text stringlengths 11 320k | source stringlengths 26 161 |
|---|---|
An orbital pole is either point at the ends of the orbital normal , an imaginary line segment that runs through a focus of an orbit (of a revolving body like a planet , moon or satellite ) and is perpendicular (or normal ) to the orbital plane . Projected onto the celestial sphere , orbital poles are similar in concept to celestial poles , but are based on the body's orbit instead of its equator .
The north orbital pole of a revolving body is defined by the right-hand rule . If the fingers of the right hand are curved along the direction of orbital motion , with the thumb extended and oriented to be parallel to the orbital axis , then the direction the thumb points is defined to be the orbital north.
The poles of Earth's orbit are referred to as the ecliptic poles . For the remaining planets, the orbital pole in ecliptic coordinates is given by the longitude of the ascending node ( ☊ ) and inclination ( i ): ℓ = ☊ − 90° , b = 90° − i . In the following table, the planetary orbit poles are given in both celestial coordinates and the ecliptic coordinates for the Earth.
When an artificial satellite orbits close to another large body, it can only maintain continuous observations in areas near its orbital poles. The continuous viewing zone (CVZ) of the Hubble Space Telescope lies inside roughly 24° of Hubble's orbital poles, which precess around the Earth's axis every 56 days. [ 2 ]
The ecliptic is the plane on which Earth orbits the Sun . The ecliptic poles are the two points where the ecliptic axis, the imaginary line perpendicular to the ecliptic, intersects the celestial sphere .
The two ecliptic poles are mapped below.
Due to axial precession , either celestial pole completes a circuit around the nearer ecliptic pole every 25,800 years.
As of 1 January 2000 [update] , the positions of the ecliptic poles expressed in equatorial coordinates , as a consequence of Earth's axial tilt , are the following:
The north ecliptic pole is located near the Cat's Eye Nebula and the south ecliptic pole is located near the Large Magellanic Cloud .
It is impossible anywhere on Earth for either ecliptic pole to be at the zenith in the night sky . By definition, the ecliptic poles are located 90° from the Sun's position . Therefore, whenever and wherever either ecliptic pole is directly overhead, the Sun must be on the horizon . The ecliptic poles can contact the zenith only within the Arctic and Antarctic circles.
The galactic coordinates of the north ecliptic pole can be calculated as ℓ = 96.38° , b = 29.81° (see celestial coordinate system ). | https://en.wikipedia.org/wiki/Orbital_normal |
In chemical bonds , an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. The general principle for orbital overlap is that, the greater the overlap between orbitals, the greater the bond strength. Linus Pauling explained the importance of orbital overlap in the molecular bond angles observed through experimentation; it is the basis for orbital hybridization . As s orbitals are spherical (and have no directionality) and p orbitals are oriented 90° to each other, a theory was needed to explain why molecules such as methane (CH 4 ) had observed bond angles of 109.5°. [ 1 ] Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp 3 in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds. [ 2 ]
A quantitative measure of the overlap of two atomic orbitals Ψ A and Ψ B on atoms A and B is their overlap integral , defined as
where the integration extends over all space. The star on the first orbital wavefunction indicates the function's complex conjugate , which in general may be complex-valued .
The overlap matrix is a square matrix , used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system, such as an atomic orbital basis set used in molecular electronic structure calculations. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an orthonormal set, the overlap matrix will be the identity matrix . The overlap matrix is always n × n , where n is the number of basis functions used. It is a kind of Gramian matrix .
In general, each overlap matrix element is defined as an overlap integral:
where
In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically 1 and the magnitude of the off-diagonal elements less than or equal to one with equality if and only if there is linear dependence in the basis set as per the Cauchy–Schwarz inequality . Moreover, the matrix is always positive definite ; that is to say, the eigenvalues are all strictly positive.
Quantum Chemistry: Fifth Edition , Ira N. Levine, 2000
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Orbital_overlap |
The orbital period (also revolution period ) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy , it usually applies to planets or asteroids orbiting the Sun , moons orbiting planets, exoplanets orbiting other stars , or binary stars . It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.
For celestial objects in general, the orbital period is determined by a 360° revolution of one body around its primary , e.g. Earth around the Sun.
Periods in astronomy are expressed in units of time , usually hours, days, or years.
Its reciprocal is the orbital frequency , a kind of revolution frequency , in units of hertz .
According to Kepler's Third Law , the orbital period T of two point masses orbiting each other in a circular or elliptic orbit is: [ 1 ]
where:
For all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity.
Inversely, for calculating the distance where a body has to orbit in order to have a given orbital period T:
For instance, for completing an orbit every 24 hours around a mass of 100 kg , a small body has to orbit at a distance of 1.08 meters from the central body's center of mass .
In the special case of perfectly circular orbits, the semimajor axis a is equal to the radius of the orbit, and the orbital velocity is constant and equal to
where:
This corresponds to 1 ⁄ √2 times (≈ 0.707 times) the escape velocity .
For a perfect sphere of uniform density , it is possible to rewrite the first equation without measuring the mass as:
where:
For instance, a small body in circular orbit 10.5 cm above the surface of a sphere of tungsten half a metre in radius would travel at slightly more than 1 mm / s , completing an orbit every hour. If the same sphere were made of lead the small body would need to orbit just 6.7 mm above the surface for sustaining the same orbital period.
When a very small body is in a circular orbit barely above the surface of a sphere of any radius and mean density ρ (in kg/m 3 ), the above equation simplifies to
(since r now nearly equals a ). Thus the orbital period in low orbit depends only on the density of the central body, regardless of its size.
So, for the Earth as the central body (or any other spherically symmetric body with the same mean density, about 5,515 kg/m 3 , [ 2 ] e.g. Mercury with 5,427 kg/m 3 and Venus with 5,243 kg/m 3 ) we get:
and for a body made of water ( ρ ≈ 1,000 kg/m 3 ), [ 3 ] or bodies with a similar density, e.g. Saturn's moons Iapetus with 1,088 kg/m 3 and Tethys with 984 kg/m 3 we get:
Thus, as an alternative for using a very small number like G , the strength of universal gravity can be described using some reference material, such as water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time if we have a unit of density. [ citation needed ] [ original research? ]
In celestial mechanics , when both orbiting bodies' masses have to be taken into account, the orbital period T can be calculated as follows: [ 4 ]
where:
In a parabolic or hyperbolic trajectory, the motion is not periodic, and the duration of the full trajectory is infinite.
For celestial objects in general, the orbital period typically refers to the sidereal period , determined by a 360° revolution of one body around its primary relative to the fixed stars projected in the sky . For the case of the Earth orbiting around the Sun , this period is referred to as the sidereal year . This is the orbital period in an inertial (non-rotating) frame of reference .
Orbital periods can be defined in several ways. The tropical period is more particularly about the position of the parent star. It is the basis for the solar year , and respectively the calendar year .
The synodic period refers not to the orbital relation to the parent star, but to other celestial objects , making it not a merely different approach to the orbit of an object around its parent, but a period of orbital relations with other objects, normally Earth, and their orbits around the Sun. It applies to the elapsed time where planets return to the same kind of phenomenon or location, such as when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.
There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics , particularly they must not be confused with other revolving periods like rotational periods . Examples of some of the common orbital ones include the following:
Periods can be also defined under different specific astronomical definitions that are mostly caused by the small complex external gravitational influences of other celestial objects. Such variations also include the true placement of the centre of gravity between two astronomical bodies ( barycenter ), perturbations by other planets or bodies, orbital resonance , general relativity , etc. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry .
One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their synodic period , which is the time between conjunctions .
An example of this related period description is the repeated cycles for celestial bodies as observed from the Earth's surface, the synodic period , applying to the elapsed time where planets return to the same kind of phenomenon or location — for example, when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.
If the orbital periods of the two bodies around the third are called T 1 and T 2 , so that T 1 < T 2 , their synodic period is given by: [ 7 ]
Table of synodic periods in the Solar System, relative to Earth: [ citation needed ]
In the case of a planet's moon , the synodic period usually means the Sun-synodic period, namely, the time it takes the moon to complete its illumination phases, completing the solar phases for an astronomer on the planet's surface. The Earth's motion does not determine this value for other planets because an Earth observer is not orbited by the moons in question. For example, Deimos 's synodic period is 1.2648 days, 0.18% longer than Deimos's sidereal period of 1.2624 d. [ citation needed ]
The concept of synodic period applies not just to the Earth, but also to other planets as well; [ citation needed ] the computation of synodic periods applies the same formula as above. [ citation needed ] The following table lists the synodic periods of some planets relative to each the Sun and each other: [ original research? ] [ citation needed ] | https://en.wikipedia.org/wiki/Orbital_period |
The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non- collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) and of an orbiting celestial body at two different times/points of its orbit.
The orbital plane is defined in relation to a reference plane by two parameters : inclination ( i ) and longitude of the ascending node (Ω).
By definition, the reference plane for the Solar System is usually considered to be Earth's orbital plane, which defines the ecliptic , the circular path on the celestial sphere that the Sun appears to follow over the course of a year.
In other cases, for instance a moon or artificial satellite orbiting another planet, it is convenient to define the inclination of the Moon's orbit as the angle between its orbital plane and the planet's equatorial plane .
The coordinate system defined that uses the orbital plane as the x y {\displaystyle xy} plane is known as the perifocal coordinate system .
For launch vehicles and artificial satellites, the orbital plane is a defining parameter of an orbit; as in general, it will take a very large amount of propellant to change the orbital plane of an object. Other parameters, such as the orbital period , the eccentricity of the orbit and the phase of the orbit are more easily changed by propulsion systems.
Orbital planes of satellites are perturbed by the non-spherical nature of the Earth's gravity . This causes the orbital plane of the satellite's orbit to slowly rotate around the Earth, depending on the angle the plane makes with the Earth's equator. For planes that are at a critical angle this can mean that the plane will track the Sun around the Earth, forming a Sun-synchronous orbit .
A launch vehicle's launch window is usually determined by the times when the target orbital plane intersects the launch site. | https://en.wikipedia.org/wiki/Orbital_plane |
In celestial mechanics , the orbital plane of reference (or orbital reference plane ) is the plane used to define orbital elements (positions). The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node .
Depending on the type of body being described, there are four different kinds of reference planes that are typically used:
On the plane of reference, a zero-point must be defined from which the angles of longitude are measured. This is usually defined as the point on the celestial sphere where the plane crosses the prime hour circle (the hour circle occupied by the First Point of Aries ), also known as the equinox . | https://en.wikipedia.org/wiki/Orbital_plane_of_reference |
An orbital pole is either point at the ends of the orbital normal , an imaginary line segment that runs through a focus of an orbit (of a revolving body like a planet , moon or satellite ) and is perpendicular (or normal ) to the orbital plane . Projected onto the celestial sphere , orbital poles are similar in concept to celestial poles , but are based on the body's orbit instead of its equator .
The north orbital pole of a revolving body is defined by the right-hand rule . If the fingers of the right hand are curved along the direction of orbital motion , with the thumb extended and oriented to be parallel to the orbital axis , then the direction the thumb points is defined to be the orbital north.
The poles of Earth's orbit are referred to as the ecliptic poles . For the remaining planets, the orbital pole in ecliptic coordinates is given by the longitude of the ascending node ( ☊ ) and inclination ( i ): ℓ = ☊ − 90° , b = 90° − i . In the following table, the planetary orbit poles are given in both celestial coordinates and the ecliptic coordinates for the Earth.
When an artificial satellite orbits close to another large body, it can only maintain continuous observations in areas near its orbital poles. The continuous viewing zone (CVZ) of the Hubble Space Telescope lies inside roughly 24° of Hubble's orbital poles, which precess around the Earth's axis every 56 days. [ 2 ]
The ecliptic is the plane on which Earth orbits the Sun . The ecliptic poles are the two points where the ecliptic axis, the imaginary line perpendicular to the ecliptic, intersects the celestial sphere .
The two ecliptic poles are mapped below.
Due to axial precession , either celestial pole completes a circuit around the nearer ecliptic pole every 25,800 years.
As of 1 January 2000 [update] , the positions of the ecliptic poles expressed in equatorial coordinates , as a consequence of Earth's axial tilt , are the following:
The north ecliptic pole is located near the Cat's Eye Nebula and the south ecliptic pole is located near the Large Magellanic Cloud .
It is impossible anywhere on Earth for either ecliptic pole to be at the zenith in the night sky . By definition, the ecliptic poles are located 90° from the Sun's position . Therefore, whenever and wherever either ecliptic pole is directly overhead, the Sun must be on the horizon . The ecliptic poles can contact the zenith only within the Arctic and Antarctic circles.
The galactic coordinates of the north ecliptic pole can be calculated as ℓ = 96.38° , b = 29.81° (see celestial coordinate system ). | https://en.wikipedia.org/wiki/Orbital_pole |
An orbital propellant depot is a cache of propellant that is placed in orbit around Earth or another body to allow spacecraft or the transfer stage of the spacecraft to be fueled in space. It is one of the types of space resource depots that have been proposed for enabling infrastructure-based space exploration . [ 1 ] Many depot concepts exist depending on the type of fuel to be supplied, location, or type of depot which may also include a propellant tanker that delivers a single load to a spacecraft at a specified orbital location and then departs. In-space fuel depots are not necessarily located near or at a space station .
Potential users of in-orbit refueling and storage facilities include space agencies , defense ministries and communications satellite or other commercial companies.
Satellite servicing depots would extend the lifetime of satellites that have nearly consumed their orbital maneuvering fuel and are likely placed in a geosynchronous orbit. The spacecraft would conduct a space rendezvous with the depot, or vice versa , and then transfer propellant to be used for subsequent orbital maneuvers . In 2011, Intelsat showed interest in an initial demonstration mission to refuel several satellites in geosynchronous orbit , but all plans have been since scrapped. [ 2 ]
A low Earth orbit (LEO) depot's primary function would be to provide propellant to a transfer stage headed to the Moon, Mars, or possibly a geosynchronous orbit. Since all or a fraction of the transfer stage propellant can be off-loaded, the separately launched spacecraft with payload and/or crew could have a larger mass or use a smaller launch vehicle. With a LEO depot or tanker fill, the size of the launch vehicle can be reduced and the flight rate increased—or, with a newer mission architecture where the beyond-Earth-orbit spacecraft also serves as the second stage, can facilitate much larger payloads—which may reduce the total launch costs since the fixed costs are spread over more flights and fixed costs are usually lower with smaller launch vehicles. A depot could also be placed at Earth-Moon Lagrange point 1 (EML-1) or behind the Moon at EML-2 to reduce costs to travel to the Moon or Mars. Placing a depot in Mars orbit has also been suggested. [ 3 ]
In 2024, on Starship’s third integrated flight , intravehicular propellant transfer in orbit was demonstrated, [ 4 ] an intervehicular propellant transfer demonstration mission is planned for 2025, [ 5 ] as this capability is critical for landing a crew on the Moon with the Starship HLS vehicle. [ 5 ]
For rockets and space vehicles, propellants usually take up 2/3 or more of their total mass.
Large upper-stage rocket engines generally use a cryogenic fuel like liquid hydrogen and liquid oxygen (LOX) as an oxidizer because of the large specific impulse possible, but must carefully consider a problem called "boil off," or the evaporation of the cryogenic propellant. The boil off from only a few days of delay may not allow sufficient fuel for higher orbit injection, potentially resulting in a mission abort. Lunar or Mars missions will require weeks to months to accumulate tens of thousands to hundreds of thousands of kilograms of propellant, so additional equipment may be required on the transfer stage or the depot to mitigate boiloff.
Non-cryogenic, earth-storable liquid rocket propellants including RP-1 ( kerosene ), hydrazine and nitrogen tetroxide (NTO), and mildly cryogenic, space-storable propellants like liquid methane and liquid oxygen , can be kept in liquid form with less boiloff than the cryogenic fuels, but also have lower specific impulse. [ 6 ] Additionally, gaseous or supercritical propellants such as those used by ion thrusters include xenon , argon , [ 7 ] [ 8 ] and bismuth . [ 9 ]
Ex-NASA administrator Mike Griffin commented at the 52nd AAS Annual Meeting in Houston, Texas, November 2005, that "at a conservatively low government price of $10,000 per kg in LEO, 250 MT of fuel for two missions per year is worth $2.5 billion, at government rates." [ 10 ]
In the depot-centric architecture, the depot is filled by tankers, and then the propellant is transferred to an upper stage prior to orbit insertion, similar to a gas station filled by tankers for automobiles. By using a depot, the launch vehicle size can be reduced and the flight rate increased. Since the accumulation of propellant may take many weeks to months, careful consideration must be given to boiloff mitigation.
In simple terms, a passive cryogenic depot is a transfer stage with stretched propellant tanks, additional insulation, and a sun shield. In one concept, hydrogen boiloff is also redirected to reduce or eliminate liquid oxygen boiloff and then used for attitude control, power, or reboost. An active cryogenic depot is a passive depot with additional power and refrigeration equipment/cryocoolers to reduce or eliminate propellant boiloff. [ 11 ] Other active cryogenic depot concepts include electrically powered attitude control equipment to conserve fuel for the end payload.
In the heavy lift architecture, propellant, which can be two-thirds or more of the total mission mass, is accumulated in fewer launches and possibly shorter time frame than the depot centric architecture. Typically the transfer stage is filled directly and no depot is included in the architecture. For cryogenic vehicles and cryogenic depots, additional boiloff mitigation equipment is typically included on the transfer stage, reducing payload fraction and requiring more propellant for the same payload unless the mitigation hardware is expended.
Heavy Lift is compared with using Commercial Launch and Propellant Depots in this power point by Dr. Alan Wilhite given at FISO Telecon. [ 12 ]
Both theoretical studies and funded development projects that are currently underway aim to provide insight into the feasibility of propellant depots. Studies have shown that a depot-centric architecture with smaller launch vehicles could be US$57 billion less expensive than a heavy-lift architecture over a 20-year time frame. [ 13 ] The cost of large launch vehicles is so high that a depot able to hold the propellant lifted by two or more medium-sized launch vehicles may be cost effective and support more payload mass on beyond-Earth orbit trajectories.
In a 2010 NASA study, an additional flight of an Ares V heavy launch vehicle was required to stage a US government Mars reference mission due to 70 tons of boiloff, assuming 0.1% boiloff/day for hydrolox propellant. [ 14 ] The study identified the need to decrease the design boiloff rate by an order of magnitude or more.
Approaches to the design of low Earth orbit (LEO) propellant depots were also discussed in the 2009 Augustine report to NASA , which "examined the [then] current concepts for in-space refueling." [ 15 ] The report determined there are essentially two approaches to refueling a spacecraft in LEO: [ 15 ]
Both approaches were considered feasible with 2009 spaceflight technology, but anticipated that significant further engineering development and in-space demonstration would be required before missions could depend on the technology. Both approaches were seen to offer the potential of long-term life-cycle savings. [ 15 ]
In 2010 United Launch Alliance (ULA) proposed their Advanced Cryogenic Evolved Stage (ACES) tanker, a concept that dates to work by Boeing in 2006, [ 16 ] sized to transport up to 73 tonnes (161,000 lb) of propellant—in early design, a first flight was proposed for no earlier than 2023, with initial usage as a propellant tanker potentially beginning in the mid-2020s. [ 17 ] [ 18 ] ACES was not funded, but some of the ideas were used in the Centaur stage of the Vulcan Centaur rocket.
Beyond theoretical studies, since at least 2017, SpaceX has undertaken funded development of an interplanetary set of technologies. While the interplanetary mission architecture consists of a combination of several elements that are considered by SpaceX to be key to making long-duration beyond Earth orbit (BEO) spaceflights possible by reducing the cost per ton delivered to Mars by multiple orders of magnitude over what NASA approaches have achieved, [ 19 ] [ 20 ] [ 21 ] refilling of propellants in orbit is one of the four key elements. In a novel mission architecture, the SpaceX design intends to enable the long-journey spacecraft to expend almost all of its propellant load during the launch to low Earth orbit while it serves as the second stage of the SpaceX Starship , and then after refilling on orbit by multiple Starship tankers, provide the large amount of energy required to put the spacecraft onto an interplanetary trajectory. The Starship tanker is designed to transport approximately 100 tonnes (220,000 lb) of propellant to low Earth orbit. [ 22 ] [ better source needed ] In April 2021, NASA selected the SpaceX Lunar Starship with in-orbit refueling for their initial lunar human landing system. [ 23 ]
Because a large portion of a rocket is propellant at time of launch, proponents point out several advantages of using a propellant depot architecture. Spacecraft could be launched unfueled and thus require less structural mass, [ 24 ] or the depot tanker itself could serve as the second-stage on launch when it is reusable. [ 22 ] An on-orbit market for refueling may be created where competition to deliver propellant for the lowest price takes place, and it may also enable an economy of scale by permitting existing rockets to fly more often to refuel the depot. [ 24 ] If used in conjunction with a mining facility on the moon , water or propellant could be exported back to the depot, further reducing the cost of propellant. [ 25 ] [ 26 ] An exploration program based on a depot architecture could be less expensive and more capable, not needing a specific rocket or a heavy lift such as the SLS [ 13 ] [ 24 ] [ 27 ] [ 28 ] [ 29 ] to support multiple destinations such as the Moon, Lagrange points, asteroids, and Mars. [ 30 ]
NASA studies in 2011 showed lower cost and faster alternatives than the Heavy Lift Launch System and listed the following advantages: [ 27 ]
Propellant depots were proposed as part of the Space Transportation System (along with nuclear "tugs" to take payloads from LEO to other destinations) in the mid-1960s. [ 31 ]
In October 2009, the U.S. Air Force and United Launch Alliance (ULA) performed an experimental on-orbit demonstration on a modified Centaur upper stage on the DMSP-18 launch to improve "understanding of propellant settling and slosh , pressure control, RL10 chilldown and RL10 two-phase shutdown operations." "The light weight of DMSP-18 allowed 12,000 pounds (5,400 kg) of remaining liquid O 2 and liquid H 2 propellant, 28% of Centaur's capacity," for the on-orbit demonstrations. The post-spacecraft mission extension ran 2.4 hours before executing the deorbit burn . [ 32 ]
NASA's Launch Services Program is working on an ongoing slosh fluid dynamics experiments with partners called CRYOTE. As of 2010 [update] , ULA is also planning additional in-space laboratory experiments to further develop cryogenic fluid management technologies using the Centaur upper stage after primary payload separation. Named CRYOTE, or CRYogenic Orbital TEstbed, it will be a testbed for demonstrating a number of technologies needed for cryogenic propellant depots, with several small-scale demonstrations planned for 2012–2014. [ 33 ] As of August 2011 [update] , ULA said this mission could launch as soon as 2012 if funded. [ 34 ] The ULA CRYOTE small-scale demonstrations are intended to lead to a ULA large-scale cryo-sat flagship technology demonstration in 2015. [ 33 ]
The Future In-Space Operations (FISO) Working Group, a consortium of participants from NASA, industry and academia, discussed propellant depot concepts and plans on several occasions in 2010, [ 35 ] with presentations of optimal depot locations for human space exploration beyond low Earth orbit, [ 36 ] a proposed simpler (single vehicle) first-generation propellant depot [ 33 ] and six important propellant-depot-related technologies for reusable cislunar transportation. [ 37 ]
NASA also has plans to mature techniques for enabling and enhancing space flights that use propellant depots in the "CRYOGENIC Propellant STorage And Transfer (CRYOSTAT) Mission". The CRYOSTAT vehicle was expected to be launched to LEO in 2015. [ 38 ]
The CRYOSTAT architecture comprises technologies in the following categories: [ 38 ]
The "Simple Depot" mission was proposed by NASA in 2011 as a potential first PTSD mission, with launch no earlier than 2015, on an Atlas V 551 . Simple Depot would use the "used" (nearly-emptied) Centaur upper stage LH2 tank for long-term storage of LO2 while LH2 would be stored in the Simple Depot LH2 module, which would be launched with only ambient-temperature gaseous Helium in it. The SD LH2 tank was to be 3 metres (9.8 ft) diameter and 16 metres (52 ft) long, 110 cubic metres (3,900 cu ft) in volume, and store 5 mT of LH2. "At a useful mixture ratio (MR) of 6:1 this quantity of LH2 can be paired with 25.7 mT of LO2, allowing for 0.7 mT of LH2 to be used for vapor cooling, for a total useful propellant mass of 30 mT. ... the described depot would have a boil-off rate approaching 0.1 percent per day, consisting entirely of hydrogen." [ 39 ]
In September 2010, ULA released a Depot-Based Space Transportation Architecture concept to propose propellant depots that could be used as way-stations for other spacecraft to stop and refuel—either in low Earth orbit (LEO) for beyond-LEO missions, or at Lagrangian point L 2 for interplanetary missions—at the AIAA Space 2010 conference. The concept proposes that waste gaseous hydrogen —an inevitable byproduct of long-term liquid hydrogen storage in the radiative heat environment of space —would be usable as a monopropellant in a solar-thermal propulsion system . The waste hydrogen would be productively used for both orbital stationkeeping and attitude control , as well as providing limited propellant and thrust to use for orbital maneuvers to better rendezvous with other spacecraft that would be inbound to receive fuel from the depot. [ 40 ] As part of the Depot-Based Space Transportation Architecture, ULA has proposed the Advanced Common Evolved Stage (ACES) upper stage rocket . ACES hardware is designed from the start as an in-space propellant depot that could be used as way-stations for other rockets to stop and refuel on the way to beyond-LEO or interplanetary missions, and to provide the high-energy technical capacity for the cleanup of space debris . [ 16 ]
In August 2011, NASA made a significant contractual commitment to the development of propellant depot technology [ 1 ] by funding four aerospace companies to "define demonstration missions that would validate the concept of storing cryogenic propellants in space to reduce the need for large launch vehicles for deep-space exploration." [ 41 ] These study contracts for storing/transferring cryogenic propellants and cryogenic depots were signed with Analytical Mechanics Associates , Boeing , Lockheed Martin and Ball Aerospace . Each company was to receive US$ 600,000 under the contract. [ 41 ] [ needs update ]
In April 2021, NASA selected the SpaceX Lunar Starship with in-orbit refuelling for their initial lunar human landing system. [ 23 ] In 2022, a larger propellant-depot Starship was being planned for Lunar Starship HLS.
The Chinese Space Agency (CNSA) performed its first satellite-to-satellite on-orbit refueling test in June 2016. [ 42 ]
There are a number of design issues with propellant depots, as well as several tasks that have not, to date, been tested in space for on-orbit servicing missions. The design issues include propellant settling and transfer, propellant usage for attitude control and reboost, the maturity of the refrigeration equipment/cryocoolers, and the power and mass required for reduced or zero boiloff depots with refrigeration.
Transfer of liquid propellants in microgravity is complicated by the uncertain distribution of liquid and gasses within a tank. Propellant settling at an in-space depot is thus more challenging than in even a slight gravity field. ULA plans to use the DMSP -18 mission to flight-test centrifugal propellant settling as a cryogenic fuel management technique that might be used in future propellant depots. [ 43 ] The proposed Simple Depot PTSD mission would use several techniques to achieve adequate settling for propellant transfer. [ 39 ]
In the absence of gravity, propellant transfer is somewhat more difficult, since liquids can float away from the inlet.
As part of the Orbital Express mission in 2007, hydrazine propellant was successfully transferred between two single-purpose designed technology demonstration spacecraft. The Boeing servicing spacecraft ASTRO transferred propellant to the Ball Aerospace serviceable client spacecraft NEXTSat . Since no crew were present on either spacecraft, this was reported as the first autonomous spacecraft-to-spacecraft fluid transfer. [ 44 ]
After propellant has been transferred to a customer, the depot's tanks will need refilling. Organizing the construction and launch of the tanker rockets bearing the new fuel is the responsibility of the propellant depot's operator. Since space agencies like NASA hope to be purchasers rather than owners, possible operators include the aerospace company that constructed the depot, manufacturers of the rockets, a specialist space depot company, or an oil/chemical company that refines the propellant. By using several tanker rockets the tankers can be smaller than the depot and larger than the spacecraft they are intended to resupply. Short range chemical propulsion tugs belonging to the depot may be used to simplify docking tanker rockets and large vehicles like Mars Transfer Vehicles.
Transfers of propellant between a LEO depot, reachable by rockets from Earth, and the possible deep space ones such as at the Lagrange Points and Phobos depots could be performed using Solar electric propulsion (SEP) tugs. [ 45 ]
Two missions are currently under development or proposed to support propellant depot refilling.
In 1962, S.T. Demetriades [ 47 ] proposed a method for refilling by collecting atmospheric gases. Moving in low Earth orbit , at an altitude of around 120 km, Demetriades' proposed depot extracts air from the fringes of the atmosphere, compresses and cools it, and extracts liquid oxygen. The remaining nitrogen is used as propellant for a nuclear-powered magnetohydrodynamic engine, which maintains the orbit, compensating for atmospheric drag . [ 47 ] This system was called "PROFAC" ( PROpulsive Fluid ACcumulator ). [ 48 ] There are, however, safety concerns with placing a nuclear reactor in low Earth orbit.
Demetriades' proposal was further refined by Christopher Jones and others [ 49 ] In this proposal, multiple collection vehicles accumulate propellant gases at around 120 km altitude, later transferring them to a higher orbit. However, Jones' proposal does require a network of orbital power-beaming satellites , to avoid placing nuclear reactors in orbit.
Asteroids can also be processed to provide liquid oxygen. [ 50 ]
Propellant depots in LEO are of little use for transfer between two low earth orbits when the depot is in a different orbital plane than the target orbit. The delta-v to make the necessary plane change is typically extremely high. On the other hand, depots are typically proposed for exploration missions, where the change over time of the depot's orbit can be chosen to align with the departure vector. This allows one well-aligned departure time minimizing fuel use that requires a very precisely-timed departure. Less efficient departure times from the same depot to the same destination exist before and after the well-aligned opportunity, but more research is required to show whether the efficiency falls off quickly or slowly. [ citation needed ] By contrast, launching directly in only one launch from the ground without orbital refueling or docking with another craft already on orbit offers daily launch opportunities though it requires larger and more expensive launchers. [ 51 ]
The restrictions on departure windows arise because low earth orbits are susceptible to significant perturbations; even over short periods they are subject to nodal regression and, less importantly, precession of perigee. Equatorial depots are more stable but also more difficult to reach. [ 51 ]
New approaches have been discovered for LEO to interplanetary orbital transfers where a three-burn orbital transfer is used, which includes a plane change at apogee in a highly-elliptical phasing orbit, in which the incremental delta-v is small—typically less than five percent of the total delta-v—"enabling departures to deep-space destinations [taking] advantage of a depot in LEO" and providing frequent departure opportunities. [ 52 ] More specifically, the 3-burn departure strategy has been shown to enable a single LEO depot in an ISS -inclination orbit (51 degrees) to dispatch nine spacecraft to "nine different interplanetary targets [where the depot need not] perform any phasing maneuvers to align with any of the departure asymptotes ... [including enabling] extending the economic benefits of dedicated smallsat launch to interplanetary missions." [ 53 ]
Boil-off of cryogenic propellants in space may be mitigated by both technological solutions as well as system-level planning and design . From a technical perspective: for a propellant depot with passive insulation system to effectively store cryogenic fluids, boil-off caused by heating from solar and other sources must be mitigated, eliminated, [ 43 ] or used for economic purposes. [ 16 ] For non-cryogenic propellants, boil-off is not a significant design problem.
Boil-off rate is governed by heat leakage and by the quantity of propellant in the tanks. With partially filled tanks, the percentage loss is higher. Heat leakage depends on surface area, while the original mass of propellant in the tanks depends on volume. So by the cube-square law , the smaller the tank, the faster the liquids will boil off. Some propellant tank designs have achieved a liquid hydrogen boil off rate as low as approximately 0.13% per day (3.8% per month) while the much higher temperature cryogenic fluid of liquid oxygen would boil off much less, about 0.016% per day (0.49% per month). [ 54 ]
It is possible to achieve zero boil-off (ZBO) with cryogenic propellant storage using an active thermal control system. Tests conducted at the NASA Lewis Research Center 's Supplemental Multilayer Insulation Research Facility (SMIRF) over the summer of 1998 demonstrated that a hybrid thermal control system could eliminate boiloff of cryogenic propellants. The hardware consisted of a pressurized 50 cu ft (1,400 litres) tank insulated with 34 layers of insulation , a condenser, and a Gifford-McMahon (GM) cryocooler that has a cooling capacity of 15 to 17.5 watts (W). Liquid hydrogen was the test fluid. The test tank was installed into a vacuum chamber, simulating space vacuum. [ 55 ]
In 2001, a cooperative effort by NASA's Ames Research Center , Glenn Research Center , and Marshall Space Flight Center (MSFC) was implemented to develop ZBO concepts for in-space cryogenic storage. The main program element was a large-scale, ZBO demonstration using the MSFC multipurpose hydrogen test bed (MHTB) – 18.10 m3 L H 2 tank (about 1300 kg of H 2 ). A commercial cryocooler was interfaced with an existing MHTB spray bar mixer and insulation system in a manner that enabled a balance between incoming and extracted thermal energy. [ 56 ]
Another NASA study in June 2003 for conceptual Mars mission showed mass savings over traditional, passive-only cryogenic storage when mission durations are 5 days in LEO for oxygen, 8.5 days for methane and 64 days for hydrogen. Longer missions equate to greater mass savings. Cryogenic xenon saves mass over passive storage almost immediately. When power to run the ZBO is already available, the break-even mission durations are even shorter, e.g. about a month for hydrogen. The larger the tank, the fewer days in LEO when ZBO has reduced mass. [ 57 ]
In addition to technical solutions to the challenge of excessive boil-off of cryogenic rocket propellants, system-level solutions have been proposed. From a systems perspective, reductions in the standby time of the liquid H2 cryogenic storage in order to achieve, effectively, a just in time delivery to each customer, matched with the balanced refinery technology to split the long-term storable feedstock—water—into the stoichiometric LOX / LH2 necessary, is theoretically capable of achieving a system-level solution to boil-off. Such proposals have been suggested as supplementing good technological techniques to reduce boil-off, but would not replace the need for efficient technological storage solutions. [ 58 ]
United Launch Alliance (ULA) has proposed a cryogenic depot which would use a conical sun shield to protect the cold propellants from solar and Earth radiation. The open end of the cone allows residual heat to radiate to the cold of deep space, while the closed cone layers attenuates the radiative heat from the Sun and Earth. [ 59 ]
Other issues are hydrogen embrittlement , a process by which some metals (including iron and titanium ) become brittle and fracture following exposure to hydrogen. The resulting leaks make storing cryogenic propellants in zero gravity conditions difficult. [ 60 ]
In the early 2010s, several in-space refueling projects got underway. Two private initiatives and a government sponsored test mission were in some level of development or testing as of 2010 [update] .
The NASA Robotic Refueling Mission (RRM) was launched in 2011 and successfully completed a series of robotically actuated propellant transfer experiments on the exposed facility platform of the International Space Station in January 2013. [ 61 ]
The set of experiments included a number of propellant valves , nozzles and seals similar to those used on many satellites and a series of four prototype tools that could be attached to the distal end of a Space Station robotic arm . Each tool was a prototype of "devices that could be used by future satellite servicing missions to refuel spacecraft in orbit. RRM is the first in-space refueling demonstration using a platform and fuel valve representative of most existing satellites, which were never designed for refueling. Other satellite servicing demos, such as the U.S. military's Orbital Express mission in 2007, transferred propellant between satellites with specially-built pumps and connections." [ 61 ]
As of March 2010 [update] , a small-scale refueling demonstration project for reaction control system (RCS) fluids was under development. Canada -based MDA Corporation announced in early 2010 that they were designing a single spacecraft that would refuel other spacecraft in orbit as a satellite-servicing demonstration. "The business model, which is still evolving, could ask customers to pay per kilogram of fuel successfully added to their satellite, with the per-kilogram price being a function of the additional revenue the operator can expect to generate from the spacecraft's extended operational life." [ 62 ]
The plan is that the fuel-depot vehicle would maneuver to an operational communications satellite , dock at the target satellite's apogee-kick motor , remove a small part of the target spacecraft's thermal protection blanket, connect to a fuel-pressure line and deliver the propellant. "MDA officials estimate the docking maneuver would take the communications satellite out of service for about 20 minutes." [ 62 ]
As of March 2011 [update] , MDA had secured a major customer for the initial demonstration project. Intelsat agreed to purchase one-half of the 2,000 kilograms (4,400 lb) of propellant payload that the MDA spacecraft would carry into geostationary orbit . Such a purchase would add somewhere between two and four years of additional service life for up to five Intelsat satellites, assuming 200 kg of fuel is delivered to each one. [ 63 ] As of March 2010 [update] , the spacecraft could be ready to begin refueling communication satellites by 2015. [ 64 ] As of January 2013 [update] , no customers had signed up for an MDA refueling mission. [ 61 ]
In 2017, MDA announced that it was restarting its satellite servicing business, with Luxembourg-based satellite owner/operator SES S.A. as its first customer. [ 65 ]
Competitive design alternatives to in-space RCS fuel transfer exist. It is possible to bring additional propellant to a space asset, and use the propellant for attitude control or orbital velocity change, without ever transferring the propellant to the target space asset.
The ViviSat Mission Extension Vehicle , also under development since the early 2010s, illustrates one alternative approach that would connect to the target satellite similarly to MDA SIS, via the kick motor, but would not transfer fuel. Rather, the Mission Extension Vehicle would use "its own thrusters to supply attitude control for the target." [ 66 ] ViviSat believes their approach is more simple and can operate at lower cost than the MDA propellant transfer approach, while having the technical ability to dock with and service a greater number (90 percent) of the approximately 450 geostationary satellites in orbit. [ 66 ] As of January 2013 [update] , no customers had signed up for a ViviSat-enabled mission extension. [ 61 ]
In 2015, Lockheed Martin proposed the Jupiter space tug . If built, Jupiter would operate in low Earth orbit shuttling cargo carriers to and from the International Space Station , remaining on orbit indefinitely, and refueling itself from subsequent transport ships carrying later cargo carrier modules. [ 67 ]
In December 2018, Orbit Fab , a silicon valley startup company founded in early 2018, flew the first of a series of experiments to the ISS in order to test and demonstrate technologies to allow for commercial in space refueling. These first rounds of testing used water as a propellant simulant. [ 68 ] In June 2021, Orbit Fab flew the first propellant depot, Tanker-001 Tenzing, carrying Hydrogen Peroxide in Sun-synchronous orbit . [ 69 ] | https://en.wikipedia.org/wiki/Orbital_propellant_depot |
An orbital ring is a concept of an artificial ring placed around a body and set rotating at such a rate that the apparent centrifugal force is large enough to counteract the force of gravity . For the Earth , the required speed is on the order of 10 km/sec, [ citation needed ] compared to a typical low Earth orbit orbital speed of 7.9 km/sec. The structure is intended to be used as a space station or as a planetary vehicle for very high-speed transportation or space launch .
Because the cable is spinning faster than orbital velocity, there is a net outward force that is countered by internal tension within the cable. This resists any attempt to bend it and allows it to carry loads. In typical conceptions, a motorized platform is placed on the cable that runs in the opposite direction at the speed that makes it appear stationary above the ground. Above Earth's equator, a platform running at 9.5 km/sec in the direction opposite the cable will appear stationary and allow a cable to be lowered to form a space elevator . This elevator is only perhaps 500 kilometres (310 mi) long, which can be built with existing materials.
The requirement to construct a planet-sized cable in low-earth orbit and accelerate it to a faster-than-orbital velocity is an obvious practical problem. Other architectures have thus been proposed that use active support in different ways and are thus able to circumvent some of these limitations. The launch loop is a partial ring, perhaps 2000 km long, that runs between two ground stations instead of encircling the world. The particle ring uses a series of separate objects that can be launched individually to produce a collection similar to a solid ring and then controlled magnetically, with the disadvantage that they have no internal tension and lifting power is derived separately. The space fountain is a vertical version of the particle ring concept that forms a space elevator. The tethered ring is a dynamic structure that uses at least one complete and continuous non-orbiting ring with a diameter that is smaller than that of the planetary body. It can be built on the planet’s surface, accelerated to operating speed, and raised to a very high altitude mechanically by tensioning its numerous tethers. [ 1 ] [ 2 ]
The orbital ring is somewhat similar to the "classic" space elevator concept. In the traditional space elevator, a large station is placed in geostationary orbit (GEO) so that it remains in a single location above the equator of Earth. A cable is then built and lowered towards Earth while a second cable, providing a counterweight, is built upward from the station and remains in place due to tidal forces . When the structure is complete, elevator-like cars can ride the cable into space. The main problem is that no known substance that can be manufactured in large quantities has the tensile strength needed to stretch from GEO to the surface. Orbital rings use a different mechanism.
In the orbital ring version, a kinetic ring is moving around the world at a higher speed than circular orbital velocity. This results in a net outward force that is countered by gravity acting on the stationary components. This can be accomplished at any altitude, although building the system above 500 kilometres (310 mi) in order to avoid most of the atmosphere is a practical requirement. A cable is then lowered from the ring to the ground and used in the same fashion as a traditional space elevator, with the difference being that the vertical cable is only 500 kilometres (310 mi) instead of 100,000 kilometres (62,000 mi) long. This length is within the capabilities of several known materials.
In order to support the elevator, the ring is not circular but slightly elliptical. Two or more stations are placed at the high ends of the path, but below the point where the orbit's apogee would be normally. The station bends the cable downward as it passes through in order to produce an upward force on the station. The resulting orbit for a two station system looks something more akin to an American football than a rounded ellipse. One can reduce the amount of tension in the orbiting ring to any required level by increasing the amount of lift generated by bending the cable. The main downside is that the elevator cable is now suspended from the high point of the system, rather than being close to the ground.
The moving ring does not need to be solid and does not need to be entirely encased by a solid sheath. Instead, a large number of individual magnetic objects can be placed in the desired orbit, and the stations deflect their path using magnets as they pass by. This version of the ring has the advantage of being much simpler to construct, as each element in the ring is completely separate and can be launched individually and requires no further working once in space. [ a ] It also does not have to be a complete ring; depending on the desired lifting power the total mass of objects might be much smaller than even the thinnest cable circling Earth. The main disadvantage is that the process of momentum exchange randomizes their velocity so some other system is required to shepherd the objects back into the correct orbits.
A detailed description of the concept was proposed and analyzed by Paul Birch in 1982, proposing a massive ring that would encircle the globe in low orbit, from which cables hang down to Earth's surface. [ 3 ] [ 4 ] [ 5 ]
In 1982, Soviet inventor Anatoly Yunitskiy also proposed an electromagnetic track encircling the equator, which he called "to space by a wheel" [ 6 ] (later, "String Transportation System"). When the velocity of the string exceeds 10 km/sec, [ failed verification ] centrifugal forces detach the (flexible and extensible) string from Earth's surface and lift the ring into space.
Andrew Meulenberg and his students, from 2008 to 2011, presented and published a number of papers based on types and applications of low-Earth-orbital rings as humanity's "stepping-stones-to-space". An overview mentions four applications of orbital rings: communication via a fiber-optic ring, surface-to-orbit transport with a "sling-on-a-ring" system, space-based solar power and climate change mitigation with a space sunshade . The "sling-on-a-ring" would involve rotating slings (made of colossal carbon tube ) attached to the orbital ring that dip down into the atmosphere. The tip of a given sling would reach an altitude of 13–15 km at its lowest, and its rotation would cause it to have near-zero tangential velocity relative to the Earth's surface below. Consequently, the sling could pick up a payload from a conventional aircraft flying at that altitude, then lift up this payload to the orbital ring. [ 7 ]
Paul Birch published a series of three articles in the Journal of the British Interplanetary Society in 1982 that laid out the mathematical basis of ring systems. [ 3 ] [ 4 ] [ 5 ]
In the simplest design of an orbital ring system, a rotating cable or possibly an inflatable space structure is placed in a low Earth orbit above the equator. Not in orbit, but riding on this ring, supported electromagnetically on superconducting magnets , are ring stations that stay in one place above some designated point on Earth. Hanging down from these ring stations are short elevator cables made from materials with high-tensile-strength-to-mass-ratio.
Although this simple model would work best above the equator, Paul Birch calculated that since the ring station can be used to accelerate the orbital ring eastwards as well as hold the tether, it is therefore possible to deliberately cause the orbital ring to precess around Earth instead of staying fixed in space while Earth rotates beneath it. By precessing the ring once every 24 hours, the Orbital Ring will hover above any meridian selected on the surface of Earth. The cables which dangle from the ring are now geostationary without having to reach geostationary altitude or be placed into the equatorial plane. This means that using the Orbital Ring concept, one or many pairs of Stations can be positioned above any points on Earth desired, or can be moved everywhere on the globe. Thus, any point on Earth can be served by a space elevator. Also, a whole network of orbital rings can be built, which, by crossing over the poles, could cover the whole planet and be capable of taking over most freight and passenger transport. By an array of elevators and several geostationary ring stations, asteroid or Moon material can be received and gently put down where land fills are needed. The electric energy generated in the process would pay for the system expansion and ultimately could pave the way for a solar-system-wide terraforming and astroengineering activity on a sound economical basis.
If built by launching the necessary materials from Earth, the cost for the system estimated by Birch in 1980s money was around $31 billion (for a "bootstrap" system intended to expand to 1000 times its initial size over the following year, which would otherwise cost 31 trillion dollars) if launched using Shuttle-derived hardware, whereas it could fall to $15 billion with space-based manufacturing, assuming a large orbital manufacturing facility is available to provide the initial 180,000 tons of steel, aluminum, and slag at a low cost, and even lower with orbital rings around the Moon. The system's cost-per-kilogram to accelerate payloads to low Earth orbit velocity would be around $0.05 in 1975 USD, assuming an energy requirement of 9 kWh/kg (roughly accurate) and an aspirational cost of electricity, provided by space-based solar power , of 0.005 US dollars per kWh. [ 8 ]
The simplest type would be a circular orbital ring in Low Earth orbit .
Two other types were also defined by Paul Birch:
In addition, he proposed the concept of "supramundane worlds" such as supra-Jovian and supra-stellar "planets". These are artificial planets that would be supported by a grid of orbital rings that would be positioned above a planet, supergiant or even a star. [ 9 ]
In the close of Arthur C. Clarke's Fountains of Paradise (1979), a reference is made to an orbital ring that is attached in the distant future to the space elevator that is the basis of the novel.
Arthur C. Clarke's 3001: The Final Odyssey (1997) features an orbital ring held aloft by four enormous inhabitable towers (assumed successors to space elevators) at the Equator.
The manga Battle Angel Alita (1990-1995) prominently features a slightly deteriorated orbital ring.
The Star Trek novel Ring Around the Sky features a decrepit ringworld in orbit above the planet, Kharzh'ulla, connected by a series of space elevators with the surface.
Orbital rings are used extensively in the collaborative fiction worldbuilding website Orion's Arm . [ 10 ]
The third part of Neal Stephenson 's Seveneves (2015) has an orbital ring around a Moon-less Earth.
In the movie Starship Troopers , an orbital ring is shown encircling the Moon.
The second iteration of the anime series Tekkaman features a complete ring, though abandoned and in disrepair due to war, and without surface tethers.
The anime series Kiddy Grade also uses orbital rings as a launch and docking bay for spaceships. These rings are connected to large towers extending from the planets surface.
The anime Mobile Suit Gundam 00 also prominently features an orbital ring, which consists primarily of linked solar panels. The ring is connected to earth via three space elevators. This ring effectively provides near unlimited power to Earth. Later in the series the ring also shows space stations mounted on its surface.
The opening battle of Star Wars: The Clone Wars ' s, Season 6, Episode 1, takes place on the ring-shaped, Ringo Vinda space station, surrounding the planet of Ringo Vinda.
Also in the Star Wars universe, Kuat shipyards, is another orbital ring around the world of Kuat. In Star Wars: Legends, Dac, the homeworld of the Calamari and the Quarrens has a massive orbital shipyard that encircles their oceanic planet.
In the Warhammer 40,000 universe, Mars has a large orbital ring called the Ring of Iron. It is primarily used as a shipyard for interstellar craft. It is the largest man made structure in the galaxy. The planet Medusa also has such a ring, called Telstarax, hailing from the Dark Age of Technology, but it is largely plundered and wrecked.
The game X3 Terran Conflict features a free-floating orbital ring around Earth, which is shattered by an explosion and subsequently de-orbited in X3: Albion Prelude
In the game Xenoblade Chronicles 2 there is a giant tree that has grown around the base of an Orbital Ring.
In Escape Velocity: Nova Earth no longer has a moon orbiting around it because it had been stripped mined for centuries and now exist as an orbital ring around the planet. Over half of it is owned by Sigma Shipyard corporation.
In the game Stellaris , orbital rings can be constructed around colonized planets. They can act as regular space stations or they can boost their planets' production through buildings and modules.
Media related to Orbital ring at Wikimedia Commons | https://en.wikipedia.org/wiki/Orbital_ring |
An orbital stretch wrapper is a means of applying stretchable plastic film to a load, consisting of a roll (or rolls) of stretch wrap supported on a vertical rotating ring and a means of passing a load through the ring's eye horizontally. Several designs are available. [ 1 ] [ 2 ]
The item or load can go through the ring on a conveyor or can be placed into the ring by a pallet truck . Small loads can be suspended within the rotating ring by hand. Stretch is achieved by creating tension between the load and film roll using a brake or gear ratio system. | https://en.wikipedia.org/wiki/Orbital_stretch_wrapper |
The Orbiting Carbon Observatory ( OCO ) was a failed NASA satellite mission intended to provide global space-based observations of atmospheric carbon dioxide ( CO 2 ). The original spacecraft was lost in a launch failure on 24 February 2009, when the payload fairing of the Taurus rocket which was carrying it failed to separate during ascent. [ 3 ] The added mass of the fairing prevented the satellite from reaching orbit . [ 4 ] It subsequently re-entered the atmosphere and crashed into the Indian Ocean near Antarctica . [ 5 ] [ 6 ] The replacement satellite, Orbiting Carbon Observatory-2 , was launched 2 July 2014 aboard a Delta II rocket. [ 7 ] [ 8 ] The Orbiting Carbon Observatory-3 , a stand-alone payload built from the spare OCO-2 flight instrument, was installed on the International Space Station 's Kibō Exposed Facility in May 2019. [ 9 ]
OCO's measurements are designed to be accurate enough to show for the first time the geographic distribution of carbon dioxide sources and sinks on a regional scale. [ 10 ] The data is planned to improve the understanding of the global carbon cycle , the natural processes and human activities that influence the abundance and distribution of the greenhouse gas . This improved understanding is expected to enable more reliable forecasts of future changes in the abundance and distribution of carbon dioxide in the atmosphere and the effect that these changes may have on Earth 's climate .
