corner. Four collections of lines go through the grommets to the risers (risers are strips of webbing joining the harness and the rigging lines of a parachute). During deployment, the slider slides down from the canopy to just above the risers. The slider is slowed by air resistance as it descends and reduces the rate at which the lines can spread. This reduces the speed at which the canopy can open and inflate.
At the same time, the overall design of a parachute still has a significant influence on the deployment speed. Modern sport parachutes' deployment speeds vary considerably. Most modern parachutes open comfortably, but individual skydivers may prefer harsher deployment. | Parachute | Wikipedia | 511 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
The deployment process is inherently chaotic. Rapid deployments can still occur even with well-behaved canopies. On rare occasions, deployment can even be so rapid that the jumper suffers bruising, injury, or death. Reducing the amount of fabric decreases the air resistance. This can be done by making the slider smaller, inserting a mesh panel, or cutting a hole in the slider.
Deployment
Reserve parachutes usually have a ripcord deployment system, which was first designed by Theodore Moscicki, but most modern main parachutes used by sports parachutists use a form of hand-deployed pilot chute. A ripcord system pulls a closing pin (sometimes multiple pins), which releases a spring-loaded pilot chute, and opens the container; the pilot chute is then propelled into the air stream by its spring, then uses the force generated by passing air to extract a deployment bag containing the parachute canopy, to which it is attached via a bridle. A hand-deployed pilot chute, once thrown into the air stream, pulls a closing pin on the pilot chute bridle to open the container, then the same force extracts the deployment bag. There are variations on hand-deployed pilot chutes, but the system described is the more common throw-out system.
Only the hand-deployed pilot chute may be collapsed automatically after deployment—by a kill line reducing the in-flight drag of the pilot chute on the main canopy. Reserves, on the other hand, do not retain their pilot chutes after deployment. The reserve deployment bag and pilot chute are not connected to the canopy in a reserve system. This is known as a free-bag configuration, and the components are sometimes not recovered after a reserve deployment.
Occasionally, a pilot chute does not generate enough force either to pull the pin or to extract the bag. Causes may be that the pilot chute is caught in the turbulent wake of the jumper (the "burble"), the closing loop holding the pin is too tight, or the pilot chute is generating insufficient force. This effect is known as "pilot chute hesitation," and, if it does not clear, it can lead to a total malfunction, requiring reserve deployment. | Parachute | Wikipedia | 462 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
Paratroopers' main parachutes are usually deployed by static lines that release the parachute, yet retain the deployment bag that contains the parachute—without relying on a pilot chute for deployment. In this configuration, the deployment bag is known as a direct-bag system, in which the deployment is rapid, consistent, and reliable.
Safety
A parachute is carefully folded, or "packed" to ensure that it will open reliably. If a parachute is not packed properly it can result in a malfunction where the main parachute fails to deploy correctly or fully. In the United States and many developed countries, emergency and reserve parachutes are packed by "riggers" who must be trained and certified according to legal standards. Sport skydivers are always trained to pack their own primary "main" parachutes.
Exact numbers are difficult to estimate because parachute design, maintenance, loading, packing technique and operator experience all have a significant impact on malfunction rates. Approximately one in a thousand sport main parachute openings malfunctions, requiring the use of the reserve parachute, although some skydivers have many thousands of jumps and never needed to use their reserve parachute.
Reserve parachutes are packed and deployed somewhat differently. They are also designed more conservatively, favouring reliability over responsiveness and are built and tested to more exacting standards, making them more reliable than main parachutes. Regulated inspection intervals, coupled with significantly less use contributes to reliability as wear on some components can adversely affect reliability. The safety advantage of a reserve parachute comes from the small probability of a main malfunction being multiplied by the even smaller probability of a reserve malfunction. This yields an even smaller probability of a double malfunction, although there is also a small possibility of a malfunctioning main parachute not being able to be released and thus interfering with the reserve parachute. In the United States, the 2017 average fatality rate is recorded to be 1 in 133,571 jumps.
Injuries and fatalities in sport skydiving are possible even under a fully functional main parachute, such as may occur if the skydiver makes an error in judgment while flying the canopy which results in a high-speed impact either with the ground or with a hazard on the ground, which might otherwise have been avoided, or results in collision with another skydiver under canopy.
Malfunctions | Parachute | Wikipedia | 477 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
Below are listed the malfunctions specific to round parachutes:
A "Mae West" or "blown periphery" is a type of round parachute malfunction that contorts the shape of the canopy into the outward appearance of a large brassiere, named after the generous proportions of the late actress Mae West. The column of nylon fabric, buffeted by the wind, rapidly heats from friction and opposite sides of the canopy can fuse together in a narrow region, removing any chance of it opening fully.
A "streamer" is the main chute which becomes entangled in its lines and fails to deploy, taking the shape of a paper streamer. The parachutist cuts it away to provide space and clean air for deploying the reserve.
An "inversion" occurs when one skirt of the canopy blows between the suspension lines on the opposite side of the parachute and then catches air. That portion then forms a secondary lobe with the canopy inverted. The secondary lobe grows until the canopy turns completely inside out.
A "barber's pole" describes having a tangle of lines behind the jumper's head, who cuts away the main and opens his reserve.
The "horseshoe" is an out-of-sequence deployment, when the parachute lines and bag are released before the bag drogue and bridle. This can cause the lines to become tangled or a situation where the parachute drogue is not released from the container.
"Jumper-In-Tow" involves a static line that does not disconnect, resulting in a jumper being towed behind the aircraft.
Records
On August 16, 1960, Joseph Kittinger, in the Excelsior III test jump, set the previous world record for the highest parachute jump. He jumped from a balloon at an altitude of (which was also a piloted balloon altitude record at the time). A small stabilizer chute deployed successfully, and Kittinger fell for 4 minutes and 36 seconds, also setting a still-standing world record for the longest parachute free-fall, if falling with a stabilizer chute is counted as free-fall. At an altitude of , Kittinger opened his main chute and landed safely in the New Mexico desert. The whole descent took 13 minutes and 45 seconds. During the descent, Kittinger experienced temperatures as low as . In the free-fall stage, he reached a top speed of 614 mph (988 km/h or 274 m/s), or Mach 0.8. | Parachute | Wikipedia | 505 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
According to Guinness World Records, Yevgeni Andreyev, a colonel in the Soviet Air Force, held the official FAI record for the longest free-fall parachute jump (without drogue chute) after falling for 24,500 m (80,380 ft) from an altitude of 25,457 m (83,523 ft) near the city of Saratov, Russia on November 1, 1962, until broken by Felix Baumgartner in 2012.
Felix Baumgartner broke Joseph Kittinger's record on October 14, 2012, with a jump from an altitude of 127,852 feet (38,969.3 m) and reaching speeds up to 833.9 mph (1,342.0 km/h or 372.8 m/s), or nearly Mach 1.1. Kittinger was an advisor for Baumgartner's jump.
Alan Eustace made a jump from the stratosphere on October 24, 2014, from an altitude of 135,889.108 feet (41,419 m). However, because Eustace's jump involved a drogue parachute while Baumgartner's did not, their vertical speed and free fall distance records remain in different record categories.
Uses
In addition to the use of a parachute to slow the descent of a person or object, a drogue parachute is used to aid horizontal deceleration of a land or air vehicle, including fixed-wing aircraft and drag racers, provide stability, as to assist certain types of light aircraft in distress, tandem free-fall; and as a pilot triggering deployment of a larger parachute.
Parachutes are also used as play equipment. | Parachute | Wikipedia | 351 | 48339 | https://en.wikipedia.org/wiki/Parachute | Technology | Aviation | null |
Pesticides are substances that are used to control pests. They include herbicides, insecticides, nematicides, fungicides, and many others (see table). The most common of these are herbicides, which account for approximately 50% of all pesticide use globally. Most pesticides are used as plant protection products (also known as crop protection products), which in general protect plants from weeds, fungi, or insects. In general, a pesticide is a chemical or biological agent (such as a virus, bacterium, or fungus) that deters, incapacitates, kills, or otherwise discourages pests. Target pests can include insects, plant pathogens, weeds, molluscs, birds, mammals, fish, nematodes (roundworms), and microbes that destroy property, cause nuisance, or spread disease, or are disease vectors. Along with these benefits, pesticides also have drawbacks, such as potential toxicity to humans and other species.
Definition
The word pesticide derives from the Latin pestis (plague) and caedere (kill).
The Food and Agriculture Organization (FAO) has defined pesticide as:
any substance or mixture of substances intended for preventing, destroying, or controlling any pest, including vectors of human or animal disease, unwanted species of plants or animals, causing harm during or otherwise interfering with the production, processing, storage, transport, or marketing of food, agricultural commodities, wood and wood products or animal feedstuffs, or substances that may be administered to animals for the control of insects, arachnids, or other pests in or on their bodies. The term includes substances intended for use as a plant growth regulator, defoliant, desiccant, or agent for thinning fruit or preventing the premature fall of fruit. Also used as substances applied to crops either before or after harvest to protect the commodity from deterioration during storage and transport.
Classifications
Pesticides can be classified by target organism (e.g., herbicides, insecticides, fungicides, rodenticides, and pediculicides – see table),
Biopesticides according to the EPA include microbial pesticides, biochemical pesticides, and plant-incorporated protectants. | Pesticide | Wikipedia | 462 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Pesticides can be classified into structural classes, with many structural classes developed for each of the target organisms listed in the table. A structural class is usually associated with a single mode of action, whereas a mode of action may encompass more than one structural class.
The pesticidal chemical (active ingredient) is mixed (formulated) with other components to form the product that is sold, and which is applied in various ways. Pesticides in gas form are fumigants.
Pesticides can be classified based upon their mode of action, which indicates the exact biological mechanism which the pesticide disrupts. The modes of action are important for resistance management, and are categorized and administered by the insecticide, herbicide, and fungicide resistance action committees.
Pesticides may be systemic or non-systemic. A systemic pesticide moves (translocates) inside the plant. Translocation may be upward in the xylem, or downward in the phloem or both. Non-systemic pesticides (contact pesticides) remain on the surface and act through direct contact with the target organism. Pesticides are more effective if they are systemic. Systemicity is a prerequisite for the pesticide to be used as a seed-treatment.
Pesticides can be classified as persistent (non-biodegradable) or non-persistent (biodegradable). A pesticide must be persistent enough to kill or control its target but must degrade fast enough not to accumulate in the environment or the food chain in order to be approved by the authorities. Persistent pesticides, including DDT, were banned many years ago, an exception being spraying in houses to combat malaria vectors. | Pesticide | Wikipedia | 340 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
History
From biblical times until the 1950s the pesticides used were inorganic compounds and plant extracts. The inorganic compounds were derivatives of copper, arsenic, mercury, sulfur, among others, and the plant extracts contained pyrethrum, nicotine, and rotenone among others. The less toxic of these are still in use in organic farming. In the 1940s the insecticide DDT, and the herbicide 2,4-D, were introduced. These synthetic organic compounds were widely used and were very profitable. They were followed in the 1950s and 1960s by numerous other synthetic pesticides, which led to the growth of the pesticide industry. During this period, it became increasingly evident that DDT, which had been sprayed widely in the environment to combat the vector, had accumulated in the food chain. It had become a global pollutant, as summarized in the well-known book Silent Spring. Finally, DDT was banned in the 1970s in several countries, and subsequently all persistent pesticides were banned worldwide, an exception being spraying on interior walls for vector control.
Resistance to a pesticide was first seen in the 1920s with inorganic pesticides, and later it was found that development of resistance is to be expected, and measures to delay it are important. Integrated pest management (IPM) was introduced in the 1950s. By careful analysis and spraying only when an economical or biological threshold of crop damage is reached, pesticide application is reduced. This became in the 2020s the official policy of international organisations, industry, and many governments. With the introduction of high yielding varieties in the 1960s in the green revolution, more pesticides were used. Since the 1980s genetically modified crops were introduced, which resulted in lower amounts of insecticides used on them. Organic agriculture, which uses only non-synthetic pesticides, has grown and in 2020 represents about 1.5 per cent of the world's total agricultural land.
Pesticides have become more effective. Application rates fell from 1,000 to 2,500 grams of active ingredient per hectare (g/ha) in the 1950s to 40–100 g/ha in the 2000s. Despite this, amounts used have increased. In high income countries over 20 years between the 1990s and 2010s amounts used increased 20%, while in the low income countries amounts increased 1623%. | Pesticide | Wikipedia | 463 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Development of new pesticides
The aim is to find new compounds or agents with improved properties such as a new mode of action or lower application rate. Another aim is to replace older pesticides which have been banned for reasons of toxicity or environmental harm or have become less effective due to development of resistance.
The process starts with testing (screening) against target organisms such as insects, fungi or plants. Inputs are typically random compounds, natural products, compounds designed to disrupt a biochemical target, compounds described in patents or literature, or biocontrol organisms.
Compounds that are active in the screening process, known as hits or leads, cannot be used as pesticides, except for biocontrol organisms and some potent natural products. These lead compounds need to be optimised by a series of cycles of synthesis and testing of analogs. For approval by regulatory authorities for use as pesticides, the optimized compounds must meet several requirements. In addition to being potent (low application rate), they must show low toxicity to non-target organisms, low environmental impact, and viable manufacturing cost. The cost of developing a pesticide in 2022 was estimated to be 350 million US dollars. It has become more difficult to find new pesticides. More than 100 new active ingredients were introduced in the 2000s and less than 40 in the 2010s. Biopesticides are cheaper to develop, since the authorities require less toxicological and environmental study. Since 2000 the rate of new biological product introduction has frequently exceeded that of conventional products.
More than 25% of existing chemical pesticides contain one or more chiral centres (stereogenic centres). Newer pesticides with lower application rates tend to have more complex structures, and thus more often contain chiral centres. In cases when most or all of the pesticidal activity in a new compound is found in one enantiomer (the eutomer), the registration and use of the compound as this single enantiomer is preferred. This reduces the total application rate and avoids the tedious environmental testing required when registering a racemate. However, if a viable enantioselective manufacturing route cannot be found, then the racemate is registered and used.
Uses | Pesticide | Wikipedia | 444 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
In addition to their main use in agriculture, pesticides have a number of other applications. Pesticides are used to control organisms that are considered to be harmful, or pernicious to their surroundings. For example, they are used to kill mosquitoes that can transmit potentially deadly diseases like West Nile virus, yellow fever, and malaria. They can also kill bees, wasps or ants that can cause allergic reactions. Insecticides can protect animals from illnesses that can be caused by parasites such as fleas. Pesticides can prevent sickness in humans that could be caused by moldy food or diseased produce. Herbicides can be used to clear roadside weeds, trees, and brush. They can also kill invasive weeds that may cause environmental damage. Herbicides are commonly applied in ponds and lakes to control algae and plants such as water grasses that can interfere with activities like swimming and fishing and cause the water to look or smell unpleasant. Uncontrolled pests such as termites and mold can damage structures such as houses. Pesticides are used in grocery stores and food storage facilities to manage rodents and insects that infest food such as grain. Pesticides are used on lawns and golf courses, partly for cosmetic reasons.
