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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 J... | 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'... | 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 approxima... | 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 small... | 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, consisten... | 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 fabr... | 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, unt... | 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 k... | 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 (activ... | 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 ... | 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 ... | 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,... | 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 ... | 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 b... | 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 estim... | 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 orga... | 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 Moni... | 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 t... | 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. Adsorpti... | 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 b... | 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 natu... | 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. Pesti... | 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 ... | 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 polyethyle... | 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 bega... | 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 blowi... | 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 hav... | 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 insul... | 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 = ... | 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... | 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 poly... | 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 pro... | 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... | 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 polyurethan... | 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 N... | 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 polyurethan... | 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, ... | 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 a... | 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... | 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 sy... | 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 o... | 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 o... | 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 celes... | 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... | 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 increasin... | 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 coor... | 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... | 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... | 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 b... | 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 spati... | 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 m... | 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 rat... | 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:
{{Equation box 1
|indent=:
|title=Navier–Stokes momentum equation (conservative form)
|equation=
|... | 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 s... | 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 ... | 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... | 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:
Consi... | 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 t... | 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 ... | 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... | 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 ... | 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 ... | 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, ... | 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 ... | 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 tha... | 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 ca... | 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 ... | 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,... | 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 ordere... | 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 h... | 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 me... | 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 ... | 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-transform... | 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... | 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 ... | 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 commutativit... | 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 squ... | 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 b... | 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... | 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 ... | 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 ... | 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 po... | 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 somewh... | 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... | 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 i... | 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 commu... | 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 t... | 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 (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 ... | 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 expone... | 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
S... | 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 ... | 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 al... | 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 supp... | 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 ho... | 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 a... | 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, associate... | 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 L... | 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... | Caesar cipher | Wikipedia | 463 | 48405 | https://en.wikipedia.org/wiki/Caesar%20cipher | Technology | Computer security | null |
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