The OCO spacecraft was provided by Orbital Sciences Corporation . [ 11 ] During its two-year mission, OCO will fly in a near polar orbit which enables the instrument to observe most of Earth's surface at least once every sixteen days. It is intended to fly in loose formation with a series of other Earth-orbiting satellites known as the Earth Observing System Afternoon Constellation, or the A-train . This coordinated flight formation was intended to enable researchers to correlate OCO data with data acquired by other instruments on other spacecraft. In particular, Earth scientists would like to compare OCO data with nearly simultaneous measurements acquired by the Atmospheric Infrared Sounder (AIRS) instrument aboard NASA's Aqua satellite and ground-based data from the Total Carbon Column Observing Network (TCCON). Alignment with the A-train demands a particularly short launch window of 30 seconds. [ 12 ]
The original cost of the mission was US$280 million . [ 13 ] It was sponsored by NASA's Earth System Science Pathfinder Program. [ 14 ] NASA's Jet Propulsion Laboratory in Pasadena, California , manages OCO for NASA's Science Mission Directorate.
The satellite will carry a single instrument designed to make the most precise measurements of atmospheric carbon dioxide ever made from space. The instrument consists of three parallel, high-resolution spectrometers , integrated into a common structure and fed by a common telescope . The spectrometers will make simultaneous measurements of the carbon dioxide and molecular oxygen absorption of sunlight reflected off the same location on Earth's surface when viewed in the near- infrared part of the electromagnetic spectrum , invisible to the human eye.
As sunlight passes through Earth's atmosphere and is reflected from Earth's surface, molecules of atmospheric gases absorb very specific colors of light. If the light is divided into a rainbow of colors, called a spectrum , the specific colors absorbed by each gas appear as dark lines. Different gases absorb different colors, so the pattern of absorption lines provides a telltale spectral "fingerprint" for that molecule. OCO's spectrometers were designed to detect these molecular fingerprints.
Each of the three spectrometers was tuned to measure the absorption in a specific range of colors. Each of these ranges includes dozens of dark absorption lines produced by either carbon dioxide or molecular oxygen. The amount of light absorbed in each spectral line increases with the number of molecules along the optical path. OCO's spectrometers measure the fraction of the light absorbed in each of these lines with very high precision. This information was then to be analyzed to determine the number of molecules along the path between the top of the atmosphere and the surface.
If the amount of carbon dioxide varies from place to place, the amount of absorption will also vary. To resolve these variations, the observatory's instrument will record an image of the spectrum produced by each spectrometer three times every second as the satellite flies over the surface at more than four miles per second. This information would then be transmitted to the ground, where carbon dioxide concentrations would be retrieved in four separate footprints for each image collected. These spatially varying carbon dioxide concentration estimates would then be analyzed using global transport models, like those used for weather prediction, to infer the locations of carbon dioxide sources and sinks. [ 15 ]
The OCO instrument was developed by Hamilton Sundstrand Sensor Systems in Pomona, California , and the Jet Propulsion Laboratory . [ 16 ]
The satellite was originally launched from Vandenberg Air Force Base in California on a dedicated Taurus XL rocket. However, the payload fairing—a clam shell-shaped covering that protects the satellite during launch—apparently failed to separate from the spacecraft. "We have not had a successful launch tonight and will not be able to have a successful OCO mission", NASA commentator George Diller said. [ 17 ]
A payload fairing is a clamshell-shaped cover that encloses and protects a payload on the pad and during early flight. Fairings are a standard component of expendable launch vehicles, and are always jettisoned as soon as possible after a rocket has climbed high enough for heating from air friction to no longer risk damaging the payload. The Taurus XL's fairing was intended to separate several seconds after stage 2 ignition. Its extra mass was not a significant factor during the flight of the larger lower stages, but kept the relatively small stage 3 from adding enough velocity to reach orbit. 17 minutes after liftoff the payload fell into the ocean near Antarctica. [ 18 ] NASA investigators later determined the cause for the launch failure to be faulty materials provided by aluminum manufacturer Sapa Profiles . [ 19 ]
Three days after the failed February 2009 launch, the OCO science team sent NASA headquarters a proposal to build and launch an OCO copy by late 2011. [ 20 ] On 1 February 2010, the FY 2011 NASA budget request did include $170 million for NASA to develop and fly a replacement for the Orbiting Carbon Observatory: OCO-2. [ 21 ]
NASA, in 2010, initially selected Orbital Sciences for launching the replacement in February 2013 on a Taurus XL 3110 from Vandenberg Air Force Base in California. [ 22 ] The launch of the Glory satellite took place on 4 March 2011 and ended in failure, like OCO. Then, in February 2012 both NASA and Orbital Sciences came to an agreement to terminate the launch contract. [ 23 ]
OCO-2 was initially scheduled for launch on 1 July 2014 at 09:56 UTC aboard a Delta II rocket, though that launch was scrubbed at 46 seconds on the countdown clock due to a faulty valve on the water suppression system that is used to flow water on the launch pad to dampen the acoustic energy during launch. The rocket launched 2 July at the same time. [ 7 ]
NASA Launch Services Program (LSP) investigators have determined the technical root cause for the Taurus XL launch failures of NASA's Orbiting Carbon Observatory (OCO) and Glory missions in 2009 and 2011, respectively: faulty materials provided by aluminium manufacturer, Sapa Profiles, Inc. (SPI). LSP's technical investigation led to the involvement of NASA's Office of the Inspector General and the U.S. Department of Justice (DOJ). The efforts of the DOJ, recently made public, resulted in the resolution of criminal charges and alleged civil claims against SPI, and its agreement to pay $46 million to the U.S. government and other commercial customers. This relates to a 19-year scheme that included falsifying thousands of certifications for aluminium extrusions to hundreds of customers. [ 24 ]
On 24 February 2009, a Taurus XL rocket (Taurus T8) carrying NASA's Orbiting Carbon Observatory (OCO) satellite failed to reach orbit. The Taurus T8 mission failed because the payload fairing did not separate during ascent, causing the rocket to not shed weight. As a result of the extra weight, the Taurus rocket failed to reach orbital velocity, resulting in a total loss of the mission. On 4 March 2011, another Taurus rocket (Taurus T9) carrying NASA's Glory scientific satellite failed to reach orbit. The Taurus T9 mission also concluded in a failure of the payload fairing to separate. The Taurus T8 and T9 missions both reentered earth's atmosphere resulting in break-up and/or burnup of the rocket and satellite, and any surviving pieces would have been dispersed in the Pacific Ocean near Antarctica. The combined cost of both mission failures was in excess of $700 million. This document's purpose is to provide a top-level outline of NASA's updated findings pertaining to the cause of both mishaps.
The Taurus T8 and T9 rockets both used 63-inch diameter payload fairings to cover and protect the spacecraft during ground operations and launch. The payload fairing halves are structurally joined and attached to the rocket using frangible joints. A frangible joint is a structural separation system that is initiated using ordnance. Initiation of the ordnance causes the ligament of the frangible joint extrusion to fracture, allowing the two payload fairing halves to be separated and subsequently jettisoned from the Taurus rocket. The frangible joints for T8 and T9 were made and assembled together, at the same time. The T8 and T9 frangible joint extrusions were manufactured by Sapa Profiles, Inc. (SPI) in its Technical Dynamics Aluminum (TDA) plant, in Portland, Oregon. [ 25 ]
This article incorporates public domain material from websites or documents of the National Aeronautics and Space Administration . | https://en.wikipedia.org/wiki/Orbiting_Carbon_Observatory |
Orbiting Carbon Observatory-2 ( OCO-2 ) is an American environmental science satellite which launched on 2 July 2014. A NASA mission, it is a replacement for the Orbiting Carbon Observatory which was lost in a launch failure in 2009. It is the second successful high-precision (better than 0.3%) CO 2 observing satellite , after GOSAT .
The OCO-2 satellite was built by Orbital Sciences Corporation , based around the LEOStar-2 bus. [ 4 ] The spacecraft is being used to study carbon dioxide concentrations and distributions in the atmosphere. [ 5 ]
OCO-2 was ordered after the original OCO spacecraft failed to achieve orbit. During the first satellite's launch atop a Taurus-XL in February 2009, the payload fairing failed to separate from around the spacecraft and the rocket did not have sufficient power to enter orbit with its additional mass. Although a Taurus launch was initially contracted for the reflight, the launch contract was cancelled after the same malfunction occurred on the launch of the Glory satellite two years later. [ 6 ]
United Launch Alliance launched OCO-2 using a Delta II rocket at the beginning of a 30-second launch window at 09:56 UTC (2:56 PDT) on 2 July 2014. Flying in the 7320-10C configuration, the rocket launched from Space Launch Complex 2W at Vandenberg Air Force Base . [ 7 ] The initial launch attempt on 1 July at 09:56:44 UTC was scrubbed at 46 seconds on the countdown clock due to a faulty valve on the water suppression system, used to flow water on the launch pad to dampen the acoustic energy during launch. [ 8 ]
OCO-2 joined the A-train satellite constellation , becoming the sixth satellite in the group. Members of the A-train fly very close together in Sun-synchronous orbit , to make nearly simultaneous measurements of Earth. A particularly short launch window of 30 seconds was necessary to achieve a proper position in the train. [ 9 ] As of 19 September 2016 it was in an orbit with a perigee of 701.10 km (435.64 mi), an apogee of 703.81 km (437.33 mi) and a 98.2° inclination . [ 2 ]
The mission is expected to cost US$467.7 million , including design, development, launch and operations. [ 1 ]
Rather than directly measuring concentrations of carbon dioxide in the atmosphere, OCO-2 records how much of the sunlight reflected off the Earth is absorbed by CO 2 molecules in an air column. [ 10 ] OCO-2 makes measurements in three different spectral bands over four to eight different footprints of approximately 1.29 km × 2.25 km (0.80 mi × 1.40 mi) each. [ 11 ] [ 12 ] About 24 soundings are collected per second while in sunlight and over 10% of these are sufficiently cloud free for further analysis. One spectral band is used for column measurements of oxygen (A-band 0.765 microns), and two are used for column measurements of carbon dioxide (weak band 1.61 microns, strong band 2.06 microns). [ 3 ]
In the retrieval algorithm measurements from the three bands are combined to yield column-averaged dry-air mole fractions of carbon dioxide. Because these are dry-air mole fractions, these measurements do not change with water content or surface pressure. Because the molecular oxygen content of the atmosphere (i.e. excluding the oxygen in water vapour) is well known to be 20.95%, oxygen is used as a measure of the total dry air column. To ensure these measurements are traceable to the World Meteorological Organization , OCO-2 measurements are carefully compared with measurements by the Total Carbon Column Observing Network (TCCON). [ 3 ]
Mission data are provided to the public by the NASA Goddard Earth Science Data and Information Services Center (GES DISC). The Level 1B data product is the least processed and contains records for all collected soundings (about 74,000 soundings per orbit). The Level 2 product contains estimates of the column-averaged dry-air mole fractions of carbon dioxide, among other parameters such as surface albedo and aerosol content. The Level 3 product consists of global maps of carbon dioxide concentrations developed by OCO-2 scientists. [ 13 ]
Media related to Orbiting Carbon Observatory-2 at Wikimedia Commons | https://en.wikipedia.org/wiki/Orbiting_Carbon_Observatory_2 |
The Orbiting Carbon Observatory-3 ( OCO-3 ) is a NASA - JPL instrument designed to measure carbon dioxide in Earth's atmosphere . The instrument is mounted on the Japanese Experiment Module-Exposed Facility on board the International Space Station (ISS). [ 4 ] OCO-3 was scheduled to be transported to space by a SpaceX Dragon from a Falcon 9 rocket on 30 April 2019, [ 5 ] but the launch was delayed to 3 May, due to problems with the space station's electrical power system. [ 6 ] This launch was further delayed to 4 May due to electrical issues aboard Of Course I Still Love You (OCISLY) , the barge used to recover the Falcon 9’s first stage. [ 7 ] OCO-3 was launched as part of CRS-17 on 4 May 2019 at 06:48 UTC. [ 8 ] The nominal mission lifetime is ten years. [ 3 ]
OCO-3 was assembled using spare materials from the Orbiting Carbon Observatory-2 satellite. [ 4 ] Because the OCO-3 instrument is similar to the OCO-2 instrument, it is expected to have similar performance with its measurements used to quantify CO 2 to 1 ppm precision or better at 3 Hz. [ 9 ]
OCO-3 is constructed from spare equipment from the OCO-2 mission. Thus its physical characteristics are similar, but with some adaptations. A 2-axis pointing mirror was added, which will allow targeting of cities and other areas on order of 100 by 100 km (62 by 62 mi) for area mapping (also called "snapshot mode"). [ 3 ] [ 17 ] [ 19 ] A 100 m (330 ft) resolution context camera was also added. [ 17 ] An onboard cryocooler will maintain detector temperatures of around −120 °C (−184 °F). [ 20 ] Entrance optics were modified to maintain a similar ground footprint to OCO-2. [ 3 ]
Similar to OCO and OCO-2, the main measurement will be of reflected near-IR sunlight. Grating spectrometers separate incoming light energy into different components of the electromagnetic spectrum (or wavelengths or "colors"). Because CO 2 and molecular oxygen absorb light at specific wavelengths, the signal or absorption levels at different wavelengths provide information on the amount of gases. [ 20 ] Three bands are used called Weak CO 2 (around 1.6 μm), Strong CO 2 (around 2.0 μm), and Oxygen-A (around 0.76 μm). [ 3 ] There are 1,016 spectral elements per band, and measurements are made simultaneously at 8 side-by-side locations or "footprints" each about 4 km 2 (1.5 sq mi) or smaller, 3 times per second.
Overall measurements from OCO-3 will help quantify sources and sinks of carbon dioxide from terrestrial ecosystems, the oceans, and from anthropogenic sources. Due to the ISS orbit, measurements will be made at latitudes less than 52°. Data from OCO-3 are expected to significantly improve understanding of global emissions from human activities, for example, using measurements over cities. [ 9 ] Near simultaneous observations from other instruments onboard the International Space Station such as ECOSTRESS (measuring plant temperatures) and Global Ecosystem Dynamics Investigation lidar (measuring forest structure) may be combined with OCO-3 observations to help improve the understanding of the terrestrial ecosystem . Similar to OCO-2, OCO-3 will also measure Solar Induced Fluorescence which is a process that occurs during plant photosynthesis . [ 3 ] [ 21 ] | https://en.wikipedia.org/wiki/Orbiting_Carbon_Observatory_3 |
Orcas or killer whales have a cosmopolitan distribution and several distinct populations or types have been documented or suggested. Three to five types of orcas may be distinct enough to be considered different races , [ 1 ] subspecies , or possibly even species [ 2 ] (see species problem ). The IUCN reported in 2008, "The taxonomy of this genus is clearly in need of review, and it is likely that O. orca will be split into a number of different species or at least subspecies over the next few years." [ 3 ] However, large variation in the ecological distinctiveness of different orca groups complicate simple differentiation into types. [ 4 ] Mammal-eating orcas in different regions were long thought likely to be closely related, but genetic testing has refuted this hypothesis. [ 5 ]
Research off the west coast of Canada and the United States in the 1970s and 1980s identified the following three types:
Separate fish-eating and mammal-eating orca communities also exist off the coast of the Russian Far East and Hokkaido , Japan. [ 30 ] [ 31 ] Russian orcas are commonly seen around the Kamchatka Peninsula and Commander Islands . Over 2,000 individual resident-like orcas and 130 transient-like orcas have been identified off Russia. [ 30 ] At least 195 individual orcas have been cataloged in the eastern tropical Pacific, ranging from Baja California and the Gulf of California in the north to the northwest coast of South America in the south and west towards Hawaii. [ 32 ] Orcas appear to regularly occur off the Galápagos Islands . [ 33 ] Orcas sighted in Hawaiian waters may belong to a greater population in the central Pacific. [ 34 ] [ 35 ]
At least 15,000 whales are estimated to inhabit the North Atlantic. [ 36 ] In the Northeast Atlantic, two orca ecotypes have been proposed. [ 37 ] Type 1 orcas consist of seven haplotypes and include herring-eating orcas of Norway and Iceland and mackerel-eating orcas of the North Sea , [ 37 ] as well as seal-eating orcas off Norway. [ 4 ] [ 38 ] Type 2 orcas consist of two haplotypes, [ 37 ] and mainly feed on baleen whales . [ 4 ] [ 37 ] These two types have now been dropped from the classification, because of a lack of samples for type 2 (5 individuals) and how little it was representative of a potential ecotype. [ 39 ]
In the Mediterranean Sea , orcas are considered "visitors", likely from the North Atlantic, and sightings become less frequent further east. [ 40 ] However, a small year-round population exists in the Strait of Gibraltar , which numbered around 39 in 2011. [ 41 ] From 2020, this population started ramming vessels and damaging their rudders . [ 42 ] Distinct populations may also exist off the west coast of tropical Africa, which have generalized diets. [ 43 ]
The northwest Atlantic population is found year-round around Labrador and Newfoundland , while some individuals seasonally travel to the waters of the eastern Canadian Arctic when the ice has melted. [ 44 ] Sightings of these whales have been documented as far south as Cape Cod and Long Island . [ 45 ] This population is possibly continuous with orcas sighted off Greenland. [ 44 ] Orcas are sighted year-round in the Caribbean Sea , [ 46 ] and an estimated 267 (as of 2020) are documented in the northern Gulf of Mexico . [ 47 ]
Recent studies using dietary tracers such as fatty acids and organic contaminants have shown how varied the diet of North Atlantic orcas is. For example, orcas in the Eastern North Atlantic (Norway, Faroe Islands, Iceland) mainly feed on fish, specifically herring. Meanwhile, those in the Central North Atlantic (Greenland) prefer to consume seals such as ringed, harp, hooded, and bearded seals. Finally, orcas in the Western North Atlantic (Eastern Canadian Arctic and Eastern Canada) tend to prey on other whale species, such as belugas and narwhals in the Arctic and baleen whales and porpoises in Eastern Canada. [ 48 ] [ 49 ]
Over 50 individual whales have been cataloged in the northern Indian Ocean, including two individuals that were sighted in the Persian Gulf in 2008 and off Sri Lanka in 2015. [ 50 ]
A small population of orcas seasonally visits the northern point of the Valdes Peninsula on the east coast of Argentina and hunt for sea lions and elephant seals on the shore, temporary stranding themselves. [ 51 ] Similar behaviors occur among orcas off the Crozet Islands , which breach to grab elephant seals. These orcas also prey on Patagonian toothfish . 65 individuals have been documented in this area. [ 52 ] Off South Africa, a distinctive "flat-tooth" morphotype exists and preys on sharks. [ 53 ] [ 54 ] A pair of male orcas, Port and Starboard , have become well known for hunting great whites and other sharks off the South African coast. [ 55 ]
Orcas occur throughout the waters of Australia, New Zealand and Papua New Guinea. They are sighted year round in New Zealand waters, while off Australia, they are seasonally concentrated off the northwest, in the inshore waters of Ningaloo Reef , and the southwest, at the Bremer region . Genetic evidence shows that the orcas of New Zealand and northwest and southwest Australia form three distinct populations. [ 56 ] New Zealand orcas mainly prey on sharks and rays . [ 57 ] [ 58 ]
Around 25,000 orcas are estimated around the Antarctic , [ 59 ] and four types have been documented. Two dwarf species, named Orcinus nanus and Orcinus glacialis , were described during the 1980s by Soviet researchers, but most cetacean researchers are skeptical about their status, and linking these directly to the types described below is difficult. [ 2 ]
Types B and C live close to the ice, and diatoms in these waters may be responsible for the yellowish colouring of both types. [ 2 ] [ 67 ] Mitochondrial DNA sequences support the theory that these are recently diverged separate species. [ 68 ] More recently, complete mitochondrial sequencing indicates the types B and C be recognized as distinct species, as should the North Pacific transients, leaving the others as subspecies pending additional data. [ 69 ] Advanced methods that sequenced the entire mitochondrial genome revealed systematic differences in DNA between different populations. [ 23 ] A 2019 study of Type D orcas also found them to be distinct from other populations and possibly even a unique species. [ 65 ] | https://en.wikipedia.org/wiki/Orca_types_and_populations |
Orcein , also called archil , orchil , lacmus and C.I. Natural Red 28 , is any dye extracted from several species of lichen , commonly known as "orchella weeds", found in various parts of the world. A major source is the archil lichen, Roccella tinctoria . [ 1 ] Orcinol is extracted from such lichens. It is then converted to orcein by ammonia and air. In traditional dye-making methods, urine was used as the ammonia source. If the conversion is carried out in the presence of potassium carbonate , calcium hydroxide , and calcium sulfate (in the form of potash , lime , and gypsum in traditional dye-making methods), the result is litmus , a more complex molecule. [ 2 ] The manufacture was described by Cocq in 1812 [ 3 ] and in the UK in 1874. [ 4 ] Edmund Roberts noted orchilla as a principal export of the Cape Verde islands, superior to the same kind of "moss" found in Italy or the Canary Islands , that in 1832 was yielding an annual revenue of $200,000. [ 5 ] : pp.14, 15 Commercial archil is either a powder (called cudbear) or a paste. It is red in acidic pH and blue in alkaline pH.
The chemical components of orcein were elucidated only in the 1950s by Hans Musso. [ 6 ] The structures are shown below. A paper originally published in 1961, embodying most of Musso's work on components of orcein and litmus, was translated into English and published in 2003 [ 7 ] in a special issue of the journal Biotechnic & Histochemistry (Vol 78, No. 6) devoted to the dye.
Orcein is a reddish-brown dye, orchil is a purple-blue dye. Orcein is also used as a stain in microscopy to visualize chromosomes , [ 8 ] elastic fibers , [ 9 ] Hepatitis B surface antigens, [ 10 ] and copper-associated proteins. [ 11 ]
Orcein is not approved as a food dye (banned in Europe since January 1977), with E number E121 before 1977 and E182 after. [ 12 ] [ 13 ] Its CAS number is 1400-62-0. [ 14 ] Its chemical formula is C 28 H 24 N 2 O 7 . It forms dark brown crystals. It is a mixture of phenoxazone derivates - hydroxyorceins, aminoorceins, and aminoorceinimines.
Cudbear is a dye extracted from orchil lichens that produces colours in the purple range. It can be used to dye wool and silk , without the use of mordant . The lichen is first boiled in a solution of ammonium carbonate . The mixture is then cooled and ammonia is added and the mixture is kept damp for 3–4 weeks. Then the lichen is dried and ground to powder.
Cudbear was the first dye to be invented in modern times, and one of the few dyes to be credited to a named individual: Dr Cuthbert Gordon of Scotland : production began in 1758, and it was patented in 1758, British patent 727. [ 15 ] [ 16 ] John Glassford invested in the new process with funds from his slave-labor tobacco business by establishing a dyeworks in Dennistoun in 1777. [ 17 ] [ 18 ] The manufacture details were carefully protected, with a ten-feet high wall being built around the manufacturing facility, and staff consisting of Highlanders sworn to secrecy. [ 17 ] The lichen consumption soon reached 250 tons per year and import from Norway and Sweden had to be arranged. [ 19 ]
A similar process was developed in France. The lichen is extracted by urine or ammonia, then the extract is acidified, the dissolved dye precipitates out and is washed. Then it is dissolved in ammonia again, the solution is heated in air until it becomes purple, then it is precipitated out with calcium chloride . The resulting insoluble purple solid is known as French purple , a fast lichen dye that was much more stable than other lichen dyes. | https://en.wikipedia.org/wiki/Orcein |
In discrete geometry , the original orchard-planting problem (or the tree-planting problem ) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane . There are also investigations into how many k -point lines there can be. Hallard T. Croft and Paul Erdős proved t k > c n 2 k 3 , {\displaystyle t_{k}>{\frac {cn^{2}}{k^{3}}},} where n is the number of points and t k is the number of k -point lines. [ 1 ] Their construction contains some m -point lines, where m > k . One can also ask the question if these are not allowed.
Define t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} to be the maximum number of 3-point lines attainable with a configuration of n points.
For an arbitrary number of n points, t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} was shown to be 1 6 n 2 − O ( n ) {\displaystyle {\tfrac {1}{6}}n^{2}-O(n)} in 1974.
The first few values of t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} are given in the following table (sequence A003035 in the OEIS ).
Since no two lines may share two distinct points, a trivial upper-bound for the number of 3-point lines determined by n points is ⌊ ( n 2 ) / ( 3 2 ) ⌋ = ⌊ n 2 − n 6 ⌋ . {\displaystyle \left\lfloor {\binom {n}{2}}{\Big /}{\binom {3}{2}}\right\rfloor =\left\lfloor {\frac {n^{2}-n}{6}}\right\rfloor .} Using the fact that the number of 2-point lines is at least 6 n 13 {\displaystyle {\tfrac {6n}{13}}} ( Csima & Sawyer 1993 ), this upper bound can be lowered to ⌊ ( n 2 ) − 6 n 13 3 ⌋ = ⌊ n 2 6 − 25 n 78 ⌋ . {\displaystyle \left\lfloor {\frac {{\binom {n}{2}}-{\frac {6n}{13}}}{3}}\right\rfloor =\left\lfloor {\frac {n^{2}}{6}}-{\frac {25n}{78}}\right\rfloor .}
Lower bounds for t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ≈ 1 8 n 2 {\displaystyle \approx {\tfrac {1}{8}}n^{2}} was given by Sylvester , who placed n points on the cubic curve y = x 3 . This was improved to 1 6 n 2 − 1 2 n + 1 {\displaystyle {\tfrac {1}{6}}n^{2}-{\tfrac {1}{2}}n+1} in 1974 by Burr , Grünbaum , and Sloane ( 1974 ), using a construction based on Weierstrass's elliptic functions . An elementary construction using hypocycloids was found by Füredi & Palásti (1984) achieving the same lower bound.
In September 2013, Ben Green and Terence Tao published a paper in which they prove that for all point sets of sufficient size, n > n 0 , there are at most n ( n − 3 ) 6 + 1 = 1 6 n 2 − 1 2 n + 1 {\displaystyle {\frac {n(n-3)}{6}}+1={\frac {1}{6}}n^{2}-{\frac {1}{2}}n+1} 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane. [ 2 ] Thus, for sufficiently large n , the exact value of t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} is known.
This is slightly better than the bound that would directly follow from their tight lower bound of n 2 {\displaystyle {\tfrac {n}{2}}} for the number of 2-point lines : n ( n − 2 ) 6 , {\displaystyle {\tfrac {n(n-2)}{6}},} proved in the same paper and solving a 1951 problem posed independently by Gabriel Andrew Dirac and Theodore Motzkin .
Orchard-planting problem has also been considered over finite fields. In this version of the problem, the n points lie in a projective plane defined over a finite field. ( Padmanabhan & Shukla 2020 ). | https://en.wikipedia.org/wiki/Orchard-planting_problem |
Orchestra Control Engine is a suite of software components (based on Linux / RTAI ) used for the planning, development and deployment of real-time control applications for industrial machines and robots.
Orchestra Control Engine has been developed by Sintesi SpA in partnership with the Italian National Research Council and in collaboration with international industrial companies in the field of robotics and production systems.
Sintesi SpA is a company that develops mechatronic components and solutions. It has specialized in measurement, control and design technologies for robotics and production systems.
Orchestra Control Engine is flexible because it can be customized. This is done visually. The solutions created are open (based on an open source framework) and are extendible. Modular components of the software allow a user to develop, debug and test control applications. For example, previously developed algorithms can be divided into functional units and reused indefinitely. All the units work together. The software can be distributed among various remote hardware devices which may be hundreds of meters apart. It also scalable in that it selects hardware which provides the best cost and performance for a particular operation. The system's parameters can be quickly reconfigured both on line and also at the time of a run.
Linux / RTAI creates Orchestra Control Engine's hard real time behaviour. Its "open source" characteristics allow changes to fit the users' requirements. Non hard real time components of orchestra Control Engine can be used with non-Linux platforms such as Microsoft Windows or Macintosh .
A hard real time multithreaded engine operates in multicore/multiprocessor architectures. Within the scheme, modules can be filled in with more or less complex algorithms which control the process. The run time engine loads the modules. The user can adapt the modules to the topology. For complex topology, multiple modules can be used or parallel loops can be implemented.
The run time manager controls the formalities of execution of the program; decides priorities within the operation; and manages the multi-thread and multiprocessor operations. It is made up of templates that define thread typologies according to the formalities of execution and from a part that manages the POU (Program Organization Unit).
The logic programming of Orchestra Control Engine assists in the use of the five contemplated languages of the IEC 61131 norm. It also assists in the use of the C/C++ language.
The path programming of Orchestra Control Engine assists in the writing of movement and workmanship mechanics. Piece manufacturing programs (part programs) can be edited according to the international ISO-DIN 60025 standard and the American EIA RS274 D standard. It is also important for the interpretation of modules and in turn for the input which allows a Motion Control Loop .
The designer is a Java IDE. It assists development of motion control applications for different environments. This involves completing new modules, using code templates, allowing the adding and shaping of new blocks and testing the modules both independently and in a control scheme. It also automatically provides XML configuration files for each module and for the control loop.
The builder is a software tool that allows Simulink models to be automatically generated into Orchestra core compatible modules. It does this by making a definition for every parameter of the Simulink model. It can generate a function which initializes the loading of a newly developed control system and, it can generate the step function which holds the code for the logic of each module.
HMI is a Java application (therefore a cross-platform one), that looks for and interacts with different parts of a control system. Orchestra HMI has a graphic interface (including a touch screen) which can run on any common PC. It can be customised to suit the user and provides user authentication. Orchestra HMI allows the user to CN configure and plan the production island and command processes such as the starting a motion program. The user can screen and edit processes. Orchestra HMI provides the visualization of signals coming from an OrchestraCore or an Orchestra Run Time Manager by means of graphic controls (indicators, 2D plots, LCD displays) and the 3D visualization of machines and anthropomorphous manipulators.
The library contains sets of modules, information from sensors, interfaces with external entities such as machines, robots, sensors and DAQ boards.
Orchestra Control Engine is a suite of programs. Using the various components in combination allows for flexibility. d
The motion control framework allows users to develop motion control applications by integrating the best modules for their purpose. The modules may be ones already available or those the user develops using the orchestra designer and builder facilities.
The modules can be run so that the process has multiple threads. Parallelisms are identified and thus algorithms are refined. The modules can be "debugged" as they are completed if specific verifications are programmed. Alternatively, the modules can be completed in "release" mode if no special verifications are required.
The modules be completed with any number of entries, parameters, states and vectorial output in double precision floating point, as well as states of any other type. These characteristics are codified through XML files.
Orchestra MultiPLC (multi programmable logic controller ) is composed of Orchestra Run Time Manager, Orchestra Logic Programming and OrchestraHMI. It allows the execution of a motion control application as one or more programs or functional blocks which may be reused. The controller's open schema accepts and translates XML files. The functional blocks can be prioritised within a series or programmed to operate periodically. New tasks may be added to the application.
Orchestra Full for Numerical Control consists of Orchestra Motion Control Framework, OrchestraMulti PLC, and some other specific components:
OrchestraGCode interprets the G-code program received by the HMI: if the G-code instruction is one of motion, then it is sent to the MotionSupervisor, if not, OrchestraGCode will write the instruction to the appropriate software.
MotionSupervisor acts as an interface between the Motion Control Loop, the Orchestra GCode, the ControllerSupervisor and the Logical Control Loop. Using information from the ControllerSupervisor, it selects either automatic or jog mode. In jog mode, MotionSupervisor provides axes to moves, direction and feed rates. In automatic and in semiautomatic mode, instructions on movement will come from the G-Code interpreter.
The MotionSupervisor also collects error messages coming from the MotionControl Loop and sends them to the ControllerSupervisor.
ControllerSupervisor centralizes all the information related to Orchestra Control Engine. It receives information from the HMI, the teach pendant and other software components. Such information is sorted to the other components even if direct channels of communication among the various components for the specific information interchange are foreseen.
ControllerSupervisor sends error messages to OrchestraHMI. Local errors are handled in the software components in which they take place. Errors beyond the local level are handled by the ControllerSupervisor instigating a safety procedure and or showing the error to the user.
Orchestra for Open Robot Controllers allows the feasibility of innovative industrial robot algorithms to be tested. It can integrate advanced sensors and functions. Its interface with a personal computer is via OrchestraCore. Its function is generally one of realization of movement rather than the logic of control and the generation of trajectory.
Orchestra Control Engine | https://en.wikipedia.org/wiki/Orchestra_Control_Engine |
Orchestrated objective reduction ( Orch OR ) is a theory postulating that consciousness originates at the quantum level inside neurons (rather than being a product of neural connections ). The mechanism is held to be a quantum process called objective reduction that is orchestrated by cellular structures called microtubules . It is proposed that the theory may answer the hard problem of consciousness and provide a mechanism for free will . [ 1 ] The hypothesis was first put forward in the early 1990s by Nobel laureate for physics Roger Penrose , and anaesthesiologist Stuart Hameroff . The hypothesis combines approaches from molecular biology , neuroscience , pharmacology , philosophy , quantum information theory , and quantum gravity . [ 2 ] [ 3 ]
While some other theories assert that consciousness emerges as the complexity of the computations performed by cerebral neurons increases, [ 4 ] [ 5 ] Orch OR posits that consciousness is based on non-computable quantum processing performed by qubits formed collectively on cellular microtubules, a process significantly amplified in the neurons. The qubits are based on oscillating dipoles forming superposed resonance rings in helical pathways throughout lattices of microtubules. The oscillations are either electric, due to charge separation from London forces , or magnetic, due to electron spin —and possibly also due to nuclear spins (that can remain isolated for longer periods) that occur in gigahertz , megahertz and kilohertz frequency ranges. [ 2 ] [ 6 ] Orchestration refers to the hypothetical process by which connective proteins, such as microtubule-associated proteins (MAPs), influence or orchestrate qubit state reduction by modifying the spacetime-separation of their superimposed states. [ 7 ] The latter is based on Penrose's objective-collapse theory for interpreting quantum mechanics, which postulates the existence of an objective threshold governing the collapse of quantum states , related to the difference of the spacetime curvature of these states in the universe's fine-scale structure. [ 8 ]
Orchestrated objective reduction has been criticized from its inception by mathematicians, philosophers, [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] and scientists. [ 14 ] [ 15 ] [ 16 ] The criticism concentrated on three issues: Penrose's interpretation of Gödel's theorem ; Penrose's abductive reasoning linking non-computability to quantum events; and the brain's unsuitability to host the quantum phenomena required by the theory, since it is considered too "warm, wet and noisy" to avoid decoherence .
In 1931, mathematician and logician Kurt Gödel proved that any effectively generated theory capable of proving basic arithmetic cannot be both consistent and complete . In other words, a mathematically sound theory lacks the means to prove itself. [ 17 ] In his first book concerning consciousness, The Emperor's New Mind (1989), Roger Penrose argued that equivalent statements to "Gödel-type propositions" had recently been put forward. [ 18 ]
Partially in response to Gödel's argument, the Penrose–Lucas argument leaves the question of the physical basis of non- computable behaviour open. Most physical laws are computable, and thus algorithmic. However, Penrose determined that wave function collapse was a prime candidate for a non-computable process. In quantum mechanics , particles are treated differently from the objects of classical mechanics . Particles are described by wave functions that evolve according to the Schrödinger equation . Non-stationary wave functions are linear combinations of the eigenstates of the system, a phenomenon described by the superposition principle . When a quantum system interacts with a classical system—i.e. when an observable is measured—the system appears to collapse to a random eigenstate of that observable from a classical vantage point.
If collapse is truly random, then no process or algorithm can deterministically predict its outcome. This provided Penrose with a candidate for the physical basis of the non-computable process that he hypothesized to exist in the brain. However, he disliked the random nature of environmentally induced collapse, as randomness was not a promising basis for mathematical understanding. Penrose proposed that isolated systems may still undergo a new form of wave function collapse, which he called objective reduction (OR). [ 7 ]
Penrose sought to reconcile general relativity and quantum theory using his own ideas about the possible structure of spacetime . [ 18 ] [ page needed ] [ 19 ] He suggested that at the Planck scale curved spacetime is not continuous, but discrete. He further postulated that each separated quantum superposition has its own piece of spacetime curvature , a blister in spacetime. Penrose suggests that gravity exerts a force on these spacetime blisters, which become unstable above the Planck scale of 10 − 35 m {\displaystyle 10^{-35}{\text{m}}} and collapse to just one of the possible states. The rough threshold for OR is given by Penrose's indeterminacy principle:
Thus, the greater the mass–energy of the object, the faster it will undergo OR and vice versa. Mesoscopic objects could collapse on a timescale relevant to neural processing. [ 7 ] [ additional citation(s) needed ]
An essential feature of Penrose's theory is that the choice of states when objective reduction occurs is selected neither randomly (as are choices following wave function collapse) nor algorithmically. Rather, states are selected by a "non-computable" influence embedded in the Planck scale of spacetime geometry. Penrose claimed that such information is Platonic , representing pure mathematical truths, which relates to Penrose's ideas concerning the three worlds: the physical, the mental, and the Platonic mathematical world. In Shadows of the Mind (1994), Penrose briefly indicates that this Platonic world could also include aesthetic and ethical values, but he does not commit to this further hypothesis. [ 19 ]
The Penrose–Lucas argument was criticized by mathematicians, [ 20 ] [ 21 ] [ 22 ] computer scientists, [ 12 ] and philosophers, [ 23 ] [ 24 ] [ 9 ] [ 10 ] [ 11 ] and the consensus among experts in these fields is that the argument fails, [ 25 ] [ 26 ] [ 27 ] with different authors attacking different aspects of the argument. [ 27 ] [ 28 ] Minsky argued that because humans can believe false ideas to be true, human mathematical understanding need not be consistent and consciousness may easily have a deterministic basis. [ 29 ] Feferman argued that mathematicians do not progress by mechanistic search through proofs, but by trial-and-error reasoning, insight and inspiration, and that machines do not share this approach with humans. [ 21 ]
Penrose outlined a predecessor to Orch OR in The Emperor's New Mind , coming to the problem from a mathematical viewpoint and in particular Gödel's theorem, but lacked a detailed proposal for how quantum processes could be implemented in the brain. Stuart Hameroff separately worked in cancer research and anesthesia , which gave him an interest in brain processes. Hameroff read Penrose's book and suggested to him that microtubules within neurons were suitable candidate sites for quantum processing, and ultimately for consciousness. [ 30 ] [ 31 ] Throughout the 1990s, the two collaborated on the Orch OR theory, which Penrose published in Shadows of the Mind (1994). [ 19 ]
Hameroff's contribution to the theory derived from his study of the neural cytoskeleton , and particularly on microtubules. [ 31 ] As neuroscience has progressed, the role of the cytoskeleton and microtubules has assumed greater importance. In addition to providing structural support, microtubule functions include axoplasmic transport and control of the cell's movement, growth and shape. [ 31 ]
Orch OR combines the Penrose–Lucas argument with Hameroff's hypothesis on quantum processing in microtubules. It proposes that when condensates in the brain undergo an objective wave function reduction, their collapse connects noncomputational decision-making to experiences embedded in spacetime's fundamental geometry. The theory further proposes that the microtubules both influence and are influenced by the conventional activity at the synapses between neurons.
Hameroff proposed that microtubules were suitable candidates for quantum processing. [ 31 ] Microtubules are made up of tubulin protein subunits. The tubulin protein dimers of the microtubules have hydrophobic pockets that may contain delocalized π electrons . Tubulin has other, smaller non-polar regions, for example 8 tryptophans per tubulin, which contain π electron-rich indole rings distributed throughout tubulin with separations of roughly 2 nm. Hameroff claims that this is close enough for the tubulin π electrons to become quantum entangled . [ 32 ] During entanglement, particle states become inseparably correlated.
Hameroff originally suggested in the fringe Journal of Cosmology that the tubulin-subunit electrons would form a Bose–Einstein condensate . [ 33 ] He then proposed a Frohlich condensate , a hypothetical coherent oscillation of dipolar molecules. However, this too was rejected by Reimers's group. [ 34 ] Hameroff and Penrose contested the conclusion, considering that Reimers's microtubule model was oversimplified. [ 35 ]
Hameroff then proposed that condensates in microtubules in one neuron can link with microtubule condensates in other neurons and glial cells via the gap junctions of electrical synapses . [ 36 ] [ 37 ] Hameroff proposed that the gap between the cells is sufficiently small that quantum objects can tunnel across it, allowing them to extend across a large area of the brain. He further postulated that the action of this large-scale quantum activity is the source of 40 Hz gamma waves , building upon the much less controversial theory that gap junctions are related to the gamma oscillation. [ 38 ]
In a study Hameroff was part of, Jack Tuszyński of the University of Alberta demonstrated that anesthetics hasten the duration of a process called delayed luminescence, in which microtubules and tubulins re-emit trapped light. Tuszyński suspects that the phenomenon has a quantum origin, with superradiance being investigated as one possibility (in a 2024 study, superradiance was confirmed to occur in networks of tryptophans , which are found in microtubules). [ 39 ] [ 40 ] Tuszyński told New Scientist that "We're not at the level of interpreting this physiologically, saying 'Yeah, this is where consciousness begins,' but it may." [ 41 ]
The 2024 study, called Ultraviolet Superradiance from Mega-Networks of Tryptophan in Biological Architectures and published in The Journal of Physical Chemistry , confirmed superradiance in networks of tryptophans. [ 39 ] [ 40 ] Large networks of tryptophans are a warm and noisy environment, in which quantum effects typically are not expected to take place. [ 39 ] The results of the study were theoretically predicted and then experimentally confirmed by the researchers. [ 39 ] [ 40 ] Majed Chergui , who led the experimental team, stated that "It's a beautiful result. It took very precise and careful application of standard protein spectroscopy methods, but guided by the theoretical predictions of our collaborators, we were able to confirm a stunning signature of superradiance in a micron-scale biological system." [ 39 ] Marlan Scully , a physicist well-known for his work in the field of theoretical quantum optics, said "We will certainly be examining closely the implications for quantum effects in living systems for years to come." [ 39 ] The study states that "by analyzing the coupling with the electromagnetic field of mega-networks of tryptophans present in these biologically relevant architectures, we find the emergence of collective quantum optical effects, namely, superradiant and subradiant eigenmodes. ... our work demonstrates that collective and cooperative UV excitations in mega-networks of tryptophans support robust quantum states in protein aggregates, with observed consequences even under thermal equilibrium conditions." [ 40 ]
In an experiment, Gregory D. Scholes and Aarat Kalra of Princeton University used lasers to excite molecules within tubulins, causing a prolonged excitation to diffuse through microtubules farther than expected, which did not occur when repeated under anesthesia. [ 42 ] However, diffusion results have to be interpreted carefully, since even classical diffusion can be very complex due to the wide range of length scales in the fluid filled extracellular space. [ 43 ]
At high concentrations (~5 MAC ) the anesthetic gas halothane causes reversible depolymerization of microtubules. [ 44 ] This cannot be the mechanism of anesthetic action, however, because human anesthesia is performed at 1 MAC . (Neither Penrose or Hameroff claim that depolymerization is the mechanism of action for Orch OR.) [ 45 ] [ 46 ] At ~1 MAC halothane, reported minor changes in tubulin protein expression (~1.3-fold) in primary cortical neurons after exposure to halothane and isoflurane are not evidence that tubulin directly interacts with general anesthetics, but rather shows that the proteins controlling tubulin production are possible anesthetic targets. [ 47 ] Further proteomic study reports 0.5 mM [ 14 C]halothane binding to tubulin monomers alongside three dozens of other proteins. [ 48 ] In addition, modulation of microtubule stability has been reported during anthracene general anesthesia of tadpoles. [ 49 ] The study called Direct Modulation of Microtubule Stability Contributes to Anthracene General Anesthesia claims to provide "strong evidence that destabilization of neuronal microtubules provides a path to achieving general anesthesia". [ 49 ]
A highly disputed theory put forth in the mid-1990s by Hameroff and Penrose posits that consciousness is based on quantum vibrations in tubulin/microtubules inside brain neurons. Computer modeling of tubulin's atomic structure [ 50 ] found that anesthetic gas molecules bind adjacent to amino acid aromatic rings of non-polar π-electrons and that collective quantum dipole oscillations among all π-electron resonance rings in each tubulin showed a spectrum with a common mode peak at 613 T Hz . [ 51 ] Simulated presence of 8 different anesthetic gases abolished the 613 THz peak, whereas the presence of 2 different nonanesthetic gases did not affect the 613 THz peak, from which it was speculated that this 613 THz peak in microtubules could be related to consciousness and anesthetic action. [ 51 ]
Another study that Hameroff was a part of claims to show "anesthetic molecules can impair π-resonance energy transfer and exciton hopping in 'quantum channels' of tryptophan rings in tubulin, and thus account for selective action of anesthetics on consciousness and memory". [ 52 ]
In a study published in August 2024, an undergraduate group led by a Wellesley College professor found that rats given epothilone B , a drug that binds to microtubules, took over a minute longer to fall unconscious when exposed to an anesthetic gas. [ 53 ]
Orch OR has been criticized both by physicists [ 14 ] [ 54 ] [ 34 ] [ 55 ] [ 56 ] and neuroscientists [ 57 ] [ 58 ] [ 59 ] who consider it to be a poor model of brain physiology . Orch OR has also been criticized for lacking explanatory power ; the philosopher Patricia Churchland wrote, "Pixie dust in the synapses is about as explanatorily powerful as quantum coherence in the microtubules." [ 60 ]
David Chalmers argues against quantum consciousness. He instead discusses how quantum mechanics may relate to dualistic consciousness . [ 61 ] Chalmers is skeptical that any new physics can resolve the hard problem of consciousness . [ 62 ] [ 63 ] [ 64 ] He argues that quantum theories of consciousness suffer from the same weakness as more conventional theories. Just as he argues that there is no particular reason why particular macroscopic physical features in the brain should give rise to consciousness, he also thinks that there is no particular reason why a particular quantum feature, such as the EM field in the brain, should give rise to consciousness either. [ 64 ]
In 2000 Max Tegmark claimed that any quantum coherent system in the brain would undergo effective wave function collapse due to environmental interaction long before it could influence neural processes (the "warm, wet and noisy" argument, as it later came to be known). [ 14 ] He determined the decoherence timescale of microtubule entanglement at brain temperatures to be on the order of femtoseconds , far too brief for neural processing. Christof Koch and Klaus Hepp also agreed that quantum coherence does not play, or does not need to play any major role in neurophysiology . [ 15 ] [ 16 ] Koch and Hepp concluded that "The empirical demonstration of slowly decoherent and controllable quantum bits in neurons connected by electrical or chemical synapses, or the discovery of an efficient quantum algorithm for computations performed by the brain, would do much to bring these speculations from the 'far-out' to the mere 'very unlikely'." [ 15 ]
In response to Tegmark's claims, Hagan, Tuszynski and Hameroff claimed that Tegmark did not address the Orch OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, although still far below 25 ms. Hameroff's group also suggested that the Debye layer of counterions could screen thermal fluctuations, and that the surrounding actin gel might enhance the ordering of water, further screening noise. They also suggested that incoherent metabolic energy could further order water, and finally that the configuration of the microtubule lattice might be suitable for quantum error correction , a means of resisting quantum decoherence. [ 65 ] [ 66 ]
In 2009, Reimers et al. and McKemmish et al., published critical assessments. Earlier versions of the theory had required tubulin-electrons to form either Bose–Einsteins or Frohlich condensates, and the Reimers group noted the lack of empirical evidence that such could occur. Additionally they calculated that microtubules could only support weak 8 MHz coherence. McKemmish et al. argued that aromatic molecules cannot switch states because they are delocalised; and that changes in tubulin protein-conformation driven by GTP conversion would result in a prohibitive energy requirement. [ 54 ] [ 34 ] [ 55 ]
In 2022, a group of Italian physicists conducted several experiments that failed to provide evidence in support of a gravity-related quantum collapse model of consciousness, weakening the possibility of a quantum explanation for consciousness. [ 67 ] [ 68 ]
Biology-based criticisms have been offered, including a lack of explanation for the probabilistic release of neurotransmitter from presynaptic axon terminals [ 69 ] [ 70 ] [ 71 ] and an error in the calculated number of the tubulin dimers per cortical neuron. [ 72 ]
In 2014, Penrose and Hameroff published responses to some criticisms and revisions to many of the theory's peripheral assumptions, while retaining the core hypothesis. [ 2 ] [ 6 ] | https://en.wikipedia.org/wiki/Orchestrated_objective_reduction |
Orchid mycorrhizae are endomycorrhizal fungi which develop symbiotic relationships with the roots and seeds of plants of the family Orchidaceae . Nearly all orchids are myco-heterotrophic at some point in their life cycle . Orchid mycorrhizae are critically important during orchid germination , as an orchid seed has virtually no energy reserve and obtains its carbon from the fungal symbiont . [ 1 ] [ 2 ]
The symbiosis starts with a structure called a protocorm . [ 3 ] During the symbiosis, the fungus develops structures called pelotons within the root cortex of the orchid. [ 4 ]
Many adult orchids retain their fungal symbionts throughout their life, although the benefits to the adult photosynthetic orchid and the fungus remain largely unexplained.