Integrated pest management, the use of multiple approaches to control pests, is becoming widespread and has been used with success in countries such as Indonesia, China, Bangladesh, the U.S., Australia, and Mexico. IPM attempts to recognize the more widespread impacts of an action on an ecosystem, so that natural balances are not upset.
Each use of a pesticide carries some associated risk. Proper pesticide use decreases these associated risks to a level deemed acceptable by pesticide regulatory agencies such as the United States Environmental Protection Agency (EPA) and the Pest Management Regulatory Agency (PMRA) of Canada.
DDT, sprayed on the walls of houses, is an organochlorine that has been used to fight malaria vectors (mosquitos) since the 1940s. The World Health Organization recommend this approach. It and other organochlorine pesticides have been banned in most countries worldwide because of their persistence in the environment and human toxicity. DDT has become less effective, as resistance was identified in Africa as early as 1955, and by 1972 nineteen species of mosquito worldwide were resistant to DDT.
Amount used | Pesticide | Wikipedia | 466 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Total pesticides use in agriculture in 2021 was 3.54 million tonnes of active ingredients (Mt), a 4 percent increase with respect to 2020, an 11 percent increase in a decade, and a doubling since 1990. Pesticides use per area of cropland in 2021 was 2.26 kg per hectare (kg/ha), an increase of 4 percent with respect to 2020; use per value of agricultural production was 0.86 kg per thousand international dollar (kg/1000 I$) (+2%); and use per person was 0.45 kg per capita (kg/cap) (+3%). Between 1990 and 2021, these indicators increased by 85 percent, 3 percent, and 33 percent, respectively. Brazil was the world's largest user of pesticides in 2021, with 720 kt of pesticides applications for agricultural use, while the USA (457 kt) was the second-largest user.
Applications per cropland area in 2021 varied widely, from 10.9 kg/hectare in Brazil to 0.8 kg/ha in the Russian Federation. The level in Brazil was about twice as high as in Argentina (5.6 kg/ha) and Indonesia (5.3 kg/ha). Insecticide use in the US has declined by more than half since 1980 (0.6%/yr), mostly due to the near phase-out of organophosphates. In corn fields, the decline was even steeper, due to the switchover to transgenic Bt corn.
Benefits
Pesticides increase agricultural yields and lower costs. One study found that not using pesticides reduced crop yields by about 10%. Another study, conducted in 1999, found that a ban on pesticides in the United States may result in a rise of food prices, loss of jobs, and an increase in world hunger.
There are two levels of benefits for pesticide use, primary and secondary. Primary benefits are direct gains from the use of pesticides and secondary benefits are effects that are more long-term.
Biological
Controlling pests and plant disease vectors
Improved crop yields
Improved crop/livestock quality
Invasive species controlled
Controlling human/livestock disease vectors and nuisance organisms
Human lives saved and disease reduced. Diseases controlled include malaria, with millions of lives having been saved or enhanced with the use of DDT alone.
Animal lives saved and disease reduced
Controlling organisms that harm other human activities and structures
Drivers view unobstructed
Tree/brush/leaf hazards prevented
Wooden structures protected | Pesticide | Wikipedia | 504 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Economics
In 2018 world pesticide sales were estimated to be $65 billion, of which 88% was used for agriculture. Generic accounted for 85% of sales in 2018. In one study, it was estimated that for every dollar ($1) that is spent on pesticides for crops results in up to four dollars ($4) in crops which would otherwise be lost to insects, fungi and weeds. In general, farmers benefit from having an increase in crop yield and from being able to grow a variety of crops throughout the year. Consumers of agricultural products also benefit from being able to afford the vast quantities of produce available year-round.
Disadvantages
On the cost side of pesticide use there can be costs to the environment and costs to human health. Pesticides safety education and pesticide applicator regulation are designed to protect the public from pesticide misuse, but do not eliminate all misuse. Reducing the use of pesticides and choosing less toxic pesticides may reduce risks placed on society and the environment from pesticide use.
Health effects
Pesticides may affect health negatively. mimicking hormones causing reproductive problems, and also causing cancer. A 2007 systematic review found that "most studies on non-Hodgkin lymphoma and leukemia showed positive associations with pesticide exposure" and thus concluded that cosmetic use of pesticides should be decreased. There is substantial evidence of associations between organophosphate insecticide exposures and neurobehavioral alterations. Limited evidence also exists for other negative outcomes from pesticide exposure including neurological, birth defects, and fetal death.
The American Academy of Pediatrics recommends limiting exposure of children to pesticides and using safer alternatives:
Pesticides are also found in majority of U.S. households with 88 million out of the 121.1 million households indicating that they use some form of pesticide in 2012. As of 2007, there were more than 1,055 active ingredients registered as pesticides, which yield over 20,000 pesticide products that are marketed in the United States.
Owing to inadequate regulation and safety precautions, 99% of pesticide-related deaths occur in developing countries that account for only 25% of pesticide usage.
One study found pesticide self-poisoning the method of choice in one third of suicides worldwide, and recommended, among other things, more restrictions on the types of pesticides that are most harmful to humans. | Pesticide | Wikipedia | 482 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
A 2014 epidemiological review found associations between autism and exposure to certain pesticides, but noted that the available evidence was insufficient to conclude that the relationship was causal.
Occupational exposure among agricultural workers
The World Health Organization and the UN Environment Programme estimate that 3 million agricultural workers in the developing world experience severe poisoning from pesticides each year, resulting in 18,000 deaths. According to one study, as many as 25 million workers in developing countries may suffer mild pesticide poisoning yearly. Other occupational exposures besides agricultural workers, including pet groomers, groundskeepers, and fumigators, may also put individuals at risk of health effects from pesticides.
Pesticide use is widespread in Latin America, as around US$3 billion are spent each year in the region. Records indicate an increase in the frequency of pesticide poisonings over the past two decades. The most common incidents of pesticide poisoning is thought to result from exposure to organophosphate and carbamate insecticides. At-home pesticide use, use of unregulated products, and the role of undocumented workers within the agricultural industry makes characterizing true pesticide exposure a challenge. It is estimated that 50–80% of pesticide poisoning cases are unreported.
Underreporting of pesticide poisoning is especially common in areas where agricultural workers are less likely to seek care from a healthcare facility that may be monitoring or tracking the incidence of acute poisoning. The extent of unintentional pesticide poisoning may be much greater than available data suggest, particularly among developing countries. Globally, agriculture and food production remain one of the largest industries. In East Africa, the agricultural industry represents one of the largest sectors of the economy, with nearly 80% of its population relying on agriculture for income. Farmers in these communities rely on pesticide products to maintain high crop yields.
Some East Africa governments are shifting to corporate farming, and opportunities for foreign conglomerates to operate commercial farms have led to more accessible research on pesticide use and exposure among workers. In other areas where large proportions of the population rely on subsistence, small-scale farming, estimating pesticide use and exposure is more difficult.
Pesticide poisoning | Pesticide | Wikipedia | 441 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Pesticides may exhibit toxic effects on humans and other non-target species, the severity of which depends on the frequency and magnitude of exposure. Toxicity also depends on the rate of absorption, distribution within the body, metabolism, and elimination of compounds from the body. Commonly used pesticides like organophosphates and carbamates act by inhibiting acetylcholinesterase activity, which prevents the breakdown of acetylcholine at the neural synapse. Excess acetylcholine can lead to symptoms like muscle cramps or tremors, confusion, dizziness and nausea. Studies show that farm workers in Ethiopia, Kenya, and Zimbabwe have decreased concentrations of plasma acetylcholinesterase, the enzyme responsible for breaking down acetylcholine acting on synapses throughout the nervous system. Other studies in Ethiopia have observed reduced respiratory function among farm workers who spray crops with pesticides. Numerous exposure pathways for farm workers increase the risk of pesticide poisoning, including dermal absorption walking through fields and applying products, as well as inhalation exposure.
Measuring exposure to pesticides
There are multiple approaches to measuring a person's exposure to pesticides, each of which provides an estimate of an individual's internal dose. Two broad approaches include measuring biomarkers and markers of biological effect. The former involves taking direct measurement of the parent compound or its metabolites in various types of media: urine, blood, serum. Biomarkers may include a direct measurement of the compound in the body before it's been biotransformed during metabolism. Other suitable biomarkers may include the metabolites of the parent compound after they've been biotransformed during metabolism. Toxicokinetic data can provide more detailed information on how quickly the compound is metabolized and eliminated from the body, and provide insights into the timing of exposure.
Markers of biological effect provide an estimation of exposure based on cellular activities related to the mechanism of action. For example, many studies investigating exposure to pesticides often involve the quantification of the acetylcholinesterase enzyme at the neural synapse to determine the magnitude of the inhibitory effect of organophosphate and carbamate pesticides. | Pesticide | Wikipedia | 451 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Another method of quantifying exposure involves measuring, at the molecular level, the amount of pesticide interacting with the site of action. These methods are more commonly used for occupational exposures where the mechanism of action is better understood, as described by WHO guidelines published in "Biological Monitoring of Chemical Exposure in the Workplace". Better understanding of how pesticides elicit their toxic effects is needed before this method of exposure assessment can be applied to occupational exposure of agricultural workers.
Alternative methods to assess exposure include questionnaires to discern from participants whether they are experiencing symptoms associated with pesticide poisoning. Self-reported symptoms may include headaches, dizziness, nausea, joint pain, or respiratory symptoms.
Challenges in assessing pesticide exposure
Multiple challenges exist in assessing exposure to pesticides in the general population, and many others that are specific to occupational exposures of agricultural workers. Beyond farm workers, estimating exposure to family members and children presents additional challenges, and may occur through "take-home" exposure from pesticide residues collected on clothing or equipment belonging to parent farm workers and inadvertently brought into the home. Children may also be exposed to pesticides prenatally from mothers who are exposed to pesticides during pregnancy. Characterizing children's exposure resulting from drift of airborne and spray application of pesticides is similarly challenging, yet well documented in developing countries. Because of critical development periods of the fetus and newborn children, these non-working populations are more vulnerable to the effects of pesticides, and may be at increased risk of developing neurocognitive effects and impaired development.
While measuring biomarkers or markers of biological effects may provide more accurate estimates of exposure, collecting these data in the field is often impractical and many methods are not sensitive enough to detect low-level concentrations. Rapid cholinesterase test kits exist to collect blood samples in the field. Conducting large scale assessments of agricultural workers in remote regions of developing countries makes the implementation of these kits a challenge. The cholinesterase assay is a useful clinical tool to assess individual exposure and acute toxicity. Considerable variability in baseline enzyme activity among individuals makes it difficult to compare field measurements of cholinesterase activity to a reference dose to determine health risk associated with exposure. Another challenge in deriving a reference dose is identifying health endpoints that are relevant to exposure. More epidemiological research is needed to identify critical health endpoints, particularly among populations who are occupationally exposed. | Pesticide | Wikipedia | 493 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Prevention
Minimizing harmful exposure to pesticides can be achieved by proper use of personal protective equipment, adequate reentry times into recently sprayed areas, and effective product labeling for hazardous substances as per FIFRA regulations. Training high-risk populations, including agricultural workers, on the proper use and storage of pesticides, can reduce the incidence of acute pesticide poisoning and potential chronic health effects associated with exposure. Continued research into the human toxic health effects of pesticides serves as a basis for relevant policies and enforceable standards that are health protective to all populations.
Environmental effects
Pesticide use raises a number of environmental concerns. Over 98% of sprayed insecticides and 95% of herbicides reach a destination other than their target species, including non-target species, air, water and soil. Pesticide drift occurs when pesticides suspended in the air as particles are carried by wind to other areas, potentially contaminating them. Pesticides are one of the causes of water pollution, and some pesticides were persistent organic pollutants (now banned), which contribute to soil and flower (pollen, nectar) contamination. Furthermore, pesticide use can adversely impact neighboring agricultural activity, as pests themselves drift to and harm nearby crops that have no pesticide used on them.
In addition, pesticide use reduces invertebrate biodiversity in streams, contributes to pollinator decline, destroys habitat (especially for birds), and threatens endangered species. Pests can develop a resistance to the pesticide (pesticide resistance), necessitating a new pesticide. Alternatively a greater dose of the pesticide can be used to counteract the resistance, although this will cause a worsening of the ambient pollution problem.
The Stockholm Convention on Persistent Organic Pollutants banned all persistent pesticides, in particular DDT and other organochlorine pesticides, which were stable and lipophilic, and thus able to bioaccumulate in the body and the food chain. and which spread throughout the planet. Persistent pesticides are no longer used for agriculture, and will not be approved by the authorities. Because the half life in soil is long (for DDT 2–15 years) residues can still be detected in humans at levels 5 to 10 times lower than found in the 1970s. | Pesticide | Wikipedia | 459 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Pesticides now have to be degradable in the environment. Such degradation of pesticides is due to both innate chemical properties of the compounds and environmental processes or conditions. For example, the presence of halogens within a chemical structure often slows down degradation in an aerobic environment. Adsorption to soil may retard pesticide movement, but also may reduce bioavailability to microbial degraders.
Pesticide contamination in the environment can be monitored through bioindicators such as bee pollinators.
Economics
In one study, the human health and environmental costs due to pesticides in the United States was estimated to be $9.6 billion: offset by about $40 billion in increased agricultural production.
Additional costs include the registration process and the cost of purchasing pesticides: which are typically borne by agrichemical companies and farmers respectively. The registration process can take several years to complete (there are 70 types of field tests) and can cost $50–70 million for a single pesticide. At the beginning of the 21st century, the United States spent approximately $10 billion on pesticides annually.
Resistance
The use of pesticides inherently entails the risk of resistance developing. Various techniques and procedures of pesticide application can slow the development of resistance, as can some natural features of the target population and surrounding environment.
Alternatives
Alternatives to pesticides are available and include methods of cultivation, use of biological pest controls (such as pheromones and microbial pesticides), genetic engineering (mostly of crops), and methods of interfering with insect breeding. Application of composted yard waste has also been used as a way of controlling pests.
These methods are becoming increasingly popular and often are safer than traditional chemical pesticides. In addition, EPA is registering reduced-risk pesticides in increasing numbers.
Cultivation practices
Cultivation practices include polyculture (growing multiple types of plants), crop rotation, planting crops in areas where the pests that damage them do not live, timing planting according to when pests will be least problematic, and use of trap crops that attract pests away from the real crop. Trap crops have successfully controlled pests in some commercial agricultural systems while reducing pesticide usage. In other systems, trap crops can fail to reduce pest densities at a commercial scale, even when the trap crop works in controlled experiments. | Pesticide | Wikipedia | 474 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Use of other organisms
Release of other organisms that fight the pest is another example of an alternative to pesticide use. These organisms can include natural predators or parasites of the pests. Biological pesticides based on entomopathogenic fungi, bacteria and viruses causing disease in the pest species can also be used.
Biological control engineering
Interfering with insects' reproduction can be accomplished by sterilizing males of the target species and releasing them, so that they mate with females but do not produce offspring. This technique was first used on the screwworm fly in 1958 and has since been used with the medfly, the tsetse fly, and the gypsy moth. This is a costly and slow approach that only works on some types of insects.
Other alternatives
Other alternatives include "laserweeding" – the use of novel agricultural robots for weed control using lasers.
Push pull strategy
Push-pull technique: intercropping with a "push" crop that repels the pest, and planting a "pull" crop on the boundary that attracts and traps it.
Effectiveness
Some evidence shows that alternatives to pesticides can be equally effective as the use of chemicals. A study of Maize fields in northern Florida found that the application of composted yard waste with high carbon to nitrogen ratio to agricultural fields was highly effective at reducing the population of plant-parasitic nematodes and increasing crop yield, with yield increases ranging from 10% to 212%; the observed effects were long-term, often not appearing until the third season of the study. Additional silicon nutrition protects some horticultural crops against fungal diseases almost completely, while insufficient silicon sometimes leads to severe infection even when fungicides are used.
Pesticide resistance is increasing and that may make alternatives more attractive.
Types
Biopesticides
Biopesticides are certain types of pesticides derived from such natural materials as animals, plants, bacteria, and certain minerals. For example, canola oil and baking soda have pesticidal applications and are considered biopesticides. Biopesticides fall into three major classes: | Pesticide | Wikipedia | 410 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Microbial pesticides which consist of bacteria, entomopathogenic fungi or viruses (and sometimes includes the metabolites that bacteria or fungi produce). Entomopathogenic nematodes are also often classed as microbial pesticides, even though they are multi-cellular.
Biochemical pesticides or herbal pesticides are naturally occurring substances that control (or monitor in the case of pheromones) pests and microbial diseases.
Plant-incorporated protectants (PIPs) have genetic material from other species incorporated into their genetic material (i.e. GM crops). Their use is controversial, especially in many European countries.
By pest type
Pesticides that are related to the type of pests are:
Regulation
International
In many countries, pesticides must be approved for sale and use by a government agency.
Worldwide, 85% of countries have pesticide legislation for the proper storage of pesticides and 51% include provisions to ensure proper disposal of all obsolete pesticides.
Though pesticide regulations differ from country to country, pesticides, and products on which they were used are traded across international borders. To deal with inconsistencies in regulations among countries, delegates to a conference of the United Nations Food and Agriculture Organization adopted an International Code of Conduct on the Distribution and Use of Pesticides in 1985 to create voluntary standards of pesticide regulation for many countries. The Code was updated in 1998 and 2002. The FAO claims that the code has raised awareness about pesticide hazards and decreased the number of countries without restrictions on pesticide use.
Three other efforts to improve regulation of international pesticide trade are the United Nations London Guidelines for the Exchange of Information on Chemicals in International Trade and the United Nations Codex Alimentarius Commission. The former seeks to implement procedures for ensuring that prior informed consent exists between countries buying and selling pesticides, while the latter seeks to create uniform standards for maximum levels of pesticide residues among participating countries.
United States
In the United States, the Environmental Protection Agency (EPA) is responsible for regulating pesticides under the Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA) and the Food Quality Protection Act (FQPA). | Pesticide | Wikipedia | 441 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Studies must be conducted to establish the conditions in which the material is safe to use and the effectiveness against the intended pest(s). The EPA regulates pesticides to ensure that these products do not pose adverse effects to humans or the environment, with an emphasis on the health and safety of children. Pesticides produced before November 1984 continue to be reassessed in order to meet the current scientific and regulatory standards. All registered pesticides are reviewed every 15 years to ensure they meet the proper standards. During the registration process, a label is created. The label contains directions for proper use of the material in addition to safety restrictions. Based on acute toxicity, pesticides are assigned to a Toxicity Class. Pesticides are the most thoroughly tested chemicals after drugs in the United States; those used on food require more than 100 tests to determine a range of potential impacts.
Some pesticides are considered too hazardous for sale to the general public and are designated restricted use pesticides. Only certified applicators, who have passed an exam, may purchase or supervise the application of restricted use pesticides. Records of sales and use are required to be maintained and may be audited by government agencies charged with the enforcement of pesticide regulations. These records must be made available to employees and state or territorial environmental regulatory agencies.
In addition to the EPA, the United States Department of Agriculture (USDA) and the United States Food and Drug Administration (FDA) set standards for the level of pesticide residue that is allowed on or in crops. The EPA looks at what the potential human health and environmental effects might be associated with the use of the pesticide.
In addition, the U.S. EPA uses the National Research Council's four-step process for human health risk assessment: (1) Hazard Identification, (2) Dose-Response Assessment, (3) Exposure Assessment, and (4) Risk Characterization.
In 2013 Kaua'i County (Hawai'i) passed Bill No. 2491 to add an article to Chapter 22 of the county's code relating to pesticides and GMOs. The bill strengthens protections of local communities in Kaua'i where many large pesticide companies test their products.
The first legislation providing federal authority for regulating pesticides was enacted in 1910.
Canada
EU | Pesticide | Wikipedia | 459 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
EU legislation has been approved banning the use of highly toxic pesticides including those that are carcinogenic, mutagenic or toxic to reproduction, those that are endocrine-disrupting, and those that are persistent, bioaccumulative and toxic (PBT) or very persistent and very bioaccumulative (vPvB) and measures have been approved to improve the general safety of pesticides across all EU member states.
In 2023 The Environment Committee of European Parliament approved a decision aiming to reduce pesticide use by 50% (the most hazardous by 65%) by the year 2030 and ensure sustainable use of pesticides (for example use them only as a last resort). The decision also includes measures for providing farmers with alternatives.
Residue
Pesticide residue refers to the pesticides that may remain on or in food after they are applied to food crops. The maximum residue limits (MRL) of pesticides in food are carefully set by the regulatory authorities to ensure, to their best judgement, no health impacts. Regulations such as pre-harvest intervals also often prevent harvest of crop or livestock products if recently treated in order to allow residue concentrations to decrease over time to safe levels before harvest. Exposure of the general population to these residues most commonly occurs through consumption of treated food sources, or being in close contact to areas treated with pesticides such as farms or lawns.
Persistent pesticides are no longer used for agriculture, and will not be approved by the authorities. Because the half life in soil is long (for DDT 2–15 years) residues can still be detected in humans at levels 5 to 10 times lower than found in the 1970s.
Residues are monitored by the authorities. In 2016, over 99% of samples of US produce had no pesticide residue or had residue levels well below the EPA tolerance levels for each pesticide. | Pesticide | Wikipedia | 374 | 48340 | https://en.wikipedia.org/wiki/Pesticide | Technology | Horticultural techniques | null |
Polyurethane (; often abbreviated PUR and PU) refers to a class of polymers composed of organic units joined by carbamate (urethane) links. In contrast to other common polymers such as polyethylene and polystyrene, polyurethane term does not refer to the single type of polymer but a group of polymers. Unlike polyethylene and polystyrene polyurethanes can be produced from a wide range of starting materials resulting various polymers within the same group. This chemical variety produces polyurethanes with different chemical structures leading to many different applications. These include rigid and flexible foams, and coatings, adhesives, electrical potting compounds, and fibers such as spandex and polyurethane laminate (PUL). Foams are the largest application accounting for 67% of all polyurethane produced in 2016.
A polyurethane is typically produced by reacting a polymeric isocyanate with a polyol. Since a polyurethane contains two types of monomers, which polymerize one after the other, they are classed as alternating copolymers. Both the isocyanates and polyols used to make a polyurethane contain two or more functional groups per molecule.
Global production in 2019 was 25 million metric tonnes, accounting for about 6% of all polymers produced in that year.
History
Otto Bayer and his coworkers at IG Farben in Leverkusen, Germany, first made polyurethanes in 1937. The new polymers had some advantages over existing plastics that were made by polymerizing olefins or by polycondensation, and were not covered by patents obtained by Wallace Carothers on polyesters. Early work focused on the production of fibers and flexible foams and PUs were applied on a limited scale as aircraft coating during World War II. Polyisocyanates became commercially available in 1952, and production of flexible polyurethane foam began in 1954 by combining toluene diisocyanate (TDI) and polyester polyols. These materials were also used to produce rigid foams, gum rubber, and elastomers. Linear fibers were produced from hexamethylene diisocyanate (HDI) and 1,4-Butanediol (BDO). | Polyurethane | Wikipedia | 481 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
DuPont introduced polyethers, specifically poly(tetramethylene ether) glycol, in 1956. BASF and Dow Chemical introduced polyalkylene glycols in 1957. Polyether polyols were cheaper, easier to handle and more water-resistant than polyester polyols. Union Carbide and Mobay, a U.S. Monsanto/Bayer joint venture, also began making polyurethane chemicals. In 1960 more than 45,000 metric tons of flexible polyurethane foams were produced. The availability of chlorofluoroalkane blowing agents, inexpensive polyether polyols, and methylene diphenyl diisocyanate (MDI) allowed polyurethane rigid foams to be used as high-performance insulation materials. In 1967, urethane-modified polyisocyanurate rigid foams were introduced, offering even better thermal stability and flammability resistance. During the 1960s, automotive interior safety components, such as instrument and door panels, were produced by back-filling thermoplastic skins with semi-rigid foam.
In 1969, Bayer exhibited an all-plastic car in Düsseldorf, Germany. Parts of this car, such as the fascia and body panels, were manufactured using a new process called reaction injection molding (RIM), in which the reactants were mixed and then injected into a mold. The addition of fillers, such as milled glass, mica, and processed mineral fibers, gave rise to reinforced RIM (RRIM), which provided improvements in flexural modulus (stiffness), reduction in coefficient of thermal expansion and better thermal stability. This technology was used to make the first plastic-body automobile in the United States, the Pontiac Fiero, in 1983. Further increases in stiffness were obtained by incorporating pre-placed glass mats into the RIM mold cavity, also known broadly as resin injection molding, or structural RIM.
Starting in the early 1980s, water-blown microcellular flexible foams were used to mold gaskets for automotive panels and air-filter seals, replacing PVC polymers. Polyurethane foams are used in many automotive applications including seating, head and arm rests, and headliners. | Polyurethane | Wikipedia | 458 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Polyurethane foam (including foam rubber) is sometimes made using small amounts of blowing agents to give less dense foam, better cushioning/energy absorption or thermal insulation. In the early 1990s, because of their impact on ozone depletion, the Montreal Protocol restricted the use of many chlorine-containing blowing agents, such as trichlorofluoromethane (CFC-11). By the late 1990s, blowing agents such as carbon dioxide, pentane, 1,1,1,2-tetrafluoroethane (HFC-134a) and 1,1,1,3,3-pentafluoropropane (HFC-245fa) were widely used in North America and the EU, although chlorinated blowing agents remained in use in many developing countries. Later, HFC-134a was also banned due to high ODP and GWP readings, and HFC-141B was introduced in early 2000s as an alternate blowing agent in developing nations.
Chemistry
Polyurethanes are produced by reacting diisocyanates with polyols, often in the presence of a catalyst, or upon exposure to ultraviolet radiation.
Common catalysts include tertiary amines, such as DABCO, DMDEE, or metallic soaps, such as dibutyltin dilaurate. The stoichiometry of the starting materials must be carefully controlled as excess isocyanate can trimerise, leading to the formation of rigid polyisocyanurates. The polymer usually has a highly crosslinked molecular structure, resulting in a thermosetting material which does not melt on heating; although some thermoplastic polyurethanes are also produced.
The most common application of polyurethane is as solid foams, which requires the presence of a gas, or blowing agent, during the polymerization step. This is commonly achieved by adding small amounts of water, which reacts with isocyanates to form CO2 gas and an amine, via an unstable carbamic acid group. The amine produced can also react with isocyanates to form urea groups, and as such the polymer will contain both these and urethane linkers. The urea is not very soluble in the reaction mixture and tends to form separate "hard segment" phases consisting mostly of polyurea. The concentration and organization of these polyurea phases can have a significant impact on the properties of the foam. | Polyurethane | Wikipedia | 510 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
The type of foam produced can be controlled by regulating the amount of blowing agent and also by the addition of various surfactants which change the rheology of the polymerising mixture. Foams can be either "closed-cell", where most of the original bubbles or cells remain intact, or "open-cell", where the bubbles have broken but the edges of the bubbles are stiff enough to retain their shape, in extreme cases reticulated foams can be formed. Open-cell foams feel soft and allow air to flow through, so they are comfortable when used in seat cushions or mattresses. Closed-cell foams are used as rigid thermal insulation. High-density microcellular foams can be formed without the addition of blowing agents by mechanically frothing the polyol prior to use. These are tough elastomeric materials used in covering car steering wheels or shoe soles.
The properties of a polyurethane are greatly influenced by the types of isocyanates and polyols used to make it. Long, flexible segments, contributed by the polyol, give soft, elastic polymer. High amounts of crosslinking give tough or rigid polymers. Long chains and low crosslinking give a polymer that is very stretchy, short chains with many crosslinks produce a hard polymer while long chains and intermediate crosslinking give a polymer useful for making foam. The choices available for the isocyanates and polyols, in addition to other additives and processing conditions allow polyurethanes to have the very wide range of properties that make them such widely used polymers.
Raw materials
The main ingredients to make a polyurethane are di- and tri-isocyanates and polyols. Other materials are added to aid processing the polymer or to modify the properties of the polymer. PU foam formulation sometimes have water added too.
Isocyanates
Isocyanates used to make polyurethane have two or more isocyanate groups on each molecule. The most commonly used isocyanates are the aromatic diisocyanates, toluene diisocyanate (TDI) and methylene diphenyl diisocyanate, (MDI). These aromatic isocyanates are more reactive than aliphatic isocyanates. | Polyurethane | Wikipedia | 470 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
TDI and MDI are generally less expensive and more reactive than other isocyanates. Industrial grade TDI and MDI are mixtures of isomers and MDI often contains polymeric materials. They are used to make flexible foam (for example slabstock foam for mattresses or molded foams for car seats), rigid foam (for example insulating foam in refrigerators) elastomers (shoe soles, for example), and so on. The isocyanates may be modified by partially reacting them with polyols or introducing some other materials to reduce volatility (and hence toxicity) of the isocyanates, decrease their freezing points to make handling easier or to improve the properties of the final polymers.