Orchids have several life stages. The first stage is the non-germinated orchid seed , the next stage is the protocorm , and the following stage is the adult orchid. Orchid seeds are very small (0.35mm to 1.50mm long), spindle-shaped, and have an opening at the pointed end. [ 5 ] Each seed has an embryo that is undifferentiated and lacks root and shoot meristems . [ 3 ] An orchid seed does not have enough nutritional support to grow on its own, and lacks endosperm . [ 2 ] Instead, it gets nutrients needed for germination from fungal symbionts in natural habitats. [ 4 ]
When an orchid seed germinates it forms an intermediate structure called protocorm, young plants which have germinated but lack leaves and which consist mainly of parenchyma cells. [ 6 ] [ 3 ] Infected protocorms tend to develop an active meristem within a few days. [ 7 ] In the adult stage, many orchids have a small amount of thick unbranched roots which results in a root system with a small surface area that is favorable to potentially mycotrophic tissue. [ 8 ]
Orchids lacking chlorophyll, called achlorophyllous mycoheterotrophs , will retain their fungal symbionts their entire lives, relying on the fungus for carbon. [ 9 ]
The debate over whether fungal symbiosis is necessary for the orchid is an old one, as Noël Bernard first proposed orchid symbiosis in 1899. In 1922 the American botanist Lewis Knudson discovered that orchid seeds could be grown on agar and fungal sugars without mycorrhizae, however modern research has found that the germination of orchids may be more successful with specific fungi. [ 10 ]
Although epiphytic orchids grow on other plants they may produce chlorophyll in their leaves, stems, and roots. As with all orchids, germination and the development into protocorms is reliant upon fungal symbionts, which decrease the time of germination and increase the vigor of the protocorm . [ 10 ] The reliance of orchids on specific fungi has been widely studied, and the populations of certain fungi which are present in the soil have proved to be of greater importance in seed germination than the orchid's proximity to older plants or their geographical location, as previously assumed. [ 11 ]
Fungi can enter at various orchid life stages. Fungal hyphae can penetrate the parenchyma cells of germinated orchid seeds, protocorms, late-staged seedlings, or adult plant roots. [ 12 ] The fungal hyphae that enter the orchid have many mitochondria and few vacuoles., [ 13 ] thus increasing their metabolic capacity when paired with an accepting symbiote. In the protocorm stage hyphae enter the chalazal (top) end of the embryo, [ 14 ] however in terrestrial orchids fungal entry into adult plant roots happens mainly through root hair tips which then take on a distorted shape. [ 2 ] The symbiosis is typically maintained throughout the lifetime of the orchid because they depend on the fungus for nutrients, sugars and minerals. [ 15 ] However, some orchids have been found to switch fungal partners during extreme conditions. [ 16 ]
Shortly after the fungus enters an orchid, the fungus produces intracellular hyphal coils called pelotons in the embryos of developing seedlings and the roots of adult plants. [ 4 ] The formation of pelotons in root cortical cells is a defining anatomical structure in orchid mycorrhiza that differentiate it from other forms of fungi. [ 17 ] The pelotons can range in size and in the arrangement and density packaging of their hyphae. [ 7 ] Pelotons are separated from the orchid's cytoplasm by an interfacial matrix and the orchid's plasma membrane. [ 18 ] The material that makes up the interfacial matrix can differ depending on the orchid mychorrizal stage of interaction. [ 19 ] Orchid cells with degenerating pelotons lack starch grains, whereas the newly invaded orchid cells contain large starch grains, suggesting the hydrolysis of starch resulting from the fungal colonization. [ 13 ] There is an enlargement of the nucleus in infected orchid cortical cells and in non-infected cortical cells near an infected area as a result of increased DNA content. [ 20 ] The increased DNA content has been correlated with the differentiation of parenchymal cells suggesting its role in orchid growth. [ 21 ] As pelotons of live fungal hyphae age and are eventually disintegrated, or lysed, they appear as brown or yellow clumps in the orchid cells. [ 6 ] The disintegrated pelotons are an area of considerable interest in current research. The disintegrated pelotons first experience a collapse where orchid microtubules surround the pelotons, which may be the mechanism behind the peloton collapse by producing physiological and structural changes of the hyphae. [ 18 ] The cortical cells of older roots tend to have more lysed pelotons than young pelotons. [ 2 ] Although pelotons are lysed, new pelotons continue to be formed, which indicates a high amount of new hyphal activity. [ 8 ]
Basidiomycetous Rhizoctonia Fungi
The fungi that form orchid mycorrhizae are typically basidiomycetes . These fungi come from a range of taxa including Ceratobasidium ( Rhizoctonia ), Sebacina , Tulasnella and Russula species. Most orchids associate with saprotrophic or pathogenic fungi, while a few associate with ectomycorrhizal fungal species. These latter associations are often called tripartite associations as they involve the orchid, the ectomycorrhizal fungus and its photosynthetic host plant. [ 22 ] [ 23 ] Some of the challenges in determining host-specificity in orchid mycorrhizae have been the methods of identifying the orchid-specific fungi from other free living fungal species in wild-sourced samples. Even with modern molecular analysis and genomic databases, this can still prove difficult, [ 24 ] partially due to the difficulty in culturing fungi from protocorms and identification of fungal samples, [ 24 ] as well as changes in evolving rDNA. [ 25 ] However it has become clearer that different fungi may associate with orchids at specific stages, whether at germination, protocorm development, or throughout the orchid's life. [ 26 ] The types of orchids and their symbiotic fungi also vary depending on the environmental niches they occupy, whether terrestrial or growing on other plants as an epiphyte . [ 27 ]
Ectomycorrhizal Basidiomycota
A number of other basidiomycetous fungi have been documented in various orchids that do not fall under the Rhizoctonia umbrella. In particular, many fully mycoheterotrophic orchids associate with ectomycorrhizal basidiomycetes belonging to genera such as Thelephora , Tomentella and Russula .
Basidiomycetes of the Atractiellales
Ascomycota
Though rare in orchids, ascomycete associations have been documented in several orchid species. The European terrestrial orchid Epipactis helleborine has a specific association with ectomycorrhizal ascomycetes in the Tuberaceae.
Orchids mycorrhiza (OM) are found in approximately 10% of the botanical diversity of earth and have unique and specialized mycorrhizal nutrient transfer interactions which define the fitness and diversity of the orchid family. [ 28 ] Orchid mycorrhizal associations involve a plethora of distinctive nutrient transport systems, structures and phenomena which have only been observed in the family Orchidaceae. These interactions are formed between basidiomycete fungi and all Orchidaceae species. [ 29 ] The basidiomycete fungi most commonly found associated with orchids, rhizoctonia, are known for their saprophytic abilities making this fungi orchid association anomalous, allowing both the plant and fungi to access a source of carbon within the mycorrhizal association not available in arbuscular mycorrhiza. [ 30 ] [ 29 ] The way and degree to which different orchid species exploit these interactions varies. [ 31 ] Orchid mycorrhizal interactions can range from wholly parasitic on the fungal partner, to a mutualistic interaction involving bidirectional nutrient transfer between the plant and mycorrhizal fungus. [ 32 ] [ 33 ] Orchid plants have an obligatory parasitic life stage at germination where all of their nutrients must be supplied by a fungus. [ 34 ] After germination, the orchid mycorrhizal interactions will become specialized to utilize the carbon and nutrients available in the environment surrounding the interaction. These associations are often thought to be dictated by the plant. [ 33 ] It has been indicated in past studies that orchid plant individuals which inhabit dense highly shaded forests may depend significantly more on their fungal partner for carbon, varying within and between species, demonstrating the variable and reactive nature of these interaction. [ 28 ] [ 35 ]
At infection of an orchid by a mycorrhizal fungus both partners are altered considerably to allow for nutrient transfer and symbiosis. Nutrient transfer mechanisms and the symbiotic mycorrhizal peloton organs start to appear only shortly after infection around 20–36 hours after initial contact. [ 28 ] There is significant genetic upregulation and downregulation of many different genes to facilitate the creation of the symbiotic organ, and the pathways with which nutrients travel. As the fungus enters the parenchyma cells of the orchid the plasma membrane invaginates to facilitate fungal infection and growth. [ 36 ] This newly invaginated plasma membrane surrounds the growing pelotons and creates a huge surface area from which nutrients can be exchanged. The pelotons of orchid mycorrhiza are intensely coiled dense fungal hyphae that are often more extensive in comparison to endomycorrhizal structures of arbuscular mycorrhiza. [ 37 ] [ 28 ] [ 38 ] The surrounding plant membrane essentially becomes rough endoplasmic reticulum with high amounts of ribosomes and a plethora of transporter proteins, and aquaporins. Additionally there is evidence from electron microscopy that indicates the occurrence of exocytosis from the plant membrane. [ 31 ] This highly convoluted and transporter rich membrane expertly performs the duties of nutrient exchange between the plant and fungus and allows for molecular manipulation by ribosomes and excreted enzymes within the interfacial apoplast. [ 33 ] [ 39 ] Pelotons are not permanent structures and are readily degraded and digested within 30 to 40 hours of their formation in orchid mycorrhiza. This happens in all endomycorrhizal associations but orchid plants readily digest fungal pelotons sooner after formation and more often than is seen in arbuscular mycorrhizal interactions. [ 40 ] It is proposed that the occurrence of this more extreme digestive pattern may have something to do with necrotorphic nutrient transfer which is the absorption of nutrients form dead cells. [ 41 ] [ 40 ] [ 28 ] The key nutrients involved in the majority of the transfer between fungi and orchid plants are carbon, nitrogen and phosphorus. [ 42 ]
Orchid mycorrhizal interactions are unique in the flow of nutrients. Typically in arbuscular mycorrhizal interactions the plants will unidirectionally supply the fungi with carbon in exchange for phosphorus or nitrogen or both depending on the environment, [ 43 ] [ 42 ] but orchid mycorrhizal nutrient transfer is less specific (but no less regulated) and there is often bidirectional flow of carbon between the fungus and plant, as well as flow of nitrogen and phosphorus from the fungus to plant. In around 400 species of plants there is no flow of carbon from plant and all of the nutrients of the plant are supplied by the fungus. [ 33 ] However, the net carbon gain by the plant in these interactions is positive in majority of the observed interactions. [ 41 ]
Phosphorus is a crucial nutrient needed by all plants and often phosphorus deficiency in soil will dictate the formation of a symbiotic relationship. Phosphorus is obtained by the mycorrhizal fungus from the surrounding soil in three different forms; organic phosphorus, phosphate, and inorganic phosphorus compounds. Often, these compounds are strongly bound to cations and need to be protonated and, or catabolized to become bioavailable. Mycorrhizal fungi are extremely efficient at doing this due to their extensive soil surface area as well as high enzymatic diversity. [ 44 ] [ 43 ] [ 42 ] Once freed from the soil the phosphorus compounds, primarily inorganic phosphate, are transferred through two proposed pathways. The first involves the active transport of inorganic phosphorus, primarily as phosphate through Pi (inorganic phosphorus) transporters out of the fungus into the interfacial apoplast, where it is protonated due to the acidic nature of the interfacial apoplast to form dihydrogen phosphate and then subsequently transferred through active Pi transporters into the plant cell. [ 42 ] [ 44 ] The second method relies on passive efflux of Pi from the fungus and active uptake by the plant, as in the prior pathway. It is observed that free living fungi ordinarily have very slight losses of Pi, thought to be due to the re-absorptive nature of the fungal hyphae, but it is proposed that during symbiosis the reabsorption of Pi is reduced thus increasing the net efflux out of the fungi. [ 42 ] Both pathways depend on relatively high concentrations of Pi in the fungal cells and low Pi concentrations inside the plant cell for efficient transport, although the second method is far more dependent on this condition. Additionally these methods are not mutually exclusive and may occur in the same individuals, but more research is required to uncover the complexities of the interaction. [ 28 ] To facilitate phosphorus exchange an array of genes are upregulated; Pi transporter genes such as MtPT4 and StPT3 are up regulated in orchids plants along with H+ ATPase transporters. [ 43 ] Fungal partners upregulated Pi transporter genes as well as alkaline phosphatase encoding genes (Perotto). [ 45 ] [ 43 ] The upregulation of phosphatase genes is significant in that it indicates that a significant portion of the phosphorus obtained by the fungi in some environments may be from organic molecule catabolism. [ 43 ] It is important to note that once a mycorrhizal symbiosis is established the only phosphorus obtained by the plant comes through the fungal tissue. [ 31 ] Additionally the large scale assisted transport of phosphorus from fungi to plant only occurs when the fungal pelotons are alive, and once these structures start to degrade significant flow of phosphorus ceases. [ 44 ] This classifies this pathway as biotrophic, meaning transfer of nutrients between two or more organisms that are both alive.
Nitrogen transport is an equally important and influential process that often occurs through mycorrhizal associations. [ 46 ] Nitrogen is significantly easier to obtain than phosphorus and far more abundant, but mycorrhizal interactions still provide a significant benefit in the allocation of nitrogen. Bioavailable nitrogen as nitrate and ammonium are absorbed from the soil media into the extraradical mycelium of the mycorrhizal fungi and assimilated into amino acids. [ 43 ] In some cases amino acids and other nitrogenous organic compounds may be present in the surrounding soil. If so organic nitrogen uptake also takes place through amino acid permeases and peptide transporters. [ 39 ] Once incorporated into the fungi as amino acids, there are a few different proposed mechanisms with which the nitrogen is transferred to the host plant. These pathways of nitrogen transfer are thought to be exclusively biotrophic, a significant amount of nitrogen may also be transferred necrotorphically but through a significantly different process. [ 46 ] In the first pathway the amino acids are transferred to the extraradical mycelium where the amino acids are broken down by the catabolic stage of the urea cycle. The primary amino acid synthesized and intra-fungally transferred is arginine , which is catabolized into ammonium. The ammonium is then shunted into the interfacial space between the peloton and the surrounding plant membrane, and transported into the plant cell via ammonium transporters and incorporated into the plant. [ 39 ]
To accommodate the transport of nitrogen via this pathway, primarily as ammonium, an array of ammonium transporter genes are up regulated in the plant partners, and additionally the mycorrhizal fungal partners often upregulate a group of protease genes as well as external amino acid permeases, nitrate transporters and ammonium transporters. [ 39 ] [ 42 ] As proposed by Cameron 2006 and Fochi 2016, nitrogen may also be transferred as amino acids such as arginine, glycine and glutamine into the cell via a select few specialized amino acid transporters. When in symbiotic relationship with orchid plants the mycorrhizal fungus T. calospora upregulates the expression of SvAAP1 and SvAAP2, these genes encode amino acid permeases, supporting the hypothesis that amino acids are an important molecule involved in nitrogen transport. [ 39 ] [ 32 ] [ 40 ] Amino acids contain a significant amount of carbon as well and the transport of carbon may be the primary driving cause of the observed upregulation of the amino acid transporter genes, nonetheless nitrogen can still be transported to the plant via this pathway. [ 41 ] The transport of inorganic nitrogen in the form of ammonium and the transport of organic nitrogen as amino acids most likely will occur simultaneously in the same species and or organism, depending on the abundance of different nitrogenous species in the soil, [ 28 ] Fochi 2016 along with Dearnaley 2007 suggest that amino acids may in fact be the primary nitrogen compound transferred to orchids, evidence for this coming from isotopic ratios of enriched C13 and N15 within plants associated with mycorrhizal fungi, but more research is needed to fully understand this transport relationship. [ 39 ] [ 31 ]
Apart from the unique peloton structures which transfer nitrogen and phosphorus from mycorrhizal fungi to orchid plants the transfer of these nutrients, as discussed above is almost identical to that observed in arbuscular mycorrhiza and ericoid mycorrhiza, but when it comes to arguably the most fundamental element involved in mycorrhizal transport carbon, orchid mycorrhiza show a distinct divergence form the traditional paradigm of unidirectional nutrient transfer in mycorrhizal associations. [ 28 ] [ 42 ] [ 41 ] Orchid mycorrhizal interactions encompass a large variety of symbiotic scenarios, ranging from fully mycoheterotrophic plants or in other words parasitic on their fungal inhabitant such as Monotropa uniflora which lacks chlorophyll, to interactions that take on a similar nature to those found in arbuscular mycorrhiza, where in bidirectional nutrient transport takes place such as in green leaved Goodyera repens . [ 47 ] [ 32 ] [ 33 ] For the documented interactions in orchid mycorrhiza it has been observed that even in photosynthetically capable species carbon will readily flow from fungi to the plant. This may or may not occur simultaneously with the transfer of carbon from plant to fungi. [ 32 ] [ 48 ] [ 41 ] The evolutionary reasoning for this phenomenon is yet to be deduced. [ 28 ]
Due to this orchid mycorrhiza in general are considered at least partially mycoheterotrophic interactions. [ 28 ] In the mycoheterotrophic orchid mycorrhiza carbon is taken up, by the fungi, as an array of peptide and carbohydrate species often the products of saprophytic reactions taking place in the soil. [ 28 ] [ 49 ] [ 50 ] These reactions are often instigated and progressed by the activity of the mycorrhizal fungus, being part of the basidiomycete phyla, orchid associated fungi have an extensive metabolic arsenal with which to pull from, and are readily able to digest cellulose and lignin to obtain the carbon. [ 50 ] Genes encoding proteases, cellulose and lignin digestive enzymes, as well as oligopeptide and carbohydrate transporters are upregulated in the mycelial soil webs to facilitate increased carbon uptake. [ 36 ] Further more the fungi which mycoheterotrophically interact with orchid plants are often also found in mycorrhizal association with beech trees, [ 51 ] and translocation of photosynthate from tree to fungus and then to orchid has been proposed, but a thorough study is still lacking. [ 52 ] Once acquired by the fungus the carbon is either converted into sugars, trehalose being extremely common for most mycorrhizal fungus, amino acids, or simply assimilated into the fungal organism. The transfer of carbon from fungi to plant happens in one of two forms either as carbohydrates primarily trehalose, but glucose and sucrose may also be involved, or as an amino acids primarily as arginine but glycine and glutamine can also be transferred. [ 33 ] [ 32 ] [ 40 ] These molecules are transported, via specialized amino acid permeases and carbohydrate transporters embedded in the fungal peloton membrane, into the interficial space where they are absorbed into the plant through similar analogous transporters in the plant endoplasmic reticulum membrane which surrounds the peloton. [ 40 ] [ 32 ] [ 49 ] Genes which encode for such transporters experience significant upregulation in both plant and fungi, similar to the upregulation seen in nitrogen and phosphorus compound transporter genes during symbiosis. [ 36 ] This active transport of carbon from fungi to plant is an exclusively biotrophic reaction but it is thought that a significant amount of carbon may also be transferred to the plant when the fungal pelotons degrade and are digested.
The transfer of nitrogen and carbon as discussed above readily occurs through live pelotons, but it has been observed by Kuga 2014 and Rasumussen 2009 that large amount of carbon and nitrogen may also be taken up by the plant during the senescence and digestion of fungal pelotons. This process is called mycophagy. [ 41 ] [ 40 ] Lysis of pelotons begins between 30 and 40 hours after contact and initial peloton formation within the majority of observed orchid plants. [ 28 ] As mentioned above as pelotons are formed they are covered by a plant derived membrane, which eventually takes on the properties of endoplasmic reticulum surrounded by Golgi apparatuses. The presence of these organelles is indicative of their purpose, which is thought to be the excretion of digestive enzymes into the interficial space between the peloton and the plant membrane to digest the fungal cells. [ 53 ] Once the fate of the peloton is decided, and degradation and digestion are to occur, a secondary membrane forms around the fungal peloton which is essentially a large vacuole which will allow the isolated degradation of the peloton. [ 53 ] Additionally peroxisomes accumulate within the digestive plant cells and undergo exosytosis into the newly formed vacuole, this process concentrates a plethora of enzymes such as uricases, chitinases, peptidases, oxidases and catalyses within the vacuole facilitating the breakdown of the peloton. [ 53 ] [ 28 ] Once degraded the fungal remnants are absorbed into the plant, transferring the nutrients and energy to the plant host. [ 41 ] Stable isotope imaging reveals that C13 and N15 applied to mycorrhizal hyphea webs was found to be readily transferred to the plant via fungal pelotons, leading to a disproportionate amount of these isotopes within the peloton containing plant cells and the peloton itself, but in senescing pelotons the concentrations of these C13 and N15 containing compound was even more pronounced, and significantly higher than in live pelotons. [ 40 ] This indicates that as plant pelotons start to senescence they are loaded with carbon and nitrogen nutritive compounds, hypothetically to be transferred to the plant during lysis. [ 40 ] Morphological changes in the hyphea further support this proposition, in that the last morphological change in the peloton before is collapses is a swelling, perhaps due to the increased nutritive compound load. [ 40 ] Once digestion is complete, reinfection of the digestive cell will occur shortly after and a new peloton will form, and eventually will be degraded. [ 41 ] Reinfection and digestion are a continuum and will cyclically occur throughout the entire life span of the cooperation. [ 28 ] Rasmussen 2009 differentiates two types of plant host cells which are involved in nutrient transfer, so called 'digestion cells' and so called 'host cells', 'digestive cells' apparently engage in dense hyphal peloton formation followed by a rapid digestion and subsequent reinfection, where as 'host cells' contain live hyphae pelotons, which will not be digested, or at least not as readily. [ 41 ] Rasmussen 2002 discusses an additional mode of nutrient transfer called pytophagy involving lysis of fungal cells but instead of lysing the entire fungal hyphae only the growing end is degraded releasing the contents of the fungal cytoplasm into the interfacial space between the plant and fungi membranes, where the plant can uptake the necessary compounds. There is little information on this mode of nutrient transfer and it is still speculative, and more research is needed. [ 54 ]
Micro-nutrient transfer is thought, for the most part, to occur by passive transport across cellular membranes, both during absorption, from soil by fungi, and transfer from fungi to host plants. [ 42 ] This is not always the case though and although research on the topic is limited, there is evidence of active transport and allocation of micro-nutrients in certain conditions. [ 55 ] The upregulation of cation transporters is observed in orchid D. officinale symbioses, suggesting fungi may assisted in the transfer of nutrients from fungi to plant. [ 56 ] Cations, especially iron, are often bound tightly to organic and clay substrates keeping them out of reach of plants, fungi and bacteria, but compounds such as siderophores are often secreted into the soil by fungi and bacteria to aid in the acquisition of these cations. [ 50 ] A siderophore is a small molecule which has extremely high affinity for Fe3+ and is commonly found being utilized by many bacteria and fungal species. [ 57 ] These compounds are released into the soil surrounding the hyphal web and strip iron from mineral compounds in the soil, the siderophore can then be reabsorbed into the fungal hyphae where the iron can be dissociated from the spiderohore and used. [ 57 ] Haselwandter investigated the presence of siderophores in orchid associated mycorrhizal fungi within the genus Rhizoctonia which utilize the siderophore basidiochrome as the major iron-chelating compound. [ 55 ] Other vital nutrients may be transferred between mycorrhizal fungi and orchid plants via specialized methods, such as chelating molecules, but more research on this subject is needed. [ 28 ]
Current molecular analysis has allowed for the identification of specific taxa forming symbiotic relationships which are of interest in the study, cultivation, and conservation of orchids. This is especially important in the trade and preservation endangered species or orchids of commercial value like the vanilla bean . [ 58 ] There have been seen trends in the type of symbioses found in orchids, depending primarily on the life-style of the orchid, as the symbiosis is primarily of benefit to the plant. Terrestrial orchids have been found to commonly associate with Tulasnellaceae , however some autotrophic and non-autotrophic orchids do associate with several ectomycorrhizal fungi. [ 59 ] [ 60 ] Epiphytic fungi , however, may associate more commonly with limited clades of rhizoctonia , a polyphyletic grouping. [ 61 ] These fungi may form significant symbioses with either an epiphytic or terrestrial orchid, but rarely do they associate with both. [ 61 ] Using seed-baiting techniques researchers have been able to isolate specific species and strains of symbiotic orchid mycorrhizae. Using this technique seeds of the epiphytic orchid Dendrobium aphyllum were found to germinate to seedlings when paired with fungal strains of Tulasnella , however, seeds of this species did not germinate when treated with Trichoderma fungi taken from adult orchids, indicating there are stage specific symbionts. These fungal symbioses, as well as their affinity towards specific symbionts, vary based on the stage of development and age of the host roots. [ 62 ] [ 9 ] [ 26 ] As an orchid ages the fungal associations become more complex.
Mixotrophic orchids like Cephalanthera longibracteata may associate generally with several fungi, most notably from Russulaceae , Tricholomataceae , Sebacinales , Thelephoraceae , [ 63 ] as they do not depend as heavily on the fungus for carbon. Some orchids can be very specific in their symbionts, preferring a single class or genus of fungi. Genotypes of Corallorhiza maculata , a myco-heterotrophic orchid, have been found to closely associate with Russulaceae regardless of geological location or the presence of other orchids. [ 64 ] The terrestrial Chilean orchids Chloraea collicensis and C. gavilu have been found to have only one key symbiont in the genus Rhizoctonia . [ 65 ] Research suggests that orchids considered to be generalists will associate with other generalist fungi, often with several species, however orchids with high degrees of specialization use a more limited range of fungal partners. [ 25 ] For instance, the photosynthetic orchid Liparis liliifolia associates with many strains of fungi in the adult stage. [ 66 ] Conversely, the photosynthetic orchid Goodyera pubescens was found to associate with only one dominate fungus, unless subjected to changes in the environment, like drought, in which case the orchid was able to change symbionts in response to stresses. [ 16 ] There is a tremendous amount of diversity in orchid fungal specificity. Interestingly, most arbuscular and ectomycorrhizal symbioses form associations with several fungi at the same time. [ 16 ] Fungal symbioses also depend on the environment so there are many differences in fungal partners of tropical versus temperate orchids.
The knowledge of species-specific fungal symbionts is crucial given the endangered status of many orchids around the world. The identification of their fungal symbionts could prove useful in the preservation and reintroduction of threatened orchids. [ 58 ] Researchers in India have used samples from adult orchids to germinate seeds from an endangered orchid Dactylorhiza hatagirea which were found to associate closely with Ceratobasidium . [ 67 ] | https://en.wikipedia.org/wiki/Orchid_mycorrhiza |
The Ordem dos Engenheiros ( OE , English: Order of Engineers ) is the regulatory and licensing body for the engineering profession in Portugal . It is headquartered in Lisbon , and has several regional branches in other Portuguese cities.
The OE was established by law in 1936. It succeeded the Portuguese Association of Civil Engineers, founded nearly 70 years earlier. The OE is a member of many international engineering organizations, including general engineering ones (e.g. FEANI ) and those for specific engineering disciplines (e.g. ECCE , EUREL , EFCE ).
The OE's mission is to contribute to the progress of engineering by supporting the efforts of its members in scientific, professional and social areas, as well as to ensure compliance with professional regulations and ethics.
It is illegal to provide engineering services or sign engineering projects in Portugal without being a member of the OE. However, many other professionals in engineering (such as technical engineers, short-cycle degree engineers, or engineers graduating from unaccredited courses) are allowed to work in the field as long as they do not provide engineering services or sign engineering projects, and they cannot officially use the title "engineer".
The OE is the entity responsible for the accreditation of engineering degrees and engineering courses in Portugal. Engineers graduating with an accredited degree are exempt from the licensing exams conducted by the Order. According to the chairman ( Portuguese : bastonário ) of the OE, only 30 to 50 percent of the candidates with an unaccredited degree pass the licensing exams, depending on the particular engineering field. Over three hundred engineering degrees are awarded in Portugal by public universities, public polytechnic schools, and private institutions. However, only about one hundred of these are accredited degrees.
A full chartered engineer ( Engenheiro ) in Portugal used to have a compulsory five-year course known as licenciatura ( licentiate ) which was granted exclusively by universities . Only engineers having the licenciatura diploma, graduated at the universities, were capacitated to develop any kind of project in engineering and were universally recognized by the Engineers Association of Portugal ( Ordem dos Engenheiros ). The polytechnic institutions of engineering, born after 1974, used to award the professional title of Engenheiro Técnico [ pt ] (Technical Engineer), a title conferred after a three years course; the degree was known as bacharelato . Polytechnic institutions conferred 3-years bacharelato degrees in several technical engineering specializations, until the late 1990s. At this time new legal decrees were adopted by Portuguese State (Administrative Rule 413A/98 of 17 July 1998), and it started to award 3 + 2 licenciaturas bietápicas ( bacharelato plus one or two extra years, conferring the licenciatura degree - a degree that had been awarded exclusively by the universities). In the mid-2000s those institutions adopted new more selective admission rules which were imposed to every Portuguese higher education institution by the State, excluding for the first time in their history the applicants with negative (less than 95/200) admission marks (in Portugal admission marks to higher education institutions are based on a combination of high school marks, and results of the entrance exams, and competition is based in a numerus clausus system). However, in many cases, polytechnic courses from several institutions across the country, started to require admission entrance exams in fields not directly related with the course (for instance, an electrical engineering or computer engineering course allows a biology entrance exam instead of mathematics and/or physics, unlike what is seen in most universities for the same engineering fields). This is the main reason many engineering courses awarded by several Portuguese polytechnic institutions and a few universities, are not currently accredited by Ordem dos Engenheiros . This is not exclusive of polytechnic engineerings since that in other polytechnic fields, like in polytechnic accountancy and management institutes or schools, history, geography, or even Portuguese language entrance exams are allowed instead of mathematics and economics, unlike what is allowed for the university courses in similar fields, although some departments of certain university institutions are using the same criteria to fight the increasing number of places left vacant every year.
Today, after many reforms and changes in higher education occurred since 1998 to the 2000s, the formal differences between polytechnic and university licenciatura degrees in engineering are in general null, and due to the Bologna process both graduates should be recognized equally all across Europe. However, there are many engineering courses whose degrees are still not recognized by the Ordem dos Engenheiros (the highest Portuguese authority in accreditation of professional engineers), especially engineering courses conferred by several polytechnical institutes and many private institutions. Among the oldest recognized and most extensively accredited engineering courses in Portugal, are those engineering degrees awarded by the state-run universities. After the large 1998 - 2000s reforms and upgrades, some polytechnic engineering licenciatura degrees started to be offered by the largest state-run polytechnic institutes, have been accredited in the same way with official recognition by Ordem dos Engenheiros . | https://en.wikipedia.org/wiki/Ordem_dos_Engenheiros |
In geometry , an order-2 apeirogonal tiling , apeirogonal dihedron , or infinite dihedron [ 1 ] is a tessellation (gap-free filling with repeated shapes) of the plane consisting of two apeirogons . It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {∞, 2}. Two apeirogons joined along all their edges can completely fill the entire plane, as an apeirogon is infinite in size and has an interior angle of 180°, which is half of a full 360°.
Similarly to the uniform polyhedra and the uniform tilings , eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism , and the apeirogonal antiprism .
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Order-2_apeirogonal_tiling |
In the geometry of hyperbolic 4-space , the order-5 icosahedral 120-cell honeycomb is one of four regular star- honeycombs . With Schläfli symbol {3,5,5/2,5}, it has five icosahedral 120-cells around each face. It is dual to the great 120-cell honeycomb .
It can be constructed by replacing the great dodecahedral cells of the great 120-cell honeycomb with their icosahedral convex hulls, thus replacing the great 120-cells with icosahedral 120-cells . It is thus analogous to the four-dimensional icosahedral 120-cell . It has density 10.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Order-5_icosahedral_120-cell_honeycomb |
In order theory , a branch of mathematics , a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order .
A partial order is a reflexive , transitive and antisymmetric relation. Given any partial orders ≤ {\displaystyle \,\leq \,} and ≤ ∗ {\displaystyle \,\leq ^{*}\,} on a set X , {\displaystyle X,} ≤ ∗ {\displaystyle \,\leq ^{*}\,} is a linear extension of ≤ {\displaystyle \,\leq \,} exactly when
It is that second property that leads mathematicians to describe ≤ ∗ {\displaystyle \,\leq ^{*}\,} as extending ≤ . {\displaystyle \,\leq .}
Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P {\displaystyle P} to a chain C {\displaystyle C} on the same ground set.
A preorder is a reflexive and transitive relation. The difference between a preorder and a partial-order is that a preorder allows two different items to be considered "equivalent", that is, both x ≾ y {\displaystyle x\precsim y} and y ≾ x {\displaystyle y\precsim x} hold, while a partial-order allows this only when x = y {\displaystyle x=y} . A relation ≾ ∗ {\displaystyle \precsim ^{*}} is called a linear extension of a preorder ≾ {\displaystyle \precsim } if:
The difference between these definitions is only in condition 3. When the extension is a partial order, condition 3 need not be stated explicitly, since it follows from condition 2. Proof : suppose that x ≾ y {\displaystyle x\precsim y} and not y ≾ x {\displaystyle y\precsim x} . By condition 2, x ≾ ∗ y {\displaystyle x\precsim ^{*}y} . By reflexivity, "not y ≾ x {\displaystyle y\precsim x} " implies that y ≠ x {\displaystyle y\neq x} . Since ≾ ∗ {\displaystyle \precsim ^{*}} is a partial order, x ≾ ∗ y {\displaystyle x\precsim ^{*}y} and y ≠ x {\displaystyle y\neq x} imply "not y ≾ ∗ x {\displaystyle y\precsim ^{*}x} ". Therefore, x ≺ ∗ y {\displaystyle x\prec ^{*}y} .
However, for general preorders, condition 3 is needed to rule out trivial extensions. Without this condition, the preorder by which all elements are equivalent ( y ≾ x {\displaystyle y\precsim x} and x ≾ y {\displaystyle x\precsim y} hold for all pairs x , y ) would be an extension of every preorder.
The statement that every partial order can be extended to a total order is known as the order-extension principle . A proof using the axiom of choice was first published by Edward Marczewski (Szpilrajin) in 1930. Marczewski writes that the theorem had previously been proven by Stefan Banach , Kazimierz Kuratowski , and Alfred Tarski , again using the axiom of choice, but that the proofs had not been published. [ 1 ]
There is an analogous statement for preorders: every preorder can be extended to a total preorder. This statement was proved by Hansson. [ 2 ] : Lemma 3
In modern axiomatic set theory the order-extension principle is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem , [ 3 ] but the reverse implication doesn't hold. [ 4 ]
Applying the order-extension principle to a partial order in which every two elements are incomparable shows that (under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the ordering principle , OP, and is a weakening of the well-ordering theorem . However, there are models of set theory in which the ordering principle holds while the order-extension principle does not. [ 5 ]
The order extension principle is constructively provable for finite sets using topological sorting algorithms, where the partial order is represented by a directed acyclic graph with the set's elements as its vertices . Several algorithms can find an extension in linear time . [ 6 ] Despite the ease of finding a single linear extension, the problem of counting all linear extensions of a finite partial order is #P-complete ; however, it may be estimated by a fully polynomial-time randomized approximation scheme . [ 7 ] [ 8 ] Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are semiorders . [ 9 ]
The order dimension of a partial order is the minimum cardinality of a set of linear extensions whose intersection is the given partial order; equivalently, it is the minimum number of linear extensions needed to ensure that each critical pair of the partial order is reversed in at least one of the extensions.
Antimatroids may be viewed as generalizing partial orders; in this view, the structures corresponding to the linear extensions of a partial order are the basic words of the antimatroid. [ 10 ]
This area also includes one of order theory's most famous open problems, the 1/3–2/3 conjecture , which states that in any finite partially ordered set P {\displaystyle P} that is not totally ordered there exists a pair ( x , y ) {\displaystyle (x,y)} of elements of P {\displaystyle P} for which the linear extensions of P {\displaystyle P} in which x < y {\displaystyle x<y} number between 1/3 and 2/3 of the total number of linear extensions of P . {\displaystyle P.} [ 11 ] [ 12 ] An equivalent way of stating the conjecture is that, if one chooses a linear extension of P {\displaystyle P} uniformly at random, there is a pair ( x , y ) {\displaystyle (x,y)} which has probability between 1/3 and 2/3 of being ordered as x < y . {\displaystyle x<y.} However, for certain infinite partially ordered sets, with a canonical probability defined on its linear extensions as a limit of the probabilities for finite partial orders that cover the infinite partial order, the 1/3–2/3 conjecture does not hold. [ 13 ]
Counting the number of linear extensions of a finite poset is a common problem in algebraic combinatorics . This number is given by the leading coefficient of the order polynomial multiplied by | P | ! . {\displaystyle |P|!.}
Young tableau can be considered as linear extensions of a finite order-ideal in the infinite poset N × N , {\displaystyle \mathbb {N} \times \mathbb {N} ,} and they are counted by the hook length formula . | https://en.wikipedia.org/wiki/Order-extension_principle |
Order (subtitled A Journal on the Theory of Ordered Sets and its Applications ) is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media . It was established in 1984 by Ivan Rival ( University of Calgary ). [ 1 ] [ 2 ] From 2010 to 2018, its editor-in-chief was Dwight Duffus ( Emory University ). He was succeeded in 2019 by Ryan R. Martin ( Iowa State University ). [ 3 ]
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2017 impact factor of 0.353. [ 7 ]
This article about a mathematics journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Order_(journal) |
Order in mathematics may refer to:
In logic, model theory and type theory: | https://en.wikipedia.org/wiki/Order_(mathematics) |
The OrderOne MANET Routing Protocol is an algorithm for computers communicating by digital radio in a mesh network to find each other, and send messages to each other along a reasonably efficient path. It was designed for, and promoted as working with wireless mesh networks .
OON's designers say it can handle thousands of nodes, where most other protocols handle less than a hundred. OON uses hierarchical algorithms to minimize the total amount of transmissions needed for routing. Routing overhead is limited to between 1% and 5% of node-to-node bandwidth in any network and does not grow as the network size grows.
The basic idea is that a network organizes itself into a tree. Nodes meet at the root of the tree to establish an initial route. The route then moves away from the root by cutting corners, as ant-trails do. When there are no more corners to cut, a nearly optimum route exists. This route is continuously maintained.
Each process can be performed with localized minimal communication, and very small router tables. OORP requires about 200K of memory. A simulated network with 500 nodes transmitting at 200 bytes/second organized itself in about 20 seconds.
As of 2004, OORP was patented or had other significant intellectual property restrictions. See the link below.
Each computer, or "node" of the network has a unique name, at least one network link, and a computer with some capacity to hold a list of neighbors.
The network nodes form a hierarchy by having each node select a parent. The parent is a neighbor node that is the next best step to the most other nodes. This method creates a hierarchy around nodes that are more likely to be present, and which have more capacity, and which are closer to the topological center of the network. The memory limitations of a small node are reflected in its small routing table, which automatically prevents it from being a preferred central node.
At the top, one or two nodes are unable to find nodes better-connected than themselves, and therefore become parents of the entire network.
The hierarchy-formation algorithm does not need a complex routing algorithm or large amounts of communication.
All nodes push a route to themselves to the root of the tree. A node wanting a connection can therefore push a request to the root of the tree, and always find a route.
The commercial protocol uses Dijkstra's algorithm to continuously optimize and maintain the route. As the network moves and changes, the path is continually adjusted.
Assuming that some nodes in the network have enough memory to know of all nodes in the network, there is no practical limitation to network size.
Since the control bandwidth is defined to be less than 5% regardless of network size, the amount of control bandwidth required is not supposed to increase as network size grows.
The system can use nodes with small amounts of memory.
The network has a reliable, low-overhead way to establish that a node is not in the network. This is a difficult, valuable property in ad hoc mesh networks.
Most routing protocols scale either by reducing proactive link-state routing information or reactively driving routing by connection requests. OORP mixes the proactive and reactive methods. Properly configured, an OORP net can scale to 100,000's of nodes and can often achieve reasonable performance even though it limits routing bandwidth to 5%.
Central nodes have an extra burden because they need to have enough memory to store information about all nodes in the network. At some number of nodes, the network will therefore cease to scale.
If all the nodes in the network are low capacity nodes the network may be overwhelmed with change. This may limit the maximum scale. However, In virtually all real world networks, the farther away from the edge nodes the more the bandwidth grows.
These critiques may have no practical effect. For example, consider a low bandwidth 9.6 Kbit/second radio. If the protocol was configured to send one packet of 180 bytes every 5 seconds, it would consume 3% of overall network bandwidth.
Public proposals for OON do not include security or authentication. Security and authentication may provided by the Integrator of the protocol. Typical security measures include encryption or signing or the protocol packets and incrementing counters to prevent replay attacks. | https://en.wikipedia.org/wiki/Order_One_Network_Protocol |
In physics , the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. [ citation needed ]
In condensed matter physics , systems typically are ordered at low temperatures ; upon heating, they undergo one or several phase transitions into less ordered states.
Examples for such an order-disorder transition are:
The degree of freedom that is ordered or disordered can be translational ( crystalline ordering), rotational ( ferroelectric ordering), or a spin state ( magnetic ordering).
The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order .
If a disordered state is not in thermodynamic equilibrium , one speaks of quenched disorder . For instance, a glass is obtained by quenching ( supercooling ) a liquid. By extension, other quenched states are called spin glass , orientational glass . In some contexts, the opposite of quenched disorder is annealed disorder .
The strictest form of order in a solid is lattice periodicity : a certain pattern (the arrangement of atoms in a unit cell ) is repeated again and again to form a translationally invariant tiling of space. This is the defining property of a crystal . Possible symmetries have been classified in 14 Bravais lattices and 230 space groups .
Lattice periodicity implies long-range order : [ 1 ] if only one unit cell is known, then by virtue of the translational symmetry it is possible to accurately predict all atomic positions at arbitrary distances. During much of the 20th century, the converse was also taken for granted – until the discovery of quasicrystals in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity.
Besides structural order, one may consider charge ordering , spin ordering, magnetic ordering , and compositional ordering. Magnetic ordering is observable in neutron diffraction .
It is a thermodynamic entropy concept often displayed by a second-order phase transition . Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low energy.
Long-range order characterizes physical systems in which remote portions of the same sample exhibit correlated behavior.
This can be expressed as a correlation function , namely the spin-spin correlation function :
where s is the spin quantum number and x is the distance function within the particular system.
This function is equal to unity when x = x ′ {\displaystyle x=x'} and decreases as the distance | x − x ′ | {\displaystyle |x-x'|} increases. Typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large | x − x ′ | {\displaystyle |x-x'|} then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also Berezinskii–Kosterlitz–Thouless transition ). Note that what constitutes a large value of | x − x ′ | {\displaystyle |x-x'|} is understood in the sense of asymptotics .
In statistical physics , a system is said to present quenched disorder when some parameters defining its behavior are random variables which do not evolve with time. These parameters are said to be quenched or frozen. Spin glasses are a typical example. Quenched disorder is contrasted with annealed disorder in which the parameters are allowed to evolve themselves.
Mathematically, quenched disorder is more difficult to analyze than its annealed counterpart as averages over thermal noise and quenched disorder play distinct roles. Few techniques to approach each are known, most of which rely on approximations. Common techniques used to analyzed systems with quenched disorder include the replica trick , based on analytic continuation , and the cavity method , where a system's response to the perturbation due to an added constituent is analyzed. While these methods yield results agreeing with experiments in many systems, the procedures have not been formally mathematically justified. Recently, rigorous methods have shown that in the Sherrington-Kirkpatrick model , an archetypal spin glass model, the replica-based solution is exact. The generating functional formalism , which relies on the computation of path integrals , is a fully exact method but is more difficult to apply than the replica or cavity procedures in practice.
A system is said to present annealed disorder when some parameters entering its definition are random variables , but whose evolution is related to that of the degrees of freedom defining the system. It is defined in opposition to quenched disorder, where the random variables may not change their values.
Systems with annealed disorder are usually considered to be easier to deal with mathematically, since the average on the disorder and the thermal average may be treated on the same footing. | https://en.wikipedia.org/wiki/Order_and_disorder |
In mathematics, specifically in order theory and functional analysis , a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle X} is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form [ a , b ] := { x ∈ X : a ≤ x and x ≤ b } {\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\}} ) and if sup { inf S : S ∈ OBound ( X ) ∩ F } = inf { sup S : S ∈ OBound ( X ) ∩ F } , {\displaystyle \sup \left\{\inf S:S\in \operatorname {OBound} (X)\cap {\mathcal {F}}\right\}=\inf \left\{\sup S:S\in \operatorname {OBound} (X)\cap {\mathcal {F}}\right\},} where OBound ( X ) {\displaystyle \operatorname {OBound} (X)} is the set of all order bounded subsets of X , in which case this common value is called the order limit of F {\displaystyle {\mathcal {F}}} in X . {\displaystyle X.} [ 1 ]
Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.
A net ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} in a vector lattice X {\displaystyle X} is said to decrease to x 0 ∈ X {\displaystyle x_{0}\in X} if α ≤ β {\displaystyle \alpha \leq \beta } implies x β ≤ x α {\displaystyle x_{\beta }\leq x_{\alpha }} and x 0 = i n f { x α : α ∈ A } {\displaystyle x_{0}=inf\left\{x_{\alpha }:\alpha \in A\right\}} in X . {\displaystyle X.} A net ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} in a vector lattice X {\displaystyle X} is said to order-converge to x 0 ∈ X {\displaystyle x_{0}\in X} if there is a net ( y α ) α ∈ A {\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}} in X {\displaystyle X} that decreases to 0 {\displaystyle 0} and satisfies | x α − x 0 | ≤ y α {\displaystyle \left|x_{\alpha }-x_{0}\right|\leq y_{\alpha }} for all α ∈ A {\displaystyle \alpha \in A} . [ 2 ]
A linear map T : X → Y {\displaystyle T:X\to Y} between vector lattices is said to be order continuous if whenever ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} is a net in X {\displaystyle X} that order-converges to x 0 {\displaystyle x_{0}} in X , {\displaystyle X,} then the net ( T ( x α ) ) α ∈ A {\displaystyle \left(T\left(x_{\alpha }\right)\right)_{\alpha \in A}} order-converges to T ( x 0 ) {\displaystyle T\left(x_{0}\right)} in Y . {\displaystyle Y.} T {\displaystyle T} is said to be sequentially order continuous if whenever ( x n ) n ∈ N {\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }} is a sequence in X {\displaystyle X} that order-converges to x 0 {\displaystyle x_{0}} in X , {\displaystyle X,} then the sequence ( T ( x n ) ) n ∈ N {\displaystyle \left(T\left(x_{n}\right)\right)_{n\in \mathbb {N} }} order-converges to T ( x 0 ) {\displaystyle T\left(x_{0}\right)} in Y . {\displaystyle Y.} [ 2 ]
In an order complete vector lattice X {\displaystyle X} whose order is regular , X {\displaystyle X} is of minimal type if and only if every order convergent filter in X {\displaystyle X} converges when X {\displaystyle X} is endowed with the order topology . [ 1 ] | https://en.wikipedia.org/wiki/Order_convergence |
In mathematics , the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992) .
The dimension of a poset P is the least integer t for which there exists a family
of linear extensions of P so that, for every x and y in P , x precedes y in P if and only if it precedes y in all of the linear extensions, if any such t exists. That is,
An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. x ≤ y {\displaystyle x\leq y} if and only if x i ≤ y i {\displaystyle x_{i}\leq y_{i}} for all i ( Hiraguti 1955 , Milner & Pouzet 1990 ).
A family R = ( < 1 , … , < t ) {\displaystyle {\mathcal {R}}=(<_{1},\dots ,<_{t})} of linear orders on X is called a realizer of a poset P = ( X , < P ) if
which is to say that for any x and y in X , x < P y precisely when x < 1 y , x < 2 y , ..., and x < t y .
Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P ."
It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair ( x , y ) of P , y < i x for some order
< i in R .
Let n be a positive integer, and let P be the partial order on the elements a i and b i (for 1 ≤ i ≤ n ) in which a i ≤ b j whenever i ≠ j , but no other pairs are comparable. In particular, a i and b i are incomparable in P ; P can be viewed as an oriented form of a crown graph . The illustration shows an ordering of this type for n = 4.
Then, for each i , any realizer must contain a linear order that begins with all the a j except a i (in some order), then includes b i , then a i , and ends with all the remaining b j . This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have a i preceding b i , which would contradict the incomparability of a i and b i in P . And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n . In fact, P is known as the standard example of a poset of dimension n , and is usually denoted by S n .
The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different partial order ( Baker, Fishburn & Roberts 1971 ). That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x , y of distinct elements is comparable in exactly one of these two partial orders. If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the permutation graphs , graphs that are both themselves comparability graphs and complementary to comparability graphs.
The partial orders of order dimension two include the series-parallel partial orders ( Valdes, Tarjan & Lawler 1982 ). They are exactly the partial orders whose Hasse diagrams have dominance drawings , which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates.
It is possible to determine in polynomial time whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether the comparability graph of the partial order is a permutation graph. However, for any k ≥ 3, it is NP-complete to test whether the order dimension is at most k ( Yannakakis 1982 ).
The incidence poset of any undirected graph G has the vertices and edges of G as its elements; in this poset, x ≤ y if either x = y or x is a vertex, y is an edge, and x is an endpoint of y . Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three ( Schnyder 1989 ).
For a complete graph on n vertices, the order dimension of the incidence poset is Θ ( log log n ) {\displaystyle \Theta (\log \log n)} ( Hoşten & Morris 1999 ). It follows that all simple n -vertex graphs have incidence posets with order dimension O ( log log n ) {\displaystyle O(\log \log n)} .
A generalization of dimension is the notion of k -dimension (written dim k {\displaystyle {\textrm {dim}}_{k}} ) which is the minimal number of chains of length at most k in whose product the partial order can be embedded. In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the inclusion order on this set. | https://en.wikipedia.org/wiki/Order_dimension |
In order theory , a branch of mathematics , an order embedding is a special kind of monotone function , which provides a way to include one partially ordered set into another. Like Galois connections , order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism . Both of these weakenings may be understood in terms of category theory .
Formally, given two partially ordered sets (posets) ( S , ≤ ) {\displaystyle (S,\leq )} and ( T , ⪯ ) {\displaystyle (T,\preceq )} , a function f : S → T {\displaystyle f:S\to T} is an order embedding if f {\displaystyle f} is both order-preserving and order-reflecting , i.e. for all x {\displaystyle x} and y {\displaystyle y} in S {\displaystyle S} , one has
Such a function is necessarily injective , since f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x ≤ y {\displaystyle x\leq y} and y ≤ x {\displaystyle y\leq x} . [ 1 ] If an order embedding between two posets S {\displaystyle S} and T {\displaystyle T} exists, one says that S {\displaystyle S} can be embedded into T {\displaystyle T} .
An order isomorphism can be characterized as a surjective order embedding. As a consequence, any order embedding f restricts to an isomorphism between its domain S and its image f ( S ), which justifies the term "embedding". [ 1 ] On the other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic.
An example is provided by the open interval ( 0 , 1 ) {\displaystyle (0,1)} of real numbers and the corresponding closed interval [ 0 , 1 ] {\displaystyle [0,1]} . The function f ( x ) = ( 94 x + 3 ) / 100 {\displaystyle f(x)=(94x+3)/100} maps the former to the subset ( 0.03 , 0.97 ) {\displaystyle (0.03,0.97)} of the latter and the latter to the subset [ 0.03 , 0.97 ] {\displaystyle [0.03,0.97]} of the former, see picture. Ordering both sets in the natural way, f {\displaystyle f} is both order-preserving and order-reflecting (because it is an affine function ). Yet, no isomorphism between the two posets can exist, since e.g. [ 0 , 1 ] {\displaystyle [0,1]} has a least element while ( 0 , 1 ) {\displaystyle (0,1)} does not.
For a similar example using arctan to order-embed the real numbers into an interval, and the identity map for the reverse direction, see e.g. Just and Weese (1996). [ 2 ]
A retract is a pair ( f , g ) {\displaystyle (f,g)} of order-preserving maps whose composition g ∘ f {\displaystyle g\circ f} is the identity. In this case, f {\displaystyle f} is called a coretraction, and must be an order embedding. [ 3 ] However, not every order embedding is a coretraction. As a trivial example, the unique order embedding f : ∅ → { 1 } {\displaystyle f:\emptyset \to \{1\}} from the empty poset to a nonempty poset has no retract, because there is no order-preserving map g : { 1 } → ∅ {\displaystyle g:\{1\}\to \emptyset } . More illustratively, consider the set S {\displaystyle S} of divisors of 6, partially ordered by x divides y , see picture. Consider the embedded sub-poset { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} . A retract of the embedding i d : { 1 , 2 , 3 } → S {\displaystyle id:\{1,2,3\}\to S} would need to send 6 {\displaystyle 6} to somewhere in { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} above both 2 {\displaystyle 2} and 3 {\displaystyle 3} , but there is no such place.
Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example: | https://en.wikipedia.org/wiki/Order_embedding |
In the mathematical field of order theory , an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections . [ 1 ]
The idea of isomorphism can be understood for finite orders in terms of Hasse diagrams . Two finite orders are isomorphic exactly when a single Hasse diagram ( up to relabeling of its elements) expresses them both, in other words when every Hasse diagram of either can be converted to a Hasse diagram of the other by simply relabeling the vertices.
Formally, given two posets ( S , ≤ S ) {\displaystyle (S,\leq _{S})} and ( T , ≤ T ) {\displaystyle (T,\leq _{T})} , an order isomorphism from ( S , ≤ S ) {\displaystyle (S,\leq _{S})} to ( T , ≤ T ) {\displaystyle (T,\leq _{T})} is a bijective function f {\displaystyle f} from S {\displaystyle S} to T {\displaystyle T} with the property that, for every x {\displaystyle x} and y {\displaystyle y} in S {\displaystyle S} , x ≤ S y {\displaystyle x\leq _{S}y} if and only if f ( x ) ≤ T f ( y ) {\displaystyle f(x)\leq _{T}f(y)} . That is, it is a bijective order-embedding . [ 2 ]
It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that f {\displaystyle f} cover all the elements of T {\displaystyle T} and that it preserve orderings, are enough to ensure that f {\displaystyle f} is also one-to-one, for if f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} then (by the assumption that f {\displaystyle f} preserves the order) it would follow that x ≤ y {\displaystyle x\leq y} and y ≤ x {\displaystyle y\leq x} , implying by the definition of a partial order that x = y {\displaystyle x=y} .
Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse. [ 3 ]
An order isomorphism from a partially ordered set to itself is called an order automorphism . [ 4 ]
When an additional algebraic structure is imposed on the posets ( S , ≤ S ) {\displaystyle (S,\leq _{S})} and ( T , ≤ T ) {\displaystyle (T,\leq _{T})} , a function from ( S , ≤ S ) {\displaystyle (S,\leq _{S})} to ( T , ≤ T ) {\displaystyle (T,\leq _{T})} must satisfy additional properties to be regarded as an isomorphism. For example, given two partially ordered groups (po-groups) ( G , ≤ G ) {\displaystyle (G,\leq _{G})} and ( H , ≤ H ) {\displaystyle (H,\leq _{H})} , an isomorphism of po-groups from ( G , ≤ G ) {\displaystyle (G,\leq _{G})} to ( H , ≤ H ) {\displaystyle (H,\leq _{H})} is an order isomorphism that is also a group isomorphism , not merely a bijection that is an order embedding . [ 5 ]
If f {\displaystyle f} is an order isomorphism, then so is its inverse function .
Also, if f {\displaystyle f} is an order isomorphism from ( S , ≤ S ) {\displaystyle (S,\leq _{S})} to ( T , ≤ T ) {\displaystyle (T,\leq _{T})} and g {\displaystyle g} is an order isomorphism from ( T , ≤ T ) {\displaystyle (T,\leq _{T})} to ( U , ≤ U ) {\displaystyle (U,\leq _{U})} , then the function composition of f {\displaystyle f} and g {\displaystyle g} is itself an order isomorphism, from ( S , ≤ S ) {\displaystyle (S,\leq _{S})} to ( U , ≤ U ) {\displaystyle (U,\leq _{U})} . [ 10 ]
Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to the other. [ 11 ] Identity functions, function inverses, and compositions of functions correspond, respectively, to the three defining characteristics of an equivalence relation : reflexivity , symmetry , and transitivity . Therefore, order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into equivalence classes , families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called order types . | https://en.wikipedia.org/wiki/Order_isomorphism |
In calculus , interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem ) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.
The problem for examination is evaluation of an integral of the form
where D is some two-dimensional area in the xy –plane. For some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D .
The method also is applicable to other multiple integrals . [ 1 ] [ 2 ]
Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes a numerical evaluation much easier and more efficient.
Consider the iterated integral
In this expression, the second integral is calculated first with respect to y and x is held constant—a strip of width dx is integrated first over the y -direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction), adding up an infinite amount of rectangles of width dy along the y -axis. This forms a three dimensional slice dx wide along the x -axis, from y = a to y = x along the y -axis, and in the z direction z = h ( y ). Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant. [ 3 ] This integration is as shown in the left panel of Figure 1, but is inconvenient especially when the function h ( y ) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z , and then the result is integrated from y = a to y = z , resulting in:
This result can be seen to be an example of the formula for integration by parts , as stated below: [ 4 ]
Substitute:
Which gives the result.
For application to principal-value integrals , see Whittaker and Watson, [ 5 ] Gakhov, [ 6 ] Lu, [ 7 ] or Zwillinger. [ 8 ] See also the discussion of the Poincaré-Bertrand transformation in Obolashvili. [ 9 ] An example where the order of integration cannot be exchanged is given by Kanwal: [ 10 ]
while:
The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski–Plemelj formula : [ 11 ]
The notation ∫ L ∗ {\displaystyle \int _{L}^{*}} indicates a Cauchy principal value . See Kanwal. [ 10 ]
A discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T.W. Körner. [ 12 ] He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example:
Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair: [ 13 ]
Theorem I — Let f ( x , y ) be a continuous function of constant sign defined for a ≤ x < ∞, c ≤ y < ∞, and let the integrals
Theorem II — Let f ( x , y ) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integrals
The most important theorem for the applications is quoted from Protter and Morrey: [ 14 ]
Theorem — Suppose F is a region given by F = { ( x , y ) : a ≤ x ≤ b , p ( x ) ≤ y ≤ q ( x ) } {\displaystyle F=\left\{(x,\ y):a\leq x\leq b,p(x)\leq y\leq q(x)\right\}\,} where p and q are continuous and p ( x ) ≤ q ( x ) for a ≤ x ≤ b . Suppose that f ( x , y ) is continuous on F . Then
In other words, both iterated integrals, when computable, are equal to the double integral and therefore equal to each other. | https://en.wikipedia.org/wiki/Order_of_integration_(calculus) |
In quantum field theory , an order operator or an order field is a quantum field version of Landau's order parameter whose expectation value characterizes phase transitions . There exists a dual version of it, the disorder operator or disorder field , whose expectation value characterizes a phase transition by indicating the prolific presence of defect or vortex lines in an ordered phase.
The disorder operator is an operator that creates a discontinuity of the ordinary order operators or a monodromy for their values. For example, a 't Hooft operator is a disorder operator. So is the Jordan–Wigner transformation . The concept of a disorder observable was first introduced in the context of 2D Ising spin lattices , where a phase transition between spin-aligned ( magnetized ) and disordered phases happens at some temperature. [ 1 ]
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Order_operator |
The order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics . The order polynomial counts the number of order-preserving maps from a poset to a chain of length n {\displaystyle n} . These order-preserving maps were first introduced by Richard P. Stanley while studying ordered structures and partitions as a Ph.D. student at Harvard University in 1971 under the guidance of Gian-Carlo Rota .
Let P {\displaystyle P} be a finite poset with p {\displaystyle p} elements denoted x , y ∈ P {\displaystyle x,y\in P} , and let [ n ] = { 1 < 2 < … < n } {\displaystyle [n]=\{1<2<\ldots <n\}} be a chain n {\displaystyle n} elements. A map ϕ : P → [ n ] {\displaystyle \phi :P\to [n]} is order-preserving if x ≤ y {\displaystyle x\leq y} implies ϕ ( x ) ≤ ϕ ( y ) {\displaystyle \phi (x)\leq \phi (y)} . The number of such maps grows polynomially with n {\displaystyle n} , and the function that counts their number is the order polynomial Ω ( n ) = Ω ( P , n ) {\displaystyle \Omega (n)=\Omega (P,n)} .
Similarly, we can define an order polynomial that counts the number of strictly order-preserving maps ϕ : P → [ n ] {\displaystyle \phi :P\to [n]} , meaning x < y {\displaystyle x<y} implies ϕ ( x ) < ϕ ( y ) {\displaystyle \phi (x)<\phi (y)} . The number of such maps is the strict order polynomial Ω ∘ ( n ) = Ω ∘ ( P , n ) {\displaystyle \Omega ^{\circ }\!(n)=\Omega ^{\circ }\!(P,n)} . [ 1 ]
Both Ω ( n ) {\displaystyle \Omega (n)} and Ω ∘ ( n ) {\displaystyle \Omega ^{\circ }\!(n)} have degree p {\displaystyle p} . The order-preserving maps generalize the linear extensions of P {\displaystyle P} , the order-preserving bijections ϕ : P ⟶ ∼ [ p ] {\displaystyle \phi :P{\stackrel {\sim }{\longrightarrow }}[p]} . In fact, the leading coefficient of Ω ( n ) {\displaystyle \Omega (n)} and Ω ∘ ( n ) {\displaystyle \Omega ^{\circ }\!(n)} is the number of linear extensions divided by p ! {\displaystyle p!} . [ 2 ]
Letting P {\displaystyle P} be a chain of p {\displaystyle p} elements, we have
Ω ( n ) = ( n + p − 1 p ) = ( ( n p ) ) {\displaystyle \Omega (n)={\binom {n+p-1}{p}}=\left(\!\left({n \atop p}\right)\!\right)} and Ω ∘ ( n ) = ( n p ) . {\displaystyle \Omega ^{\circ }(n)={\binom {n}{p}}.}
There is only one linear extension (the identity mapping), and both polynomials have leading term 1 p ! n p {\displaystyle {\tfrac {1}{p!}}n^{p}} .
Letting P {\displaystyle P} be an antichain of p {\displaystyle p} incomparable elements, we have Ω ( n ) = Ω ∘ ( n ) = n p {\displaystyle \Omega (n)=\Omega ^{\circ }(n)=n^{p}} . Since any bijection ϕ : P ⟶ ∼ [ p ] {\displaystyle \phi :P{\stackrel {\sim }{\longrightarrow }}[p]} is (strictly) order-preserving, there are p ! {\displaystyle p!} linear extensions, and both polynomials reduce to the leading term p ! p ! n p = n p {\displaystyle {\tfrac {p!}{p!}}n^{p}=n^{p}} .
There is a relation between strictly order-preserving maps and order-preserving maps: [ 3 ]
In the case that P {\displaystyle P} is a chain, this recovers the negative binomial identity . There are similar results for the chromatic polynomial and Ehrhart polynomial (see below), all special cases of Stanley's general Reciprocity Theorem . [ 4 ]
The chromatic polynomial P ( G , n ) {\displaystyle P(G,n)} counts the number of proper colorings of a finite graph G {\displaystyle G} with n {\displaystyle n} available colors. For an acyclic orientation σ {\displaystyle \sigma } of the edges of G {\displaystyle G} , there is a natural "downstream" partial order on the vertices V ( G ) {\displaystyle V(G)} implied by the basic relations u > v {\displaystyle u>v} whenever u → v {\displaystyle u\rightarrow v} is a directed edge of σ {\displaystyle \sigma } . (Thus, the Hasse diagram of the poset is a subgraph of the oriented graph σ {\displaystyle \sigma } .) We say ϕ : V ( G ) → [ n ] {\displaystyle \phi :V(G)\rightarrow [n]} is compatible with σ {\displaystyle \sigma } if ϕ {\displaystyle \phi } is order-preserving. Then we have
where σ {\displaystyle \sigma } runs over all acyclic orientations of G, considered as poset structures. [ 5 ]
The order polytope associates a polytope with a partial order. For a poset P {\displaystyle P} with p {\displaystyle p} elements, the order polytope O ( P ) {\displaystyle O(P)} is the set of order-preserving maps f : P → [ 0 , 1 ] {\displaystyle f:P\to [0,1]} , where [ 0 , 1 ] = { t ∈ R ∣ 0 ≤ t ≤ 1 } {\displaystyle [0,1]=\{t\in \mathbb {R} \mid 0\leq t\leq 1\}} is the ordered unit interval, a continuous chain poset. [ 6 ] [ 7 ] More geometrically, we may list the elements P = { x 1 , … , x p } {\displaystyle P=\{x_{1},\ldots ,x_{p}\}} , and identify any mapping f : P → R {\displaystyle f:P\to \mathbb {R} } with the point ( f ( x 1 ) , … , f ( x p ) ) ∈ R p {\displaystyle (f(x_{1}),\ldots ,f(x_{p}))\in \mathbb {R} ^{p}} ; then the order polytope is the set of points ( t 1 , … , t p ) ∈ [ 0 , 1 ] p {\displaystyle (t_{1},\ldots ,t_{p})\in [0,1]^{p}} with t i ≤ t j {\displaystyle t_{i}\leq t_{j}} if x i ≤ x j {\displaystyle x_{i}\leq x_{j}} . [ 2 ]
The Ehrhart polynomial counts the number of integer lattice points inside the dilations of a polytope . Specifically, consider the lattice L = Z n {\displaystyle L=\mathbb {Z} ^{n}} and a d {\displaystyle d} -dimensional polytope K ⊂ R d {\displaystyle K\subset \mathbb {R} ^{d}} with vertices in L {\displaystyle L} ; then we define
the number of lattice points in n K {\displaystyle nK} , the dilation of K {\displaystyle K} by a positive integer scalar n {\displaystyle n} . Ehrhart showed that this is a rational polynomial of degree d {\displaystyle d} in the variable n {\displaystyle n} , provided K {\displaystyle K} has vertices in the lattice. [ 8 ]
In fact, the Ehrhart polynomial of an order polytope is equal to the order polynomial of the original poset (with a shifted argument): [ 2 ] [ 9 ]
L ( O ( P ) , n ) = Ω ( P , n + 1 ) . {\displaystyle L(O(P),n)\ =\ \Omega (P,n{+}1).}
This is an immediate consequence of the definitions, considering the embedding of the ( n + 1 ) {\displaystyle (n{+}1)} -chain poset [ n + 1 ] = { 0 < 1 < ⋯ < n } ⊂ R {\displaystyle [n{+}1]=\{0<1<\cdots <n\}\subset \mathbb {R} } . | https://en.wikipedia.org/wiki/Order_polynomial |
In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval , its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope , meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope.
The order polytope of a partial order should be distinguished from the linear ordering polytope , a polytope defined from a number n {\displaystyle n} as the convex hull of indicator vectors of the sets of edges of n {\displaystyle n} -vertex transitive tournaments . [ 1 ]
A partially ordered set is a pair ( S , ≤ ) {\displaystyle (S,\leq )} where S {\displaystyle S} is an arbitrary set and ≤ {\displaystyle \leq } is a binary relation on pairs of elements of S {\displaystyle S} that is reflexive (for all x ∈ S {\displaystyle x\in S} , x ≤ x {\displaystyle x\leq x} ), antisymmetric (for all x , y ∈ S {\displaystyle x,y\in S} with x ≠ y {\displaystyle x\neq y} at most one of x ≤ y {\displaystyle x\leq y} and y ≤ x {\displaystyle y\leq x} can be true), and transitive (for all x , y , z ∈ S {\displaystyle x,y,z\in S} , if x ≤ y {\displaystyle x\leq y} and y ≤ z {\displaystyle y\leq z} then x ≤ z {\displaystyle x\leq z} ).
A partially ordered set ( S , ≤ ) {\displaystyle (S,\leq )} is said to be finite when S {\displaystyle S} is a finite set . In this case, the collection of all functions f {\displaystyle f} that map S {\displaystyle S} to the real numbers forms a finite-dimensional vector space , with pointwise addition of functions as the vector sum operation. The dimension of the space is just the number of elements of S {\displaystyle S} . The order polytope is defined to be the subset of this space consisting of functions f {\displaystyle f} with the following two properties: [ 2 ] [ 3 ]
For example, for a partially ordered set consisting of two elements x {\displaystyle x} and y {\displaystyle y} , with x ≤ y {\displaystyle x\leq y} in the partial order, the functions f {\displaystyle f} from these points to real numbers can be identified with points ( f ( x ) , f ( y ) ) {\displaystyle (f(x),f(y))} in the Cartesian plane . For this example, the order polytope consists of all points in the ( x , y ) {\displaystyle (x,y)} -plane with 0 ≤ x ≤ y ≤ 1 {\displaystyle 0\leq x\leq y\leq 1} . This is an isosceles right triangle with vertices at (0,0), (0,1), and (1,1).
The vertices of the order polytope consist of monotonic functions from S {\displaystyle S} to { 0 , 1 } {\displaystyle \{0,1\}} . That is, the order polytope is an integral polytope ; it has no vertices with fractional coordinates .
These functions are exactly the indicator functions of upper sets of the partial order. Therefore, the number of vertices equals the number of upper sets. [ 2 ]
The facets of the order polytope are of three types: [ 2 ]
The facets can be considered in a more symmetric way by introducing special elements ⊥ {\displaystyle \bot } below all elements in the partial order and ⊤ {\displaystyle \top } above all elements, mapped by f {\displaystyle f} to 0 and 1 respectively,
and keeping only inequalities of the third type for the resulting augmented partially ordered set. [ 2 ]
More generally, with the same augmentation by ⊥ {\displaystyle \bot } and ⊤ {\displaystyle \top } , the faces of all dimensions of the order polytope correspond 1-to-1 with quotients of the partial order. Each face is congruent to the order polytope of the corresponding quotient partial order. [ 2 ]
The order polytope of a linear order is a special type of simplex called an order simplex or orthoscheme . Each point of the unit cube whose coordinates are all distinct lies in a unique one of these orthoschemes, the order simplex for the linear order of its coordinates.
Because these order simplices are all congruent to each other and (for orders on n {\displaystyle n} elements) there are n ! {\displaystyle n!} different linear orders, the volume of each order simplex is 1 / n ! {\displaystyle 1/n!} . [ 2 ] [ 3 ] More generally, an order polytope can be partitioned into order simplices in a canonical way, with one simplex for each linear extension of the corresponding partially ordered set. [ 2 ] Therefore, the volume of any order polytope is 1 / n ! {\displaystyle 1/n!} multiplied by the number of linear extensions of the corresponding partially ordered set. [ 2 ] [ 3 ] This connection between the number of linear extensions and volume can be used to approximate the number of linear extensions of any partial order efficiently (despite the fact that computing this number exactly is #P-complete ) by applying a randomized polynomial-time approximation scheme for polytope volume. [ 4 ]
The Ehrhart polynomial of the order polytope is a polynomial whose values at integer values x {\displaystyle x} give the number of integer points in a copy of the polytope scaled by a factor of x {\displaystyle x} . For the order polytope, the Ehrhart polynomial equals (after a minor change of variables) the order polynomial of the corresponding partially ordered set. This polynomial encodes several pieces of information about the polytope including its volume (the leading coefficient of the polynomial and its number of vertices (the sum of coefficients). [ 2 ] [ 3 ]
By Birkhoff's representation theorem for finite distributive lattices , the upper sets of any partially ordered set form a finite distributive lattice, and every finite distributive lattice can be represented in this way. [ 5 ] The upper sets correspond to the vertices of the order polytope, so the mapping from upper sets to vertices provides a geometric representation of any finite distributive lattice. Under this representation, the edges of the polytope connect comparable elements of the lattice.
If two functions p {\displaystyle p} and q {\displaystyle q} both belong to the order polytope of a partially ordered set ( S , ≤ ) {\displaystyle (S,\leq )} , then the function p ∧ q {\displaystyle p\wedge q} that maps x {\displaystyle x} to min ( p ( x ) , q ( x ) ) {\displaystyle \min(p(x),q(x))} ,
and the function p ∨ q {\displaystyle p\vee q} that maps x {\displaystyle x} to max ( p ( x ) , q ( x ) ) {\displaystyle \max(p(x),q(x))} both also belong to the order polytope.
The two operations ∧ {\displaystyle \wedge } and ∨ {\displaystyle \vee } give the order polytope the structure of a continuous distributive lattice , within which the finite distributive lattice of Birkhoff's theorem is embedded.
That is, every order polytope is a distributive polytope . The distributive polytopes with all vertex coordinates equal to 0 or 1 are exactly the order polytopes. [ 6 ] | https://en.wikipedia.org/wiki/Order_polytope |
In mathematics , an order topology is a specific topology that can be defined on any totally ordered set . It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays"
for all a, b in X . Provided X has at least two elements, this is equivalent to saying that the open intervals
together with the above rays form a base for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays.
A topological space X is called orderable or linearly orderable [ 1 ] if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space .
The standard topologies on R , Q , Z , and N are the order topologies.
If Y is a subset of X , X a totally ordered set, then Y inherits a total order from X . The set Y therefore has an order topology, the induced order topology . As a subset of X , Y also has a subspace topology . The subspace topology is always at least as fine as the induced order topology, but they are not in general the same.
For example, consider the subset Y = {−1} ∪ {1/ n } n ∈ N of the rationals . Under the subspace topology, the singleton set {−1} is open in Y , but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.
Though the subspace topology of Y = {−1} ∪ {1/ n } n ∈ N in the section above is shown not to be generated by the induced order on Y , it is nonetheless an order topology on Y ; indeed, in the subspace topology every point is isolated (i.e., singleton { y } is open in Y for every y in Y ), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open), and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y , simply modify the induced order on Y by defining −1 to be the greatest element of Y and otherwise keeping the same order for the other points, so that in this new order (call it say < 1 ) we have 1/ n < 1 −1 for all n ∈ N . Then, in the order topology on Y generated by < 1 , every point of Y is isolated in Y .
We wish to define here a subset Z of a linearly ordered topological space X such that no total order on Z generates the subspace topology on Z , so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.
Let Z = { − 1 } ∪ ( 0 , 1 ) {\displaystyle Z=\{-1\}\cup (0,1)} in the real line . The same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z , but one can show that the subspace topology on Z cannot be equal to any order topology on Z .
An argument follows. Suppose by way of contradiction that there is some strict total order < on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z , but rather an arbitrarily given total order on Z that generates the subspace topology).
Let M = Z \ {−1} = (0,1), then M is connected , so M is dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element of Z , then ( − ∞ , − 1 ) {\displaystyle (-\infty ,-1)} and ( − 1 , ∞ ) {\displaystyle (-1,\infty )} separate M , a contradiction. Assume without loss of generality that −1 is the smallest element of Z . Since {−1} is open in Z , there is some point p in M such that the interval (−1, p ) is empty , so p is the minimum of M . Then M \ { p } = (0, p ) ∪ ( p ,1) is not connected with respect to the subspace topology inherited from R . On the other hand, the subspace topology of M \ { p } inherited from the order topology of Z coincides with the order topology of M \ { p } induced by <, which is connected since there are no gaps in M \ { p } and it is dense. This is a contradiction.
Several variants of the order topology can be given:
These topologies naturally arise when working with semicontinuous functions , in that a real-valued function on a topological space is lower semicontinuous if and only if it is continuous when the reals are equipped with the right order. [ 3 ] The ( natural ) compact open topology on the resulting set of continuous functions is sometimes referred to as the semicontinuous topology [ 4 ] .
Additionally, these topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff.
The left order topology is the standard topology used for many set-theoretic purposes on a Boolean algebra . [ clarification needed ]
For any ordinal number λ one can consider the spaces of ordinal numbers
together with the natural order topology. These spaces are called ordinal spaces . (Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0, λ ) and λ + 1 = [0, λ ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; for finite ordinals, the order topology is simply the discrete topology .
When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual (still discrete) topology, while [0,ω] is the one-point compactification of N .
Of particular interest is the case when λ = ω 1 , the set of all countable ordinals, and the first uncountable ordinal . The element ω 1 is a limit point of the subset [0,ω 1 ) even though no sequence of elements in [0,ω 1 ) has the element ω 1 as its limit. In particular, [0,ω 1 ] is not first-countable . The subspace [0,ω 1 ) is first-countable however, since the only point in [0,ω 1 ] without a countable local base is ω 1 . Some further properties include
Any ordinal number can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are well-ordered , so in particular totally ordered ). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept a proper class as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.
The set of limit points of an ordinal α is precisely the set of limit ordinals less than α . Successor ordinals (and zero) less than α are isolated points in α . In particular, the finite ordinals and ω are discrete topological spaces, and no ordinal beyond that is discrete. The ordinal α is compact as a topological space if and only if α is either a successor ordinal or zero.
The closed sets of a limit ordinal α are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.
Any ordinal is, of course, an open subset of any larger ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, α +1 is obtained by taking the one-point compactification of α , and for δ a limit ordinal, δ is equipped with the inductive limit topology. Note that if α is a successor ordinal, then α is compact, in which case its one-point compactification α +1 is the disjoint union of α and a point.
As topological spaces, all the ordinals are Hausdorff and even normal . They are also totally disconnected (connected components are points), scattered (every non-empty subspace has an isolated point; in this case, just take the smallest element), zero-dimensional (the topology has a clopen basis : here, write an open interval ( β , γ ) as the union of the clopen intervals ( β , γ '+1) = [ β +1, γ '] for γ '< γ ). However, they are not extremally disconnected in general (there are open sets, for example the even numbers from ω, whose closure is not open).
The topological spaces ω 1 and its successor ω 1 +1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω 1 +1, the element ω 1 is in the closure of the subset ω 1 even though no sequence of elements in ω 1 has the element ω 1 as its limit: an element in ω 1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.
The space ω 1 is first-countable but not second-countable , and ω 1 +1 has neither of these two properties, despite being compact . It is also worthy of note that any continuous function from ω 1 to R (the real line ) is eventually constant: so the Stone–Čech compactification of ω 1 is ω 1 +1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much larger than ω).
If α is a limit ordinal and X is a set, an α -indexed sequence of elements of X merely means a function from α to X . This concept, a transfinite sequence or ordinal-indexed sequence , is a generalization of the concept of a sequence . An ordinary sequence corresponds to the case α = ω.
If X is a topological space, we say that an α -indexed sequence of elements of X converges to a limit x when it converges as a net , in other words, when given any neighborhood U of x there is an ordinal β < α such that x ι is in U for all ι ≥ β .
Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω 1 is a limit point of ω 1 +1 (because it is a limit ordinal), and, indeed, it is the limit of the ω 1 -indexed sequence which maps any ordinal less than ω 1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω 1 , since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.
However, ordinal-indexed sequences are not powerful enough to replace nets (or filters ) in general: for example, on the Tychonoff plank (the product space ( ω 1 + 1 ) × ( ω + 1 ) {\displaystyle (\omega _{1}+1)\times (\omega +1)} ), the corner point ( ω 1 , ω ) {\displaystyle (\omega _{1},\omega )} is a limit point (it is in the closure) of the open subset ω 1 × ω {\displaystyle \omega _{1}\times \omega } , but it is not the limit of an ordinal-indexed sequence.
This article incorporates material from Order topology on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Order_topology |
In mathematics, specifically in order theory and functional analysis , the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally convex topological vector space (TVS) topology on X {\displaystyle X} for which every order interval is bounded, where an order interval in X {\displaystyle X} is a set of the form [ a , b ] := { z ∈ X : a ≤ z and z ≤ b } {\displaystyle [a,b]:=\left\{z\in X:a\leq z{\text{ and }}z\leq b\right\}} where a {\displaystyle a} and b {\displaystyle b} belong to X . {\displaystyle X.} [ 1 ]
The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of ( X , ≤ ) , {\displaystyle (X,\leq ),} rather than from some topology that X {\displaystyle X} starts out having.
This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of ( X , ≤ ) . {\displaystyle (X,\leq ).} For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. [ 2 ]
The family of all locally convex topologies on X {\displaystyle X} for which every order interval is bounded is non-empty (since it contains the coarsest possible topology on X {\displaystyle X} ) and the order topology is the upper bound of this family. [ 1 ]
A subset of X {\displaystyle X} is a neighborhood of the origin in the order topology if and only if it is convex and absorbs every order interval in X . {\displaystyle X.} [ 1 ] A neighborhood of the origin in the order topology is necessarily an absorbing set because [ x , x ] := { x } {\displaystyle [x,x]:=\{x\}} for all x ∈ X . {\displaystyle x\in X.} [ 1 ]
For every a ≥ 0 , {\displaystyle a\geq 0,} let X a = ⋃ n = 1 ∞ n [ − a , a ] {\displaystyle X_{a}=\bigcup _{n=1}^{\infty }n[-a,a]} and endow X a {\displaystyle X_{a}} with its order topology (which makes it into a normable space).
The set of all X a {\displaystyle X_{a}} 's is directed under inclusion and if X a ⊆ X b {\displaystyle X_{a}\subseteq X_{b}} then the natural inclusion of X a {\displaystyle X_{a}} into X b {\displaystyle X_{b}} is continuous.
If X {\displaystyle X} is a regularly ordered vector space over the reals and if H {\displaystyle H} is any subset of the positive cone C {\displaystyle C} of X {\displaystyle X} that is cofinal in C {\displaystyle C} (e.g. H {\displaystyle H} could be C {\displaystyle C} ), then X {\displaystyle X} with its order topology is the inductive limit of { X a : a ≥ 0 } {\displaystyle \left\{X_{a}:a\geq 0\right\}} (where the bonding maps are the natural inclusions). [ 3 ]
The lattice structure can compensate in part for any lack of an order unit:
Theorem [ 3 ] — Let X {\displaystyle X} be a vector lattice with a regular order and let C {\displaystyle C} denote its positive cone. Then the order topology on X {\displaystyle X} is the finest locally convex topology on X {\displaystyle X} for which C {\displaystyle C} is a normal cone ; it is also the same as the Mackey topology induced on X {\displaystyle X} with respect to the duality ⟨ X , X + ⟩ . {\displaystyle \left\langle X,X^{+}\right\rangle .}
In particular, if ( X , τ ) {\displaystyle (X,\tau )} is an ordered Fréchet lattice over the real numbers then τ {\displaystyle \tau } is the ordered topology on X {\displaystyle X} if and only if the positive cone of X {\displaystyle X} is a normal cone in ( X , τ ) . {\displaystyle (X,\tau ).} [ 3 ]
If X {\displaystyle X} is a regularly ordered vector lattice then the ordered topology is the finest locally convex TVS topology on X {\displaystyle X} making X {\displaystyle X} into a locally convex vector lattice . If in addition X {\displaystyle X} is order complete then X {\displaystyle X} with the order topology is a barreled space and every band decomposition of X {\displaystyle X} is a topological direct sum for this topology. [ 3 ] In particular, if the order of a vector lattice X {\displaystyle X} is regular then the order topology is generated by the family of all lattice seminorms on X . {\displaystyle X.} [ 3 ]
Throughout, ( X , ≤ ) {\displaystyle (X,\leq )} will be an ordered vector space and τ ≤ {\displaystyle \tau _{\leq }} will denote the order topology on X . {\displaystyle X.}
If M {\displaystyle M} is a solid vector subspace of a vector lattice X , {\displaystyle X,} then the order topology of X / M {\displaystyle X/M} is the quotient of the order topology on X . {\displaystyle X.} [ 4 ]
The order topology of a finite product of ordered vector spaces (this product having its canonical order) is identical to the product topology of the topological product of the constituent ordered vector spaces (when each is given its order topology). [ 3 ] | https://en.wikipedia.org/wiki/Order_topology_(functional_analysis) |
An order unit is an element of an ordered vector space which can be used to bound all elements from above. [ 1 ] In this way (as seen in the first example below) the order unit generalizes the unit element in the reals.
According to H. H. Schaefer , "most of the ordered vector spaces occurring in analysis do not have order units." [ 2 ]
For the ordering cone K ⊆ X {\displaystyle K\subseteq X} in the vector space X {\displaystyle X} , the element e ∈ K {\displaystyle e\in K} is an order unit (more precisely a K {\displaystyle K} -order unit) if for every x ∈ X {\displaystyle x\in X} there exists a λ x > 0 {\displaystyle \lambda _{x}>0} such that λ x e − x ∈ K {\displaystyle \lambda _{x}e-x\in K} (that is, x ≤ K λ x e {\displaystyle x\leq _{K}\lambda _{x}e} ). [ 3 ]
The order units of an ordering cone K ⊆ X {\displaystyle K\subseteq X} are those elements in the algebraic interior of K ; {\displaystyle K;} that is, given by core ( K ) . {\displaystyle \operatorname {core} (K).} [ 3 ]
Let X = R {\displaystyle X=\mathbb {R} } be the real numbers and K = R + = { x ∈ R : x ≥ 0 } , {\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} :x\geq 0\},} then the unit element 1 {\displaystyle 1} is an order unit .
Let X = R n {\displaystyle X=\mathbb {R} ^{n}} and K = R + n = { x i ∈ R : for all i = 1 , … , n : x i ≥ 0 } , {\displaystyle K=\mathbb {R} _{+}^{n}=\left\{x_{i}\in \mathbb {R} :{\text{ for all }}i=1,\ldots ,n:x_{i}\geq 0\right\},} then the unit element 1 → = ( 1 , … , 1 ) {\displaystyle {\vec {1}}=(1,\ldots ,1)} is an order unit .
Each interior point of the positive cone of an ordered topological vector space is an order unit. [ 2 ]
Each order unit of an ordered TVS is interior to the positive cone for the order topology. [ 2 ]
If ( X , ≤ ) {\displaystyle (X,\leq )} is a preordered vector space over the reals with order unit u , {\displaystyle u,} then the map p ( x ) := inf { t ∈ R : x ≤ t u } {\displaystyle p(x):=\inf\{t\in \mathbb {R} :x\leq tu\}} is a sublinear functional . [ 4 ]
Suppose ( X , ≤ ) {\displaystyle (X,\leq )} is an ordered vector space over the reals with order unit u {\displaystyle u} whose order is Archimedean and let U = [ − u , u ] . {\displaystyle U=[-u,u].} Then the Minkowski functional p U {\displaystyle p_{U}} of U , {\displaystyle U,} defined by p U ( x ) := inf { r > 0 : x ∈ r [ − u , u ] } , {\displaystyle p_{U}(x):=\inf\{r>0:x\in r[-u,u]\},} is a norm called the order unit norm .
It satisfies p U ( u ) = 1 {\displaystyle p_{U}(u)=1} and the closed unit ball determined by p U {\displaystyle p_{U}} is equal to [ − u , u ] ; {\displaystyle [-u,u];} that is, [ − u , u ] = { x ∈ X : p U ( x ) ≤ 1 } . {\displaystyle [-u,u]=\left\{x\in X:p_{U}(x)\leq 1\right\}.} [ 4 ] | https://en.wikipedia.org/wiki/Order_unit |
The ordered exponential , also called the path-ordered exponential , is a mathematical operation defined in non-commutative algebras , equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. It is a kind of product integral , or Volterra integral.
Let A be an algebra over a field K , and a ( t ) be an element of A parameterized by the real numbers,
The parameter t in a ( t ) is often referred to as the time parameter in this context.
The ordered exponential of a is denoted
where the term n = 0 is equal to 1 and where T {\displaystyle {\mathcal {T}}} is the time-ordering operator . It is a higher-order operation that ensures the exponential is time-ordered, so that any product of a ( t ) that occurs in the expansion of the exponential is ordered such that the value of t is increasing from right to left of the product. For example:
Time ordering is required, as products in the algebra are not necessarily commutative.
The operation maps a parameterized element onto another parameterized element, or symbolically,
There are various ways to define this integral more rigorously.
The ordered exponential can be defined as the left product integral of the infinitesimal exponentials, or equivalently, as an ordered product of exponentials in the limit as the number of terms grows to infinity:
where the time moments { t 0 , ..., t N } are defined as t i ≡ i Δ t for i = 0, ..., N , and Δ t ≡ t / N .
The ordered exponential is in fact a geometric integral [ broken anchor ] . [ 1 ] [ 2 ] [ 3 ]
The ordered exponential is unique solution of the initial value problem :
The ordered exponential is the solution to the integral equation :
This equation is equivalent to the previous initial value problem.
The ordered exponential can be defined as an infinite sum,
This can be derived by recursively substituting the integral equation into itself.
Given a manifold M {\displaystyle M} where for a e ∈ T M {\displaystyle e\in TM} with group transformation g : e ↦ g e {\displaystyle g:e\mapsto ge} it holds at a point x ∈ M {\displaystyle x\in M} :
Here, d {\displaystyle d} denotes exterior differentiation and J ( x ) {\displaystyle \operatorname {J} (x)} is the connection operator (1-form field) acting on e ( x ) {\displaystyle e(x)} . When integrating above equation it holds (now, J ( x ) {\displaystyle \operatorname {J} (x)} is the connection operator expressed in a coordinate basis)
with the path-ordering operator P {\displaystyle \operatorname {P} } that orders factors in order of the path γ ( t ) ∈ M {\displaystyle \gamma (t)\in M} . For the special case that J ( x ) {\displaystyle \operatorname {J} (x)} is an antisymmetric operator and γ {\displaystyle \gamma } is an infinitesimal rectangle with edge lengths | u | , | v | {\displaystyle |u|,|v|} and corners at points x , x + u , x + u + v , x + v , {\displaystyle x,x+u,x+u+v,x+v,} above expression simplifies as follows :
Hence, it holds the group transformation identity OE [ − J ] ↦ g OE [ J ] g − 1 {\displaystyle \operatorname {OE} [-\operatorname {J} ]\mapsto g\operatorname {OE} [\operatorname {J} ]g^{-1}} . If − J ( x ) {\displaystyle -\operatorname {J} (x)} is a smooth connection, expanding above quantity to second order in infinitesimal quantities | u | , | v | {\displaystyle |u|,|v|} one obtains for the ordered exponential the identity with a correction term that is proportional to the curvature tensor . | https://en.wikipedia.org/wiki/Ordered_exponential |
In mathematics , an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
An exponential E {\textstyle E} on an ordered field K {\textstyle K} is a strictly increasing isomorphism of the additive group of K {\textstyle K} onto the multiplicative group of positive elements of K {\textstyle K} . The ordered field K {\displaystyle K\,} together with the additional function E {\displaystyle E\,} is called an ordered exponential field.
A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential E {\textstyle E} . For any formally exponential field K {\textstyle K} , one can choose an exponential E {\textstyle E} on K {\textstyle K} such that 1 + 1 / n < E ( 1 ) < n {\textstyle 1+1/n<E(1)<n} for some natural number n {\textstyle n} . [ 3 ] | https://en.wikipedia.org/wiki/Ordered_exponential_field |
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry , omitting the basic notion of measurement . Ordered geometry is a fundamental geometry forming a common framework for affine , Euclidean , absolute , and hyperbolic geometry (but not for projective geometry).
Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). [ 1 ] : 176 Euclid anticipated Pasch's approach in definition 4 of The Elements : "a straight line is a line which lies evenly with the points on itself". [ 2 ]
The only primitive notions in ordered geometry are points A , B , C , ... and the ternary relation of intermediacy [ ABC ] which can be read as " B is between A and C ".
The segment AB is the set of points P such that [ APB ].
The interval AB is the segment AB and its end points A and B .
The ray A / B (read as "the ray from A away from B ") is the set of points P such that [ PAB ].
The line AB is the interval AB and the two rays A / B and B / A . Points on the line AB are said to be collinear .
An angle consists of a point O (the vertex ) and two non-collinear rays out from O (the sides ).
A triangle is given by three non-collinear points (called vertices ) and their three segments AB , BC , and CA .
If three points A , B , and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC .
If four points A , B , C , and D are non-coplanar, then a space ( 3-space ) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD .
These axioms are closely related to Hilbert's axioms of order . For a comprehensive survey of axiomatizations of ordered geometry see Pambuccian (2011). [ 3 ]
The Sylvester–Gallai theorem can be proven within ordered geometry. [ 4 ] [ 1 ] : 181, 2
Gauss , Bolyai , and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry. [ 1 ] : 189, 90
Theorem (existence of parallelism): Given a point A and a line r , not through A , there exist exactly two limiting rays from A in the plane Ar which do not meet r . So there is a parallel line through A which does not meet r .
Theorem (transmissibility of parallelism): The parallelism of a ray and a line is preserved by adding or subtracting a segment from the beginning of a ray.
The transitivity of parallelism cannot be proven in ordered geometry. [ 5 ] Therefore, the "ordered" concept of parallelism does not form an equivalence relation on lines. | https://en.wikipedia.org/wiki/Ordered_geometry |
In mathematics , an ordered pair , denoted ( a , b ), is a pair of objects in which their order is significant. The ordered pair ( a , b ) is different from the ordered pair ( b , a ), unless a = b . In contrast, the unordered pair , denoted { a , b }, always equals the unordered pair { b , a }.
Ordered pairs are also called 2-tuples , or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors . (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space .)
The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n -tuples (ordered lists of n objects). For example, the ordered triple ( a , b , c ) can be defined as ( a , ( b , c )), i.e., as one pair nested in another.
In the ordered pair ( a , b ), the object a is called the first entry , and the object b the second entry of the pair. Alternatively, the objects are called the first and second components , the first and second coordinates , or the left and right projections of the ordered pair.
Cartesian products and binary relations (and hence functions ) are defined in terms of ordered pairs, cf. picture.
Let ( a 1 , b 1 ) {\displaystyle (a_{1},b_{1})} and ( a 2 , b 2 ) {\displaystyle (a_{2},b_{2})} be ordered pairs. Then the characteristic (or defining ) property of the ordered pair is: ( a 1 , b 1 ) = ( a 2 , b 2 ) if and only if a 1 = a 2 and b 1 = b 2 . {\displaystyle (a_{1},b_{1})=(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.}
The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B , and written A × B . A binary relation between sets A and B is a subset of A × B .
The ( a , b ) notation may be used for other purposes, most notably as denoting open intervals on the real number line . In such situations, the context will usually make it clear which meaning is intended. [ 1 ] [ 2 ] For additional clarification, the ordered pair may be denoted by the variant notation ⟨ a , b ⟩ {\textstyle \langle a,b\rangle } , but this notation also has other uses.
The left and right projection of a pair p is usually denoted by π 1 ( p ) and π 2 ( p ), or by π ℓ ( p ) and π r ( p ), respectively.
In contexts where arbitrary n -tuples are considered, π n i ( t ) is a common notation for the i -th component of an n -tuple t .
In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as
For any two objects a and b , the ordered pair ( a , b ) is a notation specifying the two objects a and b , in that order. [ 3 ]
This is usually followed by a comparison to a set of two elements; pointing out that in a set a and b must be different, but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn't matter, in an ordered pair changing the order of distinct entries changes the ordered pair.
This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of order . However, as is sometimes pointed out, no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner. [ 4 ]
A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion , whose associated axiom is the characteristic property. This was the approach taken by the N. Bourbaki group in its Theory of Sets , published in 1954. However, this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed. [ 3 ]
Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the second edition of Bourbaki's Theory of Sets , published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise.
If one agrees that set theory is an appealing foundation of mathematics , then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. [ 5 ] Several set-theoretic definitions of the ordered pair are given below (see also Diepert). [ 6 ]
Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914: [ 7 ] ( a , b ) := { { { a } , ∅ } , { { b } } } . {\displaystyle \left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset \right\},\,\left\{\left\{b\right\}\right\}\right\}.} He observed that this definition made it possible to define the types of Principia Mathematica as sets. Principia Mathematica had taken types, and hence relations of all arities, as primitive .
Wiener used {{ b }} instead of { b } to make the definition compatible with type theory where all elements in a class must be of the same "type". With b nested within an additional set, its type is equal to { { a } , ∅ } {\displaystyle \{\{a\},\emptyset \}} 's.
About the same time as Wiener (1914), Felix Hausdorff proposed his definition: ( a , b ) := { { a , 1 } , { b , 2 } } {\displaystyle (a,b):=\left\{\{a,1\},\{b,2\}\right\}} "where 1 and 2 are two distinct objects different from a and b." [ 8 ]
In 1921 Kazimierz Kuratowski offered the now-accepted definition [ 9 ] [ 10 ] of the ordered pair ( a , b ): ( a , b ) K := { { a } , { a , b } } . {\displaystyle (a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.} When the first and the second coordinates are identical, the definition obtains: ( x , x ) K = { { x } , { x , x } } = { { x } , { x } } = { { x } } {\displaystyle (x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}}
Given some ordered pair p , the property " x is the first coordinate of p " can be formulated as: ∀ Y ∈ p : x ∈ Y . {\displaystyle \forall Y\in p:x\in Y.} The property " x is the second coordinate of p " can be formulated as: ( ∃ Y ∈ p : x ∈ Y ) ∧ ( ∀ Y 1 , Y 2 ∈ p : ( x ∈ Y 1 ∧ x ∈ Y 2 ) → Y 1 = Y 2 ) . {\displaystyle (\exists Y\in p:x\in Y)\land (\forall Y_{1},Y_{2}\in p:(x\in Y_{1}\land x\in Y_{2})\rightarrow Y_{1}=Y_{2}).} In the case that the left and right coordinates are identical, the right conjunct ( ∀ Y 1 , Y 2 ∈ p : ( x ∈ Y 1 ∧ x ∈ Y 2 ) → Y 1 = Y 2 ) {\displaystyle (\forall Y_{1},Y_{2}\in p:(x\in Y_{1}\land x\in Y_{2})\rightarrow Y_{1}=Y_{2})} is trivially true, since Y 1 = Y 2 {\displaystyle Y_{1}=Y_{2}} is the case.
If p = ( x , y ) = { { x } , { x , y } } {\displaystyle p=(x,y)=\{\{x\},\{x,y\}\}} then:
This is how we can extract the first coordinate of a pair (using the iterated-operation notation for arbitrary intersection and arbitrary union ): π 1 ( p ) = ⋃ ⋂ p = ⋃ { x } = x . {\displaystyle \pi _{1}(p)=\bigcup \bigcap p=\bigcup \{x\}=x.}
This is how the second coordinate can be extracted: π 2 ( p ) = ⋃ { a ∈ ⋃ p | ⋃ p ≠ ⋂ p → a ∉ ⋂ p } = ⋃ { a ∈ { x , y } | { x , y } ≠ { x } → a ∉ { x } } = ⋃ { y } = y . {\displaystyle \pi _{2}(p)=\bigcup \left\{\left.a\in \bigcup p\,\right|\,\bigcup p\neq \bigcap p\rightarrow a\notin \bigcap p\right\}=\bigcup \left\{\left.a\in \{x,y\}\,\right|\,\{x,y\}\neq \{x\}\rightarrow a\notin \{x\}\right\}=\bigcup \{y\}=y.}
(if x ≠ y {\displaystyle x\neq y} , then the set { y } {\displaystyle \{y\}} could be obtained more simply: { y } = { a ∈ { x , y } | a ∉ { x } } {\displaystyle \{y\}=\{\left.a\in \{x,y\}\,\right|\,a\notin \{x\}\}} , but the previous formula also takes into account the case when x = y {\displaystyle x=y} .)
Note that π 1 {\displaystyle \pi _{1}} and π 2 {\displaystyle \pi _{2}} are generalized functions , in the sense that their domains and codomains are proper classes .
The above Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that ( a , b ) = ( x , y ) ↔ ( a = x ) ∧ ( b = y ) {\displaystyle (a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)} . In particular, it adequately expresses 'order', in that ( a , b ) = ( b , a ) {\displaystyle (a,b)=(b,a)} is false unless b = a {\displaystyle b=a} . There are other definitions, of similar or lesser complexity, that are equally adequate:
The reverse definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definition short is so-called because it requires two rather than three pairs of braces . Proving that short satisfies the characteristic property requires the Zermelo–Fraenkel set theory axiom of regularity . [ 12 ] Moreover, if one uses von Neumann's set-theoretic construction of the natural numbers , then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0) short . Yet another disadvantage of the short pair is the fact that, even if a and b are of the same type, the elements of the short pair are not. (However, if a = b then the short version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".)
Prove: ( a , b ) = ( c , d ) if and only if a = c and b = d .
Kuratowski : If . If a = c and b = d , then {{ a }, { a , b }} = {{ c }, { c , d }}. Thus ( a, b ) K = ( c , d ) K .
Only if . Two cases: a = b , and a ≠ b .
If a = b :
If a ≠ b , then ( a , b ) K = ( c , d ) K implies {{ a }, { a , b }} = {{ c }, { c , d }}.
Reverse : ( a, b ) reverse = {{ b }, { a, b }} = {{ b }, { b, a }} = ( b, a ) K .
If . If ( a, b ) reverse = ( c, d ) reverse ,
( b, a ) K = ( d, c ) K . Therefore, b = d and a = c .
Only if . If a = c and b = d , then {{ b }, { a, b }} = {{ d }, { c, d }}.
Thus ( a, b ) reverse = ( c, d ) reverse .
Short: [ 13 ]
If : If a = c and b = d , then { a , { a, b }} = { c , { c, d }}. Thus ( a, b ) short = ( c, d ) short .
Only if : Suppose { a , { a, b }} = { c , { c, d }}.
Then a is in the left hand side, and thus in the right hand side.
Because equal sets have equal elements, one of a = c or a = { c, d } must be the case.
Again, we see that { a, b } = c or { a, b } = { c, d }.
Rosser (1953) [ 14 ] employed a definition of the ordered pair due to Quine which requires a prior definition of the natural numbers . Let N {\displaystyle \mathbb {N} } be the set of natural numbers and define first σ ( x ) := { x , if x ∉ N , x + 1 , if x ∈ N . {\displaystyle \sigma (x):={\begin{cases}x,&{\text{if }}x\notin \mathbb {N} ,\\x+1,&{\text{if }}x\in \mathbb {N} .\end{cases}}} The function σ {\displaystyle \sigma } increments its argument if it is a natural number and leaves it as is otherwise; the number 0 does not appear in the range of σ {\displaystyle \sigma } .