Aliphatic and cycloaliphatic isocyanates are used in smaller quantities, most often in coatings and other applications where color and transparency are important since polyurethanes made with aromatic isocyanates tend to darken on exposure to light. The most important aliphatic and cycloaliphatic isocyanates are 1,6-hexamethylene diisocyanate (HDI), 1-isocyanato-3-isocyanatomethyl-3,5,5-trimethyl-cyclohexane (isophorone diisocyanate, IPDI), and 4,4′-diisocyanato dicyclohexylmethane (H12MDI or hydrogenated MDI). Other more specialized isocyanates include Tetramethylxylylene diisocyanate (TMXDI).
Polyols
Polyols are polymers in their own right and have on average two or more hydroxyl groups per molecule. They can be converted to polyether polyols by co-polymerizing ethylene oxide and propylene oxide with a suitable polyol precursor. Polyester polyols are made by the polycondensation of multifunctional carboxylic acids and polyhydroxyl compounds. They can be further classified according to their end use. Higher molecular weight polyols (molecular weights from 2,000 to 10,000) are used to make more flexible polyurethanes while lower molecular weight polyols make more rigid products. | Polyurethane | Wikipedia | 479 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Polyols for flexible applications use low functionality initiators such as dipropylene glycol (f = 2), glycerine (f = 3), or a sorbitol/water solution (f = 2.75). Polyols for rigid applications use higher functionality initiators such as sucrose (f = 8), sorbitol (f = 6), toluenediamine (f = 4), and Mannich bases (f = 4). Propylene oxide and/or ethylene oxide is added to the initiators until the desired molecular weight is achieved. The order of addition and the amounts of each oxide affect many polyol properties, such as compatibility, water-solubility, and reactivity. Polyols made with only propylene oxide are terminated with secondary hydroxyl groups and are less reactive than polyols capped with ethylene oxide, which contain primary hydroxyl groups. Incorporating carbon dioxide into the polyol structure is being researched by multiple companies.
Graft polyols (also called filled polyols or polymer polyols) contain finely dispersed styrene–acrylonitrile, acrylonitrile, or polyurea (PHD) polymer solids chemically grafted to a high molecular weight polyether backbone. They are used to increase the load-bearing properties of low-density high-resiliency (HR) foam, as well as add toughness to microcellular foams and cast elastomers. Initiators such as ethylenediamine and triethanolamine are used to make low molecular weight rigid foam polyols that have built-in catalytic activity due to the presence of nitrogen atoms in the backbone. A special class of polyether polyols, poly(tetramethylene ether) glycols, which are made by polymerizing tetrahydrofuran, are used in high performance coating, wetting and elastomer applications. | Polyurethane | Wikipedia | 413 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Conventional polyester polyols are based on virgin raw materials and are manufactured by the direct polyesterification of high-purity diacids and glycols, such as adipic acid and 1,4-butanediol. Polyester polyols are usually more expensive and more viscous than polyether polyols, but they make polyurethanes with better solvent, abrasion, and cut resistance. Other polyester polyols are based on reclaimed raw materials. They are manufactured by transesterification (glycolysis) of recycled poly(ethyleneterephthalate) (PET) or dimethylterephthalate (DMT) distillation bottoms with glycols such as diethylene glycol. These low molecular weight, aromatic polyester polyols are used in rigid foam, and bring low cost and excellent flammability characteristics to polyisocyanurate (PIR) boardstock and polyurethane spray foam insulation.
Specialty polyols include polycarbonate polyols, polycaprolactone polyols, polybutadiene polyols, and polysulfide polyols. The materials are used in elastomer, sealant, and adhesive applications that require superior weatherability, and resistance to chemical and environmental attack. Natural oil polyols derived from castor oil and other vegetable oils are used to make elastomers, flexible bunstock, and flexible molded foam.
Co-polymerizing chlorotrifluoroethylene or tetrafluoroethylene with vinyl ethers containing hydroxyalkyl vinyl ether produces fluorinated (FEVE) polyols. Two-component fluorinated polyurethanes prepared by reacting FEVE fluorinated polyols with polyisocyanate have been used to make ambient cure paints and coatings. Since fluorinated polyurethanes contain a high percentage of fluorine–carbon bonds, which are the strongest bonds among all chemical bonds, fluorinated polyurethanes exhibit resistance to UV, acids, alkali, salts, chemicals, solvents, weathering, corrosion, fungi and microbial attack. These have been used for high performance coatings and paints.
Phosphorus-containing polyols are available that become chemically bonded to the polyurethane matrix for the use as flame retardants. This covalent linkage prevents migration and leaching of the organophosphorus compound. | Polyurethane | Wikipedia | 509 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Bio-derived materials
Interest in sustainable "green" products raised interest in polyols derived from vegetable oils. Various oils used in the preparation polyols for polyurethanes include soybean oil, cottonseed oil, neem seed oil, and castor oil. Vegetable oils are functionalized in various ways and modified to polyetheramides, polyethers, alkyds, etc. Renewable sources used to prepare polyols may be fatty acids or dimer fatty acids. Some biobased and isocyanate-free polyurethanes exploit the reaction between polyamines and cyclic carbonates to produce polyhydroxyurethanes. | Polyurethane | Wikipedia | 135 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Chain extenders and cross linkers
Chain extenders (f = 2) and cross linkers (f ≥ 3) are low molecular weight hydroxyl and amine terminated compounds that play an important role in the polymer morphology of polyurethane fibers, elastomers, adhesives, and certain integral skin and microcellular foams. The elastomeric properties of these materials are derived from the phase separation of the hard and soft copolymer segments of the polymer, such that the urethane hard segment domains serve as cross-links between the amorphous polyether (or polyester) soft segment domains. This phase separation occurs because the mainly nonpolar, low melting soft segments are incompatible with the polar, high melting hard segments. The soft segments, which are formed from high molecular weight polyols, are mobile and are normally present in coiled formation, while the hard segments, which are formed from the isocyanate and chain extenders, are stiff and immobile. As the hard segments are covalently coupled to the soft segments, they inhibit plastic flow of the polymer chains, thus creating elastomeric resiliency. Upon mechanical deformation, a portion of the soft segments are stressed by uncoiling, and the hard segments become aligned in the stress direction. This reorientation of the hard segments and consequent powerful hydrogen bonding contributes to high tensile strength, elongation, and tear resistance values. | Polyurethane | Wikipedia | 298 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
The choice of chain extender also determines flexural, heat, and chemical resistance properties. The most important chain extenders are ethylene glycol, 1,4-butanediol (1,4-BDO or BDO), 1,6-hexanediol, cyclohexane dimethanol and hydroquinone bis(2-hydroxyethyl) ether (HQEE). All of these glycols form polyurethanes that phase separate well and form well defined hard segment domains, and are melt processable. They are all suitable for thermoplastic polyurethanes with the exception of ethylene glycol, since its derived bis-phenyl urethane undergoes unfavorable degradation at high hard segment levels. Diethanolamine and triethanolamine are used in flex molded foams to build firmness and add catalytic activity. Diethyltoluenediamine is used extensively in RIM, and in polyurethane and polyurea elastomer formulations. | Polyurethane | Wikipedia | 221 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Catalysts
Polyurethane catalysts can be classified into two broad categories, basic and acidic amine. Tertiary amine catalysts function by enhancing the nucleophilicity of the diol component. Alkyl tin carboxylates, oxides and mercaptides oxides function as mild Lewis acids in accelerating the formation of polyurethane. As bases, traditional amine catalysts include triethylenediamine (TEDA, also called DABCO, 1,4-diazabicyclo[2.2.2]octane), dimethylcyclohexylamine (DMCHA), dimethylethanolamine (DMEA), Dimethylaminoethoxyethanol and bis-(2-dimethylaminoethyl)ether, a blowing catalyst also called A-99. A typical Lewis acidic catalyst is dibutyltin dilaurate. The process is highly sensitive to the nature of the catalyst and is also known to be autocatalytic.
Factors affecting catalyst selection include balancing three reactions: urethane (polyol+isocyanate, or gel) formation, the urea (water+isocyanate, or "blow") formation, or the isocyanate trimerization reaction (e.g., using potassium acetate, to form isocyanurate rings). A variety of specialized catalysts have been developed.
Surfactants
Surfactants are used to modify the characteristics of both foam and non-foam polyurethane polymers. They take the form of polydimethylsiloxane-polyoxyalkylene block copolymers, silicone oils, nonylphenol ethoxylates, and other organic compounds. In foams, they are used to emulsify the liquid components, regulate cell size, and stabilize the cell structure to prevent collapse and sub-surface voids. In non-foam applications they are used as air release and antifoaming agents, as wetting agents, and are used to eliminate surface defects such as pin holes, orange peel, and sink marks. | Polyurethane | Wikipedia | 446 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Production
Polyurethanes are produced by mixing two or more liquid streams. The polyol stream contains catalysts, surfactants, blowing agents (when making polyurethane foam insulation) and so on. The two components are referred to as a polyurethane system, or simply a system. The isocyanate is commonly referred to in North America as the 'A-side' or just the 'iso'. The blend of polyols and other additives is commonly referred to as the 'B-side' or as the 'poly'. This mixture might also be called a 'resin' or 'resin blend'. In Europe the meanings for 'A-side' and 'B-side' are reversed. Resin blend additives may include chain extenders, cross linkers, surfactants, flame retardants, blowing agents, pigments, and fillers. Polyurethane can be made in a variety of densities and hardnesses by varying the isocyanate, polyol or additives.
Health and safety
Fully reacted polyurethane polymer is chemically inert. No exposure limits have been established in the U.S. by OSHA (Occupational Safety and Health Administration) or ACGIH (American Conference of Governmental Industrial Hygienists). It is not regulated by OSHA for carcinogenicity.
Polyurethanes are combustible. Decomposition from fire can produce significant amounts of carbon monoxide and hydrogen cyanide, in addition to nitrogen oxides, isocyanates, and other toxic products. Due to the flammability of the material, it has to be treated with flame retardants (at least in case of furniture), almost all of which are considered harmful. California later issued Technical Bulletin 117 2013 which allowed most polyurethane foam to pass flammability tests without the use of flame retardants. Green Science Policy Institute states: "Although the new standard can be met without flame retardants, it does NOT ban their
use. Consumers who wish to reduce household exposure to flame retardants can look for a TB117-2013 tag on furniture, and verify with retailers that products do not contain flame retardants." | Polyurethane | Wikipedia | 466 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Liquid resin blends and isocyanates may contain hazardous or regulated components. Isocyanates are known skin and respiratory sensitizers. Additionally, amines, glycols, and phosphate present in spray polyurethane foams present risks.
Exposure to chemicals that may be emitted during or after application of polyurethane spray foam (such as isocyanates) are harmful to human health and therefore special precautions are required during and after this process.
In the United States, additional health and safety information can be found through organizations such as the Polyurethane Manufacturers Association (PMA) and the Center for the Polyurethanes Industry (CPI), as well as from polyurethane system and raw material manufacturers. Regulatory information can be found in the Code of Federal Regulations Title 21 (Food and Drugs) and Title 40 (Protection of the Environment). In Europe, health and safety information is available from ISOPA, the European Diisocyanate and Polyol Producers Association.
Manufacturing
The methods of manufacturing polyurethane finished goods range from small, hand pour piece-part operations to large, high-volume bunstock and boardstock production lines. Regardless of the end-product, the manufacturing principle is the same: to meter the liquid isocyanate and resin blend at a specified stoichiometric ratio, mix them together until a homogeneous blend is obtained, dispense the reacting liquid into a mold or on to a surface, wait until it cures, then demold the finished part. | Polyurethane | Wikipedia | 314 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Dispensing equipment
Although the capital outlay can be high, it is desirable to use a meter-mix or dispense unit for even low-volume production operations that require a steady output of finished parts. Dispense equipment consists of material holding (day) tanks, metering pumps, a mix head, and a control unit. Often, a conditioning or heater–chiller unit is added to control material temperature in order to improve mix efficiency, cure rate, and to reduce process variability. Choice of dispense equipment components depends on shot size, throughput, material characteristics such as viscosity and filler content, and process control. Material day tanks may be single to hundreds of gallons in size and may be supplied directly from drums, IBCs (intermediate bulk containers, such as caged IBC totes), or bulk storage tanks. They may incorporate level sensors, conditioning jackets, and mixers. Pumps can be sized to meter in single grams per second up to hundreds of pounds per minute. They can be rotary, gear, or piston pumps, or can be specially hardened lance pumps to meter liquids containing highly abrasive fillers such as chopped or hammer-milled glass fiber and wollastonite.
The pumps can drive low-pressure (10 to 30 bar, 1 to 3 MPa) or high-pressure (125 to 250 bar, 12.5 to 25.0 MPa) dispense systems. Mix heads can be simple static mix tubes, rotary-element mixers, low-pressure dynamic mixers, or high-pressure hydraulically actuated direct impingement mixers. Control units may have basic on/off and dispense/stop switches, and analogue pressure and temperature gauges, or may be computer-controlled with flow meters to electronically calibrate mix ratio, digital temperature and level sensors, and a full suite of statistical process control software. Add-ons to dispense equipment include nucleation or gas injection units, and third or fourth stream capability for adding pigments or metering in supplemental additive packages.
Tooling
Distinct from pour-in-place, bun and boardstock, and coating applications, the production of piece parts requires tooling to contain and form the reacting liquid.
The choice of mold-making material is dependent on the expected number of uses to end-of-life (EOL), molding pressure, flexibility, and heat transfer characteristics. | Polyurethane | Wikipedia | 498 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
RTV silicone is used for tooling that has an EOL in the thousands of parts. It is typically used for molding rigid foam parts, where the ability to stretch and peel the mold around undercuts is needed.
The heat transfer characteristic of RTV silicone tooling is poor. High-performance, flexible polyurethane elastomers are also used in this way.
Epoxy, metal-filled epoxy, and metal-coated epoxy is used for tooling that has an EOL in the tens of thousands of parts. It is typically used for molding flexible foam cushions and seating, integral skin and microcellular foam padding, and shallow-draft RIM bezels and fascia. The heat transfer characteristic of epoxy tooling is fair; the heat transfer characteristic of metal-filled and metal-coated epoxy is good. Copper tubing can be incorporated into the body of the tool, allowing hot water to circulate and heat the mold surface.