As x ∖ N {\displaystyle x\setminus \mathbb {N} } is the set of the elements of x {\displaystyle x} not in N {\displaystyle \mathbb {N} } go on with φ ( x ) := σ [ x ] = { σ ( α ) ∣ α ∈ x } = ( x ∖ N ) ∪ { n + 1 : n ∈ ( x ∩ N ) } . {\displaystyle \varphi (x):=\sigma [x]=\{\sigma (\alpha )\mid \alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.} This is the set image of a set x {\displaystyle x} under σ {\displaystyle \sigma } , sometimes denoted by σ ″ x {\displaystyle \sigma ''x} as well. Applying function φ {\displaystyle \varphi } to a set x simply increments every natural number in it. In particular, φ ( x ) {\displaystyle \varphi (x)} never contains contain the number 0, so that for any sets x and y , φ ( x ) ≠ { 0 } ∪ φ ( y ) . {\displaystyle \varphi (x)\neq \{0\}\cup \varphi (y).} Further, define ψ ( x ) := σ [ x ] ∪ { 0 } = φ ( x ) ∪ { 0 } . {\displaystyle \psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}.} By this, ψ ( x ) {\displaystyle \psi (x)} does always contain the number 0.
Finally, define the ordered pair ( A , B ) as the disjoint union ( A , B ) := φ [ A ] ∪ ψ [ B ] = { φ ( a ) : a ∈ A } ∪ { φ ( b ) ∪ { 0 } : b ∈ B } . {\displaystyle (A,B):=\varphi [A]\cup \psi [B]=\{\varphi (a):a\in A\}\cup \{\varphi (b)\cup \{0\}:b\in B\}.} (which is φ ″ A ∪ ψ ″ B {\displaystyle \varphi ''A\cup \psi ''B} in alternate notation).
Extracting all the elements of the pair that do not contain 0 and undoing φ {\displaystyle \varphi } yields A . Likewise, B can be recovered from the elements of the pair that do contain 0. [ 15 ]
For example, the pair ( { { a , 0 } , { b , c , 1 } } , { { d , 2 } , { e , f , 3 } } ) {\displaystyle (\{\{a,0\},\{b,c,1\}\},\{\{d,2\},\{e,f,3\}\})} is encoded as { { a , 1 } , { b , c , 2 } , { d , 3 , 0 } , { e , f , 4 , 0 } } {\displaystyle \{\{a,1\},\{b,c,2\},\{d,3,0\},\{e,f,4,0\}\}} provided a , b , c , d , e , f ∉ N {\displaystyle a,b,c,d,e,f\notin \mathbb {N} } .
In type theory and in outgrowths thereof such as the axiomatic set theory NF , the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a function , defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments. This definition works only if the set of natural numbers is infinite. This is the case in NF , but not in type theory or in NFU . J. Barkley Rosser showed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies the axiom of infinity . For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998). [ 16 ]
Early in the development of the set theory, before paradoxes were discovered, Cantor followed Frege by defining the ordered pair of two sets as the class of all relations that hold between these sets, assuming that the notion of relation is primitive: [ 17 ] ( x , y ) = { R : x R y } . {\displaystyle (x,y)=\{R:xRy\}.}
This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the cardinal of a set as the class of all sets equipotent with the given set. [ 18 ]
Morse–Kelley set theory makes free use of proper classes . [ 19 ] Morse defined the ordered pair so that its projections could be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then redefined the pair ( x , y ) = ( { 0 } × s ( x ) ) ∪ ( { 1 } × s ( y ) ) {\displaystyle (x,y)=(\{0\}\times s(x))\cup (\{1\}\times s(y))} where the component Cartesian products are Kuratowski pairs of sets and where s ( x ) = { ∅ } ∪ { { t } ∣ t ∈ x } {\displaystyle s(x)=\{\emptyset \}\cup \{\{t\}\mid t\in x\}}
This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above also admits proper classes as projections. Similarly the triple is defined as a 3-tuple as follows: ( x , y , z ) = ( { 0 } × s ( x ) ) ∪ ( { 1 } × s ( y ) ) ∪ ( { 2 } × s ( z ) ) {\displaystyle (x,y,z)=(\{0\}\times s(x))\cup (\{1\}\times s(y))\cup (\{2\}\times s(z))}
The use of the singleton set s ( x ) {\displaystyle s(x)} which has an inserted empty set allows tuples to have the uniqueness property that if a is an n -tuple and b is an m -tuple and a = b then n = m . Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs.
A category-theoretic product A × B in a category of sets represents the set of ordered pairs, with the first element coming from A and the second coming from B . In this context the characteristic property above is a consequence of the universal property of the product and the fact that elements of a set X can be identified with morphisms from 1 (a one element set) to X . While different objects may have the universal property, they are all naturally isomorphic . | https://en.wikipedia.org/wiki/Ordered_pair |
In mathematical notation , ordered set operators indicate whether an object precedes or succeeds another. These relationship operators are denoted by the unicode symbols U+227A-F, along with symbols located unicode blocks U+228x through U+22Ex.
In Political science and Decision theory , order relations are typically used in the context of an agent's choice, for example the preferences of a voter over several political candidates. | https://en.wikipedia.org/wiki/Ordered_set_operators |
In mathematics, specifically in functional analysis and order theory , an ordered topological vector space , also called an ordered TVS , is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone C := { x ∈ X : x ≥ 0 } {\displaystyle C:=\left\{x\in X:x\geq 0\right\}} is a closed subset of X . [ 1 ] Ordered TVSes have important applications in spectral theory .
If C is a cone in a TVS X then C is normal if U = [ U ] C {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C}} , where U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, [ U ] C = { [ U ] : U ∈ U } {\displaystyle \left[{\mathcal {U}}\right]_{C}=\left\{\left[U\right]:U\in {\mathcal {U}}\right\}} , and [ U ] C := ( U + C ) ∩ ( U − C ) {\displaystyle [U]_{C}:=\left(U+C\right)\cap \left(U-C\right)} is the C -saturated hull of a subset U of X . [ 2 ]
If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent: [ 2 ]
and if X is a vector space over the reals then also: [ 2 ]
If the topology on X is locally convex then the closure of a normal cone is a normal cone. [ 2 ]
If C is a normal cone in X and B is a bounded subset of X then [ B ] C {\displaystyle \left[B\right]_{C}} is bounded; in particular, every interval [ a , b ] {\displaystyle [a,b]} is bounded. [ 2 ] If X is Hausdorff then every normal cone in X is a proper cone. [ 2 ] | https://en.wikipedia.org/wiki/Ordered_topological_vector_space |
Ordered Two-Template Relay ( OTTR ) is a library preparation technique used to improve quantitation of highly modified non-coding RNA (ncRNA) species, [ 1 ] which have been difficult to characterize using traditional cDNA sequencing approaches. OTTR leverages a retroelement reverse transcriptase (RT), termed BoMoC, with template jumping properties and high processivity across modified RNA templates, [ 1 ] [ 2 ] to generate cDNA products for next-generation sequencing (NGS). Overall, OTTR offers a streamlined approach for cDNA library production of full-length and modified ncRNA targets. [ 1 ] [ 3 ]
Cellular ncRNA pools are known to be dynamically regulated and can have high degrees of variation between different cell types and developmental stages. [ 4 ] Dysregulation of transfer RNAs ( tRNAs ), a type of ncRNA, has been linked to a diverse array of detrimental physiological conditions including neurological diseases and cancer . [ 5 ] [ 6 ] While characterization of transfer RNA (tRNAs) diversity is relevant to disease, current library preparation approaches are limited in their ability to capture highly modified tRNA bases, which block reverse transcriptase and interfere with the production of full-length cDNA intermediates needed for sequencing. [ 7 ] To date, several cDNA library preparation techniques, including OTTR, [ 1 ] have attempted to overcome these problems and improve our ability to characterize ncRNA pools.
Reverse transcriptases (RTs) are polymerases capable of synthesizing complementary DNA (cDNA) using either RNA and DNA templates and have become essential biotechnology tools in both clinical and laboratory settings. [ 8 ] OTTR makes use of a unique non-long terminal repeat (LTR) retroelement RT called BoMoC, due to its specialized ability to synthesize cDNA opposite templates containing modified bases or sugar backbones and being highly processive across discontinuous RNA templates. Originally purified from the silk moth Bombyx mori , OTTR BoMoC is N-terminally truncated and modified to introduce a stabilizing active site mutation. [ 2 ]
Initially, the RNA or DNA of interest is purified and denoted as the input template (IT). The OTTR library preparation protocol require the IT is incubated with BoMoC, which uses terminal transferase or ‘tailing’ activity' [ 2 ] to add a chain-terminating dideoxynucleotide base, ddRTP (ddATP or ddGTP), to the 3’ end of the IT, in the presence of manganese (Mn 2+ ). To promote cDNA synthesis in the steps to follow, the divalent cation source is switched to magnesium (Mg 2+ ), free ddRTPs are inactivated and dNTPs are added. [ 1 ] Next, a RNA-DNA duplex, containing a 3’ +1Y (dTTP or dCTP) base overhang is added, allowing the RNA-DNA duplex to base pair with the ddRTP-containing IT. BoMoC extends from the 3’ of the +1Y base across the IT. Following this, dNTP concentrations are altered to encourage the addition of dGTP to the 3’ end of the cDNA IT through the non-template nucleotide addition (NTA) activity of BoMoC. [ 2 ] A 3’ adaptor template (AT) containing a 3’ dCTP is added to the reaction, promoting base pairing between the cDNA 3’ G overhang and the 3’C base of the AT and subsequent extension by BoMoC. When using RNA as the input template, addition of RNase A and RNase H is needed to degrade remaining RNA, leaving only the cDNA template. [ 1 ]
Based on the sequencing approach used, the 5’ and 3’ adaptor sequences used to tag the cDNA library can be altered as needed. Previously, dual adapter-tagged cDNA libraries have been characterized using Illumina NGS. [ 1 ] Low-cycle PCR can also be used to index universal adaptor cDNA libraries following the RT reaction. Alternatively, full-length adaptor sequences of choice can be included in the 5’ and 3’ adaptors used in the initial RT reaction. [ 1 ]
To date, OTTR has been used for quantifying tRNA species. This method provides a reliable and precise quantification of tRNAs and allows for the detection of changes in tRNA levels under different physiological conditions. Therefore, this method is useful for a variety of research applications in the field of molecular biology and genetics.
As tRNAs are essential components of translation, the ability to quantify tRNA levels accurately is crucial for understanding how the translation machinery is regulated. Using OTTR, researchers can determine changes in tRNA levels in response to different growth conditions, environmental stress, or genetic modifications. This information can help to identify factors that affect tRNA abundance and their potential roles in modulating translation.
In particular, the OTTR protocol has been used to characterize the small RNA composition of mammalian sperm populations. [ 9 ] This work revealed that sperm small RNA pools are composed largely of rRNA fragments and both 5' and 3' tRNA halves from the majority of tRNAs. This improved understanding of sperm payload composition has implications for our understanding of the biogenesis of structural RNA fragments in the male germline, as well as the biochemical nature of the RNAs delivered to the zygote upon fertilization.
The OTTR protocol could also be used in the study of piwi-interacting RNAs (piRNAs), which is another important classes of small RNAs. While the applications of OTTR have mainly been focused on tRNA fragment detection, OTTR has also been shown to perform well in the capture of miRNAs [ 9 ] which could be useful for the study of miRNA expression patterns in different cell types or under different conditions. [ 10 ]
Future applications of protocols similar to OTTR are also being considered in the development of diagnostic assays for small RNAs. The high fidelity of the OTTR protocol in capturing small RNAs could make it an attractive option when paired with liquid biopsy assays, where circulating small RNAs are analyzed as biomarkers for various diseases. [ 10 ] | https://en.wikipedia.org/wiki/Ordered_two-template_relay |
This page lists examples of the orders of magnitude of molar concentration . Source values are parenthesized where unit conversions were performed.
M denotes the non-SI unit molar : | https://en.wikipedia.org/wiki/Orders_of_magnitude_(molar_concentration) |
To help compare different orders of magnitude , the following list describes various speed levels between approximately 2.2 × 10 −18 m/s and 3.0 × 10 8 m/s (the speed of light ). Values in bold are exact. | https://en.wikipedia.org/wiki/Orders_of_magnitude_(speed) |
In proof theory , ordinal analysis assigns ordinals (often large countable ordinals ) to mathematical theories as a measure of their strength.
If theories have the same proof-theoretic ordinal they are often equiconsistent , and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or Δ 2 1 {\displaystyle \Delta _{2}^{1}} functions of the theory. [ 1 ]
The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε 0 . See Gentzen's consistency proof .
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations.
The proof-theoretic ordinal of such a theory T {\displaystyle T} is the supremum of the order types of all ordinal notations (necessarily recursive , see next section) that the theory can prove are well founded —the supremum of all ordinals α {\displaystyle \alpha } for which there exists a notation o {\displaystyle o} in Kleene's sense such that T {\displaystyle T} proves that o {\displaystyle o} is an ordinal notation. Equivalently, it is the supremum of all ordinals α {\displaystyle \alpha } such that there exists a recursive relation R {\displaystyle R} on ω {\displaystyle \omega } (the set of natural numbers) that well-orders it with ordinal α {\displaystyle \alpha } and such that T {\displaystyle T} proves transfinite induction of arithmetical statements for R {\displaystyle R} .
Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem T {\displaystyle T} of Z 2 to "prove α {\displaystyle \alpha } well-ordered", we instead construct an ordinal notation ( A , < ~ ) {\displaystyle (A,{\tilde {<}})} with order type α {\displaystyle \alpha } . T {\displaystyle T} can now work with various transfinite induction principles along ( A , < ~ ) {\displaystyle (A,{\tilde {<}})} , which substitute for reasoning about set-theoretic ordinals.
However, some pathological notation systems exist that are unexpectedly difficult to work with. For example, Rathjen gives a primitive recursive notation system ( N , < T ) {\displaystyle (\mathbb {N} ,<_{T})} that is well-founded iff PA is consistent, [ 2 ] p. 3 despite having order type ω {\displaystyle \omega } - including such a notation in the ordinal analysis of PA would result in the false equality P T O ( P A ) = ω {\displaystyle {\mathsf {PTO(PA)}}=\omega } .
Since an ordinal notation must be recursive, the proof-theoretic ordinal of any theory is less than or equal to the Church–Kleene ordinal ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} . In particular, the proof-theoretic ordinal of an inconsistent theory is equal to ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} , because an inconsistent theory trivially proves that all ordinal notations are well-founded.
For any theory that's both Σ 1 1 {\displaystyle \Sigma _{1}^{1}} -axiomatizable and Π 1 1 {\displaystyle \Pi _{1}^{1}} -sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the Σ 1 1 {\displaystyle \Sigma _{1}^{1}} bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by Π 1 1 {\displaystyle \Pi _{1}^{1}} -soundness. Thus the proof-theoretic ordinal of a Π 1 1 {\displaystyle \Pi _{1}^{1}} -sound theory that has a Σ 1 1 {\displaystyle \Sigma _{1}^{1}} axiomatization will always be a (countable) recursive ordinal , that is, strictly less than ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} . [ 2 ] Theorem 2.21
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations.
Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes Π 2 1 − C A 0 {\displaystyle \Pi _{2}^{1}-CA_{0}} , full second-order arithmetic ( Π ∞ 1 − C A 0 {\displaystyle \Pi _{\infty }^{1}-CA_{0}} ) and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF.
A U T − K P l r {\displaystyle \mathbf {AUT-KPl} ^{r}} , A U T − K P l r + K P i r {\displaystyle \mathbf {AUT-KPl} ^{r}+\mathbf {KPi} ^{r}} [ 38 ] : 72
This is a list of symbols used in this table:
This is a list of the abbreviations used in this table:
In general, a subscript 0 means that the induction scheme is restricted to a single set induction axiom.
A superscript zero indicates that ∈ {\displaystyle \in } -induction is removed (making the theory significantly weaker). | https://en.wikipedia.org/wiki/Ordinal_analysis |
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. [ 1 ] : 2 These data exist on an ordinal scale , one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a ranking . [ 2 ] It also differs from the interval scale and ratio scale by not having category widths that represent equal increments of the underlying attribute. [ 3 ]
A well-known example of ordinal data is the Likert scale . An example of a Likert scale is: [ 4 ] : 685
Examples of ordinal data are often found in questionnaires: for example, the survey question "Is your general health poor, reasonable, good, or excellent?" may have those answers coded respectively as 1, 2, 3, and 4. Sometimes data on an interval scale or ratio scale are grouped onto an ordinal scale: for example, individuals whose income is known might be grouped into the income categories $0–$19,999, $20,000–$39,999, $40,000–$59,999, ..., which then might be coded as 1, 2, 3, 4, .... Other examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework. [ 5 ]
Ordinal data analysis requires a different set of analyses than other qualitative variables. These methods incorporate the natural ordering of the variables in order to avoid loss of power. [ 1 ] : 88 Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including the median or mode, are generally more appropriate. [ 6 ]
Stevens (1946) argued that, because the assumption of equal distance between categories does not hold for ordinal data, the use of means and standard deviations for description of ordinal distributions and of inferential statistics based on means and standard deviations was not appropriate. Instead, positional measures like the median and percentiles, in addition to descriptive statistics appropriate for nominal data (number of cases, mode, contingency correlation), should be used. [ 3 ] : 678 Nonparametric methods have been proposed as the most appropriate procedures for inferential statistics involving ordinal data (e.g, Kendall's W , Spearman's rank correlation coefficient , etc.), especially those developed for the analysis of ranked measurements. [ 5 ] : 25–28 However, the use of parametric statistics for ordinal data may be permissible with certain caveats to take advantage of the greater range of available statistical procedures. [ 7 ] [ 8 ] [ 4 ] : 90
In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, [ 9 ] : 59–61 other percentiles (such as quartiles and deciles), [ 9 ] : 71 and the quartile deviation. [ 9 ] : 77 One-sample tests for ordinal data include the Kolmogorov-Smirnov one-sample test , [ 5 ] : 51–55 the one-sample runs test , [ 5 ] : 58–64 and the change-point test. [ 5 ] : 64–71
In lieu of testing differences in means with t -tests , differences in distributions of ordinal data from two independent samples can be tested with Mann-Whitney , [ 9 ] : 259–264 runs , [ 9 ] : 253–259 Smirnov , [ 9 ] : 266–269 and signed-ranks [ 9 ] : 269–273 tests. Test for two related or matched samples include the sign test [ 5 ] : 80–87 and the Wilcoxon signed ranks test . [ 5 ] : 87–95 Analysis of variance with ranks [ 9 ] : 367–369 and the Jonckheere test for ordered alternatives [ 5 ] : 216–222 can be conducted with ordinal data in place of independent samples ANOVA . Tests for more than two related samples includes the Friedman two-way analysis of variance by ranks [ 5 ] : 174–183 and the Page test for ordered alternatives . [ 5 ] : 184–188 Correlation measures appropriate for two ordinal-scaled variables include Kendall's tau , [ 9 ] : 436–439 gamma , [ 9 ] : 442–443 r s , [ 9 ] : 434–436 and d yx /d xy . [ 9 ] : 443
Ordinal data can be considered as a quantitative variable. In logistic regression , the equation
is the model and c takes on the assigned levels of the categorical scale. [ 1 ] : 189 In regression analysis , outcomes ( dependent variables ) that are ordinal variables can be predicted using a variant of ordinal regression , such as ordered logit or ordered probit .
In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks. [ 10 ]
Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a contingency tables . A correlation r is found between the variables where r lies between -1 and 1. To test the trend, a test statistic :
is used where n is the sample size. [ 1 ] : 87
R can be found by letting u 1 ≤ u 2 ≤ . . . ≤ u I {\displaystyle u_{1}\leq u_{2}\leq ...\leq u_{I}} be the row scores and v 1 ≤ v 2 ≤ . . . ≤ v I {\displaystyle v_{1}\leq v_{2}\leq ...\leq v_{I}} be the column scores. Let u ¯ = ∑ i u i p i + {\displaystyle {\bar {u}}\ =\sum _{i}u_{i}p_{i+}} be the mean of the row scores while v ¯ = ∑ j v j p j + . {\displaystyle {\bar {v}}\ =\sum _{j}v_{j}p_{j+}.} . Then p i + {\displaystyle p_{i+}} is the marginal row probability and p + j {\displaystyle p_{+j}} is the marginal column probability. R is calculated by:
Classification methods have also been developed for ordinal data. The data are divided into different categories such that each observation is similar to others. Dispersion is measured and minimized in each group to maximize classification results. The dispersion function is used in information theory . [ 11 ]
There are several different models that can be used to describe the structure of ordinal data. [ 12 ] Four major classes of model are described below, each defined for a random variable Y {\displaystyle Y} , with levels indexed by k = 1 , 2 , … , q {\displaystyle k=1,2,\dots ,q} .
Note that in the model definitions below, the values of μ k {\displaystyle \mu _{k}} and β {\displaystyle \mathbf {\beta } } will not be the same for all the models for the same set of data, but the notation is used to compare the structure of the different models.
The most commonly-used model for ordinal data is the proportional odds model, defined by log [ Pr ( Y ≤ k ) P r ( Y > k ) ] = log [ Pr ( Y ≤ k ) 1 − Pr ( Y ≤ k ) ] = μ k + β T x {\displaystyle \log \left[{\frac {\Pr(Y\leq k)}{Pr(Y>k)}}\right]=\log \left[{\frac {\Pr(Y\leq k)}{1-\Pr(Y\leq k)}}\right]=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} } where the parameters μ k {\displaystyle \mu _{k}} describe the base distribution of the ordinal data, x {\displaystyle \mathbf {x} } are the covariates and β {\displaystyle \mathbf {\beta } } are the coefficients describing the effects of the covariates.
This model can be generalized by defining the model using μ k + β k T x {\displaystyle \mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} } instead of μ k + β T x {\displaystyle \mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} } , and this would make the model suitable for nominal data (in which the categories have no natural ordering) as well as ordinal data. However, this generalization can make it much more difficult to fit the model to the data.
The baseline category model is defined by log [ Pr ( Y = k ) Pr ( Y = 1 ) ] = μ k + β k T x {\displaystyle \log \left[{\frac {\Pr(Y=k)}{\Pr(Y=1)}}\right]=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} }
This model does not impose an ordering on the categories and so can be applied to nominal data as well as ordinal data.
The ordered stereotype model is defined by log [ Pr ( Y = k ) Pr ( Y = 1 ) ] = μ k + ϕ k β T x {\displaystyle \log \left[{\frac {\Pr(Y=k)}{\Pr(Y=1)}}\right]=\mu _{k}+\phi _{k}\mathbf {\beta } ^{T}\mathbf {x} } where the score parameters are constrained such that 0 = ϕ 1 ≤ ϕ 2 ≤ ⋯ ≤ ϕ q = 1 {\displaystyle 0=\phi _{1}\leq \phi _{2}\leq \dots \leq \phi _{q}=1} .
This is a more parsimonious, and more specialised, model than the baseline category logit model: ϕ k β {\displaystyle \phi _{k}\mathbf {\beta } } can be thought of as similar to β k {\displaystyle \mathbf {\beta } _{k}} .
The non-ordered stereotype model has the same form as the ordered stereotype model, but without the ordering imposed on ϕ k {\displaystyle \phi _{k}} . This model can be applied to nominal data.
Note that the fitted scores, ϕ ^ k {\displaystyle {\hat {\phi }}_{k}} , indicate how easy it is to distinguish between the different levels of Y {\displaystyle Y} . If ϕ ^ k ≈ ϕ ^ k − 1 {\displaystyle {\hat {\phi }}_{k}\approx {\hat {\phi }}_{k-1}} then that indicates that the current set of data for the covariates x {\displaystyle \mathbf {x} } do not provide much information to distinguish between levels k {\displaystyle k} and k − 1 {\displaystyle k-1} , but that does not necessarily imply that the actual values k {\displaystyle k} and k − 1 {\displaystyle k-1} are far apart. And if the values of the covariates change, then for that new data the fitted scores ϕ ^ k {\displaystyle {\hat {\phi }}_{k}} and ϕ ^ k − 1 {\displaystyle {\hat {\phi }}_{k-1}} might then be far apart.
The adjacent categories model is defined by log [ Pr ( Y = k ) Pr ( Y = k + 1 ) ] = μ k + β k T x {\displaystyle \log \left[{\frac {\Pr(Y=k)}{\Pr(Y=k+1)}}\right]=\mu _{k}+\mathbf {\beta } _{k}^{T}\mathbf {x} } although the most common form, referred to in Agresti (2010) [ 12 ] as the "proportional odds form" is defined by log [ Pr ( Y = k ) Pr ( Y = k + 1 ) ] = μ k + β T x {\displaystyle \log \left[{\frac {\Pr(Y=k)}{\Pr(Y=k+1)}}\right]=\mu _{k}+\mathbf {\beta } ^{T}\mathbf {x} }
This model can only be applied to ordinal data, since modelling the probabilities of shifts from one category to the next category implies that an ordering of those categories exists.
The adjacent categories logit model can be thought of as a special case of the baseline category logit model, where β k = β ( k − 1 ) {\displaystyle \mathbf {\beta } _{k}=\mathbf {\beta } (k-1)} . The adjacent categories logit model can also be thought of as a special case of the ordered stereotype model, where ϕ k ∝ k − 1 {\displaystyle \phi _{k}\propto k-1} , i.e. the distances between the ϕ k {\displaystyle \phi _{k}} are defined in advance, rather than being estimated based on the data.
The proportional odds model has a very different structure to the other three models, and also a different underlying meaning. Note that the size of the reference category in the proportional odds model varies with k {\displaystyle k} , since Y ≤ k {\displaystyle Y\leq k} is compared to Y > k {\displaystyle Y>k} , whereas in the other models the size of the reference category remains fixed, as Y = k {\displaystyle Y=k} is compared to Y = 1 {\displaystyle Y=1} or Y = k + 1 {\displaystyle Y=k+1} .
There are variants of all the models that use different link functions, such as the probit link or the complementary log-log link.
Differences in ordinal data can be tested using rank tests .
Ordinal data can be visualized in several different ways. Common visualizations are the bar chart or a pie chart . Tables can also be useful for displaying ordinal data and frequencies. Mosaic plots can be used to show the relationship between an ordinal variable and a nominal or ordinal variable. [ 13 ] A bump chart—a line chart that shows the relative ranking of items from one time point to the next—is also appropriate for ordinal data. [ 14 ]
Color or grayscale gradation can be used to represent the ordered nature of the data. A single-direction scale, such as income ranges, can be represented with a bar chart where increasing (or decreasing) saturation or lightness of a single color indicates higher (or lower) income. The ordinal distribution of a variable measured on a dual-direction scale, such as a Likert scale, could also be illustrated with color in a stacked bar chart. A neutral color (white or gray) might be used for the middle (zero or neutral) point, with contrasting colors used in the opposing directions from the midpoint, where increasing saturation or darkness of the colors could indicate categories at increasing distance from the midpoint. [ 15 ] Choropleth maps also use color or grayscale shading to display ordinal data. [ 16 ]
The use of ordinal data can be found in most areas of research where categorical data are generated. Settings where ordinal data are often collected include the social and behavioral sciences and governmental and business settings where measurements are collected from persons by observation, testing, or questionnaires . Some common contexts for the collection of ordinal data include survey research ; [ 17 ] [ 18 ] and intelligence , aptitude , personality testing and decision-making . [ 2 ] [ 4 ] : 89–90
Calculation of 'Effect Size' (Cliff's Delta d ) using ordinal data has been recommended as a measure of statistical dominance. [ 19 ] | https://en.wikipedia.org/wiki/Ordinal_data |
In mathematics , ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. [ 1 ] [ 2 ] The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems . [ 3 ] [ 1 ]
While Gödel showed that every recursively enumerable axiomatic system that can interpret basic arithmetic suffers from some form of incompleteness, Turing focused on a method so that a complete system of logic may be constructed from a given system of logic. By repeating the process, a sequence L1, L2, … of logic is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate logic with any constructive ordinal . [ 3 ]
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ordinal_logic |
In mathematical logic and set theory , an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals . A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties:
There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann , Heinz Bachmann , Wilfried Buchholz, Georg Cantor , Solomon Feferman , Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte , Gaisi Takeuti (called ordinal diagrams ), Oswald Veblen . Stephen Cole Kleene has a system of notations, called Kleene's O , which includes ordinal notations but it is not as well behaved as the other systems described here.
Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals. There are several such desirable properties. Unfortunately, no one system can have all of them since they contradict each other.
As usual, we must start off with a constant symbol for zero, "0", which we may consider to be a function of arity zero. This is necessary because there are no smaller ordinals in terms of which zero can be described. The most obvious next step would be to define a unary function, "S", which takes an ordinal to the smallest ordinal greater than it; in other words, S is the successor function. In combination with zero, successor allows one to name any natural number.
The third function might be defined as one that maps each ordinal to the smallest ordinal that cannot yet be described with the above two functions and previous values of this function. This would map β to ω·β except when β is a fixed point of that function plus a finite number in which case one uses ω·(β+1).
The fourth function would map α to ω ω ·α except when α is a fixed point of that plus a finite number in which case one uses ω ω ·(α+1).
One could continue in this way, but it would give us an infinite number of functions. So instead let us merge the unary functions into a binary function. By transfinite recursion on α, we can use transfinite recursion on β to define ξ(α,β) = the smallest ordinal γ such that α < γ and β < γ and γ is not the value of ξ for any smaller α or for the same α with a smaller β.
Thus, define ξ-notations as follows:
The function ξ is defined for all pairs of ordinals and is one-to-one. It always gives values larger than its arguments and its range is all ordinals other than 0 and the epsilon numbers (ε=ω ε ) .
One has ξ(α, β) < ξ(γ, δ) if and only if either (α = γ and β < δ) or (α < γ and β < ξ(γ, δ)) or (α > γ and ξ(α, β) ≤ δ).
With this definition, the first few ξ-notations are:
In general, ξ(0,β) = β+1. While ξ(1+α,β) = ω ω α ·(β+k) for k = 0 or 1 or 2 depending on special situations: k = 2 if α is an epsilon number and β is finite. Otherwise, k = 1 if β is a multiple of ω ω α+1 plus a finite number. Otherwise, k = 0.
The ξ-notations can be used to name any ordinal less than ε 0 with an alphabet of only two symbols ("0" and "ξ"). If these notations are extended by adding functions that enumerate epsilon numbers, then they will be able to name any ordinal less than the first epsilon number that cannot be named by the added functions. This last property, adding symbols within an initial segment of the ordinals gives names within that segment, is called repleteness (after Solomon Feferman ).
There are many different systems for ordinal notation introduced by various authors. It is often quite hard to convert between the different systems.
"Exponential polynomials" in 0 and ω gives a system of ordinal notation for ordinals less than ε 0 . There are many equivalent ways to write these; instead of exponential polynomials, one can use rooted trees, or nested parentheses, or the system described above.
The 2-variable Veblen functions ( Veblen 1908 ) can be used to give a system of ordinal notation for ordinals less than the Feferman-Schutte ordinal . The Veblen functions in a finite or transfinite number of variables give systems of ordinal notations for ordinals less than the small and large Veblen ordinals .
Ackermann (1951) described a system of ordinal notation rather weaker than the system described earlier by Veblen. The limit of his system is sometimes called the Ackermann ordinal .
Bachmann (1950) introduced the key idea of using uncountable ordinals to produce new countable ordinals. His original system was rather cumbersome to use as it required choosing a special sequence converging to each ordinal. Later systems of notation introduced by Feferman and others avoided this complication.
Takeuti (1987) described a system of ordinal notation known as "ordinal diagrams", whose limit is the Takeuti–Feferman–Buchholz ordinal . [ 1 ] The system was later simplified by Feferman.
Feferman introduced theta functions, described in Buchholz (1986) as follows. For an ordinal α, θ α is a function mapping ordinals to ordinals. Often θ α (β) is written as θαβ. The set C (α, β) is defined by induction on α to be the set of ordinals that can be generated from 0, ω 1 , ω 2 , ..., ω ω , together with the ordinals less than β by the operations of ordinal addition and the functions θ ξ for ξ<α. And the function θ γ is defined to be the function enumerating the ordinals δ with δ∉ C (γ,δ). The problem with this system is that ordinal notations and collapsing functions are not identical, and therefore this function does not qualify as an ordinal notation. An associated ordinal notation is not known.
Buchholz (1986) described the following system of ordinal notation as a simplification of Feferman's theta functions. Define:
The functions ψ v (α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows:
where C v (α) is the smallest set such that
This system has about the same strength as Fefermans system, as θ ε Ω v + 1 0 = ψ 0 ( ε Ω v + 1 ) {\displaystyle \theta \varepsilon _{\Omega _{v}+1}0=\psi _{0}(\varepsilon _{\Omega _{v}+1})} for v ≤ ω. Yet, while this system is powerful, it does not qualify as an ordinal notation. Buchholz did create an associated ordinal notation, yet it is complicated: the definition is in the main article.
Kleene (1938) described a system of notation for all recursive ordinals (those less than the Church–Kleene ordinal ). Unfortunately, unlike the other systems described above there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations that represent the ordinal sum, product, and power (see ordinal arithmetic ) of any two given notations in Kleene's O {\displaystyle {\mathcal {O}}} ; and given any notation for an ordinal, there is a recursively enumerable set of notations that contains one element for each smaller ordinal and is effectively ordered. Kleene's O {\displaystyle {\mathcal {O}}} denotes a canonical (and very non-computable) set of notations. It uses a subset of the natural numbers instead of finite strings of symbols, and is not recursive, therefore, once again, not qualifying as an ordinal notation. | https://en.wikipedia.org/wiki/Ordinal_notation |
In human developmental psychology or non-human primate experiments , ordinal numerical competence or ordinal numerical knowledge is the ability to count objects in order and to understand the greater than and less than relationships between numbers. It has been shown that children as young as two can make some ordinal numerical decisions. There are studies indicating that some non-human primates, like chimpanzees and rhesus monkeys have some ordinal numerical competence.
There is no evidence to support prenatal ordinal numerical competence. Teratogens such as stress [ 1 ] can alter prenatal neural development, leading to diminished competence after birth. Physical effects of teratogens are common, but endocrine effects are harder to measure. These are the factors that influence neural development and by extension the development of ordinal numerical competence. Premature birth is also a risk factor for developmental problems including reduced brain activity. [ 2 ] Brain activity is measured from outside the body with electroencephalography .
There have been a vast number of studies done on infants and their knowledge of numbers. Most research confirms that infants do in fact have a profound innate sense of number, both in abstract and finite ways. Infants as young as 49 hours can accurately match up images with a certain number of objects, with sounds that contain the same number ("ra, ra, ra, ra") as the number of objects in the image. [ 3 ] Because the sounds are abstract, or visibly there, we can see that infants as young as 49 hours have some abstract numerical sense as well as concrete numerical sense shown by their recognition of the image with the corresponding number of objects. [ 3 ] Similarly, infants around the age of 7 months can also match up images of random objects. [ 4 ]
Although children as young as 49 hours can match up the number of sounds with the number of objects, they can only do so at certain ratios. [ 3 ] When 1:3 ratios were used (4 sounds and 4 objects or 12 objects), around 90% of the infants paid more attention to the corresponding image thus showing their recognition. However, when 1:2 ratios were used, only 68% of infants showed recognition of the correct corresponding image. [ 3 ] This tells us that although infants can recognize corresponding numbers of sounds and objects, the two images of objects must be visibly different - one must have a much larger number of objects, or a much smaller number of objects. [ 3 ]
Although there has to be a stark difference in the choices for infants to recognize the correct matching set of numbers (1:3 vs 1:2), this seems to prove that infants have an innate numerical sense, but it may not be the same numerical sense as older children. Around the age of three and a half years children lose some of their numerical sense. Whereas children younger than three can recognize that four pebbles spread out in a line is less than six pebbles scrunched together in a line, children around the age of three and a half mysteriously lose this ability. [ 5 ] Researchers believe that this is because children around this age begin to rely heavily on the physical properties of the world and objects within it, [ 5 ] such that longer equals more. Although the ability to recognize that six pebbles closely lined up together is more than four pebbles spread out farther from one another goes away around that age, it comes back around four years of age when children begin to count. [ 5 ]
Both behavioral research and brain-imaging research show distinct differences in the way "exact" arithmetic and "approximate" arithmetic are processed. Exact arithmetic is information that is precise and follows specific rules and patterns such as multiplication tables or geometric formulas, and approximate arithmetic is a general comparison between numbers such as the comparisons of greater than or less than. Research shows that exact arithmetic is language-based and processed in the left inferior frontal lobe. Approximate arithmetic is processed much differently in a different part of the brain. Approximate arithmetic is processed in the bilateral areas of the parietal lobes. This part of the brain processes visual information to understand how objects are spatially related to each other, for example, understanding that 10 of something is more than two of something. This difference in brain function can create a difference in how we experience certain types of arithmetic. Approximate arithmetic can be experienced as intuitive and exact arithmetic experienced as recalled knowledge. [ 6 ]
The conclusions from behavioral research and brain-imaging research are supported by observations of patients with injuries to certain parts of the brain. People with left parietal injuries can lose the ability to understand quantities of things, but keep at least some ability to do exact arithmetic, such as multiplication. [ 7 ] [ 8 ] [ 9 ] [ 10 ] People with left-hemisphere brain damage can lose the ability to do exact arithmetic, but keep a sense of quantity, including the ability to compare larger and smaller numbers. [ 7 ] This information confirms that distinct parts of the brain are used to know and use approximate and exact arithmetic. [ 6 ]
Various researchers suggest that the processing of approximate arithmetic could be related to the numerical abilities that have been independently established in various animal species [ 11 ] [ 12 ] [ 13 ] [ 14 ] and in preverbal human infants. [ 15 ] This may mean that approximate arithmetic is an adaptive train that humans developed through evolution. [ 16 ] The combination of this potential evolutionary trait and language-based exact arithmetic may be the reason that humans are able to do advanced mathematics like physics. [ 6 ]
Animals share a non-verbal system for representing number as analogue magnitudes. [ 17 ] Animals have been known to base their rationality on Weber’s Law . This historically important psychological law quantifies the perception of change in a given stimulus. The law states that the change in a stimulus that will be just noticeable is a constant ratio of the original stimulus. Weber’s Law describes discriminability between values based on perceptual continua such as line length, brightness, and weight. [ 18 ]
Studies of rhesus monkeys' foraging decisions indicate that animals spontaneously, and without training, exhibit rudimentary numerical abilities. Most animals can determine numbers in the values 1 through 9, but recent experiments have discovered that rhesus monkeys can quantify values from 1 to 30. Monkeys' numerical discrimination capacity is imposed by the ratio of the values compared, rather than absolute set size. [ 12 ] This computation process focuses around Weber’s Law and the expectation violation procedure. This suggests that rhesus monkeys have access to a spontaneous system of representation, which encodes the numerical differences between sets of one, two and three objects, and contrasts three objects from either four or five objects as well. These representations indicate the semantics of an encoded natural language. These encoded natural languages are also seen in experiments with many animals including pigeons and rats.
Experiments have shown that rats are able to be trained to press one lever after hearing two bursts of white noise, then press another lever after four bursts of white noise. The interburst interval is varied between trials so the discrimination is based on number of bursts and not time duration of the sequence. Studies show that rats as well as pigeons learned to make different responses to both short and long durations of signals. During testing, rats exhibited a pattern called break-run-break ; when it came to responding after a stint of little to no response, they would suddenly respond in high frequency, then return to little or no response activity. [ 19 ] Data suggests that rats and pigeons are able to process time and number information at the same time. The Mode Control Model shows that these animals can process number and time information by transmission pulses to accumulators controlled by switches that operate different modes. [ 19 ] | https://en.wikipedia.org/wiki/Ordinal_numerical_competence |
All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In mathematics , especially order theory , a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders , in which every pair is comparable.
Formally, a partial order is a homogeneous binary relation that is reflexive , antisymmetric , and transitive . A partially ordered set ( poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting of a set X {\displaystyle X} (called the ground set of P {\displaystyle P} ) and a partial order ≤ {\displaystyle \leq } on X {\displaystyle X} . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
The term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders. Strict and non-strict partial orders can be put into a one-to-one correspondence , so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.
A reflexive , weak , [ 1 ] or non-strict partial order , [ 2 ] commonly referred to simply as a partial order , is a homogeneous relation ≤ on a set P {\displaystyle P} that is reflexive , antisymmetric , and transitive . That is, for all a , b , c ∈ P , {\displaystyle a,b,c\in P,} it must satisfy:
A non-strict partial order is also known as an antisymmetric preorder .
An irreflexive , strong , [ 1 ] or strict partial order is a homogeneous relation < on a set P {\displaystyle P} that is irreflexive , asymmetric and transitive ; that is, it satisfies the following conditions for all a , b , c ∈ P : {\displaystyle a,b,c\in P:}
A transitive relation is asymmetric if and only if it is irreflexive. [ 3 ] So the definition is the same if it omits either irreflexivity or asymmetry (but not both).
A strict partial order is also known as an asymmetric strict preorder .
Strict and non-strict partial orders on a set P {\displaystyle P} are closely related. A non-strict partial order ≤ {\displaystyle \leq } may be converted to a strict partial order by removing all relationships of the form a ≤ a ; {\displaystyle a\leq a;} that is, the strict partial order is the set < := ≤ ∖ Δ P {\displaystyle <\;:=\ \leq \ \setminus \ \Delta _{P}} where Δ P := { ( p , p ) : p ∈ P } {\displaystyle \Delta _{P}:=\{(p,p):p\in P\}} is the identity relation on P × P {\displaystyle P\times P} and ∖ {\displaystyle \;\setminus \;} denotes set subtraction . Conversely, a strict partial order < on P {\displaystyle P} may be converted to a non-strict partial order by adjoining all relationships of that form; that is, ≤ := Δ P ∪ < {\displaystyle \leq \;:=\;\Delta _{P}\;\cup \;<\;} is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ≠ b . {\displaystyle a<b{\text{ if }}a\leq b{\text{ and }}a\neq b.} Conversely, if < is a strict partial order, then the corresponding non-strict partial order ≤ {\displaystyle \leq } is the reflexive closure given by: a ≤ b if a < b or a = b . {\displaystyle a\leq b{\text{ if }}a<b{\text{ or }}a=b.}
The dual (or opposite ) R op {\displaystyle R^{\text{op}}} of a partial order relation R {\displaystyle R} is defined by letting R op {\displaystyle R^{\text{op}}} be the converse relation of R {\displaystyle R} , i.e. x R op y {\displaystyle xR^{\text{op}}y} if and only if y R x {\displaystyle yRx} . The dual of a non-strict partial order is a non-strict partial order, [ 4 ] and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.
Given a set P {\displaystyle P} and a partial order relation, typically the non-strict partial order ≤ {\displaystyle \leq } , we may uniquely extend our notation to define four partial order relations ≤ , {\displaystyle \leq ,} < , {\displaystyle <,} ≥ , {\displaystyle \geq ,} and > {\displaystyle >} , where ≤ {\displaystyle \leq } is a non-strict partial order relation on P {\displaystyle P} , < {\displaystyle <} is the associated strict partial order relation on P {\displaystyle P} (the irreflexive kernel of ≤ {\displaystyle \leq } ), ≥ {\displaystyle \geq } is the dual of ≤ {\displaystyle \leq } , and > {\displaystyle >} is the dual of < {\displaystyle <} . Strictly speaking, the term partially ordered set refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, ( P , ≤ ) {\displaystyle (P,\leq )} or ( P , < ) {\displaystyle (P,<)} , or, in rare instances, the non-strict and strict relations together, ( P , ≤ , < ) {\displaystyle (P,\leq ,<)} . [ 5 ]
The term ordered set is sometimes used as a shorthand for partially ordered set , as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than ≤ {\displaystyle \leq } such as ⊑ {\displaystyle \sqsubseteq } [ 6 ] or ⪯ {\displaystyle \preceq } [ 7 ] to distinguish partial orders from total orders.
When referring to partial orders, ≤ {\displaystyle \leq } should not be taken as the complement of > {\displaystyle >} . The relation > {\displaystyle >} is the converse of the irreflexive kernel of ≤ {\displaystyle \leq } , which is always a subset of the complement of ≤ {\displaystyle \leq } , but > {\displaystyle >} is equal to the complement of ≤ {\displaystyle \leq } if, and only if , ≤ {\displaystyle \leq } is a total order. [ a ]
Another way of defining a partial order, found in computer science , is via a notion of comparison . Specifically, given ≤ , < , ≥ , and > {\displaystyle \leq ,<,\geq ,{\text{ and }}>} as defined previously, it can be observed that two elements x and y may stand in any of four mutually exclusive relationships to each other: either x < y , or x = y , or x > y , or x and y are incomparable . This can be represented by a function compare : P × P → { < , > , = , | } {\displaystyle {\text{compare}}:P\times P\to \{<,>,=,\vert \}} that returns one of four codes when given two elements. [ 8 ] [ 9 ] This definition is equivalent to a partial order on a setoid , where equality is taken to be a defined equivalence relation rather than set equality. [ 10 ]
Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric . This includes both reflexive and irreflexive partial orders as subtypes. [ 1 ]
A finite poset can be visualized through its Hasse diagram . [ 11 ] Specifically, taking a strict partial order relation ( P , < ) {\displaystyle (P,<)} , a directed acyclic graph (DAG) may be constructed by taking each element of P {\displaystyle P} to be a node and each element of < {\displaystyle <} to be an edge. The transitive reduction of this DAG [ b ] is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.
Standard examples of posets arising in mathematics include:
One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig. 4):
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to ordered vector spaces over the same field , the result is in each case also an ordered vector space.
See also orders on the Cartesian product of totally ordered sets .
Another way to combine two (disjoint) posets is the ordinal sum [ 12 ] (or linear sum ), [ 13 ] Z = X ⊕ Y , defined on the union of the underlying sets X and Y by the order a ≤ Z b if and only if:
If two posets are well-ordered , then so is their ordinal sum. [ 14 ]
Series-parallel partial orders are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the disjoint union of two partially ordered sets, with no order relation between elements of one set and elements of the other set.
The examples use the poset ( P ( { x , y , z } ) , ⊆ ) {\displaystyle ({\mathcal {P}}(\{x,y,z\}),\subseteq )} consisting of the set of all subsets of a three-element set { x , y , z } , {\displaystyle \{x,y,z\},} ordered by set inclusion (see Fig. 1).
There are several notions of "greatest" and "least" element in a poset P , {\displaystyle P,} notably:
As another example, consider the positive integers , ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any g divides for instance 2 g , which is distinct from it, so g is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset { 2 , 3 , 5 , 10 } , {\displaystyle \{2,3,5,10\},} which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig. 6).
Given two partially ordered sets ( S , ≤) and ( T , ≼) , a function f : S → T {\displaystyle f:S\to T} is called order-preserving , or monotone , or isotone , if for all x , y ∈ S , {\displaystyle x,y\in S,} x ≤ y {\displaystyle x\leq y} implies f ( x ) ≼ f ( y ) .
If ( U , ≲) is also a partially ordered set, and both f : S → T {\displaystyle f:S\to T} and g : T → U {\displaystyle g:T\to U} are order-preserving, their composition g ∘ f : S → U {\displaystyle g\circ f:S\to U} is order-preserving, too.
A function f : S → T {\displaystyle f:S\to T} is called order-reflecting if for all x , y ∈ S , {\displaystyle x,y\in S,} f ( x ) ≼ f ( y ) implies x ≤ y . {\displaystyle x\leq y.} If f is both order-preserving and order-reflecting, then it is called an order-embedding of ( S , ≤) into ( T , ≼) .
In the latter case, f is necessarily injective , since f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x ≤ y and y ≤ x {\displaystyle x\leq y{\text{ and }}y\leq x} and in turn x = y {\displaystyle x=y} according to the antisymmetry of ≤ . {\displaystyle \leq .} If an order-embedding between two posets S and T exists, one says that S can be embedded into T . If an order-embedding f : S → T {\displaystyle f:S\to T} is bijective , it is called an order isomorphism , and the partial orders ( S , ≤) and ( T , ≼) are said to be isomorphic . Isomorphic orders have structurally similar Hasse diagrams (see Fig. 7a). It can be shown that if order-preserving maps f : S → T {\displaystyle f:S\to T} and g : T → U {\displaystyle g:T\to U} exist such that g ∘ f {\displaystyle g\circ f} and f ∘ g {\displaystyle f\circ g} yields the identity function on S and T , respectively, then S and T are order-isomorphic. [ 15 ]
For example, a mapping f : N → P ( N ) {\displaystyle f:\mathbb {N} \to \mathbb {P} (\mathbb {N} )} from the set of natural numbers (ordered by divisibility) to the power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors . It is order-preserving: if x divides y , then each prime divisor of x is also a prime divisor of y . However, it is neither injective (since it maps both 12 and 6 to { 2 , 3 } {\displaystyle \{2,3\}} ) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its prime power divisors defines a map g : N → P ( N ) {\displaystyle g:\mathbb {N} \to \mathbb {P} (\mathbb {N} )} that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set { 4 } {\displaystyle \{4\}} ), but it can be made one by restricting its codomain to g ( N ) . {\displaystyle g(\mathbb {N} ).} Fig. 7b shows a subset of N {\displaystyle \mathbb {N} } and its isomorphic image under g . The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices ; see Birkhoff's representation theorem .
Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:
Note that S ( n , k ) refers to Stirling numbers of the second kind .
The number of strict partial orders is the same as that of partial orders.
If the count is made only up to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... (sequence A000112 in the OEIS ) is obtained.
A poset P ∗ = ( X ∗ , ≤ ∗ ) {\displaystyle P^{*}=(X^{*},\leq ^{*})} is called a subposet of another poset P = ( X , ≤ ) {\displaystyle P=(X,\leq )} provided that X ∗ {\displaystyle X^{*}} is a subset of X {\displaystyle X} and ≤ ∗ {\displaystyle \leq ^{*}} is a subset of ≤ {\displaystyle \leq } . The latter condition is equivalent to the requirement that for any x {\displaystyle x} and y {\displaystyle y} in X ∗ {\displaystyle X^{*}} (and thus also in X {\displaystyle X} ), if x ≤ ∗ y {\displaystyle x\leq ^{*}y} then x ≤ y {\displaystyle x\leq y} .
If P ∗ {\displaystyle P^{*}} is a subposet of P {\displaystyle P} and furthermore, for all x {\displaystyle x} and y {\displaystyle y} in X ∗ {\displaystyle X^{*}} , whenever x ≤ y {\displaystyle x\leq y} we also have x ≤ ∗ y {\displaystyle x\leq ^{*}y} , then we call P ∗ {\displaystyle P^{*}} the subposet of P {\displaystyle P} induced by X ∗ {\displaystyle X^{*}} , and write P ∗ = P [ X ∗ ] {\displaystyle P^{*}=P[X^{*}]} .
A partial order ≤ ∗ {\displaystyle \leq ^{*}} on a set X {\displaystyle X} is called an extension of another partial order ≤ {\displaystyle \leq } on X {\displaystyle X} provided that for all elements x , y ∈ X , {\displaystyle x,y\in X,} whenever x ≤ y , {\displaystyle x\leq y,} it is also the case that x ≤ ∗ y . {\displaystyle x\leq ^{*}y.} A linear extension is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order ( order-extension principle ). [ 16 ]
In computer science , algorithms for finding linear extensions of partial orders (represented as the reachability orders of directed acyclic graphs ) are called topological sorting .
Every poset (and every preordered set ) may be considered as a category where, for objects x {\displaystyle x} and y , {\displaystyle y,} there is at most one morphism from x {\displaystyle x} to y . {\displaystyle y.} More explicitly, let hom( x , y ) = {( x , y )} if x ≤ y (and otherwise the empty set ) and ( y , z ) ∘ ( x , y ) = ( x , z ) . {\displaystyle (y,z)\circ (x,y)=(x,z).} Such categories are sometimes called posetal .
Posets are equivalent to one another if and only if they are isomorphic . In a poset, the smallest element, if it exists, is an initial object , and the largest element, if it exists, is a terminal object . Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed .
If P {\displaystyle P} is a partially ordered set that has also been given the structure of a topological space , then it is customary to assume that { ( a , b ) : a ≤ b } {\displaystyle \{(a,b):a\leq b\}} is a closed subset of the topological product space P × P . {\displaystyle P\times P.} Under this assumption partial order relations are well behaved at limits in the sense that if lim i → ∞ a i = a , {\displaystyle \lim _{i\to \infty }a_{i}=a,} and lim i → ∞ b i = b , {\displaystyle \lim _{i\to \infty }b_{i}=b,} and for all i , {\displaystyle i,} a i ≤ b i , {\displaystyle a_{i}\leq b_{i},} then a ≤ b . {\displaystyle a\leq b.} [ 17 ]
A convex set in a poset P is a subset I of P with the property that, for any x and y in I and any z in P , if x ≤ z ≤ y , then z is also in I . This definition generalizes the definition of intervals of real numbers . When there is possible confusion with convex sets of geometry , one uses order-convex instead of "convex".
A convex sublattice of a lattice L is a sublattice of L that is also a convex set of L . Every nonempty convex sublattice can be uniquely represented as the intersection of a filter and an ideal of L .
An interval in a poset P is a subset that can be defined with interval notation:
Whenever a ≤ b does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set {1, 2, 4, 5, 8} is convex, but not an interval.
An interval I is bounded if there exist elements a , b ∈ P {\displaystyle a,b\in P} such that I ⊆ [ a , b ] . Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let P = (0, 1) ∪ (1, 2) ∪ (2, 3) as a subposet of the real numbers. The subset (1, 2) is a bounded interval, but it has no infimum or supremum in P , so it cannot be written in interval notation using elements of P .
A poset is called locally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product N × N {\displaystyle \mathbb {N} \times \mathbb {N} } is not locally finite, since (1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1) .
Using the interval notation, the property " a is covered by b " can be rephrased equivalently as [ a , b ] = { a , b } . {\displaystyle [a,b]=\{a,b\}.}
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders .
Media related to Hasse diagrams at Wikimedia Commons; each of which shows an example for a partial order | https://en.wikipedia.org/wiki/Ordinal_sum |
In mathematics , an ordinary differential equation ( ODE ) is a differential equation (DE) dependent on only a single independent variable . As with any other DE, its unknown(s) consists of one (or more) function (s) and involves the derivatives of those functions. [ 1 ] The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable, [ 2 ] and, less commonly, in contrast with stochastic differential equations (SDEs) where the progression is random. [ 3 ]
A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
where a 0 ( x ) , … , a n ( x ) {\displaystyle a_{0}(x),\ldots ,a_{n}(x)} and b ( x ) {\displaystyle b(x)} are arbitrary differentiable functions that do not need to be linear, and y ′ , … , y ( n ) {\displaystyle y',\ldots ,y^{(n)}} are the successive derivatives of the unknown function y {\displaystyle y} of the variable x {\displaystyle x} . [ 4 ]
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function ). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation ). [ 5 ]
Some ODEs can be solved explicitly in terms of known functions and integrals . When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.
Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences . Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations. [ 6 ]
Specific mathematical fields include geometry and analytical mechanics . Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modeling), chemistry (reaction rates), [ 7 ] biology (infectious diseases, genetic variation), ecology and population modeling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton , Leibniz , the Bernoulli family , Riccati , Clairaut , d'Alembert , and Euler .
A simple example is Newton's second law of motion—the relationship between the displacement x {\displaystyle x} and the time t {\displaystyle t} of an object under the force F {\displaystyle F} , is given by the differential equation
which constrains the motion of a particle of constant mass m {\displaystyle m} . In general, F {\displaystyle F} is a function of the position x ( t ) {\displaystyle x(t)} of the particle at time t {\displaystyle t} . The unknown function x ( t ) {\displaystyle x(t)} appears on both sides of the differential equation, and is indicated in the notation F ( x ( t ) ) {\displaystyle F(x(t))} . [ 8 ] [ 9 ] [ 10 ] [ 11 ]
In what follows, y {\displaystyle y} is a dependent variable representing an unknown function y = f ( x ) {\displaystyle y=f(x)} of the independent variable x {\displaystyle x} . The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the Leibniz's notation d y d x , d 2 y d x 2 , … , d n y d x n {\displaystyle {\frac {dy}{dx}},{\frac {d^{2}y}{dx^{2}}},\ldots ,{\frac {d^{n}y}{dx^{n}}}} is more useful for differentiation and integration , whereas Lagrange's notation y ′ , y ″ , … , y ( n ) {\displaystyle y',y'',\ldots ,y^{(n)}} is more useful for representing higher-order derivatives compactly, and Newton's notation ( y ˙ , y ¨ , y . . . ) {\displaystyle ({\dot {y}},{\ddot {y}},{\overset {...}{y}})} is often used in physics for representing derivatives of low order with respect to time.
Given F {\displaystyle F} , a function of x {\displaystyle x} , y {\displaystyle y} , and derivatives of y {\displaystyle y} . Then an equation of the form
is called an explicit ordinary differential equation of order n {\displaystyle n} . [ 12 ] [ 13 ]
More generally, an implicit ordinary differential equation of order n {\displaystyle n} takes the form: [ 14 ]
There are further classifications:
A number of coupled differential equations form a system of equations. If y {\displaystyle \mathbf {y} } is a vector whose elements are functions; y ( x ) = [ y 1 ( x ) , y 2 ( x ) , … , y m ( x ) ] {\displaystyle \mathbf {y} (x)=[y_{1}(x),y_{2}(x),\ldots ,y_{m}(x)]} , and F {\displaystyle \mathbf {F} } is a vector-valued function of y {\displaystyle \mathbf {y} } and its derivatives, then
is an explicit system of ordinary differential equations of order n > {\displaystyle n>} and dimension m {\displaystyle m} . In column vector form:
These are not necessarily linear. The implicit analogue is:
where 0 = ( 0 , 0 , … , 0 ) {\displaystyle {\boldsymbol {0}}=(0,0,\ldots ,0)} is the zero vector . In matrix form
For a system of the form F ( x , y , y ′ ) = 0 {\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}} , some sources also require that the Jacobian matrix ∂ F ( x , u , v ) ∂ v {\displaystyle {\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}} be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. [ 18 ] [ 19 ] [ 20 ] Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, [ citation needed ] although note that any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order , [ 21 ] which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.
The behavior of a system of ODEs can be visualized through the use of a phase portrait .
Given a differential equation
a function u : I ⊂ R → R {\displaystyle u:I\subset \mathbb {R} \to \mathbb {R} } , where I {\displaystyle I} is an interval, is called a solution or integral curve for F {\displaystyle F} , if u {\displaystyle u} is n {\displaystyle n} -times differentiable on I {\displaystyle I} , and
Given two solutions u : J ⊂ R → R {\displaystyle u:J\subset \mathbb {R} \to \mathbb {R} } and v : I ⊂ R → R {\displaystyle v:I\subset \mathbb {R} \to \mathbb {R} } , u {\displaystyle u} is called an extension of v {\displaystyle v} if I ⊂ J {\displaystyle I\subset J} and
A solution that has no extension is called a maximal solution . A solution defined on all of R {\displaystyle \mathbb {R} } is called a global solution .
A general solution of an n {\displaystyle n} th-order equation is a solution containing n {\displaystyle n} arbitrary independent constants of integration . A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set ' initial conditions or boundary conditions '. [ 22 ] A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution. [ 23 ]
In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the homogeneous solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the guessing method section in this article, and is frequently used when discussing the method of undetermined coefficients and variation of parameters .
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, [ 24 ] meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of Lipschitz differential equations.
As example, the equation:
Admits the finite duration solution:
The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century has it received special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably Casorati and Cayley . To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
The primitive attempt in dealing with differential equations had in view a reduction to quadratures , that is, expressing the solutions in terms of known function and their integrals. This is possible for linear equations with constant coefficients, it appeared in the 19th century that this is generally impossible in other cases. Hence, analysts began the study (for their own) of functions that are solutions of differential equations, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by quadratures, but whether a given differential equation suffices for the definition of a function, and, if so, what are the characteristic properties of such functions.
Two memoirs by Fuchs [ 25 ] inspired a novel approach, subsequently elaborated by Thomé and Frobenius . Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those in his theory of Abelian integrals . As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces f = 0 {\displaystyle f=0} under rational one-to-one transformations.
From 1870, Sophus Lie 's work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using Lie groups , be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable integration difficulties. He also emphasized the subject of transformations of contact .
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations. [ 26 ]
A general solution approach uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions ( Lie theory ). Continuous group theory , Lie algebras , and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs , recursion operators, Bäcklund transform , and finally finding exact analytic solutions to DE.
Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
Sturm–Liouville theory is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined via second-order homogeneous linear equations . The problems are identified as Sturm–Liouville problems (SLP) and are named after J. C. F. Sturm and J. Liouville , who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. [ 27 ] SLPs are also useful in the analysis of certain partial differential equations.
There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are
In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of Grönwall's inequality are met.
Also, uniqueness theorems like the Lipschitz one above do not apply to DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone. [ 28 ]
The theorem can be stated simply as follows. [ 29 ] For the equation and initial value problem: y ′ = F ( x , y ) , y 0 = y ( x 0 ) {\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})} if F {\displaystyle F} and ∂ F / ∂ y {\displaystyle \partial F/\partial y} are continuous in a closed rectangle R = [ x 0 − a , x 0 + a ] × [ y 0 − b , y 0 + b ] {\displaystyle R=[x_{0}-a,x_{0}+a]\times [y_{0}-b,y_{0}+b]} in the x − y {\displaystyle x-y} plane, where a {\displaystyle a} and b {\displaystyle b} are real (symbolically: a , b ∈ R {\displaystyle a,b\in \mathbb {R} } ) and x {\displaystyle x} denotes the Cartesian product , square brackets denote closed intervals , then there is an interval I = [ x 0 − h , x 0 + h ] ⊂ [ x 0 − a , x 0 + a ] {\displaystyle I=[x_{0}-h,x_{0}+h]\subset [x_{0}-a,x_{0}+a]} for some h ∈ R {\displaystyle h\in \mathbb {R} } where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F {\displaystyle F} to be linear, this applies to non-linear equations that take the form F ( x , y ) {\displaystyle F(x,y)} , and it can also be applied to systems of equations.
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely: [ 30 ]
For each initial condition ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} there exists a unique maximum (possibly infinite) open interval
such that any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain I max {\displaystyle I_{\max }} .
In the case that x ± ≠ ± ∞ {\displaystyle x_{\pm }\neq \pm \infty } , there are exactly two possibilities
where Ω {\displaystyle \Omega } is the open set in which F {\displaystyle F} is defined, and ∂ Ω ¯ {\displaystyle \partial {\bar {\Omega }}} is its boundary.
Note that the maximum domain of the solution
This means that F ( x , y ) = y 2 {\displaystyle F(x,y)=y^{2}} , which is C 1 {\displaystyle C^{1}} and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.
Even in such a simple setting, the maximum domain of solution cannot be all R {\displaystyle \mathbb {R} } since the solution is
which has maximum domain:
This shows clearly that the maximum interval may depend on the initial conditions. The domain of y {\displaystyle y} could be taken as being R ∖ ( x 0 + 1 / y 0 ) , {\displaystyle \mathbb {R} \setminus (x_{0}+1/y_{0}),} but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it.
The maximum domain is not R {\displaystyle \mathbb {R} } because
which is one of the two possible cases according to the above theorem.
Differential equations are usually easier to solve if the order of the equation can be reduced.
Any explicit differential equation of order n {\displaystyle n} ,
can be written as a system of n {\displaystyle n} first-order differential equations by defining a new family of unknown functions
for i = 1 , 2 , … , n {\displaystyle i=1,2,\ldots ,n} . The n {\displaystyle n} -dimensional system of first-order coupled differential equations is then
more compactly in vector notation:
where
Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.
In the table below, P ( x ) {\displaystyle P(x)} , Q ( x ) {\displaystyle Q(x)} , P ( y ) {\displaystyle P(y)} , Q ( y ) {\displaystyle Q(y)} , and M ( x , y ) {\displaystyle M(x,y)} , N ( x , y ) {\displaystyle N(x,y)} are any integrable functions of x {\displaystyle x} , y {\displaystyle y} ; b {\displaystyle b} and c {\displaystyle c} are real given constants; C 1 , C 2 , … {\displaystyle C_{1},C_{2},\ldots } are arbitrary constants ( complex in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.
In the integral solutions, λ {\displaystyle \lambda } and ε {\displaystyle \varepsilon } are dummy variables of integration (the continuum analogues of indices in summation ), and the notation ∫ x F ( λ ) d λ {\displaystyle \int ^{x}F(\lambda )\,d\lambda } just means to integrate F ( λ ) {\displaystyle F(\lambda )} with respect to λ {\displaystyle \lambda } , then after the integration substitute λ = x {\displaystyle \lambda =x} , without adding constants (explicitly stated).
P 1 ( x ) Q 1 ( y ) + P 2 ( x ) Q 2 ( y ) d y d x = 0 P 1 ( x ) Q 1 ( y ) d x + P 2 ( x ) Q 2 ( y ) d y = 0 {\displaystyle {\begin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac {dy}{dx}}&=0\\P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy&=0\end{aligned}}}
d y d x = F ( x ) d y = F ( x ) d x {\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(x)\\dy&=F(x)\,dx\end{aligned}}}
d y d x = F ( y ) d y = F ( y ) d x {\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(y)\\dy&=F(y)\,dx\end{aligned}}}
P ( y ) d y d x + Q ( x ) = 0 P ( y ) d y + Q ( x ) d x = 0 {\displaystyle {\begin{aligned}P(y){\frac {dy}{dx}}+Q(x)&=0\\P(y)\,dy+Q(x)\,dx&=0\end{aligned}}}
d y d x = F ( y x ) {\displaystyle {\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)}
y M ( x y ) + x N ( x y ) d y d x = 0 y M ( x y ) d x + x N ( x y ) d y = 0 {\displaystyle {\begin{aligned}yM(xy)+xN(xy)\,{\frac {dy}{dx}}&=0\\yM(xy)\,dx+xN(xy)\,dy&=0\end{aligned}}}
ln ( C x ) = ∫ x y N ( λ ) d λ λ [ N ( λ ) − M ( λ ) ] {\displaystyle \ln(Cx)=\int ^{xy}{\frac {N(\lambda )\,d\lambda }{\lambda [N(\lambda )-M(\lambda )]}}}
If N = M {\displaystyle N=M} , the solution is x y = C {\displaystyle xy=C} .
M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
where ∂ M ∂ y = ∂ N ∂ x {\displaystyle {\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}}
where Y ( y ) = N ( x , y ) − ∂ ∂ y ∫ x M ( λ , y ) d λ {\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}M(\lambda ,y)\,d\lambda } and X ( x ) = M ( x , y ) − ∂ ∂ x ∫ y N ( x , λ ) d λ {\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}N(x,\lambda )\,d\lambda }
M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
where ∂ M ∂ y ≠ ∂ N ∂ x {\displaystyle {\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}}
∂ ( μ M ) ∂ y = ∂ ( μ N ) ∂ x {\displaystyle {\frac {\partial (\mu M)}{\partial y}}={\frac {\partial (\mu N)}{\partial x}}}
F ( x , y ) = ∫ x μ ( λ , y ) M ( λ , y ) d λ + ∫ y Y ( λ ) d λ = ∫ y μ ( x , λ ) N ( x , λ ) d λ + ∫ x X ( λ ) d λ = C {\displaystyle {\begin{aligned}F(x,y)=&\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\=&\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}}
where Y ( y ) = N ( x , y ) − ∂ ∂ y ∫ x μ ( λ , y ) M ( λ , y ) d λ {\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda } and X ( x ) = M ( x , y ) − ∂ ∂ x ∫ y μ ( x , λ ) N ( x , λ ) d λ {\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda }
d 2 y d x 2 = F ( y ) {\displaystyle {\frac {d^{2}y}{dx^{2}}}=F(y)}
d y d x + P ( x ) y = Q ( x ) {\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)}
y = e − ∫ x P ( λ ) d λ [ ∫ x e ∫ λ P ( ε ) d ε Q ( λ ) d λ + C ] {\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left[\int ^{x}e^{\int ^{\lambda }P(\varepsilon )\,d\varepsilon }Q(\lambda )\,d\lambda +C\right]}
d 2 y d x 2 + 2 p ( x ) d y d x + ( p ( x ) 2 + p ′ ( x ) ) y = q ( x ) {\displaystyle {\frac {d^{2}y}{dx^{2}}}+2p(x){\frac {dy}{dx}}+\left(p(x)^{2}+p'(x)\right)y=q(x)}
d 2 y d x 2 + b d y d x + c y = r ( x ) {\displaystyle {\frac {d^{2}y}{dx^{2}}}+b{\frac {dy}{dx}}+cy=r(x)}
Particular integral y p {\displaystyle y_{p}} : in general the method of variation of parameters , though for very simple r ( x ) {\displaystyle r(x)} inspection may work. [ 29 ]
If b 2 > 4 c {\displaystyle b^{2}>4c} , then
y c = C 1 e − x 2 ( b + b 2 − 4 c ) + C 2 e − x 2 ( b − b 2 − 4 c ) {\displaystyle y_{c}=C_{1}e^{-{\frac {x}{2}}\,\left(b+{\sqrt {b^{2}-4c}}\right)}+C_{2}e^{-{\frac {x}{2}}\,\left(b-{\sqrt {b^{2}-4c}}\right)}}
If b 2 = 4 c {\displaystyle b^{2}=4c} , then
y c = ( C 1 x + C 2 ) e − b x 2 {\displaystyle y_{c}=(C_{1}x+C_{2})e^{-{\frac {bx}{2}}}}
If b 2 < 4 c {\displaystyle b^{2}<4c} , then
y c = e − b x 2 [ C 1 sin ( x 4 c − b 2 2 ) + C 2 cos ( x 4 c − b 2 2 ) ] {\displaystyle y_{c}=e^{-{\frac {bx}{2}}}\left[C_{1}\sin \left(x\,{\frac {\sqrt {4c-b^{2}}}{2}}\right)+C_{2}\cos \left(x\,{\frac {\sqrt {4c-b^{2}}}{2}}\right)\right]}
∑ j = 0 n b j d j y d x j = r ( x ) {\displaystyle \sum _{j=0}^{n}b_{j}{\frac {d^{j}y}{dx^{j}}}=r(x)}
Particular integral y p {\displaystyle y_{p}} : in general the method of variation of parameters , though for very simple r ( x ) {\displaystyle r(x)} inspection may work. [ 29 ]
Since α j {\displaystyle \alpha _{j}} are the solutions of the polynomial of degree n {\displaystyle n} : ∏ j = 1 n ( α − α j ) = 0 {\textstyle \prod _{j=1}^{n}(\alpha -\alpha _{j})=0} , then:
for α j {\displaystyle \alpha _{j}} all different, y c = ∑ j = 1 n C j e α j x {\displaystyle y_{c}=\sum _{j=1}^{n}C_{j}e^{\alpha _{j}x}} for each root α j {\displaystyle \alpha _{j}} repeated k j {\displaystyle k_{j}} times, y c = ∑ j = 1 n ( ∑ ℓ = 1 k j C j , ℓ x ℓ − 1 ) e α j x {\displaystyle y_{c}=\sum _{j=1}^{n}\left(\sum _{\ell =1}^{k_{j}}C_{j,\ell }x^{\ell -1}\right)e^{\alpha _{j}x}} for some α j {\displaystyle \alpha _{j}} complex, then setting α j = χ j + i γ j {\displaystyle \alpha _{j}=\chi _{j}+i\gamma _{j}} , and using Euler's formula , allows some terms in the previous results to be written in the form C j e α j x = C j e χ j x cos ( γ j x + φ j ) {\displaystyle C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\varphi _{j})} where φ j {\displaystyle \varphi _{j}} is an arbitrary constant (phase shift).
When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form: y = A e α t {\displaystyle y=Ae^{\alpha t}} since this is a very common solution that physically behaves in a sinusoidal way.
In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions together to obtain the general solution to the ODE, that is:
general solution = general solution of the associated homogeneous equation + particular solution {\displaystyle {\text{general solution}}={\text{general solution of the associated homogeneous equation}}+{\text{particular solution}}} | https://en.wikipedia.org/wiki/Ordinary_differential_equation |
Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore . It gives a sufficient condition for a graph to be Hamiltonian , essentially stating that a graph with sufficiently many edges must contain a Hamilton cycle . Specifically, the theorem considers the sum of the degrees of pairs of non-adjacent vertices : if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is Hamiltonian.
Let G be a (finite and simple) graph with n ≥ 3 vertices. We denote by deg v the degree of a vertex v in G , i.e. the number of incident edges in G to v . Then, Ore's theorem states that if
then G is Hamiltonian .
It is equivalent to show that every non-Hamiltonian graph G does not obey condition (∗) . Accordingly, let G be a graph on n ≥ 3 vertices that is not Hamiltonian, and let H be formed from G by adding edges one at a time that do not create a Hamiltonian cycle, until no more edges can be added. Let x and y be any two non-adjacent vertices in H . Then adding edge xy to H would create at least one new Hamiltonian cycle, and the edges other than xy in such a cycle must form a Hamiltonian path v 1 v 2 ... v n in H with x = v 1 and y = v n . For each index i in the range 2 ≤ i ≤ n , consider the two possible edges in H from v 1 to v i and from v i − 1 to v n . At most one of these two edges can be present in H , for otherwise the cycle v 1 v 2 ... v i − 1 v n v n − 1 ... v i would be a Hamiltonian cycle. Thus, the total number of edges incident to either v 1 or v n is at most equal to the number of choices of i , which is n − 1 . Therefore, H does not obey property (∗) , which requires that this total number of edges ( deg v 1 + deg v n ) be greater than or equal to n . Since the vertex degrees in G are at most equal to the degrees in H , it follows that G also does not obey property (∗) .
Palmer (1997) describes the following simple algorithm for constructing a Hamiltonian cycle in a graph meeting Ore's condition.
Each step increases the number of consecutive pairs in the cycle that are adjacent in the graph, by one or two pairs (depending on whether v j and v j + 1 are already adjacent), so the outer loop can only happen at most n times before the algorithm terminates, where n is the number of vertices in the given graph. By an argument similar to the one in the proof of the theorem, the desired index j must exist, or else the nonadjacent vertices v i and v i + 1 would have too small a total degree. Finding i and j , and reversing part of the cycle, can all be accomplished in time O( n ). Therefore, the total time for the algorithm is O( n 2 ), matching the number of edges in the input graph.
Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least n /2 , the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least n .
In turn Ore's theorem is generalized by the Bondy–Chvátal theorem . One may define a closure operation on a graph in which, whenever two nonadjacent vertices have degrees adding to at least n , one adds an edge connecting them; if a graph meets the conditions of Ore's theorem, its closure is a complete graph . The Bondy–Chvátal theorem states that a graph is Hamiltonian if and only if its closure is Hamiltonian; since the complete graph is Hamiltonian, Ore's theorem is an immediate consequence.
Woodall (1972) found a version of Ore's theorem that applies to directed graphs . Suppose a digraph G has the property that, for every two vertices u and v , either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G . Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every edge in a given undirected graph by a pair of directed edges. A closely related theorem by Meyniel (1973) states that an n -vertex strongly connected digraph with the property that, for every two nonadjacent vertices u and v , the total number of edges incident to u or v is at least 2 n − 1 must be Hamiltonian.
Ore's theorem may also be strengthened to give a stronger conclusion than Hamiltonicity as a consequence of the degree condition in the theorem. Specifically, every graph satisfying the conditions of Ore's theorem is either a regular complete bipartite graph or is pancyclic ( Bondy 1971 ). | https://en.wikipedia.org/wiki/Ore's_theorem |
In computer algebra , an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. [ 1 ] The concept is named after Øystein Ore .
Let K {\displaystyle K} be a (commutative) field and A = K [ x 1 , … , x s ] {\displaystyle A=K[x_{1},\ldots ,x_{s}]} be a commutative polynomial ring (with A = K {\displaystyle A=K} when s = 0 {\displaystyle s=0} ). The iterated skew polynomial ring A [ ∂ 1 ; σ 1 , δ 1 ] ⋯ [ ∂ r ; σ r , δ r ] {\displaystyle A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]} is called an Ore algebra when the σ i {\displaystyle \sigma _{i}} and δ j {\displaystyle \delta _{j}} commute for i ≠ j {\displaystyle i\neq j} , and satisfy σ i ( ∂ j ) = ∂ j {\displaystyle \sigma _{i}(\partial _{j})=\partial _{j}} , δ i ( ∂ j ) = 0 {\displaystyle \delta _{i}(\partial _{j})=0} for i > j {\displaystyle i>j} .
Ore algebras satisfy the Ore condition , and thus can be embedded in a (skew) field of fractions.
The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ore_algebra |
An ore dock is a large structure used for loading ore (typically from railway cars or ore jennies ) onto ships, which then carry the ore to steelworks or to transshipment points. Most known ore docks were constructed near iron mines on the upper Great Lakes and served the lower Great Lakes. Ore docks still in existence are typically about 60 feet (18 m) wide, 80 feet (24 m) high, and vary from 900 feet (270 m) to 2,400 feet (730 m) in length. They are commonly constructed from wood , steel , reinforced concrete , or combinations of these materials. [ 1 ]
They are commonly used for loading bulk ore carriers with high mass, low-value ore, such as iron ore , in raw or taconite form.
Ore docks are typically long, high structures, with a railway track or tracks along the top and a number of "pockets" into which ore is unloaded from railcars, typically by gravity. Each pocket has a chute that can be lowered to discharge the ore into the hold of a ship berthed alongside. The use of pockets and chutes allows the dock itself to be loaded with ore before it is transferred into the freighter . [ 2 ]
The docks' storage bins or pockets are typically wider at the top than the bottom, and they lead to movable steel chutes. These chutes project out over the water at a slight angle from the sides of the docks. The hinged chutes, which are lowered to drop ore from the pockets into ships, are located at twelve-foot intervals over the length of the dock.
This spacing is not coincidental, as the docks and the lakers they could load evolved together, and laker hatch spacing is typically 12, or 24-foot (7.3 m) on center.
Since the late 1800s, ore docks have been a common sight in many Lake Superior ports of Minnesota , Wisconsin and especially Michigan . Rich iron ore deposits were first discovered in the Upper Peninsula in the 1840s and remain a significant source of wealth for the state. By the 1890s, Michigan was the largest supplier of iron ore in the United States. Railroads would haul ore from the mines to the ore docks on the Great Lakes in places such as Escanaba and Marquette , where the cargo would be loaded onto ore freighters and transported to the rest of the country.
Iron ore was first discovered in Michigan's Upper Peninsula on September 19, 1844, by William A. Burt , who was then employed to lead a party engaged in surveying the Upper Peninsula on behalf of the United States government. Variations in the readings of magnetic compasses employed suggested that something was disturbing the local magnetic fields, and, on investigation of part of the Marquette Iron Range , ore was found as close to the surface as just below the sod. [ 3 ]
Development of this find was ongoing throughout the 1840s and early 1850s. The initial focus was on producing iron locally, as shipping was difficult. Early shipping attempts using mule teams, plank roads , and barrels loaded as deck cargo on schooners required transshipment after portaging the St. Marys River rapids, and costs proved prohibitive. However, as the mines continued to develop and railways were put in place, the volume of ore increased, far outstripping the local production capacity. In 1855, the Soo Locks opened, and the volume of ore shipped increased, with a total of 1447 tons shipped on various brigs and schooners. The first dock specifically for the ore trade was built in 1857 in Marquette. It was flat rather than elevated, and the vessels were loaded by men using wheelbarrows. [ 3 ]
By 1860, the volume of ore shipped through the Soo Locks had increased to 114,401 tons; it fell to 49,909 tons the next year after the American Civil War broke out. But by 1862, an additional wooden dock had been constructed at Marquette, this time featuring an elevated railway trestle for ore jennies to discharge ore into pockets.
In 1911 another iron ore loading dock was built: the wooden frame work was replaced with a concrete and steel frame work. The new iron ore loading dock at Marquette was 1200 feet long; 75 feet high; and 60 feet wide. Four railroad tracks ran along the top and there was storage space inside the bottom concrete part for 60,000 tons of iron ore. The new ore loading dock incorporated a concrete bottom half with a light steel frame work on top, replacing the wooden supports previously used. While this structure was more expensive to build than previous iron ore loading docks, the builders claimed the new construction would hold up far longer than the old construction method. [ 4 ] Schooners started to feature regularly spaced hatch covers, which sped up loading. [ 3 ] But steamers of the day were not well adapted for bulk cargoes such as iron ore. They did not have hatches through their decks. Instead they had gangways through the sides. So ore shipments were loaded via wheelbarrows through the gangways. By having a flat surface on one side and moorings directly under the pockets, schooners could receive the ore directly. Steamers moored on the opposite side of the dock for the manual method using wheelbarrows. [ 3 ] However, unloading was still a laborious hand-powered affair.
During this period, the iron ore trade was dwarfed by the grain and lumber trade. For example, in 1866 about 1,500,000 tons of grain were received at the port of Buffalo, New York , alone. Chicago received about 400,000 tons of lumber. The amount of iron ore delivered to all Lake Erie ports amounted to only 278,976 gross tons. [ 3 ] Until 1876, Marquette was the only port on Lake Superior that shipped iron ore to the east. [ 5 ]
After the Civil War, advancement was rapid. The Cleveland Iron Mining Company 's dock was 30 feet (9.1 m) above the lake level and was originally built with 29 schooner pockets and 6 steam boat pockets. By 1872, it had been extended an additional 350 feet (110 m), providing 54 additional pockets. During the 1873 season, the total tonnage of iron ore shipped from the port of Marquette was 1,175,000 tons. [ 5 ]
As additional ore ranges (the Gogebic, Menominee East, and Menominee West) began to be developed, and the US developed more rapidly as an industrial power, other ports also constructed ore docks, primarily on Lake Superior. For example, Ashland, Wisconsin , the natural port for the Gogebic Range, had three docks by 1916, the first built by the Milwaukee, Lake Shore and Western Railway in 1884-1885. It was 1,400 feet (430 m) long, with 234 pockets, and four tracks on the dock, and could hold 25,000 tons of ore. [ 6 ]
The natural port for the Menominee East range was Escanaba, Michigan ; the first docks there were built in 1865.
By 1884, locomotives played a vital part in hauling ore from the mines to the docks. The pockets and chutes were filled using the hopper-type ore cars. In 1889, the Duluth, South Shore and Atlantic Railway operated a dock with 284 pockets. The railroads also provided alternate transportation for people, goods and eventually, iron ore.
The length and capacity of docks corresponded with the capacity of the ore freighters which they were designed to accommodate. As ore carriers increased in height and width, higher pocket openings were required and new docks had to be constructed, and older ones rebuilt. In 1867, vessels carried ore cargoes of 550 tons. The first steel ore freighters introduced in the 1880s had a capacity of 2,500 tons. By 1898, ships were carrying 6,400 tons and by 1938, 15,000 tons.
The demand for iron ore grew, faster dispatch of vessels was necessary, and larger rail cars and the number of more powerful locomotives for transporting the ore from mines to the docks increased with improved technology. These developments caused the old wooden Duluth, South Shore and Atlantic Railway ore dock built in 1898 to become obsolete. A new dock was constructed by the Lake Superior and Ishpeming Railroad Company of reinforced concrete and steel in 1911 and 1912 in the Upper Harbor near Presque Isle Park. This dock is 75 feet (23 m) high, 1,200 feet (370 m) long and has 200 pockets, with a total capacity of 50,000 tons of ore pellets. By 1929, more than one and one half times the combined yearly tonnage of the Panama and Suez canals passed through the Soo Locks. The dock is still in use today.
Activity also continued in Marquette's Lower Harbor. The Duluth, South Short and Atlantic Railway, in order to remain competitive, completed construction of a new dock in 1932. This dock was constructed of steel and concrete, 85.5 feet (26.1 m) high, 969 feet (295 m) long, with 150 pockets. The total capacity was 47,500 tons. The D.S.S. & A. Railway merged with other companies to form the Soo Line. The dock remained in operation until the late 1960s, when a decrease in demand for iron ore forced it to close. In 1987, the Soo Line sold its Lake States Division to Wisconsin Central Ltd . The ownership passed to the latter railroad, where it remains today.
Many docks have been torn down or abandoned, but a few remain in operation. In 2008, the Lake Superior & Ishpeming dock, one of the docks at Marquette, Michigan , loaded its 400 millionth ton of ore after 90 years of service. [ 7 ] Another dock, built later by Merritt-Chapman & Scott for the Duluth, South Shore and Atlantic Railway (DSS&A), was taken out of service in 1971. [ citation needed ] As of 2022 [update] , there are plans to redevelop this ore dock. The Mining Journal , a local newspaper, has stated that these include "ecological education facilities, year-round indoor botanical gardens, historical preservation and education, and community spaces." [ 8 ]
The largest dock of the type in the world exist in Superior, Wisconsin as part of BNSF's Allouez Taconite Facility, however, these docks have been abandoned since 1978. Ore docks in Duluth, MN, [ citation needed ] and Two Harbors, MN, [ citation needed ] are still in service and operated by Canadian National Railway and Lake Superior and Ishpeming Railroad .
Ore docks are mentioned in popular culture, at least one high school athletic team, the Ashland, Wisconsin Oredockers takes their name from them, [ 9 ] and they have been mentioned in song. [ 10 ] | https://en.wikipedia.org/wiki/Ore_dock |
Various theories of ore genesis explain how the various types of mineral deposits form within Earth's crust . Ore-genesis theories vary depending on the mineral or commodity examined.
Ore-genesis theories generally involve three components: source, transport or conduit, and trap. (This also applies to the petroleum industry: petroleum geologists originated this analysis.)
The biggest deposits form when the source is large, the transport mechanism is efficient, and the trap is active and ready at the right time.
These processes are the physicochemical phenomena and reactions caused by movement of hydrothermal water within the crust, often as a consequence of magmatic intrusion or tectonic upheavals. The foundations of hydrothermal processes are the source-transport-trap mechanism.
Sources of hydrothermal solutions include seawater and meteoric water circulating through fractured rock, formational brines (water trapped within sediments at deposition), and metamorphic fluids created by dehydration of hydrous minerals during metamorphism .
Metal sources may include a plethora of rocks. However most metals of economic importance are carried as trace elements within rock-forming minerals, and so may be liberated by hydrothermal processes. This happens because of:
Transport by hydrothermal solutions usually requires a salt or other soluble species which can form a metal-bearing complex. These metal-bearing complexes facilitate transport of metals within aqueous solutions, generally as hydroxides , but also by processes similar to chelation .
This process is especially well understood in gold metallogeny where various thiosulfate, chloride, and other gold-carrying chemical complexes (notably tellurium -chloride/sulfate or antimony-chloride/sulfate). The majority of metal deposits formed by hydrothermal processes include sulfide minerals , indicating sulfur is an important metal-carrying complex.
Sulfide deposition within the trap zone occurs when metal-carrying sulfate, sulfide, or other complexes become chemically unstable due to one or more of the following processes;
Metal can also precipitate when temperature and pressure or oxidation state favour different ionic complexes in the water, for instance the change from sulfide to sulfate, oxygen fugacity , exchange of metals between sulfide and chloride complexes, et cetera.
Ore deposits formed by lateral secretion are formed by metamorphic reactions during shearing , which liberate mineral constituents such as quartz, sulfides, gold, carbonates, and oxides from deforming rocks, and focus these constituents into zones of reduced pressure or dilation such as faults . This may occur without much hydrothermal fluid flow, and this is typical of podiform chromite deposits.
Metamorphic processes also control many physical processes which form the source of hydrothermal fluids, outlined above.
Surficial processes are the physical and chemical phenomena which cause concentration of ore material within the regolith , generally by the action of the environment. This includes placer deposits, laterite deposits, and residual or eluvial deposits. Superficial deposits processes of ore formation include;
Classification of hydrothermal ore deposits is also achieved by classifying according to the temperature of formation, which roughly also correlates with particular mineralising fluids, mineral associations and structural styles. [ 2 ] This scheme, proposed by Waldemar Lindgren (1933) classified hydrothermal deposits as follows: [ 2 ]
Ore deposits are usually classified by ore formation processes and geological setting. For example, sedimentary exhalative deposits (SEDEX), are a class of ore deposit formed on the sea floor (sedimentary) by exhalation of brines into seawater (exhalative), causing chemical precipitation of ore minerals when the brine cools, mixes with sea water, and loses its metal carrying capacity.
Ore deposits rarely fit neatly into the categories in which geologists wish to place them. Many may be formed by one or more of the basic genesis processes above, creating ambiguous classifications and much argument and conjecture. Often ore deposits are classified after examples of their type, for instance Broken Hill type lead-zinc-silver deposits or Carlin–type gold deposits .
As they require the conjunction of specific environmental conditions to form, particular mineral deposit types tend to occupy specific geodynamic niches, [ 6 ] therefore, this page has been organised by metal commodity . It is also possible to organise theories the other way, namely according to geological criteria of formation. Often ores of the same metal can be formed by multiple processes, and this is described here under each metal or metal complex.
Iron ores are overwhelmingly derived from ancient sediments known as banded iron formations (BIFs). These sediments are composed of iron oxide minerals deposited on the sea floor. Particular environmental conditions are needed to transport enough iron in sea water to form these deposits, such as acidic and oxygen-poor atmospheres within the Proterozoic Era.
Often, more recent weathering is required to convert the usual magnetite minerals into more easily processed hematite . Some iron deposits within the Pilbara of Western Australia are placer deposits , formed by accumulation of hematite gravels called pisolites which form channel-iron deposits . These are preferred because they are cheap to mine.
Lead - zinc deposits are generally accompanied by silver , hosted within the lead sulfide mineral galena or within the zinc sulfide mineral sphalerite .
Lead and zinc deposits are formed by discharge of deep sedimentary brine onto the sea floor (termed sedimentary exhalative or SEDEX), or by replacement of limestone , in skarn deposits, some associated with submarine volcanoes (called volcanogenic massive sulfide ore deposits or VMS), or in the aureole of subvolcanic intrusions of granite. The vast majority of SEDEX lead and zinc deposits are Proterozoic in age, although there are significant Jurassic examples in Canada and Alaska.
The carbonate replacement type deposit is exemplified by the Mississippi valley type (MVT) ore deposits. MVT and similar styles occur by replacement and degradation of carbonate sequences by hydrocarbons , which are thought important for transporting lead.
Gold deposits are formed via a very wide variety of geological processes. Deposits are classified as primary, alluvial or placer deposits, or residual or laterite deposits. Often a deposit will contain a mixture of all three types of ore.
Plate tectonics is the underlying mechanism for generating gold deposits. The majority of primary gold deposits fall into two main categories: lode gold deposits or intrusion -related deposits.
Lode gold deposits , also referred to as orogenic gold are generally high-grade, thin, vein and fault hosted. They are primarily made up of quartz veins also known as lodes or reefs , which contain either native gold or gold sulfides and tellurides . Lode gold deposits are usually hosted in basalt or in sediments known as turbidite , although when in faults , they may occupy intrusive igneous rocks such as granite .
Lode-gold deposits are intimately associated with orogeny and other plate collision events within geologic history. It is thought that most lode gold deposits are sourced from metamorphic rocks by the dehydration of basalt during metamorphism. The gold is transported up faults by hydrothermal waters and deposited when the water cools too much to retain gold in solution.
Intrusive related gold (Lang & Baker, 2001) is generally hosted in granites, porphyry , or rarely dikes . Intrusive related gold usually also contains copper , and is often associated with tin and tungsten , and rarely molybdenum , antimony , and uranium . Intrusive-related gold deposits rely on gold existing in the fluids associated with the magma (White, 2001), and the inevitable discharge of these hydrothermal fluids into the wall-rocks (Lowenstern, 2001). Skarn deposits are another manifestation of intrusive-related deposits.
Placer deposits are sourced from pre-existing gold deposits and are secondary deposits. Placer deposits are formed by alluvial processes within rivers and streams, and on beaches . Placer gold deposits form via gravity , with the density of gold causing it to sink into trap sites within the river bed, or where water velocity drops, such as bends in rivers and behind boulders. Often placer deposits are found within sedimentary rocks and can be billions of years old, for instance the Witwatersrand deposits in South Africa . Sedimentary placer deposits are known as 'leads' or 'deep leads'.
Placer deposits are often worked by fossicking , and panning for gold is a popular pastime.
Laterite gold deposits are formed from pre-existing gold deposits (including some placer deposits) during prolonged weathering of the bedrock. Gold is deposited within iron oxides in the weathered rock or regolith , and may be further enriched by reworking by erosion. Some laterite deposits are formed by wind erosion of the bedrock leaving a residuum of native gold metal at surface.
A bacterium, Cupriavidus metallidurans , plays a vital role in the formation of gold nuggets by precipitating metallic gold from a solution of gold (III) tetrachloride , a compound highly toxic to most other microorganisms. [ 8 ] Similarly, Delftia acidovorans can form gold nuggets. [ 9 ]
Platinum and palladium are precious metals generally found in ultramafic rocks. The source of platinum and palladium deposits is ultramafic rocks which have enough sulfur to form a sulfide mineral while the magma is still liquid. This sulfide mineral (usually pentlandite , pyrite , chalcopyrite , or pyrrhotite ) gains platinum by mixing with the bulk of the magma because platinum is chalcophile and is concentrated in sulfides. Alternatively, platinum occurs in association with chromite either within the chromite mineral itself or within sulfides associated with it.
Sulfide phases only form in ultramafic magmas when the magma reaches sulfur saturation. This is generally thought to be nearly impossible by pure fractional crystallisation, so other processes are usually required in ore genesis models to explain sulfur saturation. These include contamination of the magma with crustal material, especially sulfur-rich wall-rocks or sediments; magma mixing; volatile gain or loss.
Often platinum is associated with nickel , copper , chromium , and cobalt deposits.
Nickel deposits are generally found in two forms, either as sulfide or laterite.
Sulfide type nickel deposits are formed in essentially the same manner as platinum deposits. Nickel is a chalcophile element which prefers sulfides, so an ultramafic or mafic rock which has a sulfide phase in the magma may form nickel sulfides. The best nickel deposits are formed where sulfide accumulates in the base of lava tubes or volcanic flows — especially komatiite lavas.
Komatiitic nickel-copper sulfide deposits are considered to be formed by a mixture of sulfide segregation, immiscibility, and thermal erosion of sulfidic sediments. The sediments are considered to be necessary to promote sulfur saturation.
Some subvolcanic sills in the Thompson Belt of Canada host nickel sulfide deposits formed by deposition of sulfides near the feeder vent. Sulfide was accumulated near the vent due to the loss of magma velocity at the vent interface. The massive Voisey's Bay nickel deposit is considered to have formed via a similar process.
The process of forming nickel laterite deposits is essentially similar to the formation of gold laterite deposits, except that ultramafic or mafic rocks are required. Generally nickel laterites require very large olivine -bearing ultramafic intrusions. Minerals formed in laterite nickel deposits include gibbsite .
Copper is found in association with many other metals and deposit styles. Commonly, copper is either formed within sedimentary rocks, or associated with igneous rocks.
The world's major copper deposits are formed within the granitic porphyry copper style. Copper is enriched by processes during crystallisation of the granite and forms as chalcopyrite — a sulfide mineral, which is carried up with the granite.
Sometimes granites erupt to surface as volcanoes , and copper mineralisation forms during this phase when the granite and volcanic rocks cool via hydrothermal circulation .
Sedimentary copper forms within ocean basins in sedimentary rocks. Generally this forms by brine from deeply buried sediments discharging into the deep sea, and precipitating copper and often lead and zinc sulfides directly onto the sea floor. This is then buried by further sediment. This is a process similar to SEDEX zinc and lead, although some carbonate-hosted examples exist.
Often copper is associated with gold , lead , zinc , and nickel deposits.
Uranium deposits are usually sourced from radioactive granites, where certain minerals such as monazite are leached during hydrothermal activity or during circulation of groundwater . The uranium is brought into solution by acidic conditions and is deposited when this acidity is neutralised. Generally this occurs in certain carbon-bearing sediments, within an unconformity in sedimentary strata. The majority of the world's nuclear power is sourced from uranium in such deposits.
Uranium is also found in nearly all coal at several parts per million , and in all granites. Radon is a common problem during mining of uranium as it is a radioactive gas.
Uranium is also found associated with certain igneous rocks, such as granite and porphyry . The Olympic Dam deposit in Australia is an example of this type of uranium deposit. It contains 70% of Australia's share of 40% of the known global low-cost recoverable uranium inventory.
Mineral sands are the predominant type of titanium , zirconium , and thorium deposit. They are formed by accumulation of such heavy minerals within beach systems, and are a type of placer deposits . The minerals which contain titanium are ilmenite, rutile, and leucoxene , zirconium is contained within zircon , and thorium is generally contained within monazite . These minerals are sourced from primarily granite bedrock by erosion and transported to the sea by rivers where they accumulate within beach sands. Rarely, but importantly, gold , tin , and platinum deposits can form in beach placer deposits.
These three metals generally form in a certain type of granite , via a similar mechanism to intrusive-related gold and copper. They are considered together because the process of forming these deposits is essentially the same. Skarn type mineralisation related to these granites is a very important type of tin, tungsten, and molybdenum deposit. Skarn deposits form by reaction of mineralised fluids from the granite reacting with wall rocks such as limestone . Skarn mineralisation is also important in lead , zinc , copper , gold , and occasionally uranium mineralisation.
Greisen granite is another related tin-molybdenum and topaz mineralisation style.
The overwhelming majority of rare-earth elements , tantalum , and lithium are found within pegmatite . Ore genesis theories for these ores are wide and varied, but most involve metamorphism and igneous activity. [ 10 ] Lithium is present as spodumene or lepidolite within pegmatite.
Carbonatite intrusions are an important source of these elements. Ore minerals are essentially part of the unusual mineralogy of carbonatite.
Phosphate is used in fertilisers. Immense quantities of phosphate rock or phosphorite occur in sedimentary shelf deposits, ranging in age from the Proterozoic to currently forming environments. [ 11 ] Phosphate deposits are thought to be sourced from the skeletons of dead sea creatures which accumulated on the seafloor. Similar to iron ore deposits and oil, particular conditions in the ocean and environment are thought to have contributed to these deposits within the geological past.
Phosphate deposits are also formed from alkaline igneous rocks such as nepheline syenites , carbonatites , and associated rock types. The phosphate is, in this case, contained within magmatic apatite , monazite , or other rare-earth phosphates.
Due to the presence of vanabins , concentration of vanadium found in the blood cells of Ascidia gemmata belonging to the suborder Phlebobranchia is 10,000,000 times higher than that in the surrounding seawater. A similar biological process might have played a role in the formation of vanadium ores.
Vanadium is also present in fossil fuel deposits such as crude oil , coal , oil shale , and oil sands . In crude oil, concentrations up to 1200 ppm have been reported.
Precious metals such as gold and platinum , but also many other rare and noble metals , largely originated within neutron star collisions - collisions between exceedingly heavy massive and dense remnants of supernovas . In the final moments of the collision, the physical conditions are so extreme that these heavy rare elements can be formed, and are sprayed into space. Interstellar dust and gas clouds contain some of these elements, as did the dust cloud from which the Solar System formed.
Those heavy metals fell to the centre of the molten core of Earth, and are no longer accessible. However about 200 million years after Earth formed, a late heavy bombardment of meteors impacted Earth. As Earth had already begun to cool and solidify, the material (including heavy metals) in that bombardment became part of Earth's crust , rather than falling deep into the core. They became processed and exposed by geological processes over billions of years. It is believed that this represents the origin of many elements, and all heavy metals, that are now found on Earth. [ 12 ] [ 13 ] | https://en.wikipedia.org/wiki/Ore_genesis |
Oregrounds iron was a grade of iron that was regarded as the best grade available in 18th century England . The term was derived from the small Swedish city of Öregrund , the port from which the bar iron was shipped. It was produced using the Walloon process .
Oregrounds iron is the equivalent of the Swedish vallonjärn , which literally translates as Walloon iron . The Swedish name derives from the iron being produced by the Walloon version of the finery forge process, the Walloon process [ 1 ] as opposed to the German method, which was more common in Sweden. Actually, the term is more specialised, as all the Swedish Walloon forges made iron from ore ultimately derived from the Dannemora mine . It was made in about 20 forges mainly in Uppland .
Many of the ironworks were founded by Louis de Geer and other Dutch entrepreneurs who set up ironworks in Sweden in the 1610s and 1620s, with blast furnaces and finery forges . Most of the early forgemen were also from Wallonia . [ 2 ]
The technique was developed in Wallonia in present-day Belgium during the Middle Ages . The Walloon method [ 3 ] consisted of making pig iron in a blast furnace , followed by refining it in a finery forge . The process was devised in the Liège region, and spread [ 4 ] into France and thence from the Pays de Bray to England before the end of the 15th century. [ 5 ] [ 6 ] Louis de Geer took it to Roslagen in Sweden in the early 17th century, where he employed Walloon ironmakers. [ 7 ] Iron made there by this method was known in England as oregrounds iron. [ 8 ]
Swedish law required bars of iron to have the forge's mark stamped into it for quality control reasons. In Britain, the iron was known by these 'marks', and the quality of each brand was well-known to the buyers in London , Sheffield , Birmingham and elsewhere. It was divided into two grades:
Its special property was its purity. The manganese content of the Dannemora ore caused impurities, which would otherwise have remained in the iron, to react preferentially with the manganese and to be carried off into the slag . This level of purity meant that the iron was particularly suitable for conversion to steel by being re- carburized , using the cementation process . This made it particularly suitable for making steel, oregrounds iron was an indispensable raw material for metal manufactures, particularly the Sheffield cutlery industry. Substantial quantities were also (until about 1808) bought for use by the British Navy .