Aluminum is used for tooling that has an EOL in the hundreds of thousands of parts. It is typically used for molding microcellular foam gasketing and cast elastomer parts, and is milled or extruded into shape.
Mirror-finish stainless steel is used for tooling that imparts a glossy appearance to the finished part. The heat transfer characteristic of metal tooling is excellent.
Finally, molded or milled polypropylene is used to create low-volume tooling for molded gasket applications. Instead of many expensive metal molds, low-cost plastic tooling can be formed from a single metal master, which also allows greater design flexibility. The heat transfer characteristic of polypropylene tooling is poor, which must be taken into consideration during the formulation process.
Applications
In 2008, the global consumption of polyurethane raw materials was above 12 million metric tons, and the average annual growth rate was about 5%. Revenues generated with PUR on the global market are expected to rise to approximately US$75 billion by 2022. As they are such an important class of materials, research is constantly taking place and papers published.
Degradation and environmental fate
Effects of visible light | Polyurethane | Wikipedia | 454 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
Polyurethanes, especially those made using aromatic isocyanates, contain chromophores that interact with light. This is of particular interest in the area of polyurethane coatings, where light stability is a critical factor and is the main reason that aliphatic isocyanates are used in making polyurethane coatings. When PU foam, which is made using aromatic isocyanates, is exposed to visible light, it discolors, turning from off-white to yellow to reddish brown. It has been generally accepted that apart from yellowing, visible light has little effect on foam properties. This is especially the case if the yellowing happens on the outer portions of a large foam, as the deterioration of properties in the outer portion has little effect on the overall bulk properties of the foam itself.
It has been reported that exposure to visible light can affect the variability of some physical property test results.
Higher-energy UV radiation promotes chemical reactions in foam, some of which are detrimental to the foam structure.
Hydrolysis and biodegradation
Polyurethanes may degrade due to hydrolysis. This is a common problem with shoes left in a closet, and reacting with moisture in the air.
Microbial degradation of polyurethane is believed to be due to the action of esterase, urethanase, hydrolase and protease enzymes. The process is slow as most microbes have difficulty moving beyond the surface of the polymer. Susceptibility to fungi is higher due to their release of extracellular enzymes, which are better able to permeate the polymer matrix. Two species of the Ecuadorian fungus Pestalotiopsis are capable of biodegrading polyurethane in aerobic and anaerobic conditions such as found at the bottom of landfills. Degradation of polyurethane items at museums has been reported. Polyester-type polyurethanes are more easily biodegraded by fungus than polyether-type. | Polyurethane | Wikipedia | 407 | 48366 | https://en.wikipedia.org/wiki/Polyurethane | Physical sciences | Polymers | Chemistry |
In astronomy, coordinate systems are used for specifying positions of celestial objects (satellites, planets, stars, galaxies, etc.) relative to a given reference frame, based on physical reference points available to a situated observer (e.g. the true horizon and north to an observer on Earth's surface). Coordinate systems in astronomy can specify an object's relative position in three-dimensional space or plot merely by its direction on a celestial sphere, if the object's distance is unknown or trivial.
Spherical coordinates, projected on the celestial sphere, are analogous to the geographic coordinate system used on the surface of Earth. These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, have the same fundamental () plane and primary (-axis) direction, such as an axis of rotation. Each coordinate system is named after its choice of fundamental plane.
Coordinate systems
The following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the celestial sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane. The primary direction is the starting point of the longitudinal coordinates. The origin is the zero distance point, the "center of the celestial sphere", although the definition of celestial sphere is ambiguous about the definition of its center point.
Horizontal system
The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, which revolves around its own axis once per sidereal day (23 hours, 56 minutes and 4.091 seconds) in relation to the star background. The positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of stars relative to an observer's ideal horizon.
Equatorial system
The equatorial coordinate system is centered at Earth's center, but fixed relative to the celestial poles and the March equinox. The coordinates are based on the location of stars relative to Earth's equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the Solar System, and modern star maps almost exclusively use equatorial coordinates. | Astronomical coordinate systems | Wikipedia | 477 | 48381 | https://en.wikipedia.org/wiki/Astronomical%20coordinate%20systems | Physical sciences | Celestial sphere | null |
The equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescope's or other instrument's scales so that they match the equatorial coordinates of the selected object to observe.
Popular choices of pole and equator are the older B1950 and the modern J2000 systems, but a pole and equator "of date" can also be used, meaning one appropriate to the date under consideration, such as when a measurement of the position of a planet or spacecraft is made. There are also subdivisions into "mean of date" coordinates, which average out or ignore nutation, and "true of date," which include nutation.
Ecliptic system
The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane. There are two principal variants of the ecliptic coordinate system: geocentric ecliptic coordinates centered on the Earth and heliocentric ecliptic coordinates centered on the center of mass of the Solar System.
The geocentric ecliptic system was the principal coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon, and planets. It was used to define the twelve astrological signs of the zodiac, for instance.
The heliocentric ecliptic system describes the planets' orbital movement around the Sun, and centers on the barycenter of the Solar System (i.e. very close to the center of the Sun). The system is primarily used for computing the positions of planets and other Solar System bodies, as well as defining their orbital elements.
Galactic system
The galactic coordinate system uses the approximate plane of the Milky Way Galaxy as its fundamental plane. The Solar System is still the center of the coordinate system, and the zero point is defined as the direction towards the Galactic Center. Galactic latitude resembles the elevation above the galactic plane and galactic longitude determines direction relative to the center of the galaxy.
Supergalactic system
The supergalactic coordinate system corresponds to a fundamental plane that contains a higher than average number of local galaxies in the sky as seen from Earth.
Converting coordinates
Conversions between the various coordinate systems are given. See the notes before using these equations.
Notation | Astronomical coordinate systems | Wikipedia | 458 | 48381 | https://en.wikipedia.org/wiki/Astronomical%20coordinate%20systems | Physical sciences | Celestial sphere | null |
Horizontal coordinates
, azimuth
, altitude
Equatorial coordinates
, right ascension
, declination
, hour angle
Ecliptic coordinates
, ecliptic longitude
, ecliptic latitude
Galactic coordinates
, galactic longitude
, galactic latitude
Miscellaneous
, observer's longitude
, observer's latitude
, obliquity of the ecliptic (about 23.4°)
, local sidereal time
, Greenwich sidereal time
Hour angle ↔ right ascension
Equatorial ↔ ecliptic
The classical equations, derived from spherical trigonometry, for the longitudinal coordinate are presented to the right of a bracket; dividing the first equation by the second gives the convenient tangent equation seen on the left. The rotation matrix equivalent is given beneath each case. This division is ambiguous because tan has a period of 180° () whereas cos and sin have periods of 360° (2).
Equatorial ↔ horizontal
Azimuth () is measured from the south point, turning positive to the west.
Zenith distance, the angular distance along the great circle from the zenith to a celestial object, is simply the complementary angle of the altitude: .
In solving the equation for , in order to avoid the ambiguity of the arctangent, use of the two-argument arctangent, denoted , is recommended. The two-argument arctangent computes the arctangent of , and accounts for the quadrant in which it is being computed. Thus, consistent with the convention of azimuth being measured from the south and opening positive to the west,
,
where
.
If the above formula produces a negative value for , it can be rendered positive by simply adding 360°.
Again, in solving the equation for , use of the two-argument arctangent that accounts for the quadrant is recommended. Thus, again consistent with the convention of azimuth being measured from the south and opening positive to the west,
,
where
Equatorial ↔ galactic
These equations are for converting equatorial coordinates to Galactic coordinates.
run_going
are the equatorial coordinates of the North Galactic Pole and is the Galactic longitude of the North Celestial Pole. Referred to J2000.0 the values of these quantities are:
If the equatorial coordinates are referred to another equinox, they must be precessed to their place at J2000.0 before applying these formulae.
These equations convert to equatorial coordinates referred to B2000.0.
>laft_spasse>11.3 | Astronomical coordinate systems | Wikipedia | 483 | 48381 | https://en.wikipedia.org/wiki/Astronomical%20coordinate%20systems | Physical sciences | Celestial sphere | null |
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary direction towards the March equinox, and a right-handed convention.
The origin at the centre of Earth means the coordinates are geocentric, that is, as seen from the centre of Earth as if it were transparent. The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates increase northward from and eastward around the fundamental plane.
Primary direction
This description of the orientation of the reference frame is somewhat simplified; the orientation is not quite fixed. A slow motion of Earth's axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and a small oscillation of the Earth's axis, nutation.
In order to fix the exact primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch, when giving a position. The three most commonly used are:
Mean equinox of a standard epoch (usually J2000.0, but may include B1950.0, B1900.0, etc.) is a fixed standard direction, allowing positions established at various dates to be compared directly.
Mean equinox of date is the intersection of the ecliptic of "date" (that is, the ecliptic in its position at "date") with the mean equator (that is, the equator rotated by precession to its position at "date", but free from the small periodic oscillations of nutation). Commonly used in planetary orbit calculation.
True equinox of date is the intersection of the ecliptic of "date" with the true equator (that is, the mean equator plus nutation). This is the actual intersection of the two planes at any particular moment, with all motions accounted for. | Equatorial coordinate system | Wikipedia | 490 | 48384 | https://en.wikipedia.org/wiki/Equatorial%20coordinate%20system | Physical sciences | Celestial sphere | null |
A position in the equatorial coordinate system is thus typically specified true equinox and equator of date, mean equinox and equator of J2000.0, or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.
Spherical coordinates
Use in astronomy
A star's spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of sufficiently distant objects is the same for all observers, and it is convenient to specify this direction with the same coordinates for all. In contrast, in the horizontal coordinate system, a star's position differs from observer to observer based on their positions on the Earth's surface, and is continuously changing with the Earth's rotation.
Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects. Setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere.
Declination
The declination symbol , (lower case "delta", abbreviated DEC) measures the angular distance of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north celestial pole has a declination of +90°. The origin for declination is the celestial equator, which is the projection of the Earth's equator onto the celestial sphere. Declination is analogous to terrestrial latitude.
Right ascension
The right ascension symbol , (lower case "alpha", abbreviated RA) measures the angular distance of an object eastward along the celestial equator from the March equinox to the hour circle passing through the object. The March equinox point is one of the two points where the ecliptic intersects the celestial equator. Right ascension is usually measured in sidereal hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates. There are = 15° in one hour of right ascension, and 24h of right ascension around the entire celestial equator.
When used together, right ascension and declination are usually abbreviated RA/Dec.
Hour angle | Equatorial coordinate system | Wikipedia | 453 | 48384 | https://en.wikipedia.org/wiki/Equatorial%20coordinate%20system | Physical sciences | Celestial sphere | null |
Alternatively to right ascension, hour angle (abbreviated HA or LHA, local hour angle), a left-handed system, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of Earth. Hour angle may be considered a means of measuring the time since upper culmination, the moment when an object contacts the meridian overhead.
A culminating star on the observer's meridian is said to have a zero hour angle (0h). One sidereal hour (approximately 0.9973 solar hours) later, Earth's rotation will carry the star to the west of the meridian, and its hour angle will be 1h. When calculating topocentric phenomena, right ascension may be converted into hour angle as an intermediate step.
Rectangular coordinates: geocentric equatorial coordinates
There are a number of rectangular variants of equatorial coordinates. All have:
The origin at the centre of the Earth.
The fundamental plane in the plane of the Earth's equator.
The primary direction (the axis) toward the March equinox, that is, the place where the Sun crosses the celestial equator in a northward direction in its annual apparent circuit around the ecliptic.
A right-handed convention, specifying a axis 90° to the east in the fundamental plane and a axis along the north polar axis. | Equatorial coordinate system | Wikipedia | 289 | 48384 | https://en.wikipedia.org/wiki/Equatorial%20coordinate%20system | Physical sciences | Celestial sphere | null |
The reference frames do not rotate with the Earth (in contrast to Earth-centred, Earth-fixed frames), remaining always directed toward the equinox, and drifting over time with the motions of precession and nutation.
In astronomy:
The position of the Sun is often specified in the geocentric equatorial rectangular coordinates , , and a fourth distance coordinate, , in units of the astronomical unit.
The positions of the planets and other Solar System bodies are often specified in the geocentric equatorial rectangular coordinates , , and a fourth distance coordinate, (equal to ), in units of the astronomical unit.These rectangular coordinates are related to the corresponding spherical coordinates by
In astrodynamics:
The positions of artificial Earth satellites are specified in geocentric equatorial coordinates, also known as geocentric equatorial inertial (GEI), Earth-centred inertial (ECI), and conventional inertial system (CIS), all of which are equivalent in definition to the astronomical geocentric equatorial rectangular frames, above. In the geocentric equatorial frame, the , and axes are often designated , and , respectively, or the frame's basis is specified by the unit vectors , and .
The Geocentric Celestial Reference Frame (GCRF) is the geocentric equivalent of the International Celestial Reference Frame (ICRF). Its primary direction is the equinox of J2000.0, and does not move with precession and nutation, but it is otherwise equivalent to the above systems.
Generalization: heliocentric equatorial coordinates
In astronomy, there is also a heliocentric rectangular variant of equatorial coordinates, designated , , , which has:
The origin at the centre of the Sun.
The fundamental plane in the plane of the Earth's equator.
The primary direction (the axis) toward the March equinox.
A right-handed convention, specifying a axis 90° to the east in the fundamental plane and a axis along Earth's north polar axis.
This frame is similar to the , , frame above, except that the origin is removed to the centre of the Sun. It is commonly used in planetary orbit calculation. The three astronomical rectangular coordinate systems are related by | Equatorial coordinate system | Wikipedia | 442 | 48384 | https://en.wikipedia.org/wiki/Equatorial%20coordinate%20system | Physical sciences | Celestial sphere | null |
The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way Galaxy, and the fundamental plane parallel to an approximation of the galactic plane but offset to its north. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane.
Spherical coordinates
Galactic longitude
Longitude (symbol ) measures the angular distance of an object eastward along the galactic equator from the Galactic Center. Analogous to terrestrial longitude, galactic longitude is usually measured in degrees (°).
Galactic latitude
Latitude (symbol ) measures the angle of an object northward of the galactic equator (or midplane) as viewed from Earth. Analogous to terrestrial latitude, galactic latitude is usually measured in degrees (°).
Definition
The first galactic coordinate system was used by William Herschel in 1785. A number of different coordinate systems, each differing by a few degrees, were used until 1932, when Lund Observatory assembled a set of conversion tables that defined a standard galactic coordinate system based on a galactic north pole at RA , dec +28° (in the B1900.0 epoch convention) and a 0° longitude at the point where the galactic plane and equatorial plane intersected.