This and other uses absorbed substantially the whole output of the industry. The trade in oregrounds iron was controlled from the 1730s to the 1850s by a cartel of merchants, of whom the longest enduring members were the Sykes family of Hull . Other participants were resident in (or controlling imports through) London and Bristol. These merchants advanced money to Swedish exporting houses, which in turn advanced it to the ironmasters, thus buying up the output of the forges several years in advance. | https://en.wikipedia.org/wiki/Oregrounds_iron |
Orgalim (derived originally from the French Organisme de Liaison des Industries Métalliques Européennes ) is a European trade association represents Europe’s engineering and technology industries. It represent over 35 trade associations from 28 European countries that have over 770,000 member companies between them.
Orgalim is a European-level federation that engages with EU policymakers on behalf of its membership, speaking for 28 national industry associations and 20 European sector associations. Orgalim is registered under the European Union Transparency Register as part of European Union lobbying rules. [ 1 ] – ID number: 20210641335-88.
Orgalim was formally created in late 1954, therefore pre-dating the official European Union project. Founding associations came from Austria , Belgium , France , West Germany , Italy , the Netherlands , Switzerland , the UK , Sweden , Finland , Denmark and Norway . Meetings and informal collaboration between industries had begun in 1948, and although initially created as an informal club without any financial demands, the organisation became increasingly structured and eventually developed a secretariat in the early 1950s.
Various other engineering groups had been created at the same time as the European Coal and Steel Community developed, such as MEFTA and COLIME. Orgalim members decided in 1960 to incorporate the groups into Orgalim as working groups. [ 2 ]
Orgalim's advocacy work addresses a broad spectrum of policy and regulatory issues from digital transformation and trade to Internal Market and environment policies. [ 3 ] [ 4 ]
As part of its service, Orgalim publishes legal publications to provide companies with contractual solutions for business-to-business relations – with use cases ranging from product supply, product installation, repair and maintenance, to agency contracts and distributor abroad contracts. [ 5 ] [ 6 ]
Orgalim is a member of the Alliance for a Competitive European Industry (ACEI) and the European Forum for Manufacturing (EFM). [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Orgalim |
An organ-on-a-chip ( OOC ) is a multi-channel 3D microfluidic cell culture , integrated circuit (chip) that simulates the activities, mechanics and physiological response of an entire organ or an organ system . [ 1 ] [ 2 ] It constitutes the subject matter of significant biomedical engineering research, more precisely in bio-MEMS . The convergence of labs-on-chips (LOCs) and cell biology has permitted the study of human physiology in an organ-specific context. By acting as a more sophisticated in vitro approximation of complex tissues than standard cell culture , they provide the potential as an alternative to animal models for drug development and toxin testing.
Although multiple publications claim to have translated organ functions onto this interface, the development of these microfluidic applications is still in its infancy. Organs-on-chips vary in design and approach between different researchers. Organs that have been simulated by microfluidic devices include brain , lung , heart , kidney , liver , prostate , vessel ( artery ), skin , bone , cartilage and more. [ 3 ]
A limitation of the early organ-on-a-chip approach is that simulation of an isolated organ may miss significant biological phenomena that occur in the body's complex network of physiological processes, and that this oversimplification limits the inferences that can be drawn. Many aspects of subsequent microphysiometry aim to address these constraints by modeling more sophisticated physiological responses under accurately simulated conditions via microfabrication , microelectronics and microfluidics. [ 4 ]
The development of organ chips has enabled the study of the complex pathophysiology of human viral infections . An example is the liver chip platform that has enabled studies of viral hepatitis . [ 5 ]
A lab-on-a-chip is a device that integrates one or several laboratory functions on a single chip that deals with handling particles in hollow microfluidic channels. It has been developed for over a decade. Advantages in handling particles at such a small scale include lowering fluid volume consumption (lower reagents costs, less waste), increasing portability of the devices, increasing process control (due to quicker thermo-chemical reactions) and decreasing fabrication costs. Additionally, microfluidic flow is entirely laminar (i.e., no turbulence ). Consequently, there is virtually no mixing between neighboring streams in one hollow channel. In cellular biology convergence, this rare property in fluids has been leveraged to better study complex cell behaviors, such as cell motility in response to chemotactic stimuli , stem cell differentiation , axon guidance , subcellular propagation of biochemical signaling and embryonic development . [ 6 ]
3D cell-culture models exceed 2D culture systems by promoting higher levels of cell differentiation and tissue organization. 3D culture systems are more successful because the flexibility of the ECM gels accommodates shape changes and cell-cell connections – formerly prohibited by rigid 2D culture substrates. Nevertheless, even the best 3D culture models fail to mimic an organ's cellular properties in many aspects, [ 6 ] including tissue-to-tissue interfaces (e.g., epithelium and vascular endothelium ), spatiotemporal gradients of chemicals, and the mechanically active microenvironments (e.g. arteries' vasoconstriction and vasodilator responses to temperature differentials). The application of microfluidics in organs-on-chips enables the efficient transport and distribution of nutrients and other soluble cues throughout the viable 3D tissue constructs. Organs-on-chips are referred to as the next wave of 3D cell-culture models that mimic whole living organs' biological activities, dynamic mechanical properties and biochemical functionalities. [ 7 ]
Brain-on-a-chip devices are devices that allow the culturing and manipulation of brain-related tissues through microfabrication and microfluidics by: 1) improving culture viability; 2) supporting high-throughput screening for simple models; 3) modeling tissue or organ-level physiology and disease in vitro /ex vivo , and 4) adding high precision and tunability of microfluidic devices. [ 8 ] [ 9 ] Brain-on-a-chip devices can span multiple levels of complexity in terms of cell culture methodology and can include brain parenchyma and/or blood-brain barrier tissues. [ 10 ] Devices have been made using platforms that range from traditional 2D cell culture to 3D tissues in the form of organotypic brain slices and more recently organoids.
Organotypic brain slices are an in vitro model that replicates in vivo physiology with additional throughput and optical benefits, [ 8 ] thus pairing well with microfluidic devices. Brain slices have advantages over primary cell culture in that tissue architecture is preserved and multicellular interactions can still occur. [ 11 ] [ 12 ] There is flexibility in their use, as slices can be used acutely (less than 6 hours after slice harvesting) or cultured for later experimental use. Because organotypic brain slices can maintain viability for weeks, they allow for long-term effects to be studied. [ 11 ] Slice-based systems also provide experimental access with precise control of extracellular environments, making it a suitable platform for correlating disease with neuropathological outcomes. [ 12 ] Organotypic brain slices can be extracted and cultured from multiple animal species (e.g. rats), but also from humans. [ 13 ]
Microfluidic devices have been paired with organotypic slices to improve culture viability. The standard procedure for culturing organotypic brain slices (around 300 microns in thickness) uses semi-porous membranes to create an air-medium interface, [ 14 ] but this technique results in diffusion limitations of nutrients and dissolved gases. Because microfluidic systems introduce laminar flow of these necessary nutrients and gases, transport is improved and higher tissue viability can be achieved. [ 9 ] In addition to keeping standard slices viable, brain-on-a-chip platforms have allowed the successful culturing of thicker brain slices (approximately 700 microns), despite a significant transport barrier due to thickness. [ 15 ] As thicker slices retain more native tissue architecture, this allows brain-on-a-chip devices to achieve more " in vivo -like" characteristics without sacrificing cell viability. Microfluidic devices support high-throughput screening and toxicological assessments in both 2D and slice cultures, leading to the development of novel therapeutics targeted for the brain. [ 8 ] One device was able to screen the drugs pitavastatin and irinotecan combinatorically in glioblastoma multiform (the most common form of human brain cancer). [ 16 ] These screening approaches have been combined with the modeling of the blood-brain barrier (BBB), a significant hurdle for drugs to overcome when treating the brain, allowing for drug efficacy across this barrier to be studied in vitro . [ 17 ] [ 18 ] [ 19 ] Microfluidic probes have been used to deliver dyes with high regional precision, making way for localized microperfusion in drug applications. [ 20 ] [ 21 ] Microfluidic BBB in vitro models replicate a 3D environment for embedded cells (which provides precise control of cellular and extracellular environment), replicate shear stress, have more physiologically relevant morphology in comparison to 2D models, and provide easy incorporation of different cell types into the device. [ 22 ] Because microfluidic devices can be designed with optical accessibility, this also allows for the visualization of morphology and processes in specific regions or individual cells. Brain-on-a-chip systems can model organ-level physiology in neurological diseases, such as Alzheimer's disease , Parkinson's disease , and multiple sclerosis more accurately than with traditional 2D and 3D cell culture techniques. [ 23 ] [ 24 ] The ability to model these diseases in a way that is indicative of in vivo conditions is essential for the translation of therapies and treatments. [ 9 ] [ 8 ] Additionally, brain-on-a-chip devices have been used for medical diagnostics, such as in biomarker detection for cancer in brain tissue slices. [ 25 ]
Brain-on-a-chip devices can cause shear stress on cells or tissue due to flow through small channels, which can result in cellular damage. [ 9 ] These small channels also introduce susceptibility to the trapping of air bubbles that can disrupt flow and potentially cause damage to the cells. [ 26 ] The widespread use of PDMS ( polydimethylsiloxane ) in brain-on-a-chip devices has some drawbacks. Although PDMS is cheap, malleable, and transparent, proteins and small molecules can be absorbed by it and later leech at uncontrolled rates. [ 9 ]
Despite the progress in microfluidic BBB devices, these devices are often too technically complex, require highly specialized setups and equipment, and are unable to detect temporal and spatial differences in the transport kinetics of substances that migrate across cellular barriers. Also, direct measurements of permeability in these models are limited due to the limited perfusion and complex, poorly defined geometry of the newly formed microvascular network. [ 22 ]
The human gut-on-a-chip contains two microchannels that are separated by the flexible porous Extracellular Matrix (ECM)-coated membrane lined by the gut epithelial cells: Caco-2, which has been used extensively as the intestinal barrier. [ 27 ] [ 28 ] Caco-2 cells are cultured under spontaneous differentiation of its parental cell, a human colon adenocarcinoma , that represent the model of protective and absorptive properties of the gut. [ 27 ] The microchannels are fabricated from polydimethylsiloxane (PDMS) polymer. [ 28 ] In order to mimic the gut microenvironment, peristalsis-like fluid flow is designed. [ 28 ] By inducing suction in the vacuum chambers along both sides of the main cell channel bilayer, cyclic mechanical strain of stretching and relaxing are developed to mimic the gut behaviors. [ 28 ] Furthermore, cells undergo spontaneous villus morphogenesis and differentiation, which generalizes characteristics of intestinal cells. [ 28 ] [ 29 ] Under the three-dimensional villi scaffold, cells not only proliferate, but metabolic activities are also enhanced. [ 30 ] Another important player in the gut is the microbes, namely gut microbiota . Many microbial species in the gut microbiota are strict anaerobes. In order to co-culture these oxygen intolerant anaerobes with the oxygen favorable intestinal cells, a polysulfone fabricated gut-on-a-chip is designed. [ 31 ] The system maintained the co-culture of colon epithelial cells, goblet-like cells, and bacteria Faecalibacterium prausnitzii , Eubacterium rectale , and Bacteroides thetaiotaomicron . [ 31 ]
Oral administration is one of the most common methods for drug administration. It allows patients, especially out-patients, to self-serve the drugs with minimal possibility of experiencing acute drug reactions and in most cases: pain-free. However, the drug's action in the body can be largely influenced by the first pass effect . The gut, which plays an important role in the human digestive system, determines the effectiveness of a drug by absorbing its chemical and biological properties selectively. [ 32 ] While it is costly and time-consuming to develop new drugs, the fact that the gut-on-a-chip technology attains a high level of throughput has significantly decreased research and development costs and time for new drugs. [ 33 ]
Even though the cause for inflammatory bowel disease (IBD) is elusive, its pathophysiology involves the gut microbiota . [ 34 ] Current methods of inducing IBD are using inflammatory cues to activate Caco-2. It was found that the intestinal epithelium experienced a reduction in barrier function and increased cytokine concentrations. [ 33 ] The gut-on-a-chip allowed for the assessment on drug transport, absorption and toxicity as well as potential developments in studying pathogenesis and interactions in the microenvironment overall. [ 35 ] Immune cells are essential in mediating inflammatory processes in many gastrointestinal disorders, a recent gut-on-a-chip system also includes multiple immune cells, e.g., macrophages, dendritic cells, and CD4+ T cells in the system. [ 36 ] Additionally, the gut-on-a-chip allows the testing of anti-inflammatory effects of bacterial species. [ 31 ]
The chip was used to model human radiation-induced injury to the intestine in vitro as it recapitulated the injuries at both cellular and tissue levels. Injuries include but not limited to: inhabitation of mucus production, promotion of villus blunting, and distortion of microvilli. [ 37 ]
Lung-on-a-chips are being designed in an effort to improve the physiological relevance of existing in vitro alveolar - capillary interface models. [ 38 ] Such a multifunctional microdevice can reproduce key structural, functional and mechanical properties of the human alveolar-capillary interface (i.e., the fundamental functional unit of the living lung).
Dongeun Huh from Wyss Institute for Biologically Inspired Engineering at Harvard describes their fabrication of a system containing two closely apposed microchannels separated by a thin (10 μm) porous flexible membrane made of PDMS . [ 39 ] The device largely comprises three microfluidic channels, and only the middle one holds the porous membrane. Culture cells were grown on either side of the membrane: human alveolar epithelial cells on one side, and human pulmonary microvascular endothelial cells on the other.
The compartmentalization of the channels facilitates not only the flow of air as a fluid which delivers cells and nutrients to the apical surface of the epithelium, but also allows for pressure differences to exist between the middle and side channels. During normal inspiration in a human's respiratory cycle , intrapleural pressure decreases, triggering an expansion of the alveoli. As air is pulled into the lungs, alveolar epithelium and the coupled endothelium in the capillaries are stretched. Since a vacuum is connected to the side channels, a decrease in pressure will cause the middle channel to expand, thus stretching the porous membrane and subsequently, the entire alveolar-capillary interface. The pressure-driven dynamic motion behind the stretching of the membrane, also described as a cyclic mechanical strain (valued at approximately 10%), significantly increases the rate of nanoparticle translocation across the porous membrane, when compared to a static version of this device, and to a Transwell culture system.
In order to fully validate the biological accuracy of a device, its whole-organ responses must be evaluated. In this instance, researchers inflicted injuries to the cells:
Additionally, researchers believe the potential value of this lung-on-a-chip system will aid in toxicology applications. By investigating the pulmonary response to nanoparticles , researchers hope to learn more about health risks in certain environments, and correct previously oversimplified in vitro models. Because a microfluidic lung-on-a-chip can more exactly reproduce the mechanical properties of a living human lung, its physiological responses will be quicker and more accurate than a Transwell culture system. Nevertheless, published studies admit that responses of a lung-on-a-chip do not yet fully reproduce the responses of native alveolar epithelial cells.
Past efforts to replicate in vivo cardiac tissue environments have proven to be challenging due to difficulties when mimicking contractility and electrophysiological responses. Such features would greatly increase the accuracy of in vitro experiments.
Microfluidics has already contributed to in vitro experiments on cardiomyocytes , which generate the electrical impulses that control the heart rate . [ 41 ] For instance, researchers have built an array of PDMS microchambers, aligned with sensors and stimulating electrodes as a tool that will electrochemically and optically monitor the cardiomyocytes' metabolism. [ 42 ] Another lab-on-a-chip similarly combined a microfluidic network in PDMS with planar microelectrodes, this time to measure extracellular potentials from single adult murine cardiomyocytes. [ 43 ]
A reported design of a heart-on-a-chip claims to have built "an efficient means of measuring structure-function relationships in constructs that replicate the hierarchical tissue architectures of laminar cardiac muscle." [ 44 ] This chip determines that the alignment of the myocytes in the contractile apparatus made of cardiac tissue and the gene expression profile (affected by shape and cell structure deformation) contributes to the force produced in cardiac contractility. This heart-on-a-chip is a biohybrid construct: an engineered anisotropic ventricular myocardium is an elastomeric thin film .
The design and fabrication process of this particular microfluidic device entails first covering the edges of a glass surface with tape (or any protective film) such as to contour the substrate's desired shape. A spin coat layer of PNIPA is then applied. After its dissolution, the protective film is peeled away, resulting in a self-standing body of PNIPA. The final steps involve the spin coating of protective surface of PDMS over the cover slip and curing. Muscular thin films (MTF) enable cardiac muscle monolayers to be engineered on a thin flexible substrate of PDMS. [ 45 ] In order to properly seed the 2D cell culture, a microcontact printing technique was used to lay out a fibronectin "brick wall" pattern on the PDMS surface. Once the ventricular myocytes were seeded on the functionalized substrate, the fibronectin pattern oriented them to generate an anisotropic monolayer.
After the cutting of the thin films into two rows with rectangular teeth, and subsequent placement of the whole device in a bath, electrodes stimulate the contraction of the myocytes via a field-stimulation – thus curving the strips/teeth in the MTF. Researchers have developed a correlation between tissue stress and the radius of curvature of the MTF strips during the contractile cycle, validating the demonstrated chip as a "platform for quantification of stress, electrophysiology and cellular architecture." [ 44 ]
While researchers have focused on 2D cell cultures , [ 46 ] [ 47 ] [ 48 ] 3D cell constructs mimic the in vivo environment [ 49 ] and the interactions (e.g., cell to cell ) occurring in the human body [ 50 ] better. Hence, they are considered promising models for studies such as toxicology [ 51 ] and response to drugs . [ 50 ] Based on the study of Chen et al., [ 52 ] the interactions of valvular endothelial / interstitial cells ( V ECs / V ICs ) are studied via a 3D PDMS - glass microfluidic device with a top channel flowed with V ECs under shear stress , a membrane with uniform pores, and a bottom channel containing V IC - hydrogel . [ 52 ] V ECs are verified to restrain the differentiation of morbid V IC myofibroblast , with reinforced suppression by shear stress . [ 52 ]
Another PDMS 3D microfluidic heart-on-a-chip design [ 50 ] is measured to generate 10% to 15% of uniaxial cyclic mechanical strains . The device consists of a cell culture with hanging posts for caging and an actuation compartment with scaffolding posts to avoid buckling of PDMS , along with the cardiac cycle pressure signal imitation. [ 50 ] The neonatal rat micro-engineered cardiac tissues (μECTs) stimulated by this design show improved synchronous beating, proliferation , maturation, and viability compared to the unstimulated control. [ 50 ] The contraction rate of human induced pluripotent stem cell -derived cardiomyocytes (hiPSC-CM) is observed to accelerate with 100-fold less isoprenaline , a heart block treatment, when having electrical pacing signal (+ES) compared to that without ES. [ 50 ]
3D microfluidic heart-on-a-chips have also facilitated the research of heart diseases . For instance, cardiac hypertrophy and fibrosis are studied via the respective biomarker level of the mechanically stimulated μECTs, such as atrial natriuretic peptide (ANP) for the former [ 53 ] and transforming growth factor-β (TGF-β) for the latter. [ 54 ] Also, the knowledge of ischaemia is gained by action potential observations. [ 55 ]
The microfluidic approaches utilized for teasing apart specific mechanisms at the single-cell level and at the tissue-level are becoming increasingly sophisticated and so are the fabrication methods. Rapid dissemination and availability of low cost, high resolution 3D printing technology is revolutionizing this space and opening new possibilities for building patient specific heart and cardiovascular systems. The confluence of high resolution 3D printing, patient derived iPSCs with artificial intelligence is posed to make significant strides towards truly personalized heart modelling and ultimately, patient care. [ 56 ]
Renal cells and nephrons have already been simulated by microfluidic devices. "Such cell cultures can lead to new insights into cell and organ function and be used for drug screening". [ 57 ] A kidney-on-a-chip device has the potential to accelerate research encompassing artificial replacement for lost kidney function . Nowadays, dialysis requires patients to go to a clinic up to three times per week. A more transportable and accessible form of treatment would not only increase the patient's overall health (by increasing frequency of treatment), but the whole process would become more efficient and tolerable. [ 58 ] Artificial kidney research is striving to bring transportability, wearability and perhaps implantation capability to the devices through innovative disciplines: microfluidics, miniaturization and nanotechnology. [ 59 ]
The nephron is the functional unit of the kidney and is composed of a glomerulus and a tubular component. [ 60 ] Researchers at MIT claim to have designed a bioartificial device that replicates the function of the nephron's glomerulus, proximal convoluted tubule and loop of Henle .
Each part of the device has its unique design, generally consisting of two microfabricated layers separated by a membrane. The only inlet to the microfluidic device is designed for the entering blood sample. In the glomerulus' section of the nephron, the membrane allows certain blood particles through its wall of capillary cells, composed by the endothelium, basement membrane and the epithelial podocytes. The fluid that is filtered from the capillary blood into Bowman's space is called filtrate or primary urine . [ 61 ]
In the tubules, some substances are added to the filtrate as part of the urine formation, and some substances reabsorbed out of the filtrate and back into the blood. The first segment of these tubules is the proximal convoluted tubule. This is where the almost complete absorption of nutritionally important substances takes place. In the device, this section is merely a straight channel, but blood particles going to the filtrate have to cross the previously mentioned membrane and a layer of renal proximal tubule cells. The second segment of the tubules is the loop of Henle where the reabsorption of water and ions from the urine takes place. The device's looping channels strives to simulate the countercurrent mechanism of the loop of Henle. Likewise, the loop of Henle requires a number of different cell types because each cell type has distinct transport properties and characteristics. These include the descending limb cells, thin ascending limb cells, thick ascending limb cells, cortical collecting duct cells and medullary collecting duct cells. [ 60 ]
One step towards validating the microfluidic device's simulation of the full filtration and reabsorption behavior of a physiological nephron would include demonstrating that the transport properties between blood and filtrate are identical with regards to where they occur and what is being let in by the membrane. For example, the large majority of passive transport of water occurs in the proximal tubule and the descending thin limb, or the active transport of NaCl largely occurs in the proximal tubule and the thick ascending limb. The device's design requirements would require the filtration fraction in the glomerulus to vary between 15 and 20%, or the filtration reabsorption in the proximal convoluted tubule to vary between 65 and 70%, and finally the urea concentration in urine (collected at one of the two outlets of the device) to vary between 200 and 400 mM. [ 62 ]
One recent report illustrates a biomimic nephron on hydrogel microfluidic devices with establishing the function of passive diffusion. [ 63 ] The complex physiological function of nephron is achieved on the basis of interactions between vessels and tubules (both are hollow channels). [ 64 ] However, conventional laboratory techniques usually focus on 2D structures, such as petri-dish that lacks capability to recapitulate real physiology that occurs in 3D. Therefore, the authors developed a new method to fabricate functional, cell-lining and perfusable microchannels inside 3D hydrogel. The vessel endothelial and renal epithelial cells are cultured inside hydrogel microchannel and form cellular coverage to mimic vessels and tubules, respectively. They employed confocal microscope to examine the passive diffusion of one small organic molecule (usually drugs) between the vessels and tubules in hydrogel. The study demonstrates the beneficial potential to mimic renal physiology for regenerative medicine and drug screening.
The liver is a major organ of metabolism , and it is related to glycogen storage, decomposition of red blood cells, certain protein and hormone synthesis, and detoxification . [ 66 ] Within these functions, its detoxification response is essential for new drug development and clinical trials . In addition, because of its multi-functions, the liver is prone to many diseases, and liver diseases have become a global challenge. [ 67 ]
Liver-on-a-chip devices utilize microfluidic techniques to simulate the hepatic system by imitating complex hepatic lobules that involve liver functions. Liver-on-a-chip devices provide a good model to help researchers work on dysfunction and pathogenesis of the liver with relatively low cost. Researchers use primary rat hepatocytes and other nonparenchymal cells. [ 68 ] [ 69 ] [ 70 ] This coculture method is extensively studied and is proved to be beneficial for extension of hepatocytes survival time and support the performance of liver-specific functions. [ 69 ] Many liver-on-a-chip systems are made of poly(dimethylsiloxane) (PDMS) with multiple channels and chambers based on specific design and objective. [ 68 ] [ 69 ] [ 70 ] PDMS is used and has become popular because it has relatively low price for raw materials, and it is also easily molded for microfluidic devices. [ 71 ] But PDMS can absorb important signaling molecules including proteins and hormones. Other more inert materials such as polysulfone or polycarbonate are used in liver-chips. [ 72 ]
A study by Emulate researchers assessed advantages of using liver-chips predicting drug-induced liver injury which could reduce the high costs and time needed in drug development workflows / pipelines , sometimes described as the pharmaceutical industry 's "productivity crisis". [ 73 ] [ 65 ] Zaher Nahle subsequently outlined 12 "reasons why micro-physiological systems (MPS) like organ-chips are better at modeling human diseases". [ 73 ]
One design from Kane et al. cocultures primary rat hepatocytes and 3T3-J2 fibroblasts in an 8*8 element array of microfluidic wells. [ 68 ] Each well is separated into two chambers. The primary chamber contains rat hepatocytes and 3T3-J2 fibroblasts and is made of glass for cells adhesion. Each of primary chamber is connected to a microfluidic network that supply metabolic substrate and remove metabolic byproducts. A 100 μm thick membrane of PDMS separates the primary and secondary chamber, allowing the secondary chamber to be connected to another microfluidic network that perfuses 37 °C room air with 10% carbon dioxide, and producing air exchange for rat hepatocytes. The production of urea and steady-state protein proves the viability of this device for use in high-throughput toxicity studies. [ 68 ]
Another design from Kang et al. cocultures primary rat hepatocytes and endothelial cells . [ 69 ] A single-channel is made first. Hepatocytes and endothelial cells are then planted on the device and are separated by a thin Matrigel layer in between. The metabolic substrate and metabolic byproducts share this channel to be supplied or removed. Later, a dual-channel is made, and endothelial cells and hepatocytes cells have their own channels to supply the substrate or remove the byproduct. The production of urea and positive result on hepatitis B virus (HBV) replication test shows its potential to study hepatotropic viruses. [ 69 ]
There are a few other applications on liver-on-a-chip. Lu et al. developed a liver tumor-on-a-chip model. The decellularized liver matrix (DLM)-gelatin methacryloyl (GelMA)-based biomimetic liver tumor -on-a-chip proved to be a suitable design for further anti-tumor studies. [ 70 ] Zhou et al. analyzed alcohol injures on the hepatocytes and the signaling and recovery. [ 74 ]
The liver-on-a-chip has shown its great potential for liver-related research. Future goals for liver-on-a-chip devices focus on recapitulating a more realistic hepatic environment, including reagents in fluids, cell types, extending survival time, etc. [ 69 ]
Recreation of the prostate epithelium is motivated by evidence suggesting it to be the site of nucleation in cancer metastasis. [ 75 ] [ 76 ] These systems essentially serve as the next step in the development of cells cultured from mice to two and subsequently three-dimensional human cell culturing. [ 77 ] [ 6 ] PDMS developments have enabled the creation of microfluidic systems that offer the benefit of adjustable topography, gas and liquid exchange , as well as an ease of observation via conventional microscopy. [ 78 ]
Researchers at the University of Grenoble Alpes have outlined a methodology that utilizes such a microfluidic system in the attempt to construct a viable Prostate epithelium model. The approach focuses on a cylindrical microchannel configuration, mimicking the morphology of a human secretory duct, within which the epithelium is located. [ 79 ] [ 80 ] Various microchannel diameters were assessed for successful promotion of cell cultures, and it was observed that diameters of 150-400 μm were the most successful. Furthermore, cellular adhesion endured throughout this experimentation, despite the introduction of physical stress through variations in microfluidic currents.
The objective of these constructions is to facilitate the collection of prostatic fluid, along with gauging cellular reactions to microenvironmental changes . [ 81 ] [ 82 ] Additionally, prostate-on-a-chip enables the recreation of metastasis scenarios, which allows the assessment of drug candidates and other therapeutic approaches. [ 83 ] [ 84 ] Scalability of this method is also attractive to researchers, as the reusable mold approach ensures a low-cost of production. [ 85 ]
Cardiovascular diseases are often caused by changes in structure and function of small blood vessels. For instance, self-reported rates of hypertension suggest that the rate is increasing, says a 2003 report from the National Health and Nutrition Examination Survey . [ 86 ] A microfluidic platform simulating the biological response of an artery could not only enable organ-based screens to occur more frequently throughout a drug development trial, but also yield a comprehensive understanding of the underlying mechanisms behind pathologic changes in small arteries and develop better treatment strategies. Axel Gunther from the University of Toronto argues that such MEMS -based devices could potentially help in the assessment of a patient's microvascular status in a clinical setting ( personalized medicine ). [ 87 ]
Conventional methods used to examine intrinsic properties of isolated resistance vessels (arterioles and small arteries with diameters varying between 30 μm and 300 μm) include the pressure myography technique. However, such methods currently require manually skilled personnel and are not scalable. An artery-on-a-chip could overcome several of these limitations by accommodating an artery onto a platform which would be scalable, inexpensive and possibly automated in its manufacturing.
An organ-based microfluidic platform has been developed as a lab-on-a-chip onto which a fragile blood vessel can be fixed, allowing for determinants of resistance artery malfunctions to be studied.
The artery microenvironment is characterized by surrounding temperature, transmural pressure , and luminal & abluminal drug concentrations. The multiple inputs from a microenvironment cause a wide range of mechanical or chemical stimuli on the smooth muscle cells (SMCs) and endothelial cells (ECs) that line the vessel's outer and luminal walls, respectively. Endothelial cells are responsible for releasing vasoconstriction and vasodilator factors, thus modifying tone. Vascular tone is defined as the degree of constriction inside a blood vessel relative to its maximum diameter. [ 88 ] Pathogenic concepts currently believe that subtle changes to this microenvironment have pronounced effects on arterial tone and can severely alter peripheral vascular resistance . The engineers behind this design believe that a specific strength lies in its ability to control and simulate heterogeneous spatiotemporal influences found within the microenvironment, whereas myography protocols have, by virtue of their design, only established homogeneous microenvironments. [ 87 ] They proved that by delivering phenylephrine through only one of the two channels providing superfusion to the outer walls, the drug-facing side constricted much more than the drug opposing side.
The artery-on-a-chip is designed for reversible implantation of the sample. The device contains a microchannel network, an artery loading area and a separate artery inspection area. There is a microchannel used for loading the artery segment, and when the loading well is sealed, it is also used as a perfusion channel, to replicate the process of nutritive delivery of arterial blood to a capillary bed in the biological tissue. [ 89 ] Another pair of microchannels serves to fix the two ends of the arterial segment. Finally, the last pair of microchannels is used to provide superfusion flow rates, in order to maintain the physiological and metabolic activity of the organ by delivering a constant sustaining medium over the abluminal wall. A thermoelectric heater and a thermoresistor are connected to the chip and maintain physiological temperatures at the artery inspection area.
The protocol of loading and securing the tissue sample into the inspection zone helps understand how this approach acknowledges whole organ functions. After immersing the tissue segment into the loading well, the loading process is driven by a syringe withdrawing a constant flow rate of buffer solution at the far end of the loading channel. This causes the transport of the artery towards its dedicated position. This is done with closed fixation and superfusion in/outlet lines. After stopping the pump , sub-atmospheric pressure is applied through one of the fixation channels. Then after sealing the loading well shut, the second fixation channel is subjected to a sub-atmospheric pressure. Now the artery is symmetrically established in the inspection area, and a transmural pressure is felt by the segment. The remaining channels are opened and constant perfusion and superfusion are adjusted using separate syringe pumps. [ 87 ]
Vessel-on-chips have been applied to study many disease processes. For example, Alireza Mashaghi and his co-workers developed a model to study viral hemorrhagic syndrome, which involves virus induced vascular integrity loss. The model was used to study Ebola virus disease and to study anti-Ebola drugs. [ 90 ] In 2021, the approach has been adapted to model Lassa fever and to show the therapeutic effects of peptide FX-06 for Lassa virus disease. [ 91 ]
Human skin is the first line of defense against many pathogens and can itself be subject to a variety of diseases and issues, such as cancers and inflammation. As such, skin-on-a-chip (SoC) applications include testing of topical pharmaceuticals and cosmetics, studying the pathology of skin diseases and inflammation, [ 92 ] and "creating noninvasive automated cellular assays " to test for the presence of antigens or antibodies that could denote the presence of a pathogen. [ 93 ] Despite the wide variety of potential applications, relatively little research has gone into developing a skin-on-a-chip compared to many other organ-on-a-chips, such as lungs and kidneys. [ 94 ] Issues such as detachment of the collagen scaffolding from microchannels, [ 94 ] incomplete cellular differentiation, [ 95 ] and predominant use of poly(dimethysiloxane) (PDMS) for device fabrication, which has been shown to leach chemicals into biological samples and cannot be mass-produced [ 96 ] stymie standardization of a platform. One additional difficulty is the variability of cell-culture scaffolding, or the base substance in which to culture cells, that is used in skin-on-chip devices. In the human body, this substance is known as the extracellular matrix.
The extracellular matrix (ECM) is composed primarily of collagen, and various collagen-based scaffolding has been tested in SoC models. Collagen tends to detach from the microfluidic backbone during culturing due to the contraction of fibroblasts . One study attempted to address this problem by comparing the qualities of collagen scaffolding from three different animal sources: pig skin, rat tail, and duck feet. [ 94 ] Other studies also faced detachment issues due to contraction, which can problematic considering that the process of full skin differentiation can take up to several weeks. [ 94 ] Contraction issues have been avoided by replacing collagen scaffolding with a fibrin -based dermal matrix, which did not contract. [ 96 ] Greater differentiation and formation of cell layers was also reported in microfluidic culture when compared to traditional static culture, agreeing with earlier findings of improved cell-cell and cell-matrix interactions due to dynamic perfusion, or increased permeation through interstitial spaces due to the pressure from continuous media flow. [ 8 ] [ 97 ] This improved differentiation and growth is thought to be in part a product of shear stress created by the pressure gradient along a microchannel due to fluid flow, [ 98 ] which may also improve nutrient supply to cells not directly adjacent to the medium . In static cultures, used in traditional skin equivalents, cells receive nutrients in the medium only through diffusion, whereas dynamic perfusion can improve nutrient flow through interstitial spaces, or gaps between cells. [ 98 ] This perfusion has also been demonstrated to improve tight junction formation of the stratum corneum , the tough outer layer of the epidermis, which is the main barrier to penetration of the surface layer of the skin. [ 99 ]
Dynamic perfusion may also improve cell viability, demonstrated by placing a commercial skin equivalent in a microfluidic platform that extended the expected lifespan by several weeks. [ 100 ] This early study also demonstrated the importance of hair follicles in skin equivalent models. Hair follicles are the primary route into the subcutaneous layer for topical creams and other substances applied to the surface of the skin, a feature that more recent studies have often not accounted for. [ 100 ]
One study developed a SoC consisting of three layers, the epidermis , dermis , and endothelial layer, separated by porous membranes, to study edema , swelling due to extracellular fluid accumulation, a common response to infection or injury and an essential step for cellular repair. It was demonstrated that pre-application of Dex, a steroidal cream with anti-inflammatory properties, reduced this swelling in the SoC. [ 92 ]
The endometrium has been modeled for its role in implantation and other stages of pregnancy . [ 101 ]
Researchers are working towards building a multi-channel 3D microfluidic cell culture system that compartmentalizes microenvironments in which 3D cellular aggregates are cultured to mimic multiple organs in the body. [ 102 ] Most organ-on-a-chip models today only culture one cell type, so even though they may be valid models for studying whole organ functions, the systemic effect of a drug on the human body is not verified.
In particular, an integrated cell culture analog (μCCA) was developed and included lung cells, drug-metabolizing liver and fat cells . The cells were linked in a 2D fluidic network with culture medium circulating as a blood surrogate, thus efficiently providing a nutritional delivery transport system, while simultaneously removing wastes from the cells. [ 103 ] "The development of the μCCA laid the foundation for a realistic in vitro pharmacokinetic model and provided an integrated biomimetic system for culturing multiple cell types with high fidelity to in vivo situations", claim C. Zhang et al. They have developed a microfluidic human-on-a-chip, culturing four different cell types to mimic four human organs: liver, lung, kidney and fat. [ 104 ] They focused on developing a standard serum -free culture media that would be valuable to all cell types included in the device. Optimized standard media are generally targeted to one specific cell-type, whereas a human-on-a-chip will evidently require a common medium (CM). In fact, they claim to have identified a cell culture CM that, when used to perfuse all cell cultures in the microfluidic device, maintains the cells' functional levels. Heightening the sensitivity of the in vitro cultured cells ensures the validity of the device, or that any drug injected into the microchannels will stimulate an identical physiological and metabolic reaction from the sample cells as whole organs in humans.
A human-on-a-chip design that allows tuning microfluidic transport to multiple tissues using a single fluidic actuator was designed and evaluated for modelling prediabetic hyperglycaemia using liver and pancreatic tissues. [ 105 ]
With more extensive development of these kinds of chips, pharmaceutical companies will potentially be able to measure direct effects of one organ's reaction on another. For instance, the delivery of biochemical substances would be screened to confirm that even though it may benefit one cell type, it does not compromise the functions of others. It is probably already possible to print these organs with 3D printers, but the cost is too high. Designing whole body biomimetic devices addresses a major reservation that pharmaceutical companies have towards organs-on-chips, namely the isolation of organs. [ citation needed ] As these devices become more and more accessible, the complexity of the design increases exponentially. Systems will soon have to simultaneously provide mechanical perturbation and fluid flow through a circulatory system . "Anything that requires dynamic control rather than just static control is a challenge", says Takayama from the University of Michigan . [ 106 ] This challenge has been partially tackled by tissue engineering Linda Griffith group from MIT. A complex multi-organ-on-a-chip was developed to have 4, 7, or 10 organs interconnected through fluidic control. [ 107 ] The system is able to maintain the function of these organs for weeks.
In the early phase of drug development, animal models were the only way of obtaining in vivo data that would predict the human pharmacokinetic responses. However, experiments on animals are lengthy, expensive and controversial. For example, animal models are often subjected to mechanical or chemical techniques that simulate human injuries. There are also concerns with regards to the validity of such animal models, due to deficiency in cross-species extrapolation. [ 108 ] Moreover, animal models offer very limited control of individual variables and it can be cumbersome to harvest specific information.
Therefore, mimicking a human's physiological responses in an in vitro model needs to be made more affordable, and needs to offer cellular level control in biological experiments: biomimetic microfluidic systems could replace animal testing . The development of MEMS-based biochips that reproduce complex organ-level pathological responses could revolutionize many fields, including toxicology and the developmental process of pharmaceuticals and cosmetics that rely on animal testing and clinical trials . [ 109 ]
Recently, physiologically based perfusion in vitro systems have been developed to provide cell culture environment close to in vivo cell environment. A new testing platforms based on multi-compartmental perfused systems have gained a remarkable interest in pharmacology and toxicology. It aims to provide a cell culture environment close to the in vivo situation to reproduce more reliably in vivo mechanisms or ADME processes that involve its absorption, distribution, metabolism, and elimination. Perfused in vitro systems combined with kinetic modelling are promising tools for studying in vitro the different processes involved in the toxicokinetics of xenobiotics.
Efforts made toward the development of micro fabricated cell culture systems that aim to create models that replicate aspects of the human body as closely as possible and give examples that demonstrate their potential use in drug development, such as identifying synergistic drug interactions as well as simulating multi-organ metabolic interactions. Multi compartment micro fluidic-based devices, particularly those that are physical representations of physiologically based pharmacokinetic ( PBPK ) models that represent the mass transfer of compounds in compartmental models of the mammalian body, may contribute to improving the drug development process. Some emerging technologies have the ability to measure multiple biological processes in a co-culture of mixed cell types, cells from different parts of the body, which is suggested to provide more similarity to in Vivo models. [ 110 ]
Mathematical pharmacokinetic (PK) models aim to estimate concentration-time profiles within each organ on the basis of the initial drug dose. Such mathematical models can be relatively simple, treating the body as a single compartment in which the drug distribution reaches a rapid equilibrium after administration. Mathematical models can be highly accurate when all parameters involved are known. Models that combine PK or PBPK models with PD models can predict the time-dependent pharmacological effects of a drug. We can nowadays predict with PBPK models the PK of about any chemical in humans, almost from first principles. These models can be either very simple, like statistical dose-response models, or sophisticated and based on systems biology, according to the goal pursued and the data available. All we need for those models are good parameter values for the molecule of interest.
Microfluidic cell culture systems such as micro cell culture analogs (μCCAs) could be used in conjunction with PBPK models. These μCCAs scaled-down devices, termed also body-on-a-chip devices, can simulate multi-tissue interactions under near-physiological fluid flow conditions and with realistic tissue-to-tissue size ratios . Data obtained with these systems may be used to test and refine mechanistic hypotheses. Microfabricating devices also allows us to custom-design them and scale the organs' compartments correctly with respect to one another.
Because the device can be used with both animal and human cells, it can facilitate cross-species extrapolation. Used in conjunction with PBPK models, the devices permit an estimation of effective concentrations that can be used for studies with animal models or predict the human response. In the development of multicompartment devices, representations of the human body such as those in used PBPK models can be used to guide the device design with regard to the arrangement of chambers and fluidic channel connections to augment the drug development process, resulting in increased success in clinical trials. | https://en.wikipedia.org/wiki/Organ-on-a-chip |
In a multicellular organism , an organ is a collection of tissues joined in a structural unit to serve a common function. [ 1 ] In the hierarchy of life , an organ lies between tissue and an organ system . Tissues are formed from same type cells to act together in a function. Tissues of different types combine to form an organ which has a specific function. The intestinal wall for example is formed by epithelial tissue and smooth muscle tissue . [ 2 ] Two or more organs working together in the execution of a specific body function form an organ system, also called a biological system or body system.
An organ's tissues can be broadly categorized as parenchyma , the functional tissue, and stroma , the structural tissue with supportive, connective, or ancillary functions. For example, the gland 's tissue that makes the hormones is the parenchyma , whereas the stroma includes the nerves that innervate the parenchyma, the blood vessels that oxygenate and nourish it and carry away its metabolic wastes, and the connective tissues that provide a suitable place for it to be situated and anchored. The main tissues that make up an organ tend to have common embryologic origins, such as arising from the same germ layer . Organs exist in most multicellular organisms . In single-celled organisms such as members of the eukaryotes , the functional analogue of an organ is known as an organelle . In plants, there are three main organs. [ 3 ]
The number of organs in any organism depends on the definition used. There are approximately 79 organs in the human body; the precise count is debated. [ 4 ]
Except for placozoans , multicellular animals including humans have a variety of organ systems . These specific systems are widely studied in human anatomy . The functions of these organ systems often share significant overlap. For instance, the nervous and endocrine system both operate via a shared organ, the hypothalamus . For this reason, the two systems are combined and studied as the neuroendocrine system . The same is true for the musculoskeletal system because of the relationship between the muscular and skeletal systems .
In the study of anatomy , viscera ( sg. : viscus ) refers to the internal organs of the abdominal , thoracic , and pelvic cavities . [ 5 ] The abdominal organs may be classified as solid organs or hollow organs . The solid organs are the liver , pancreas , spleen , kidneys , and adrenal glands . The hollow organs of the abdomen are the stomach , intestines , gallbladder , bladder , and rectum . [ 6 ] In the thoracic cavity , the heart is a hollow, muscular organ. [ 7 ] Splanchnology is the study of the viscera. [ 8 ] The term "visceral" is contrasted with the term " parietal ", meaning "of or relating to the wall of a body part, organ or cavity ". [ 9 ] The two terms are often used in describing a membrane or piece of connective tissue, referring to the opposing sides. [ 10 ]
The organ level of organisation in animals can be first detected in flatworms and the more derived phyla , i.e. the bilaterians . The less-advanced taxa (i.e. Placozoa , Porifera , Ctenophora and Cnidaria ) do not show unification of their tissues into organs.
More complex animals are composed of different organs, which have evolved over time. For example, the liver and heart evolved in the chordates about 550-500 million years ago, while the gut and brain are even more ancient, arising in the ancestor of vertebrates, insects, molluscs , and worms about 700–650 million years ago.
Given the ancient origin of most vertebrate organs, researchers have looked for model systems, where organs have evolved more recently, and ideally have evolved multiple times independently. An outstanding model for this kind of research is the placenta , which has evolved more than 100 times independently in vertebrates, has evolved relatively recently in some lineages, and exists in intermediate forms in extant taxa. [ 11 ] Studies on the evolution of the placenta have identified a variety of genetic and physiological processes that contribute to the origin and evolution of organs, these include the re-purposing of existing animal tissues, the acquisition of new functional properties by these tissues, and novel interactions of distinct tissue types. [ 11 ]
The study of plant organs is covered in plant morphology . Organs of plants can be divided into vegetative and reproductive. Vegetative plant organs include roots , stems , and leaves . The reproductive organs are variable. In flowering plants , they are represented by the flower , seed and fruit . [ citation needed ] In conifers , the organ that bears the reproductive structures is called a cone . In other divisions ( phyla ) of plants, the reproductive organs are called strobili , in Lycopodiophyta , or simply gametophores in mosses . Common organ system designations in plants include the differentiation of shoot and root. All parts of the plant above ground (in non- epiphytes ), including the functionally distinct leaf and flower organs, may be classified together as the shoot organ system. [ 12 ]
The vegetative organs are essential for maintaining the life of a plant. While there can be 11 organ systems in animals, there are far fewer in plants, where some perform the vital functions, such as photosynthesis , while the reproductive organs are essential in reproduction . However, if there is asexual vegetative reproduction , the vegetative organs are those that create the new generation of plants (see clonal colony ).
Many societies have a system for organ donation , in which a living or deceased donor's organ are transplanted into a person with a failing organ. The transplantation of larger solid organs often requires immunosuppression to prevent organ rejection or graft-versus-host disease .
There is considerable interest throughout the world in creating laboratory-grown or artificial organs . [ citation needed ]
Beginning in the 20th century, [ 13 ] organ transplants began to take place as scientists knew more about the anatomy of organs. These came later in time as procedures were often dangerous and difficult. [ 14 ] Both the source and method of obtaining the organ to transplant are major ethical issues to consider, and because organs as resources for transplant are always more limited than demand for them, various notions of justice, including distributive justice , are developed in the ethical analysis. This situation continues as long as transplantation relies upon organ donors rather than technological innovation, testing, and industrial manufacturing. [ citation needed ]
Animal donor organs and tissue have been subjects of study since the 1960s, and some xenotransplant tissues, particularly heart valves, have been commonly utilized. Xenotransplant has the potential to address the critical shortage in organ grafts. The science behind [xenotransplant] trials has advanced considerably and more human clinical trials utilizing porcine xenografts are quickly approaching. [ 15 ]
The English word "organ" dates back to the twelfth century and refers to any musical instrument. By the late 14th century, the musical term's meaning had narrowed to refer specifically to the keyboard-based instrument . At the same time, a second meaning arose, in reference to a "body part adapted to a certain function". [ 16 ]
Plant organs are made from tissue composed of different types of tissue. The three tissue types are ground, vascular, and dermal. [ 17 ] When three or more organs are present, it is called an organ system. [ 18 ]
The adjective visceral , also splanchnic , is used for anything pertaining to the internal organs. Historically, viscera of animals were examined by Roman pagan priests like the haruspices or the augurs in order to divine the future by their shape, dimensions or other factors. [ 19 ] This practice remains an important ritual in some remote, tribal societies.
The term "visceral" is contrasted with the term " parietal ", meaning "of or relating to the wall of a body part, organ or cavity " [ 9 ] The two terms are often used in describing a membrane or piece of connective tissue, referring to the opposing sides. [ 20 ]
Aristotle used the word frequently in his philosophy, both to describe the organs of plants or animals (e.g. the roots of a tree, the heart or liver of an animal) because, in ancient Greek, the word ' organon ' means 'tool', and Aristotle believed that the organs of the body were tools for us by means of which we can do things. For similar reasons, his logical works, taken as a whole, are referred to as the Organon because logic is a tool for philosophical thinking. [ 21 ] Earlier thinkers, such as those who wrote texts in the Hippocratic corpus , generally did not believe that there were organs of the body but only different parts of the body. [ 22 ]
Some alchemists (e.g. Paracelsus ) adopted the Hermetic Qabalah assignment between the seven vital organs and the seven classical planets as follows: [ 23 ]
Chinese traditional medicine recognizes eleven organs, associated with the five Chinese traditional elements and with yin and yang , as follows:
The Chinese associated the five elements with the five planets (Jupiter, Mars, Venus, Saturn, and Mercury) similar to the way the classical planets were associated with different metals. The yin and yang distinction approximates the modern notion of solid and hollow organs. | https://en.wikipedia.org/wiki/Organ_(biology) |
Organ culture is the cultivation of either whole organs or parts of organs in vitro . [ 1 ] It is a development from tissue culture methods of research, as the use of the actual in vitro organ itself allows for more accurate modelling of the functions of an organ in various states and conditions. [ 2 ]
A key objective of organ culture is to maintain the architecture of the tissue and direct it towards normal development. In this technique, it is essential that the tissue is never disrupted or damaged. It thus requires careful handling. The media used for a growing organ culture are generally the same as those used for tissue culture. The techniques for organ culture can be classified into (i) those employing a solid medium and (ii) those employing liquid medium.