In 1958, the International Astronomical Union (IAU) defined the galactic coordinate system in reference to radio observations of galactic neutral hydrogen through the hydrogen line, changing the definition of the Galactic longitude by 32° and the latitude by 1.5°. In the equatorial coordinate system, for equinox and equator of 1950.0, the north galactic pole is defined at right ascension , declination +27.4°, in the constellation Coma Berenices, with a probable error of ±0.1°. Longitude 0° is the great semicircle that originates from this point along the line in position angle 123° with respect to the equatorial pole. The galactic longitude increases in the same direction as right ascension. Galactic latitude is positive towards the north galactic pole, with a plane passing through the Sun and parallel to the galactic equator being 0°, whilst the poles are ±90°. Based on this definition, the galactic poles and equator can be found from spherical trigonometry and can be precessed to other epochs; see the table.
The IAU recommended that during the transition period from the old, pre-1958 system to the new, the old longitude and latitude should be designated and while the new should be designated and . This convention is occasionally seen. | Galactic coordinate system | Wikipedia | 511 | 48389 | https://en.wikipedia.org/wiki/Galactic%20coordinate%20system | Physical sciences | Celestial sphere: General | Astronomy |
Radio source Sagittarius A*, which is the best physical marker of the true Galactic Center, is located at , (J2000). Rounded to the same number of digits as the table, , −29.01° (J2000), there is an offset of about 0.07° from the defined coordinate center, well within the 1958 error estimate of ±0.1°. Due to the Sun's position, which currently lies north of the midplane, and the heliocentric definition adopted by the IAU, the galactic coordinates of Sgr A* are latitude south, longitude . Since as defined the galactic coordinate system does not rotate with time, Sgr A* is actually decreasing in longitude at the rate of galactic rotation at the sun, , approximately 5.7 milliarcseconds per year (see Oort constants).
Conversion between equatorial and galactic coordinates
An object's location expressed in the equatorial coordinate system can be transformed into the galactic coordinate system. In these equations, is right ascension, is declination. NGP refers to the coordinate values of the north galactic pole and NCP to those of the north celestial pole.
The reverse (galactic to equatorial) can also be accomplished with the following conversion formulas.
Where:
Rectangular coordinates
In some applications use is made of rectangular coordinates based on galactic longitude and latitude and distance. In some work regarding the distant past or future the galactic coordinate system is taken as rotating so that the -axis always goes to the centre of the galaxy.
There are two major rectangular variations of galactic coordinates, commonly used for computing space velocities of galactic objects. In these systems the -axes are designated , but the definitions vary by author. In one system, the axis is directed toward the Galactic Center ( = 0°), and it is a right-handed system (positive towards the east and towards the north galactic pole); in the other, the axis is directed toward the galactic anticenter ( = 180°), and it is a left-handed system (positive towards the east and towards the north galactic pole).
In the constellations
The galactic equator runs through the following constellations: | Galactic coordinate system | Wikipedia | 444 | 48389 | https://en.wikipedia.org/wiki/Galactic%20coordinate%20system | Physical sciences | Celestial sphere: General | Astronomy |
Sagittarius
Serpens
Scutum
Aquila
Sagitta
Vulpecula
Cygnus
Cepheus
Cassiopeia
Camelopardalis
Perseus
Auriga
Taurus
Gemini
Orion
Monoceros
Canis Major
Puppis
Vela
Carina
Crux
Centaurus
Circinus
Norma
Ara
Scorpius
Ophiuchus | Galactic coordinate system | Wikipedia | 72 | 48389 | https://en.wikipedia.org/wiki/Galactic%20coordinate%20system | Physical sciences | Celestial sphere: General | Astronomy |
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are an elliptic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).
The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other problems. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics.
The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the domain. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample. | Navier–Stokes equations | Wikipedia | 470 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Flow velocity
The solution of the equations is a flow velocity. It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.
General continuum equations
The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is:
By setting the Cauchy stress tensor to be the sum of a viscosity term (the deviatoric stress) and a pressure term (volumetric stress), we arrive at:
where
is the material derivative, defined as ,
is the (mass) density,
is the flow velocity,
is the divergence,
is the pressure,
is time,
is the deviatoric stress tensor, which has order 2,
represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on.
In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations. | Navier–Stokes equations | Wikipedia | 437 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Assuming conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative ) of any finite volume (V) to represent the change of velocity in fluid media:
where
is the material derivative of mass per unit volume (density, ),
is the mathematical operation for the integration throughout the volume (V),
is the partial derivative mathematical operator,
is the divergence of the flow velocity (), which is a scalar field, Note 1
is the gradient of density (), which is the vector derivative of a scalar field, Note 1
Note 1 - Refer to the mathematical operator del represented by the nabla () symbol.
to arrive at the conservation form of the equations of motion. This is often written:
where is the outer product of the flow velocity ():
The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).
All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.
Convective acceleration
A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.
Compressible flow
Remark: here, the deviatoric stress tensor is denoted as it was in the general continuum equations and in the incompressible flow section.
The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: | Navier–Stokes equations | Wikipedia | 496 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient , or more simply the rate-of-strain tensor:
the deviatoric stress is linear in this variable: , where is independent on the strain rate tensor, is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product.
the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the second viscosity and the dynamic viscosity , as it is usual in linear elasticity:
where is the identity tensor, and is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as:
Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow:
Given this relation, and since the trace of the identity tensor in three dimensions is three:
the trace of the stress tensor in three dimensions becomes:
So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:
Introducing the bulk viscosity ,
we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:
which can also be arranged in the other usual form:
Note that in the compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term:
and the deviatoric stress tensor is still coincident with the shear stress tensor (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:
Both bulk viscosity and dynamic viscosity need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.
The most general of the Navier–Stokes equations become | Navier–Stokes equations | Wikipedia | 505 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
in index notation, the equation can be written as
The corresponding equation in conservation form can be obtained by considering that, given the mass continuity equation, the left side is equivalent to:
To give finally:
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Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity can be assumed to be constant in which case, the effect of the volume viscosity is that the mechanical pressure is not equivalent to the thermodynamic pressure: as demonstrated below.
However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming . The assumption of setting is called as the Stokes hypothesis. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become
If the dynamic and bulk viscosities are assumed to be uniform in space, the equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor is and the divergence of tensor is , one finally arrives to the compressible Navier–Stokes momentum equation:
where is the material derivative. is the shear kinematic viscosity and is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation.
By bringing the operator on the flow velocity on the left side, one also has:
The convective acceleration term can also be written as
where the vector is known as the Lamb vector. | Navier–Stokes equations | Wikipedia | 478 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with .
Incompressible flow
The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:
the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient .
the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity :
where
is the rate-of-strain tensor. So this decomposition can be made explicit as:
This is constitutive equation is also called the Newtonian law of viscosity.
Dynamic viscosity need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.
The divergence of the deviatoric stress in case of uniform viscosity is given by:
because for an incompressible fluid.
Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures. the incompressible Navier–Stokes equations are best visualized by dividing for the density:
where is called the kinematic viscosity.
By isolating the fluid velocity, one can also state:
If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, , then we have
where is called the unit pressure head.
In incompressible flows, the pressure field satisfies the Poisson equation,
which is obtained by taking the divergence of the momentum equations.
It is well worth observing the meaning of each term (compare to the Cauchy momentum equation): | Navier–Stokes equations | Wikipedia | 469 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
The higher-order term, namely the shear stress divergence , has simply reduced to the vector Laplacian term . This Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the heat conduction. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of convection–diffusion equations.
In the usual case of an external field being a conservative field:
by defining the hydraulic head:
one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:
The incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the fundamental equation of hydraulics. The domain for these equations is commonly a 3 or less dimensional Euclidean space, for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar partial differential equations to be solved. In 3-dimensional orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical. Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier–Stokes equations from Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems.
A special case of the fundamental equation of hydraulics is the Bernoulli's equation.
The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,
where and are solenoidal and irrotational projection operators satisfying , and and are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation. | Navier–Stokes equations | Wikipedia | 512 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:
with a similar structure in 2D. Thus the governing equation is an integro-differential equation similar to Coulomb and Biot–Savart law, not convenient for numerical computation.
An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation, is given by,
for divergence-free test functions satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There one will be able to address the question "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?".
The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation equation. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.
Weak form of the incompressible Navier–Stokes equations
Strong form
Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density in a domain
with boundary
being and portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ():
is the fluid velocity, the fluid pressure, a given forcing term, the outward directed unit normal vector to , and the viscous stress tensor defined as:
Let be the dynamic viscosity of the fluid, the second-order identity tensor and the strain-rate tensor defined as:
The functions and are given Dirichlet and Neumann boundary data, while is the initial condition. The first equation is the momentum balance equation, while the second represents the mass conservation, namely the continuity equation.
Assuming constant dynamic viscosity, using the vectorial identity
and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:
Moreover, note that the Neumann boundary conditions can be rearranged as:
Weak form
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation
multiply it for a test function , defined in a suitable space , and integrate both members with respect to the domain : | Navier–Stokes equations | Wikipedia | 509 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:
Using these relations, one gets:
In the same fashion, the continuity equation is multiplied for a test function belonging to a space and integrated in the domain :
The space functions are chosen as follows:
Considering that the test function vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:
Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:
Discrete velocity
With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is
It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' theorem. Discussion will be restricted to 2D in the following.
We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,
Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.
Taking the curl of the scalar stream function elements gives divergence-free velocity elements. The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.
Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces.
Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions. | Navier–Stokes equations | Wikipedia | 486 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations.
Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.
Pressure recovery
Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is,
where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions one would choose the irrotational vector elements obtained from the gradient of the pressure element.
Non-inertial frame of reference
The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term. Consider a stationary inertial frame of reference , and a non-inertial frame of reference , which is translating with velocity and rotating with angular velocity with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes
Here and are measured in the non-inertial frame. The first term in the parenthesis represents Coriolis acceleration, the second term is due to centrifugal acceleration, the third is due to the linear acceleration of with respect to and the fourth term is due to the angular acceleration of with respect to .
Other equations
The Navier–Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass, balance of energy, and/or an equation of state.
Continuity equation for incompressible fluid
Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, as discussed above in the "General continuum equations" within this article, as follows: | Navier–Stokes equations | Wikipedia | 441 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
A fluid media for which the density () is constant is called incompressible. Therefore, the rate of change of density () with respect to time and the gradient of density are equal to zero . In this case the general equation of continuity, , reduces to: . Furthermore, assuming that density () is a non-zero constant means that the right-hand side of the equation is divisible by density (). Therefore, the continuity equation for an incompressible fluid reduces further to:This relationship, , identifies that the divergence of the flow velocity vector () is equal to zero , which means that for an incompressible fluid the flow velocity field is a solenoidal vector field or a divergence-free vector field. Note that this relationship can be expanded upon due to its uniqueness with the vector Laplace operator , and vorticity which is now expressed like so, for an incompressible fluid:
Stream function for incompressible 2D fluid
Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with and no dependence of anything on ), where the equations reduce to:
Differentiating the first with respect to , the second with respect to and subtracting the resulting equations will eliminate pressure and any conservative force.
For incompressible flow, defining the stream function through
results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation:
where is the 2D biharmonic operator and is the kinematic viscosity, . We can also express this compactly using the Jacobian determinant:
This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero.
In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function. | Navier–Stokes equations | Wikipedia | 451 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
The incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.
Properties
Nonlinearity
The Navier–Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation. In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.
The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.
Turbulence
Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly. | Navier–Stokes equations | Wikipedia | 377 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart–Allmaras, –, –, and SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales.
Applicability
Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.
The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, capillarity of internal layers in fluids appears for flow with high gradients. For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement.
Failing that, one may have to resort to molecular dynamics or various hybrid methods.
Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist. | Navier–Stokes equations | Wikipedia | 504 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Application to specific problems
The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension.
Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by scale analysis to further simplify the problem.
Parallel flow
Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) boundary value problem is:
The boundary condition is the no slip condition. This problem is easily solved for the flow field:
From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.
Radial flow
Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function that must satisfy:
This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for (approximately; this is not ), the parameter being the Reynolds number with appropriately chosen scales. This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.
Convection
A type of natural convection that can be described by the Navier–Stokes equation is the Rayleigh–Bénard convection. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. | Navier–Stokes equations | Wikipedia | 406 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Exact solutions of the Navier–Stokes equations
Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also, more interesting examples, solutions to the full non-linear equations, exist, such as Jeffery–Hamel flow, Von Kármán swirling flow, stagnation point flow, Landau–Squire jet, and Taylor–Green vortex.Landau & Lifshitz (1987) pp. 75–88. Time-dependent self-similar solutions of the three-dimensional non-compressible Navier-Stokes equations in Cartesian coordinate can be given with the help of the Kummer's functions with quadratic arguments. For the compressible Navier-Stokes equations the time-dependent self-similar solutions are however the Whittaker functions again with quadratic arguments when the polytropic equation of state is used as a closing condition. Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.
Under additional assumptions, the component parts can be separated.
A three-dimensional steady-state vortex solution
A steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let be a constant radius of the inner coil. One set of solutions is given by:
for arbitrary constants and . This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where is a constant, and neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any turbulence properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field:
Viscous three-dimensional periodic solutions
Two examples of periodic fully-three-dimensional viscous solutions are described in.
These solutions are defined on a three-dimensional torus and are characterized by positive and negative helicity respectively.
The solution with positive helicity is given by: | Navier–Stokes equations | Wikipedia | 498 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
where is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is at .
The pressure field is obtained from the velocity field as (where and are reference values for the pressure and density fields respectively).
Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by .
These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green Taylor–Green vortex.
Wyld diagrams
Wyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via a perturbation expansion of the fundamental continuum mechanics. Similar to the Feynman diagrams in quantum field theory, these diagrams are an extension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words, these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated and interacting fluid particles to obey stochastic processes associated to pseudo-random functions in probability distributions.
Representations in 3D
Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. means the partial derivative of with respect to , and means the second-order partial derivative of with respect to .
A 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows. On suitably short time scales, the dynamics of turbulence is deterministic.
Cartesian coordinates
From the general form of the Navier–Stokes, with the velocity vector expanded as , sometimes respectively named , , , we may write the vector equation explicitly,
Note that gravity has been accounted for as a body force, and the values of , , will depend on the orientation of gravity with respect to the chosen set of coordinates.
The continuity equation reads:
When the flow is incompressible, does not change for any fluid particle, and its material derivative vanishes: . The continuity equation is reduced to:
Thus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see Incompressible flow).
This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain. | Navier–Stokes equations | Wikipedia | 499 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Cylindrical coordinates
A change of variables on the Cartesian equations will yield the following momentum equations for , , and
The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is:
This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity (), and the remaining quantities are independent of :
Spherical coordinates
In spherical coordinates, the , , and momentum equations are (note the convention used: is polar angle, or colatitude, ):
Mass continuity will read:
These equations could be (slightly) compacted by, for example, factoring from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.
Navier–Stokes equations use in games
The Navier–Stokes equations are used extensively in video games in order to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games" by Jos Stam, which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids" from 1999. Stam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992.
More recent implementations based upon this work run on the game systems graphics processing unit (GPU) as opposed to the central processing unit (CPU) and achieve a much higher degree of performance.
Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.
An introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation. | Navier–Stokes equations | Wikipedia | 469 | 48395 | https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes%20equations | Physical sciences | Fluid mechanics | null |
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.
Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
History
Ancient
Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE.
Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in .
Medieval
Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem.
In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of Taylor series of trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.
Modern | Mathematical analysis | Wikipedia | 511 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Foundations
The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.
Modernization
In the 18th century, Euler introduced the notion of a mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis. | Mathematical analysis | Wikipedia | 405 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.
Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling curves), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.
Important concepts
Metric spaces
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.
Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).
Formally, a metric space is an ordered pair where is a set and is a metric on , i.e., a function
such that for any , the following holds:
, with equality if and only if (identity of indiscernibles),
(symmetry), and
(triangle inequality).
By taking the third property and letting , it can be shown that (non-negative).
Sequences and limits | Mathematical analysis | Wikipedia | 466 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted
Main branches
Calculus
Real analysis
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
Complex analysis
Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. | Mathematical analysis | Wikipedia | 511 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Harmonic analysis
Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves. This includes the study of the notions of Fourier series and Fourier transforms (Fourier analysis), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory, number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience.
Differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
Measure theory
A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the -dimensional Euclidean space . For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word – specifically, 1. | Mathematical analysis | Wikipedia | 390 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set . It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a -algebra. This means that the empty set, countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Vector analysis
Vector analysis, also called vector calculus, is a branch of mathematical analysis dealing with vector-valued functions.
Scalar analysis
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.
Tensor analysis | Mathematical analysis | Wikipedia | 482 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Other topics
Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves.
Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
Stochastic calculus – analytical notions developed for stochastic processes.
Set-valued analysis – applies ideas from analysis and topology to set-valued functions.
Convex analysis, the study of convex sets and functions.
Idempotent analysis – analysis in the context of an idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.
Tropical analysis – analysis of the idempotent semiring called the tropical semiring (or max-plus algebra/min-plus algebra).
Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory.
Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
Smooth infinitesimal analysis, which is developed in a smooth topos.
Applications
Techniques from analysis are also found in other areas such as:
Physical sciences
The vast majority of classical mechanics, relativity, and quantum mechanics is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law, the Schrödinger equation, and the Einstein field equations.
Functional analysis is also a major factor in quantum mechanics. | Mathematical analysis | Wikipedia | 467 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
Signal processing
When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
Other areas of mathematics
Techniques from analysis are used in many areas of mathematics, including:
Analytic number theory
Analytic combinatorics
Continuous probability
Differential entropy in information theory
Differential games
Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
Differentiable manifolds
Differential topology
Partial differential equations
Famous Textbooks
Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau
Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin
Differential and Integral Calculus (3 volumes), by Grigorii Fichtenholz
The Fundamentals of Mathematical Analysis (2 volumes), by Grigorii Fichtenholz
A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky
Mathematical Analysis (2 volumes), by Vladimir Zorich
A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov
Differential And Integral Calculus, by Nikolai Piskunov
A Course of Mathematical Analysis, by Aleksandr Khinchin
Mathematical Analysis: A Special Course, by Georgiy Shilov
Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson
Problems in Mathematical Analysis, by Boris Demidovich
Problems and Theorems in Analysis (2 volumes), by George Pólya, Gábor Szegő
Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol
Principles of Mathematical Analysis, by Walter Rudin
Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein
Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors
Complex Analysis, by Elias Stein
Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein
Analysis (2 volumes), by Terence Tao
Analysis (3 volumes), by Herbert Amann, Joachim Escher
Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov
Real and Functional Analysis, by Serge Lang | Mathematical analysis | Wikipedia | 475 | 48396 | https://en.wikipedia.org/wiki/Mathematical%20analysis | Mathematics | Analysis | null |
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Formally, a ring is a set endowed with two binary operations called addition and multiplication such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity. See .)
Whether a ring is commutative has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.
Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of real square matrices with , group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.
The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis. | Ring (mathematics) | Wikipedia | 426 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Definition
A ring is a set equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms:
is an abelian group under addition, meaning that:
for all in (that is, is associative).
for all in (that is, is commutative).
There is an element in such that for all in (that is, is the additive identity).
For each in there exists in such that (that is, is the additive inverse of ).
is a monoid under multiplication, meaning that:
for all in (that is, is associative).
There is an element in such that and for all in (that is, is the multiplicative identity).
Multiplication is distributive with respect to addition, meaning that:
for all in (left distributivity).
for all in (right distributivity).
In notation, the multiplication symbol is often omitted, in which case is written as .
Variations on the definition
In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a "" (IPA: ) with a missing "i". For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in below, many authors apply the term "ring" without requiring a multiplicative identity.
Although ring addition is commutative, ring multiplication is not required to be commutative: need not necessarily equal . Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology.
In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field. | Ring (mathematics) | Wikipedia | 438 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: .)
There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring".
Illustration
The most familiar example of a ring is the set of all integers consisting of the numbers
The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.
Some properties
Some basic properties of a ring follow immediately from the axioms:
The additive identity is unique.
The additive inverse of each element is unique.
The multiplicative identity is unique.
For any element in a ring , one has (zero is an absorbing element with respect to multiplication) and .
If in a ring (or more generally, is a unit element), then has only one element, and is called the zero ring.
If a ring contains the zero ring as a subring, then itself is the zero ring.
The binomial formula holds for any and satisfying .
Example: Integers modulo 4
Equip the set with the following operations:
The sum in is the remainder when the integer is divided by (as is always smaller than , this remainder is either or ). For example, and
The product in is the remainder when the integer is divided by . For example, and
Then is a ring: each axiom follows from the corresponding axiom for If is an integer, the remainder of when divided by may be considered as an element of and this element is often denoted by "" or which is consistent with the notation for . The additive inverse of any in is For example,
has a subring , and if is prime, then has no subrings.
Example: 2-by-2 matrices
The set of 2-by-2 square matrices with entries in a field is | Ring (mathematics) | Wikipedia | 466 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
With the operations of matrix addition and matrix multiplication, satisfies the above ring axioms. The element is the multiplicative identity of the ring. If and then while this example shows that the ring is noncommutative.
More generally, for any ring , commutative or not, and any nonnegative integer , the square matrices of dimension with entries in form a ring; see Matrix ring.
History
Dedekind
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.
Hilbert
The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if then:
and so on; in general, is going to be an integral linear combination of , , and .
Fraenkel and Noether
The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen. | Ring (mathematics) | Wikipedia | 488 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Multiplicative identity and the term "ring"
Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not.
Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of in the definition of "ring", especially in advanced books by notable authors such as Artin, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2022 that use the term without the requirement for a . Likewise, the Encyclopedia of Mathematics does not require unit elements in rings. In a research article, the authors often specify which definition of ring they use in the beginning of that article.
Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a , then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.
Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:
to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit", or "ring with 1".
to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.
Basic examples | Ring (mathematics) | Wikipedia | 464 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Commutative rings
The prototypical example is the ring of integers with the two operations of addition and multiplication.
The rational, real and complex numbers are commutative rings of a type called fields.
A unital associative algebra over a commutative ring is itself a ring as well as an -module. Some examples:
The algebra of polynomials with coefficients in .
The algebra of formal power series with coefficients in .
The set of all continuous real-valued functions defined on the real line forms a commutative -algebra. The operations are pointwise addition and multiplication of functions.
Let be a set, and let be a ring. Then the set of all functions from to forms a ring, which is commutative if is commutative.
The ring of quadratic integers, the integral closure of in a quadratic extension of It is a subring of the ring of all algebraic integers.
The ring of profinite integers the (infinite) product of the rings of -adic integers over all prime numbers .
The Hecke ring, the ring generated by Hecke operators.
If is a set, then the power set of becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring. | Ring (mathematics) | Wikipedia | 261 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Noncommutative rings
For any ring and any natural number , the set of all square -by- matrices with entries from , forms a ring with matrix addition and matrix multiplication as operations. For , this matrix ring is isomorphic to itself. For (and not the zero ring), this matrix ring is noncommutative.
If is an abelian group, then the endomorphisms of form a ring, the endomorphism ring of . The operations in this ring are addition and composition of endomorphisms. More generally, if is a left module over a ring , then the set of all -linear maps forms a ring, also called the endomorphism ring and denoted by .
The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
If is a group and is a ring, the group ring of over is a free module over having as basis. Multiplication is defined by the rules that the elements of commute with the elements of and multiply together as they do in the group .
The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras are noncommutative.
Non-rings
The set of natural numbers with the usual operations is not a ring, since is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to to get as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers The natural numbers (including ) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
Let be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution: Then is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of .
Basic concepts
Products and powers
For each nonnegative integer , given a sequence of elements of , one can define the product recursively: let and let for . | Ring (mathematics) | Wikipedia | 494 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
As a special case, one can define nonnegative integer powers of an element of a ring: and for . Then for all .
Elements in a ring
A left zero divisor of a ring is an element in the ring such that there exists a nonzero element of such that . A right zero divisor is defined similarly.
A nilpotent element is an element such that for some . One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.
An idempotent is an element such that . One example of an idempotent element is a projection in linear algebra.
A unit is an element having a multiplicative inverse; in this case the inverse is unique, and is denoted by . The set of units of a ring is a group under ring multiplication; this group is denoted by or or . For example, if is the ring of all square matrices of size over a field, then consists of the set of all invertible matrices of size , and is called the general linear group.
Subring
A subset of is called a subring if any one of the following equivalent conditions holds:
the addition and multiplication of restrict to give operations making a ring with the same multiplicative identity as .
; and for all in , the elements , , and are in .
can be equipped with operations making it a ring such that the inclusion map is a ring homomorphism.
For example, the ring of integers is a subring of the field of real numbers and also a subring of the ring of polynomials (in both cases, contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers does not contain the identity element and thus does not qualify as a subring of one could call a subrng, however.
An intersection of subrings is a subring. Given a subset of , the smallest subring of containing is the intersection of all subrings of containing , and it is called the subring generated by . | Ring (mathematics) | Wikipedia | 435 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
For a ring , the smallest subring of is called the characteristic subring of . It can be generated through addition of copies of and . It is possible that ( times) can be zero. If is the smallest positive integer such that this occurs, then is called the characteristic of . In some rings, is never zero for any positive integer , and those rings are said to have characteristic zero.
Given a ring , let denote the set of all elements in such that commutes with every element in : for any in . Then is a subring of , called the center of . More generally, given a subset of , let be the set of all elements in that commute with every element in . Then is a subring of , called the centralizer (or commutant) of . The center is the centralizer of the entire ring . Elements or subsets of the center are said to be central in ; they (each individually) generate a subring of the center.
Ideal
Let be a ring. A left ideal of is a nonempty subset of such that for any in and in , the elements and are in . If denotes the -span of , that is, the set of finite sums
then is a left ideal if . Similarly, a right ideal is a subset such that . A subset is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of . If is a subset of , then is a left ideal, called the left ideal generated by ; it is the smallest left ideal containing . Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of .
If is in , then and are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by . The principal ideal is written as . For example, the set of all positive and negative multiples of along with form an ideal of the integers, and this ideal is generated by the integer . In fact, every ideal of the ring of integers is principal.
Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. | Ring (mathematics) | Wikipedia | 470 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.
For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal of is called a prime ideal if for any elements we have that implies either or Equivalently, is prime if for any ideals , we have that implies either or . This latter formulation illustrates the idea of ideals as generalizations of elements.
Homomorphism
A homomorphism from a ring to a ring is a function from to that preserves the ring operations; namely, such that, for all , in the following identities hold:
If one is working with , then the third condition is dropped.
A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to (that is, a ring homomorphism that is an inverse function), or equivalently if it is bijective.
Examples:
The function that maps each integer to its remainder modulo (a number in ) is a homomorphism from the ring to the quotient ring ("quotient ring" is defined below).
If is a unit element in a ring , then is a ring homomorphism, called an inner automorphism of .
Let be a commutative ring of prime characteristic . Then is a ring endomorphism of called the Frobenius homomorphism.
The Galois group of a field extension is the set of all automorphisms of whose restrictions to are the identity.
For any ring , there are a unique ring homomorphism and a unique ring homomorphism .
An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map is an epimorphism.
An algebra homomorphism from a -algebra to the endomorphism algebra of a vector space over is called a representation of the algebra. | Ring (mathematics) | Wikipedia | 478 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Given a ring homomorphism , the set of all elements mapped to 0 by is called the kernel of . The kernel is a two-sided ideal of . The image of , on the other hand, is not always an ideal, but it is always a subring of .
To give a ring homomorphism from a commutative ring to a ring with image contained in the center of is the same as to give a structure of an algebra over to (which in particular gives a structure of an -module).
Quotient ring
The notion of quotient ring is analogous to the notion of a quotient group. Given a ring and a two-sided ideal of , view as subgroup of ; then the quotient ring is the set of cosets of together with the operations
for all in . The ring is also called a factor ring.
As with a quotient group, there is a canonical homomorphism , given by . It is surjective and satisfies the following universal property:
If is a ring homomorphism such that , then there is a unique homomorphism such that
For any ring homomorphism , invoking the universal property with produces a homomorphism that gives an isomorphism from to the image of .
Module
The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring , an -module is an abelian group equipped with an operation (associating an element of to every pair of an element of and an element of ) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all , in and all , in ,
is an abelian group under addition.
When the ring is noncommutative these axioms define left modules; right modules are defined similarly by writing instead of . This is not only a change of notation, as the last axiom of right modules (that is ) becomes , if left multiplication (by ring elements) is used for a right module.
Basic examples of modules are ideals, including the ring itself. | Ring (mathematics) | Wikipedia | 458 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.
The axioms of modules imply that , where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
Any ring homomorphism induces a structure of a module: if is a ring homomorphism, then is a left module over by the multiplication: . If is commutative or if is contained in the center of , the ring is called a -algebra. In particular, every ring is an algebra over the integers.
Constructions
Direct product
Let and be rings. Then the product can be equipped with the following natural ring structure:
for all in and in . The ring with the above operations of addition and multiplication and the multiplicative identity is called the direct product of with . The same construction also works for an arbitrary family of rings: if are rings indexed by a set , then is a ring with componentwise addition and multiplication.
Let be a commutative ring and be ideals such that whenever . Then the Chinese remainder theorem says there is a canonical ring isomorphism:
A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let be rings, the inclusions with the images (in particular are rings though not subrings). Then are ideals of and
as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to . Equivalently, the above can be done through central idempotents. Assume that has the above decomposition. Then we can write
By the conditions on one has that are central idempotents and , (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let which are two-sided ideals. If each is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to . | Ring (mathematics) | Wikipedia | 483 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).
Polynomial ring
Given a symbol (called a variable) and a commutative ring , the set of polynomials
forms a commutative ring with the usual addition and multiplication, containing as a subring. It is called the polynomial ring over . More generally, the set of all polynomials in variables forms a commutative ring, containing as subrings.
If is an integral domain, then is also an integral domain; its field of fractions is the field of rational functions. If is a Noetherian ring, then is a Noetherian ring. If is a unique factorization domain, then is a unique factorization domain. Finally, is a field if and only if is a principal ideal domain.
Let be commutative rings. Given an element of , one can consider the ring homomorphism
(that is, the substitution). If and , then . Because of this, the polynomial is often also denoted by . The image of the map is denoted by ; it is the same thing as the subring of generated by and .
Example: denotes the image of the homomorphism
In other words, it is the subalgebra of generated by and .
Example: let be a polynomial in one variable, that is, an element in a polynomial ring . Then is an element in and is divisible by in that ring. The result of substituting zero to in is , the derivative of at .
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism and an element in there exists a unique ring homomorphism such that and restricts to . For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.
To give an example, let be the ring of all functions from to itself; the addition and the multiplication are those of functions. Let be the identity function. Each in defines a constant function, giving rise to the homomorphism . The universal property says that this map extends uniquely to
( maps to ) where is the polynomial function defined by . The resulting map is injective if and only if is infinite.
Given a non-constant monic polynomial in , there exists a ring containing such that is a product of linear factors in . | Ring (mathematics) | Wikipedia | 497 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Let be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in and the set of closed subvarieties of . In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)
There are some other related constructions. A formal power series ring consists of formal power series
together with multiplication and addition that mimic those for convergent series. It contains as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).
Matrix ring and endomorphism ring
Let be a ring (not necessarily commutative). The set of all square matrices of size with entries in forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by . Given a right -module , the set of all -linear maps from to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of and is denoted by .
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: This is a special case of the following fact: If is an -linear map, then may be written as a matrix with entries in , resulting in the ring isomorphism:
Any ring homomorphism induces .
Schur's lemma says that if is a simple right -module, then is a division ring. If is a direct sum of -copies of simple -modules then
The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.
A ring and the matrix ring over it are Morita equivalent: the category of right modules of is equivalent to the category of right modules over . In particular, two-sided ideals in correspond in one-to-one to two-sided ideals in . | Ring (mathematics) | Wikipedia | 455 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Limits and colimits of rings
Let be a sequence of rings such that is a subring of for all . Then the union (or filtered colimit) of is the ring defined as follows: it is the disjoint union of all 's modulo the equivalence relation if and only if in for sufficiently large .
Examples of colimits:
A polynomial ring in infinitely many variables:
The algebraic closure of finite fields of the same characteristic
The field of formal Laurent series over a field : (it is the field of fractions of the formal power series ring )
The function field of an algebraic variety over a field is where the limit runs over all the coordinate rings of nonempty open subsets (more succinctly it is the stalk of the structure sheaf at the generic point.)
Any commutative ring is the colimit of finitely generated subrings.
A projective limit (or a filtered limit) of rings is defined as follows. Suppose we are given a family of rings , running over positive integers, say, and ring homomorphisms , such that are all the identities and is whenever . Then is the subring of consisting of such that maps to under , .
For an example of a projective limit, see .
Localization
The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring and a subset of , there exists a ring together with the ring homomorphism that "inverts" ; that is, the homomorphism maps elements in to unit elements in and, moreover, any ring homomorphism from that "inverts" uniquely factors through The ring is called the localization of with respect to . For example, if is a commutative ring and an element in , then the localization consists of elements of the form (to be precise, )
The localization is frequently applied to a commutative ring with respect to the complement of a prime ideal (or a union of prime ideals) in . In that case one often writes for is then a local ring with the maximal ideal This is the reason for the terminology "localization". The field of fractions of an integral domain is the localization of at the prime ideal zero. If is a prime ideal of a commutative ring , then the field of fractions of is the same as the residue field of the local ring and is denoted by | Ring (mathematics) | Wikipedia | 498 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
If is a left -module, then the localization of with respect to is given by a change of rings
The most important properties of localization are the following: when is a commutative ring and a multiplicatively closed subset
is a bijection between the set of all prime ideals in disjoint from and the set of all prime ideals in
running over elements in with partial ordering given by divisibility.
The localization is exact: is exact over whenever is exact over .
Conversely, if is exact for any maximal ideal then is exact.
A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)
In category theory, a localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring may be thought of as an endomorphism of any -module. Thus, categorically, a localization of with respect to a subset of is a functor from the category of -modules to itself that sends elements of viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, then maps to and -modules map to -modules.)
Completion
Let be a commutative ring, and let be an ideal of .
The completion of at is the projective limit it is a commutative ring. The canonical homomorphisms from to the quotients induce a homomorphism The latter homomorphism is injective if is a Noetherian integral domain and is a proper ideal, or if is a Noetherian local ring with maximal ideal , by Krull's intersection theorem. The construction is especially useful when is a maximal ideal. | Ring (mathematics) | Wikipedia | 404 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
The basic example is the completion of at the principal ideal generated by a prime number ; it is called the ring of -adic integers and is denoted The completion can in this case be constructed also from the -adic absolute value on The -adic absolute value on is a map from to given by where denotes the exponent of in the prime factorization of a nonzero integer into prime numbers (we also put and ). It defines a distance function on and the completion of as a metric space is denoted by It is again a field since the field operations extend to the completion. The subring of consisting of elements with is isomorphic to
Similarly, the formal power series ring is the completion of at (see also Hensel's lemma)
A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.
Rings with generators and relations
The most general way to construct a ring is by specifying generators and relations. Let be a free ring (that is, free algebra over the integers) with the set of symbols, that is, consists of polynomials with integral coefficients in noncommuting variables that are elements of . A free ring satisfies the universal property: any function from the set to a ring factors through so that is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.
Now, we can impose relations among symbols in by taking a quotient. Explicitly, if is a subset of , then the quotient ring of by the ideal generated by is called the ring with generators and relations . If we used a ring, say, as a base ring instead of then the resulting ring will be over . For example, if then the resulting ring will be the usual polynomial ring with coefficients in in variables that are elements of (It is also the same thing as the symmetric algebra over with symbols .) | Ring (mathematics) | Wikipedia | 497 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
In the category-theoretic terms, the formation is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.)
Let , be algebras over a commutative ring . Then the tensor product of -modules is an -algebra with multiplication characterized by
Special kinds of rings
Domains
A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.
Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let be a finite-dimensional vector space over a field and a linear map with minimal polynomial . Then, since is a unique factorization domain, factors into powers of distinct irreducible polynomials (that is, prime elements):
Letting we make a -module. The structure theorem then says is a direct sum of cyclic modules, each of which is isomorphic to the module of the form Now, if then such a cyclic module (for ) has a basis in which the restriction of is represented by a Jordan matrix. Thus, if, say, is algebraically closed, then all 's are of the form and the above decomposition corresponds to the Jordan canonical form of .
In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.
The following is a chain of class inclusions that describes the relationship between rings, domains and fields: | Ring (mathematics) | Wikipedia | 482 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Division ring
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.
The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.
Semisimple rings
A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself.
Examples
A division ring is semisimple (and simple).
For any division ring and positive integer , the matrix ring is semisimple (and simple).
For a field and finite group , the group ring is semisimple if and only if the characteristic of does not divide the order of (Maschke's theorem).
Clifford algebras are semisimple.
The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.
Properties
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)
For a ring , the following are equivalent:
is semisimple.
is artinian and semiprimitive.
is a finite direct product where each is a positive integer, and each is a division ring (Artin–Wedderburn theorem). | Ring (mathematics) | Wikipedia | 449 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Semisimplicity is closely related to separability. A unital associative algebra over a field is said to be separable if the base extension is semisimple for every field extension . If happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)
Central simple algebra and Brauer group
For a field , a -algebra is central if its center is and is simple if it is a simple ring. Since the center of a simple -algebra is a field, any simple -algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a -algebra. The matrix ring of size over a ring will be denoted by .
The Skolem–Noether theorem states any automorphism of a central simple algebra is inner.
Two central simple algebras and are said to be similar if there are integers and such that Since the similarity is an equivalence relation. The similarity classes with the multiplication form an abelian group called the Brauer group of and is denoted by . By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.
For example, is trivial if is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). has order 2 (a special case of the theorem of Frobenius). Finally, if is a nonarchimedean local field (for example, then through the invariant map.
Now, if is a field extension of , then the base extension induces . Its kernel is denoted by . It consists of such that is a matrix ring over (that is, is split by .) If the extension is finite and Galois, then is canonically isomorphic to
Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.
Valuation ring
If is a field, a valuation is a group homomorphism from the multiplicative group to a totally ordered abelian group such that, for any , in with nonzero, The valuation ring of is the subring of consisting of zero and all nonzero such that . | Ring (mathematics) | Wikipedia | 487 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Examples:
The field of formal Laurent series over a field comes with the valuation such that is the least degree of a nonzero term in ; the valuation ring of is the formal power series ring
More generally, given a field and a totally ordered abelian group , let be the set of all functions from to whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by convolution: It also comes with the valuation such that is the least element in the support of . The subring consisting of elements with finite support is called the group ring of (which makes sense even if is not commutative). If is the ring of integers, then we recover the previous example (by identifying with the series whose th coefficient is .)
Rings with extra structure
A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
An associative algebra is a ring that is also a vector space over a field such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of -by- matrices over the real field has dimension as a real vector space.
A ring is a topological ring if its set of elements is given a topology which makes the addition map () and the multiplication map to be both continuous as maps between topological spaces (where inherits the product topology or any other product in the category). For example, -by- matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
A λ-ring is a commutative ring together with operations that are like th exterior powers:
For example, is a λ-ring with the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.
A totally ordered ring is a ring with a total ordering that is compatible with ring operations.
Some examples of the ubiquity of rings
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.
Cohomology ring of a topological space
To any topological space one can associate its integral cohomology ring | Ring (mathematics) | Wikipedia | 482 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
a graded ring. There are also homology groups of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a -multilinear form and an -multilinear form to get a ()-multilinear form.
The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
Burnside ring of a group
To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
Representation ring of a group ring
To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure. | Ring (mathematics) | Wikipedia | 507 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Function field of an irreducible algebraic variety
To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
Face ring of a simplicial complex
Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
Category-theoretic description
Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of -modules). The monoid action of a ring on an abelian group is simply an -module. Essentially, an -module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring". | Ring (mathematics) | Wikipedia | 278 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Let be an abelian group and let be its endomorphism ring (see above). Note that, essentially, is the set of all morphisms of , where if is in , and is in , the following rules may be used to compute and :
where as in is addition in , and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, , is an abelian group. Furthermore, for every in , right (or left) multiplication by gives rise to a morphism of , by right (or left) distributivity. Let . Consider those endomorphisms of , that "factor through" right (or left) multiplication of . In other words, let be the set of all morphisms of , having the property that . It was seen that every in gives rise to a morphism of : right multiplication by . It is in fact true that this association of any element of , to a morphism of , as a function from to , is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian -group (by -group, it is meant a group with being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian -group.
Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
Generalization
Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.
Rng
A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed. | Ring (mathematics) | Wikipedia | 443 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
Nonassociative ring
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
Semiring
A semiring (sometimes rig) is obtained by weakening the assumption that is an abelian group to the assumption that is a commutative monoid, and adding the axiom that for all a in (since it no longer follows from the other axioms).
Examples:
the non-negative integers with ordinary addition and multiplication;
the tropical semiring.
Other ring-like objects
Ring object in a category
Let be a category with finite products. Let pt denote a terminal object of (an empty product). A ring object in is an object equipped with morphisms (addition), (multiplication), (additive identity), (additive inverse), and (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object equipped with a factorization of its functor of points through the category of rings:
Ring scheme
In algebraic geometry, a ring scheme over a base scheme is a ring object in the category of -schemes. One example is the ring scheme over , which for any commutative ring returns the ring of -isotypic Witt vectors of length over .
Ring spectrum
In algebraic topology, a ring spectrum is a spectrum together with a multiplication and a unit map from the sphere spectrum , such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra. | Ring (mathematics) | Wikipedia | 377 | 48404 | https://en.wikipedia.org/wiki/Ring%20%28mathematics%29 | Mathematics | Algebra | null |
In cryptography, a Caesar cipher, also known as Caesar's cipher, the shift cipher, Caesar's code, or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is replaced by a letter some fixed number of positions down the alphabet. For example, with a left shift of 3, would be replaced by , would become , and so on. The method is named after Julius Caesar, who used it in his private correspondence.
The encryption step performed by a Caesar cipher is often incorporated as part of more complex schemes, such as the Vigenère cipher, and still has modern application in the ROT13 system. As with all single-alphabet substitution ciphers, the Caesar cipher is easily broken and in modern practice offers essentially no communications security.
Example
The transformation can be represented by aligning two alphabets; the cipher alphabet is the plain alphabet rotated left or right by some number of positions. For instance, here is a Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 (the shift parameter is used as the key):
When encrypting, a person looks up each letter of the message in the "plain" line and writes down the corresponding letter in the "cipher" line.
Plaintext: THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG
Ciphertext: QEB NRFZH YOLTK CLU GRJMP LSBO QEB IXWV ALD
Deciphering is done in reverse, with a right shift of 3.
The encryption can also be represented using modular arithmetic by first transforming the letters into numbers, according to the scheme, A → 0, B → 1, ..., Z → 25. Encryption of a letter x by a shift n can be described mathematically as,
Decryption is performed similarly,
(Here, "mod" refers to the modulo operation. The value x is in the range 0 to 25, but if or are not in this range then 26 should be added or subtracted.)
The replacement remains the same throughout the message, so the cipher is classed as a type of monoalphabetic substitution, as opposed to polyalphabetic substitution.
History and usage | Caesar cipher | Wikipedia | 463 | 48405 | https://en.wikipedia.org/wiki/Caesar%20cipher | Technology | Computer security | null |
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