Organ culture technology has contributed to advances in embryology , [ 3 ] inflammation , cancer , and stem cell biology research. [ 2 ]
In April 2006, scientists reported a successful trial of seven bladders grown in-vitro and given to humans. [ 4 ] [ 5 ] A bladder has been cultured by Anthony Atala of the Wake Forest Institute for Regenerative Medicine in Winston-Salem, North Carolina. A jawbone has been cultured at Columbia University, a lung has been cultured at Yale. A beating rat heart has been cultured by Doris Taylor at the University of Minnesota. An artificial kidney has been cultured by H. David Humes at the University of Michigan. [ 6 ]
Silk cut from silkworm cocoons has been successfully used as growth scaffolding for heart tissue production. Heart tissue does not regenerate if damaged, so producing replacement patches is of great interest. The experiment used rat heart cells and produced functional heart tissue. In order to further test applications to humans as a cure, a way to transform human stem cells into heart tissue would have to be found. [ 7 ]
In 2015, Harald Ott was able to grow a rat forelimb. [ 8 ] He now works at Ott Lab which focuses on the creation of bioartificial hearts, lungs, tracheas and kidneys. [ 9 ]
In 2016, another test was done in which human cells were used to assemble intricately structured hearts. The hearts ultimately proved immature but proved we were yet one step further to making a heart from stem cells. [ 10 ] [ 11 ]
In January 2017, scientists from Salk Institute for Biological Studies managed to create a pig embryo that had part of its DNA, critical for the growth of organs, edited out. They then introduced human stem cells inside the pig embryo to have the human DNA fill in the gaps. [ 12 ] [ 13 ]
In March 2022, research was published that demonstrated the tentative success of a corneal implant called BPCDX that was shown to have significant tissue attachment and host cell migration once implanted. [ 14 ]
Embryonic organ culture is an easier alternative to normal organ culture derived from adult animals. The following are four techniques employed for embryonic organ culture.
The following are general steps in organ culture on plasma clots.
The technique has now been modified, and a raft of lens paper or rayon net is used on which the tissue is placed. Transfer of the tissue can then be achieved by raft easily. Excessive fluid is removed and the net with the tissue placed again on the fresh pool of medium.
Media solidified with agar are also used for organ culture and these media consist of 7 parts 1% agar in BSS, 3 parts chick embryo extract and 3 parts of horse serum. Defined media with or without serum are also used with agar. The medium with agar provides the mechanical support for organ culture. It does not liquefy. Embryonic organs generally grow well on agar, but adult organ culture will not survive on this medium.
The culture of adult organs or parts from adult animals is more difficult due to their greater requirement of oxygen . A variety of adult organs (e.g. the liver ) have been cultured using special media with special apparatus (Towell's II culture chamber). Since serum was found to be toxic, serum-free media were used, and the special apparatus permitted the use of 95% oxygen.
In this approach the explant is placed onto a raft of lens paper or rayon acetate, which is floated on serum in a watch glass. Rayon acetate rafts are made to float on the serum by treating their 4 corners with silicone.
Similarly, floatability of lens paper is enhanced by treating it with silicone. On each raft, 4 or more explants are usually placed.
In a combination of raft and clot techniques, the explants are first placed on a suitable raft, which is then kept on a plasma clot. This modification makes media changes easy, and prevents the sinking of explants into liquefied plasma.
Initially devised by Trowell in 1954, the grid method utilizes 25 mm x 25 mm pieces of a suitable wire mesh or perforated stainless steel sheet whose edges are bent to form 4 legs of about 4 mm height.
Skeletal tissues are generally placed directly on the grid but softer tissues like glands or skin are first placed on rafts, which are then kept on the grids.
The grids themselves are placed in a culture chamber filled with fluid medium up to the grid; the chamber is supplied with a mixture of O 2 and CO 2 to meet the high O 2 requirements of adult mammalian organs. A modification of the original grid method is widely used to study the growth and differentiation of adult and embryonic tissues.
Cultured organs can be an alternative for organs from other (living or deceased) people.
This is useful as the availability of transplantable organs (derived from other people) is declining in developed countries.
Another advantage is that cultured organs, created using the patients own stem cell, allows for organ transplants where the patient would no longer require immunosuppressive drugs . [ 15 ] | https://en.wikipedia.org/wiki/Organ_culture |
Organ futures is the short term used in academic proposals for futures contracts on organs from human cadavers . They are not legal anywhere at this time.
Organ futures would be used as an economic means to encourage organ donation by compensating transplant organ donors . Financial futures contracts are essentially agreements to pay a specified sum at a specified time.
The four key academic papers describing proposals for organ futures were published between the mid-1980s and mid-1990s.
The explanations below focus on donation of cadaveric organs and ignore living donation.
Schwindt & Vining (1986) [ 1 ] suggest that the organ donor is paid at the time they agree to enter the life-time futures contract. The agreement is mutually revocable. They propose a single government broker as the buyer. Organ recipients would pay the supply price plus a load factor to the broker.
Hansmann (1989) [ 2 ] also suggests payment at the time of contract. Instead of direct payment , he proposes reductions to health insurance premiums as indirect incentive. The hospital where the organ donor dies is expected to verify a seller registry and determine the buyer. Buyers may be health insurance providers or specialist traders.
Cohen (1989) [ 3 ] introduces a significant change to previous proposals by making payment conditional on organ extraction . Thus, the donor is not directly compensated during their lifetime. However, the payment is allocated to their estate or a designee. Hospitals are expected to notify buyers and preserve cadavers . They can be made liable for consequences of negligence. Buyers may be public or private organizations.
Crespi (1994) [ 4 ] aims to integrate what he deems the most useful aspects of previous models into his own. Payment would be either guaranteed upon death or dependent on organ extraction. The money would go to the seller's estate ; rights would not be assignable, and creditors would not have any claim on it. Hospitals are expected to notify the buyer, preserve the body and be prepared to harvest the organs if required. Any legally competent person can be a buyer and assign their rights freely.
Currently, organ futures are legally unfeasible because most countries follow international guidance that requires financial neutrality from the donor. [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Organ_futures |
Organ printing utilizes techniques similar to conventional 3D printing where a computer model is fed into a printer that lays down successive layers of plastics or wax until a 3D object is produced. [ 1 ] In the case of organ printing, the material being used by the printer is a biocompatible plastic. [ 1 ] The biocompatible plastic forms a scaffold that acts as the skeleton for the organ that is being printed. [ 1 ] As the plastic is being laid down, it is also seeded with human cells from the patient's organ that is being printed for. [ 1 ] After printing, the organ is transferred to an incubation chamber to give the cells time to grow. [ 1 ] After a sufficient amount of time, the organ is implanted into the patient. [ 1 ]
To many researchers the ultimate goal of organ printing is to create organs that can be fully integrated into the human body. [ 1 ] Successful organ printing has the potential to impact several industries, notably artificial organs organ transplants , [ 2 ] pharmaceutical research, [ 3 ] and the training of physicians and surgeons . [ 4 ]
The field of organ printing stemmed from research in the area of stereolithography , the basis for the practice of 3D printing that was invented in 1984. [ 5 ] In this early era of 3D printing, it was not possible to create lasting objects because of the material used for the printing process was not durable. [ 6 ] 3D printing was instead used as a way to model potential end products that would eventually be made from different materials under more traditional techniques. [ 5 ] In the beginning of the 1990s, nanocomposites were developed that allowed 3D printed objects to be more durable, permitting 3D printed objects to be used for more than just models. [ 6 ] It was around this time that those in the medical field began considering 3D printing as an avenue for generating artificial organs. [ 5 ] By the late 1990s, medical researchers were searching for biocompatible materials that could be used in 3D printing. [ 5 ]
The concept of bioprinting was first demonstrated in 1988. [ 7 ] At this time, a researcher used a modified HP inkjet printer to deposit cells using cytoscribing technology. [ 7 ] Progress continued in 1999 when the first artificial organ made using bioprinting was printed by a team of scientist leads by Dr. Anthony Atala at the Wake Forest Institute for Regenerative Medicine . [ 8 ] The scientists at Wake Forest printed an artificial scaffold for a human bladder and then seeded the scaffold with cells from their patient. [ 5 ] Using this method, they were able to grow a functioning organ and ten years after implantation the patient had no serious complications. [ 9 ]
After the bladder at Wake Forest, strides were taken towards printing other organs. In 2002, a miniature, fully functional kidney was printed. [ 6 ] In 2003, Dr. Thomas Boland from Clemson University patented the use of inkjet printing for cells. [ 10 ] This process utilized a modified spotting system for the deposition of cells into organized 3D matrices placed on a substrate . [ 10 ] This printer allowed for extensive research into bioprinting and suitable biomaterials. [ 9 ] For instance, since these initial findings, the 3D printing of biological structures has been further developed to encompass the production of tissue and organ structures, as opposed to cell matrices . [ 11 ] Additionally, more techniques for printing, such as extrusion bioprinting, have been researched and subsequently introduced as a means of production . [ 11 ]
In 2004, the field of bioprinting was drastically changed by yet another new bioprinter. [ 9 ] This new printer was able to use live human cells without having to build an artificial scaffold first. [ 9 ] In 2009, Organovo used this novel technology to create the first commercially available bioprinter. [ 9 ] Soon after, Organovo's bioprinter was used to develop a biodegradable blood vessel , the first of its kind, without a cell scaffold. [ 9 ]
In the 2010s and beyond, further research has been put forth into producing other organs, such as the liver and heart valves , and tissues , such as a blood-borne network, via 3D printing. [ 9 ] In 2019, scientists in Israel made a major breakthrough when they were able to print a rabbit-sized heart with a network of blood vessels that were capable of contracting like natural blood vessels. [ 12 ] The printed heart had the correct anatomical structure and function compared to real hearts. [ 12 ] This breakthrough represented a real possibility of printing fully functioning human organs. [ 9 ] In fact, scientists at the Warsaw Foundation for Research and Development of Science in Poland have been working on creating a fully artificial pancreas using bioprinting technology. [ 9 ] As of today, these scientists have been able to develop a functioning prototype. [ 9 ] This is a growing field and much research is still being conducted.
3D printing for the manufacturing of artificial organs has been a major topic of study in biological engineering . As the rapid manufacturing techniques entailed by 3D printing become increasingly efficient, their applicability in artificial organ synthesis has grown more evident. Some of the primary benefits of 3D printing lie in its capability of mass-producing scaffold structures, as well as the high degree of anatomical precision in scaffold products. This allows for the creation of constructs that more effectively resemble the microstructure of a natural organ or tissue structure . [ 13 ] Organ printing using 3D printing can be conducted using a variety of techniques, each of which confers specific advantages that can be suited to particular types of organ production.
Sacrificial writing into function tissue (SWIFT) is a method of organ printing where living cells are packed tightly to mimic the density that occurs in the human body. While packing, tunnels are carved to mimic blood vessels and oxygen and essential nutrients are delivered via these tunnels. This technique pieces together other methods that only packed cells or created vasculature . SWIFT combines both and is an improvement that brings researchers closer to creating functional artificial organs. [ 2 ]
This method of organ printing uses spatially controlled light or laser to create a 2D pattern that is layered through a selective photopolymerization in the bio-ink reservoir. A 3D structure can then be built in layers using the 2D pattern. Afterwards the bio-ink is removed from the final product. SLA bioprinting allows for the creation of complex shapes and internal structures. The feature resolution for this method is extremely high and the only disadvantage is the scarcity of resins that are biocompatible. [ 14 ]
Drop-based bioprinting makes cellular developments utilizing droplets of an assigned material, which has oftentimes been combined with a cell line. Cells themselves can also be deposited in this manner with or without polymer. When printing polymer scaffolds using these methods, each drop starts to polymerize upon contact with the substrate surface and merge into a larger structure as droplets start to coalesce. Polymerization can happen through a variety of methods depending on the polymer used. For instance, alginate polymerization is started by calcium ions in the substrate, which diffuse into the liquified bioink and permit for the arrangement of a strong gel. Drop-based bioprinting is commonly utilized due to its productive speed. However, this may make it less appropriate for more complicated organ structures. [ 15 ]
Extrusion bioprinting includes the consistent statement of a specific printing fabric and cell line from an extruder , a sort of portable print head. This tends to be a more controlled and gentler handle for fabric or cell statement, and permits for more noteworthy cell densities to be utilized within the development of 3D tissue or organ structures. In any case, such benefits are set back by the slower printing speeds involved by this procedure. Extrusion bioprinting is frequently coupled with UV light, which photopolymerizes the printed fabric to create a more steady, coordinated construct. [ 11 ]
Fused deposition modeling (FDM) is more common and inexpensive compared to selective laser sintering. This printer uses a printhead that is similar in structure to an inkjet printer; however, ink is not used. Plastic beads are heated at high temperature and released from the printhead as it moves, building the object in thin layers. [ 3 ] A variety of plastics can be used with FDM printers. Additionally, most of the parts printed by FDM are typically composed from the same thermoplastics that are utilized in tradition injection molding or machining techniques. [ 3 ] Due to this, these parts have analogous durability, mechanical properties, and stability characteristics. [ 3 ] Precision control allows for a consistent release amount and specific location deposition for each layer contributing to the shape. [ 3 ] As the heated plastic is deposited from the printhead, it fuses or bonds to the layers below. As each layer cools, they harden and gradually take hold of the solid shape intended to be created as more layers are contributed to the structure.
Selective laser sintering (SLS) uses powdered material as the substrate for printing new objects. SLS can be used to create metal, plastic, and ceramic objects. This technique uses a laser controlled by a computer as the power source to sinter powdered material. [ 16 ] The laser traces a cross-section of the shape of the desired object in the powder, which fuses it together into a solid form. [ 16 ] A new layer of powder is then laid down and the process repeats itself, building each layer with every new application of powder, one by one, to form the entirety of the object. One of the advantages of SLS printing is that it requires very little additional tooling, i.e. sanding, once the object is printed. [ 16 ] Recent advances in organ printing using SLS include 3D constructs of craniofacial implants as well as scaffolds for cardiac tissue engineering. [ 16 ]
Printing materials must fit a broad spectrum of criteria, one of the foremost being biocompatibility . The resulting scaffolds formed by 3D printed materials should be physically and chemically appropriate for cell proliferation . Biodegradability is another important factor, and insures that the artificially formed structure can be broken down upon successful transplantation, to be replaced by a completely natural cellular structure. Due to the nature of 3D printing, materials used must be customizable and adaptable, being suited to wide array of cell types and structural conformations. [ 17 ]
Materials for 3D printing usually consist of alginate or fibrin polymers that have been integrated with cellular adhesion molecules, which support the physical attachment of cells. Such polymers are specifically designed to maintain structural stability and be receptive to cellular integration. The term bio-ink has been used as a broad classification of materials that are compatible with 3D bioprinting. [ 18 ] Hydrogel alginates have emerged as one of the most commonly used materials in organ printing research, as they are highly customizable, and can be fine-tuned to simulate certain mechanical and biological properties characteristic of natural tissue. The ability of hydrogels to be tailored to specific needs allows them to be used as an adaptable scaffold material, that are suited for a variety of tissue or organ structures and physiological conditions . [ 19 ] A major challenge in the use of alginate is its stability and slow degradation, which makes it difficult for the artificial gel scaffolding to be broken down and replaced with the implanted cells' own extracellular matrix . [ 20 ] Alginate hydrogel that is suitable for extrusion printing is also often less structurally and mechanically sound; however, this issue can be mediated by the incorporation of other biopolymers , such as nanocellulose , to provide greater stability. The properties of the alginate or mixed-polymer bioink are tunable and can be altered for different applications and types of organs. [ 20 ]
Other natural polymers that have been used for tissue and 3D organ printing include chitosan , hydroxyapatite (HA) , collagen , and gelatin . Gelatin is a thermosensitive polymer with properties exhibiting excellent wear solubility , biodegradability, biocompatibility, as well as a low immunologic rejection . [ 21 ] These qualities are advantageous and result in high acceptance of the 3D bioprinted organ when implanted in vivo. [ 21 ]
Synthetic polymers are human made through chemical reactions of monomers . Their mechanical properties are favorable in that their molecular weights can be regulated from low to high based on differing requirements. [ 21 ] However, their lack of functional groups and structural complexity has limited their usage in organ printing. Current synthetic polymers with excellent 3D printability and in vivo tissue compatibility, include polyethylene glycol (PEG) , poly(lactic-glycolic acid) (PLGA) , and polyurethane (PU) . PEG is a biocompatible, nonimmunogenic synthetic polyether that has tunable mechanical properties for use in 3D bioprinting. [ 21 ] Though PEG has been utilized in various 3D printing applications, the lack of cell-adhesive domains has limited further use in organ printing. PLGA, a synthetic copolymer , is widely familiar in living creatures, such as animals, humans, plants, and microorganisms . PLGA is used in conjunction with other polymers to create different material systems, including PLGA-gelatin, PLGA-collagen, all of which enhance mechanical properties of the material, biocompatible when placed in vivo , and have tunable biodegradability. [ 21 ] PLGA has most often been used in printed constructs for bone , liver, and other large organ regeneration efforts. Lastly, PU is unique in that it can be classified into two groups: biodegradable or non-biodegradable. [ 21 ] It has been used in the field of bioprinting due to its excellent mechanical and bioinert properties. An application of PU would be inanimate artificial hearts ; however, using existing 3D bioprinters, this polymer cannot be printed. [ 21 ] A new elastomeric PU was created composed of PEG and polycaprolactone (PCL) monomers . [ 21 ] This new material exhibits excellent biocompatibility, biodegradability, bioprintability, and biostability for use in complex bioartificial organ printing and manufacturing. [ 21 ] Due to high vascular and neural network construction, this material can be applied to organ printing in a variety of complex ways, such as the brain , heart, lung , and kidney.
Natural-synthetic hybrid polymers are based on the synergic effect between synthetic and biopolymeric constituents. [ 21 ] Gelatin-methacryloyl (GelMA) has become a popular biomaterial in the field of bioprinting. GelMA has shown it has viable potential as a bioink material due to its suitable biocompatibility and readily tunable psychochemical properties. [ 21 ] Hyaluronic acid (HA) -PEG is another natural-synthetic hybrid polymer that has proven to be very successful in bioprinting applications. HA combined with synthetic polymers aid in obtaining more stable structures with high cell viability and limited loss in mechanical properties after printing. [ 21 ] A recent application of HA-PEG in bioprinting is the creation of artificial liver. Lastly, a series of biodegradable polyurethane (PU)-gelatin hybrid polymers with tunable mechanical properties and efficient degradation rates have been implemented in organ printing. [ 21 ] This hybrid has the ability to print complicated structures such as a nose -shaped construct.
All of the polymers described above have the potential to be manufactured into implantable, bioartificial organs for purposes including, but not limited to, customized organ restoration, drug screening , as well as metabolic model analysis.
The creation of a complete organ often requires incorporation of a variety of different cell types, arranged in distinct and patterned ways. One advantage of 3D-printed organs, compared to traditional transplants , is the potential to use cells derived from the patient to make the new organ. This significantly decreases the likelihood of transplant rejection, and may remove the need for immunosuppressive drugs after transplant, which would reduce the health risks of transplants. However, since it may not always be possible to collect all the needed cell types, it may be necessary to collect adult stem cells or induce pluripotency in collected tissue. [ 19 ] This involves resource-intensive cell growth and differentiation and comes with its own set of potential health risks, since cell proliferation in a printed organ occurs outside the body and requires external application of growth factors. However, the ability of some tissues to self-organize into differentiated structures may provide a way to simultaneously construct the tissues and form distinct cell populations, improving the efficacy and functionality of organ printing. [ 22 ]
The types of printers used for organ printing include: [ 14 ]
These printers are used in the methods described previously. Each printer requires different materials and has its own advantages and limitations.
Currently, the sole method for treatment for those in organ failure is to await a transplant from a living or recently deceased donor. [ 23 ] In the United States alone, there are over 100,000 patients on the organ transplant list waiting for donor organs to become available. [ 24 ] Patients on the donor list can wait days, weeks, months, or even years for a suitable organ to become available. The average wait time for some common organ transplants are as follows: four months for a heart or lung, eleven months for a liver, two years for a pancreas, and five years for a kidney. [ 25 ] This is a significant increase from the 1990s, when a patient could wait as little as five weeks for a heart. [ 23 ] These extensive wait times are due to a shortage of organs as well as the requirement for finding an organ that is suitable for the recipient. [ 25 ] An organ is deemed suitable for a patient based on blood type , comparable body size between donor and recipient, the severity of the patient's medical condition, the length of time the patient has been waiting for an organ, patient availability (i.e. ability to contact patient, if patient has an infection), the proximity of the patient to the donor, and the viability time of the donor organ. [ 26 ] In the United States, 20 people die everyday waiting for organs. [ 24 ] 3D organ printing has the potential to remove both these issues; if organs could be printed as soon as there is need, there would be no shortage. Additionally, seeding printed organs with a patient's own cells would eliminate the need to screen donor organs for compatibility.
Surgical usage of 3D printing has evolved from printing surgical instrumentation to the development of patient-specific technologies for total joint replacements, dental implants, and hearing aids . [ 27 ] In the field of organ printing, applications can be applied for patients and surgeons. For instance, printed organs have been used to model structure and injury to better understand the anatomy and discuss a treatment regime with patients. [ 28 ] For these cases, the functionality of the organ is not required and is used for proof-of-concept. These model organs provide advancement for improving surgical techniques, training inexperienced surgeons, and moving towards patient-specific treatments. [ 28 ]
3D organ printing technology permits the fabrication of high degrees of complexity with great reproducibility, in a fast and cost-effective manner. [ 3 ] 3D printing has been used in pharmaceutical research and fabrication, providing a transformative system allowing precise control of droplet size and dose, personalized medicine , and the production of complex drug-release profiles. [ 3 ] This technology calls for implantable drug delivery devices, in which the drug is injected into the 3D printed organ and is released once in vivo. [ 3 ] Also, organ printing has been used as a transformative tool for in vitro testing. [ 3 ] The printed organ can be utilized in discovery and dosage research upon drug-release factors. [ 3 ]
Organ printing technology can also be combined with microfluidic technology to develop organs-on-chips . [ 29 ] These organs-on-chips have the potential to be used for disease models, aiding in drug discovery, and performing high-throughput assays . [ 29 ] Organ-on-chips work by providing a 3D model that imitates the natural extracellular matrix, allowing them to display realistic responses to drugs. [ 29 ] Thus far, research has been focused on developing liver-on-a-chip and heart-on-a-chip, but there exists the potential to develop an entire body-on-a-chip model. [ 29 ]
By combining 3D printed organs, researchers are able to create a body-on-a-chip. The heart-on-a-chip model has already been used to investigate how several drugs with heart rate-based negative side effects, such as the chemotherapeutic drug doxorubicin could affect people on an individual basis. [ 30 ] The new body-on-a-chip platform includes liver, heart, lungs, and kidney-on-a-chip. The organs-on-a-chip are separately printed or constructed and then integrated together. Using this platform drug toxicity studies are performed in high throughput, lowering the cost and increasing the efficiency in the drug-discovery pipeline. [ 29 ]
3D-printing techniques have been used in a variety of industries for the overall goal of fabricating a product. Organ printing, on the other hand, is a novel industry that utilizes biological components to develop therapeutic applications for organ transplants. Due to the increased interest in this field, regulation and ethical considerations desperately need to be established. [ 31 ] Specifically, there can be legal complications from pre-clinical to clinical translation for this treatment method. [ 32 ]
The current American regulation for organ matching is centered on the national registry of organ donors after the National Organ Transplant Act was passed in 1984. [ 1 ] This act was set in place to ensure equal and honest distribution, although it has been proven insufficient due to the large demand for organ transplants. Organ printing can assist in diminishing the imbalance between supply and demand by printing patient-specific organ replacements, all of which is unfeasible without regulation. The Food and Drug Administration (FDA) is responsible for regulation of biologics , devices, and drugs in the United States. [ 31 ] [ 32 ] Due to the complexity of this therapeutic approach, the location of organ printing on the spectrum has not been discerned. Studies have characterized printed organs as multi-functional combination products, meaning they fall between the biologics and devices sectors of the FDA; this leads to more extensive processes for review and approval. [ 31 ] [ 32 ] [ 33 ] In 2016, the FDA issued draft guidance on the Technical Considerations for Additive Manufactured Devices and is currently [ as of? ] evaluating new submissions for 3D printed devices. [ 34 ] However, the technology itself is not advanced enough for the FDA to mainstream it directly. [ 33 ] Currently, the 3D printers, rather than the finished products, are the main focus in safety and efficacy evaluations in order to standardize the technology for personalized treatment approaches. From a global perspective, only South Korea and Japan's medical device regulation administrations have provided guidelines that are applicable to 3D bio-printing. [ 31 ]
There are also concerns with intellectual property and ownership. These can have a large impact on more consequential matters such as piracy, quality control for manufacturing, and unauthorized use on the black market. [ 32 ] [ 33 ] These considerations are focused more on the materials and fabrication processes; they are more extensively explained in the legal aspects subsection of 3D printing.
From an ethical standpoint, there are concerns with respect to the availability of organ printing technologies, the cell sources, and public expectations. Although this approach may be less expensive than traditional surgical transplantation, there is skepticism in regards to social availability of these 3D printed organs. Contemporary research has found that there is potential social stratification for the wealthier population to have access to this therapy while the general population remains on the organ registry. [ 35 ] The cell sources mentioned previously also need to be considered. Organ printing can decrease or eliminate animal studies and trials, but also raises questions on the ethical implications of autologous and allogenic sources. [ 35 ] [ 36 ] More specifically, studies have begun to examine future risks for humans undergoing experimental testing. [ 31 ] Generally, this application can give rise to social, cultural, and religious differences, making it more difficult for worldwide integration and regulation. [ 32 ] Overall, the ethical considerations of organ printing are similar to those of general ethics of bioprinting , but are extrapolated from tissue to organ. Altogether, organ printing possesses short- and long-term legal and ethical consequences that need to be considered before mainstream production can be feasible.
Organ printing for medical applications is still in the developmental stages. Thus, the long term impacts of organ printing have yet to be determined. Researchers hope that organ printing could decrease the organ transplant shortage. [ 37 ] There is currently a shortage of available organs, including liver, kidneys, and lungs. [ 38 ] The lengthy wait time to receive life saving organs is one of the leading causes of death in the United States, with nearly one third of deaths each year in the United States that could be delayed or prevented with organ transplants. [ 38 ] Currently the only organ that has been 3D bioprinted and successfully transplanted into a human is a bladder. [ 39 ] The bladder was formed from the host's bladder tissue. [ 39 ] Researchers have proposed that a potential positive impact of 3D printed organs is the ability to customize organs for the recipient. [ 3 ] Developments enabling an organ recipient’s host cells to be used to synthesize organs decreases the risk of organ rejection. [ 38 ]
The ability to print organs has decreased the demand for animal testing. [ 40 ] Animal testing is used to determine the safety of products ranging from makeup to medical devices. Cosmetic companies are already using smaller tissue models to test new products on skin. [ 40 ] The ability to 3D print skin reduces the need for animal trials for makeup testing. [ 38 ] In addition, the ability to print models of human organs to test the safety and efficacy of new drugs further reduces the necessity for animal trials. [ 40 ] Researchers at Harvard University determined that drug safety can be accurately tested on smaller tissue models of lungs. [ 40 ] The company Organovo, which designed one of the initial commercial bioprinters in 2009, has displayed that biodegradable 3D tissue models can be used to research and develop new drugs, including those to treat cancer. [ 41 ] An additional impact of organ printing includes the ability to rapidly create tissue models, therefore increasing productivity. [ 3 ]
One of the challenges of 3D printing organs is to recreate the vasculature required to keep the organs alive. [ 42 ] Designing a correct vasculature is necessary for the transport of nutrients, oxygen, and waste. [ 42 ] Blood vessels, especially capillaries, are difficult due to the small diameter. [ 38 ] Progress has been made in this area at Rice University, where researchers designed a 3D printer to make vessels in biocompatible hydrogels and designed a model of lungs that can oxygenate blood. [ 42 ] However, accompanied with this technique is the challenge of replicating the other minute details of organs. [ 42 ] It is difficult to replicate the entangled networks of airways, blood vessels, and bile ducts and complex geometry of organs. [ 42 ]
The challenges faced in the organ printing field extends beyond the research and development of techniques to solve the issues of multivascularization and difficult geometries. Before organ printing can become widely available, a source for sustainable cell sources must be found and large-scale manufacturing processes need to be developed. [ 43 ] Additional challenges include designing clinical trials to test the long-term viability and biocompatibility of synthetic organs. [ 43 ] While many developments have been made in the field of organ printing, more research must be conducted. | https://en.wikipedia.org/wiki/Organ_printing |
An organ system is a biological system consisting of a group of organs that work together to perform one or more functions. [ 1 ] Each organ has a specialized role in an organism body, and is made up of distinct tissues .
There are 11 distinct organ systems in human beings, [ 2 ] which form the basis of human anatomy and physiology . The 11 organ systems: the respiratory system, digestive and excretory system, circulatory system, urinary system, integumentary system, skeletal system, muscular system, endocrine system, lymphatic system, nervous system, and reproductive system. There are other systems in the body that are not organ systems—for example, the immune system protects the organism from infection, but it is not an organ system since it is not composed of organs. Some organs are in more than one system—for example, the nose is in the respiratory system and also serves as a sensory organ in the nervous system; the testes and ovaries are both part of the reproductive and endocrine systems.
Other animals have similar organ systems to humans although simpler animals may have a lot of organs in an organ system or even fewer organ systems.
Plants have two major organs systems. Vascular plants have two distinct organ systems: a shoot system , and a root system . The shoot system consists stems, leaves, and the reproductive parts of the plant (flowers and fruits). The shoot system generally grows above ground, where it absorbs the light needed for photosynthesis . The root system, which supports the plants and absorbs water and minerals, is usually underground. [ 3 ] | https://en.wikipedia.org/wiki/Organ_system |
Organelle biogenesis is the biogenesis , or creation, of cellular organelles in cells . Organelle biogenesis includes the process by which cellular organelles are split between daughter cells during mitosis ; this process is called organelle inheritance. [ 1 ]
Following the discovery of cellular organelles in the nineteenth century, little was known about their function and synthesis until the development of electron microscopy and subcellular fractionation in the twentieth century. This allowed experiments on the function, structure, and biogenesis of these organelles to commence.
Mechanisms of protein sorting and retrieval have been found to give organelles their characteristic composition . It is known that cellular organelles can come from preexisting organelles; however, it is a subject of controversy whether organelles can be created without a preexisting one. [ 1 ]
Several processes are known to have developed for organelle biogenesis. These can range from de novo synthesis to the copying of a template organelle; the formation of an organelle 'from scratch' and using a preexisting organelle as a template to manufacture an organelle, respectively. The distinct structures of each organelle are thought to be caused by the different mechanisms of the processes which create them and the proteins that they are made up of. Organelles may also be 'split' between two cells during the process of cellular division (known as organelle inheritance), where the organelle of the parent cell doubles in size and then splits with each half being delivered to their respective daughter cells. [ 1 ] [ 2 ]
The process of organelle biogenesis is known to be regulated by specialized transcription networks that modulate the expression of the genes that code for specific organellar proteins. In order for organelle biogenesis to be carried out properly, the specific genes coding for the organellar proteins must be transcribed properly and the translation of the resulting mRNA must be successful. In addition to this, the process requires the transfer of polypeptides to their site of function, guided by signaling peptides . If proteins are not directed to their respective sites of subcellular function, a defective organelle that fails to fulfill its tasks within the cell properly may result. [ 3 ]
Several metabolic diseases are known to be caused by a fault in the process of organelle biogenesis. These may include mitochondrial biogenesis defects, peroxisome biogenesis disorders, and lysosomal storage disorders. [ 3 ] | https://en.wikipedia.org/wiki/Organelle_biogenesis |
Organic Geochemistry is a monthly peer-reviewed scientific journal published by Elsevier covering research on all aspects of organic geochemistry . It is an official journal of the European Association of Organic Geochemists . The editors-in-chief are Bart van Dongen ( University of Manchester ), Elizabeth Minor ( University of Minnesota Duluth ), and Clifford Walters ( University of Texas at Austin ).
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2023 impact factor of 2.6. [ 8 ]
According to the Web of Science , the journal's two most cited papers (as of 22 June 2013 [update] ) are: [ 9 ] | https://en.wikipedia.org/wiki/Organic_Geochemistry |
Organic Letters is a biweekly peer-reviewed scientific journal covering research in organic chemistry . It was established in 1999 and is published by the American Chemical Society . In 2014, the journal moved to a hybrid open access publishing model. The founding editor-in-chief was Amos Smith . The current editor-in-chief is Marisa C. Kozlowski. [ 1 ] The journal is abstracted and indexed in: the Science Citation Index Expanded , Scopus , Academic Search Premier , BIOSIS Previews , Chemical Abstracts Service , EMBASE , and MEDLINE . [ 2 ]
This article about a chemistry journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Organic_Letters |
Organic Preparations and Procedures International is a bimonthly scientific journal focusing on organic chemists engaged in synthesis. Topics include original preparative chemistry in association with the synthesis of organic and organometallic compounds.
This article about a chemistry journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Organic_Preparations_and_Procedures_International |
Organic Reactions is a peer-reviewed book series that was established in 1942. It publishes detailed descriptions of useful organic reactions . Each article (called a chapter) is an invited review of the primary source material for the given reaction, and is written under tight editorial control, making it a secondary to tertiary‑level source. Each chapter explores the practical and theoretical aspects of the reaction, including its selectivity and reproducibility. The longest chapter runs to 1,303 pages. [ 1 ] While individual articles are not open access , the journal's wiki maintains a repository of summaries of reactions. [ 2 ] [ 3 ] The series is abstracted and indexed in Scopus . [ 4 ]
Prior to World War II , the center of organic chemistry research and industrial production was Germany. Students interested in pursuing a career in organic chemistry needed to learn German to read articles and textbooks, and often went to graduate school in Germany. When the war broke out, an effort to jump‑start a native US organic chemical industry and academic network was initiated. As part of this effort, the journal was launched. The first volume was published in 1942, with Roger Adams as editor-in-chief . [ 5 ] In the early years a volume would come out every two years or so, but the pace of publishing has accelerated, with volume 100 issued in 2019. [ 1 ] | https://en.wikipedia.org/wiki/Organic_Reactions |
Organic Syntheses is a peer-reviewed scientific journal that was established in 1921. It publishes detailed and checked procedures for the synthesis of organic compounds . A unique feature of the review process is that all of the data and experiments reported in an article must be successfully repeated in the laboratory of a member of the editorial board as a check for reproducibility prior to publication. [ 1 ] [ 2 ] The journal is published by Organic Syntheses, Inc., a non-profit corporation. [ 3 ] An annual print version is published by John Wiley & Sons on behalf of Organic Syntheses, Inc.
Prior to World War I , work on synthetic organic chemistry in the United States had been quite limited, and most of the reagents used in laboratories had to be imported from Europe. When export stoppages and trade embargoes cut off this source, Clarence Derick, a professor of chemistry at University of Illinois at Urbana-Champaign , began an effort to synthesize these needed chemicals in industrial quantities in a university laboratory with the help of a few graduate students. This work was performed during the summer break and came to be known as the "summer prep". Students who worked in the laboratory were paid and received credit. [ 4 ]
The basic procedures were often obtained from textbooks, and the procedures were sketchy. Reproducibility was important in summer preps, so students were required to keep meticulous record books. The procedures were finally collected and published for the first time in a four-pamphlet set called Organic Chemical Reagents , which quickly sold out. The publishers received submissions from other chemists, which spawned the idea for serial publication, and the first annual volume of Organic Syntheses was thus published in 1921. By then, chemists from other universities and industry were also contributing. [ 4 ]
One example of much needed chemicals were dyes for sensitizing photographic film. Research efforts in this field led to the foundation of Eastman Kodak Organic Chemicals Division. [ 4 ]
The summer preps contributed to the war effort in World War II but were discontinued in 1950 because by then an infrastructure of chemical companies with their own research had been established.
Until 1998, Organic Syntheses was published only as an annual printed volume. In that year, all past volumes were made available on an open access website and new articles are now published online as soon as they are accepted. [ 4 ] | https://en.wikipedia.org/wiki/Organic_Syntheses |
An organic acid anhydride is an acid anhydride that is also an organic compound . An acid anhydride is a compound that has two acyl groups bonded to the same oxygen atom. [ 1 ] A common type of organic acid anhydride is a carboxylic anhydride , where the parent acid is a carboxylic acid , the formula of the anhydride being (RC(O)) 2 O. Symmetrical acid anhydrides of this type are named by replacing the word acid in the name of the parent carboxylic acid by the word anhydride . [ 2 ] Thus, (CH 3 CO) 2 O is called acetic anhydride. Mixed (or unsymmetrical ) acid anhydrides, such as acetic formic anhydride (see below), are known, whereby reaction occurs between two different carboxylic acids. Nomenclature of unsymmetrical acid anhydrides list the names of both of the reacted carboxylic acids before the word "anhydride" (for example, the dehydration reaction between benzoic acid and propanoic acid would yield "benzoic propanoic anhydride"). [ 3 ]
One or both acyl groups of an acid anhydride may also be derived from another type of organic acid , such as sulfonic acid or a phosphonic acid . One of the acyl groups of an acid anhydride can be derived from an inorganic acid such as phosphoric acid . The mixed anhydride 1,3-bisphosphoglyceric acid , an intermediate in the formation of ATP via glycolysis , [ 4 ] is the mixed anhydride of 3-phosphoglyceric acid and phosphoric acid . Acidic oxides are also classified as acid anhydrides.
The nomenclature of organic acid anhydrides is derived from the names of the constituent carboxylic acids. In symmetrical acid anhydrides, only the prefix of the original carboxylic acid is used and the suffix "anhydride" is added. For most unsymmetrical acid anhydrides - also called mixed anhydrides- the prefixes from both acids reacted are listed before the suffix, e.g., benzoic propanoic anhydride. [ 5 ]
Organic acid anhydrides are prepared in industry by diverse means. Acetic anhydride is mainly produced by the carbonylation of methyl acetate . [ 6 ] Maleic anhydride is produced by the oxidation of benzene or butane . Laboratory routes emphasize the dehydration of the corresponding acids. The conditions vary from acid to acid, but acetic anhydride (itself an acid anhydride) is a common dehydrating agent as illustrated by the preparation of 3-nitro phthalic anhydride : [ 7 ]
In addition to symmetrical, acyclic anhydrides, other classes are recognized as discussed in the following sections.
Mixed anhydrides have the formula RC(O)OC(O)R'. They tend to redistribute upon heating although acetic formic anhydride can be distilled at one atmosphere. [ 8 ] Those containing the acetyl group can be prepared using ketene as an acetylating agent:
Acid chlorides are also effective precursors as illustrated by the reaction with sodium formate : [ 9 ]
Cyclic anhydrides, a kind of a heterocycle , are derived from some dicarboxylic acids. Commercially important examples are those that form five- and six-membered rings, succinic anhydride , maleic anhydride]], and phthalic anhydride . Although these five-membered rings form readily, they can be opened. [ 10 ] [ 11 ] [ 12 ]
Cyclic anhydrides with smaller rings are only of academic interest, e.g., malonic anhydride . [ 13 ] Cyclic anhydrides that would give larger rings, e.g. from adipic acid, are no importance.
The following xamples are mostly cyclic anhydrides:
These compounds are sometimes useful crosslinking agents .
Acid anhydrides are a source of reactive acyl groups, and their reactions and uses resemble those of acyl halides . For example, they are useful in Friedel-Crafts acylation of arenes (ArH):
Unlike acid halides, however, anhydrides do not react with Gilman reagents. [ citation needed ] Acid anhydrides tend to be less electrophilic than acyl chlorides . Usually only one acyl group is transferred per molecule of acid anhydride, which generall leads to a lower atom efficiency . The low cost, however, of acetic anhydride makes it a common choice for acetylation reactions.
Acylations with cyclic anhydrides can be distinctive vs the acyclic anhydrides, e.g. for the preparation of keto acids. [ 14 ] The action of phthalic anhydride on the electron-rich phenols results in diacylation. [ 15 ]
In reactions with alcohols and amines, the reactions afford equal amounts of the acylated product and the carboxylic acid:
The reactivity of anhydrides can be enhanced by using a catalytic amount of N,N-dimethylaminopyridine ("DMAP") or even pyridine . [ 16 ]
First, DMAP ( 2 ) attacks the anhydride ( 1 ) to form a tetrahedral intermediate, which collapses to eliminate a carboxylate ion to give amide 3 . This intermediate amide is more activated towards nucleophilic attack than the original anhydride, because dimethylaminopyridine is a better leaving group than a carboxylate. In the final set of steps, a nucleophile (Nuc) attacks 3 to give another tetrahedral intermediate. When this intermediate collapses to give the product 4 , the pyridine group is eliminated and its aromaticity is restored – a powerful driving force, and the reason why the pyridine compound is a better leaving group than a carboxylate ion.
For prochiral cyclic anhydrides, the alcoholysis reaction can be conducted asymmetrically. [ 17 ]
Acetic anhydride is a major industrial chemical widely used for preparing acetate esters, e.g. cellulose acetate . Maleic anhydride is the precursor to various resins by copolymerization with styrene . Maleic anhydride is a dienophile in the Diels-Alder reaction . [ 8 ]
Dianhydrides, molecules containing two acid anhydride functions, are used to synthesize polyimides and sometimes polyesters [ 18 ] and polyamides . [ 19 ] Examples of dianhydrides: pyromellitic dianhydride (PMDA), 3,3', 4,4' - oxydiphtalic dianhydride (ODPA), 3,3', 4,4'-benzophenone tetracarboxylic dianhydride (BTDA) , 4,4'-diphtalic (hexafluoroisopropylidene) anhydride (6FDA), benzoquinonetetracarboxylic dianhydride , ethylenetetracarboxylic dianhydride . Polyanhydrides are a class of polymers characterized by anhydride bonds that connect repeat units of the polymer backbone chain . [ citation needed ]
Natural products containing acid anhydrides have been isolated from animals, bacteria and fungi. [ 20 ] [ 21 ] [ 22 ] Examples include cantharidin from species of blister beetle, including the Spanish fly , Lytta vesicatoria , and tautomycin , from the bacterium Streptomyces spiroverticillatus . The maleidride family of fungal secondary metabolites, which possess a wide range of antibiotic and antifungal activity, are alicyclic compounds with maleic anhydride functional groups. [ 23 ] A number of proteins in prokaryotes [ 24 ] and eukaryotes [ 25 ] undergo spontaneous cleavage between the amino acid residues aspartic acid and proline via an acid anhydride intermediate. In some cases, the anhydride may then react with nucleophiles of other cellular components, such as at the surface of the bacterium Neisseria meningitidis or on proteins localized nearby. [ 26 ]
Imides are structurally related analogues, where the bridging oxygen is replaced by nitrogen. They are similarly formed by the condensation of dicarboxylic acids with ammonia. The replacement of all oxygen atoms with nitrogen gives imidines , these are a rare functional group which are very prone to hydrolysis. [ citation needed ]
Sulfur can replace oxygen, either in the carbonyl group or in the bridge. In the former case, the name of the acyl group is enclosed in parentheses to avoid ambiguity in the name, [ 2 ] e.g., (thioacetic) anhydride (CH 3 C(S)OC(S)CH 3 ). When two acyl groups are attached to the same sulfur atom, the resulting compound is called a thioanhydride, [ 2 ] e.g., acetic thioanhydride ((CH 3 C(O)) 2 S). | https://en.wikipedia.org/wiki/Organic_acid_anhydride |
An organic azide is an organic compound that contains an azide (– N 3 ) functional group . [ 1 ] Because of the hazards associated with their use, few azides are used commercially although they exhibit interesting reactivity for researchers. Low molecular weight azides are considered especially hazardous and are avoided. In the research laboratory, azides are precursors to amines . They are also popular for their participation in the " click reaction " between an azide and an alkyne and in Staudinger ligation . These two reactions are generally quite reliable, lending themselves to combinatorial chemistry .
Phenyl azide ("diazoamidobenzol"), was prepared in 1864 by Peter Griess by the reaction of ammonia and phenyldiazonium . [ 2 ] [ 3 ] In the 1890s, Theodor Curtius , who had discovered hydrazoic acid ( HN 3 ), described the rearrangement of acyl azides to isocyanates subsequently named the Curtius rearrangement . [ 4 ] Rolf Huisgen described the eponymous 1,3-dipolar cycloaddition . [ 5 ] [ 6 ]
The interest in azides among organic chemists has been relatively modest due to the reported instability of these compounds. [ 7 ] The situation has changed dramatically with the discovery by Sharpless et al. of Cu-catalysed (3+2)-cycloadditions between organic azides and terminal alkynes. [ 8 ] [ 9 ] The azido- and the alkyne groups are " bioorthogonal ", which means they do not interact with living systems, and at the same time they undergo an impressively fast and selective coupling. This type of formal 1,3-dipolar cycloaddition became the most famous example of so-called " click chemistry " [ 10 ] [ 11 ] (perhaps, the only one known to a non-specialist), and the field of organic azides exploded.
Myriad methods exist, most often using preformed azide-containing reagent.
As a pseudohalide , azide generally displaces many leaving group, e.g. Br − , I − , Ts O − , sulfonate , [ 13 ] [ 14 ] and others to give the azido compound. [ 15 ] The azide source is most often sodium azide ( NaN 3 ), although lithium azide ( LiN 3 ) has been demonstrated.
Aliphatic alcohols give azides via a variant of the Mitsunobu reaction , with the use of hydrazoic acid . [ 1 ] Hydrazines may also form azides by reaction with sodium nitrite : [ 16 ] Alcohols can be converted into azides in one step using 2-azido-1,3-dimethylimidazolinium hexafluorophosphate (ADMP) [ 17 ] or under Mitsunobu conditions [ 18 ] with diphenylphosphoryl azide (DPPA).
Trimethylsilyl azide (CH 3 ) 3 SiN 3 , and tributyltin azide (CH 3 CH 2 CH 2 CH 2 ) 3 SnN 3 , have all been used, [ 7 ] including enantioselective [ 19 ] modifications of the reaction are also known. Aminoazides are accessible by the epoxide and aziridine ring cleavage, respectively. [ 20 ] [ 21 ]
The azo transfer compounds, trifluoromethanesulfonyl azide and imidazole-1-sulfonyl azide , react with amines to give the corresponding azides. Diazo transfer onto amines using trifluoromethanesulfonyl azide ( Tf N 3 ) and Tosyl azide ( Ts N 3 ) has been reported. [ 22 ]
Hydroazidation of alkenes has been demonstrated [ 23 ]
Aryl azides may be prepared by displacement of the appropriate diazonium salt with sodium azide or trimethylsilyl azide . Nucleophilic aromatic substitution is also possible, even with chlorides . Anilines and aromatic hydrazines undergo diazotization , as do alkyl amines and hydrazines. [ 1 ]
Alkyl or aryl acyl chlorides react with sodium azide in aqueous solution to give acyl azides , [ 24 ] [ 25 ] which give isocyanates in the Curtius rearrangement .
A classic method for the synthesis of azides is the Dutt–Wormall reaction [ 26 ] in which a diazonium salt reacts with a sulfonamide first to a diazoaminosulfinate and then on hydrolysis the azide and a sulfinic acid . [ 27 ]
Organic azides engage in useful organic reactions . The terminal nitrogen is mildly nucleophilic. Generally, nucleophiles attack the azide at the terminal nitrogen N γ , while electrophiles react at the internal atom N α . [ 28 ] Azides easily extrude diatomic nitrogen , a tendency that is exploited in many reactions such as the Staudinger ligation or the Curtius rearrangement . [ 29 ]
Azides may be reduced to amines by hydrogenolysis [ 30 ] or with a phosphine (e.g., triphenylphosphine ) in the Staudinger reaction . This reaction allows azides to serve as protected -NH 2 synthons, as illustrated by the synthesis of 1,1,1-tris(aminomethyl)ethane :
In the azide alkyne Huisgen cycloaddition , organic azides react as 1,3-dipoles , reacting with alkynes to give substituted 1,2,3-triazoles .
Some azide reactions are shown in the following scheme. Probably the most famous is the reaction with phosphines , which leads to iminophosphoranes 22; these can be hydrolysed into primary amines 23 (the Staudinger reaction ), [ 31 ] react with carbonyl compounds to give imines 24 (the aza-Wittig reaction), [ 32 ] [ 33 ] [ 34 ] or undergo other transformations. Thermal decomposition of azides gives nitrenes, which participate in a variety of reactions; vinyl azides 19 decompose into 2H-azirines 20. [ 28 ] [ 35 ] Alkyl azides with low nitrogen-content (( n C + n O) / n N ≥ 3) are relatively stable and decompose only above ca. 175 °C. [ 36 ]
Direct photochemical decomposition of alkyl azides leads almost exclusively to imines (e.g. 25 and 26). [ 28 ] It is proposed that the azide group is promoted to the singlet excited state and then undergoes concerted rearrangement without the intermediacy of nitrenes. The presence of triplet sensitisers, however, may change the reaction mechanism and result in the formation of triplet nitrenes. The latter were observed directly by ESR spectroscopy at −269 °C as well as inferred in some photolyses. [ 37 ] [ 38 ] Triplet methyl nitrene is 31 kJ/mol more stable than its singlet form, and thus is most likely the ground state. [ 28 ] [ 39 ]
The (3+2)-cycloaddition of azides to double or triple bonds is one of the most utilised cycloadditions in organic chemistry and affords triazolines (e.g. 17) or triazoles , respectively. [ 40 ] [ 41 ] [ 42 ] The uncatalysed reaction is a concerted pericyclic process , in which the configuration of the alkene component is transferred to the triazoline product. The Woodward–Hoffmann denomination is [π4s+π2s] and the reaction is symmetry-allowed. According to Sustmann, this is a Type II cycloaddition, which means the two HOMOs and the two LUMOs have comparable energies, and thus both electron-withdrawing and electron-donating substituents may lead to an increase in the reaction rate. [ 43 ] [ 44 ] The reaction is generally free from significant solvent effects because both the reactants and the transition state (TS) are non-polar. [ 45 ]
Another azide regular is tosyl azide here in reaction with norbornadiene in a nitrogen insertion reaction: [ 46 ]
Some azides are valuable as bioorthogonal chemical reporters , molecules that can be "clicked" to see the metabolic path it has taken inside a living system.
The antiviral drug zidovudine (AZT) contains an azido group.
This article incorporates text by Oleksandr Zhurakovskyi available under the CC BY 2.5 license. | https://en.wikipedia.org/wiki/Organic_azide |
Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials , i.e., matter in its various forms that contain carbon atoms . [ 1 ] Study of structure determines their structural formula . Study of properties includes physical and chemical properties , and evaluation of chemical reactivity to understand their behavior. The study of organic reactions includes the chemical synthesis of natural products , drugs , and polymers , and study of individual organic molecules in the laboratory and via theoretical ( in silico ) study.
The range of chemicals studied in organic chemistry includes hydrocarbons (compounds containing only carbon and hydrogen ) as well as compounds based on carbon, but also containing other elements, [ 1 ] [ 2 ] [ 3 ] especially oxygen , nitrogen , sulfur , phosphorus (included in many biochemicals ) and the halogens . Organometallic chemistry is the study of compounds containing carbon– metal bonds.
Organic compounds form the basis of all earthly life and constitute the majority of known chemicals. The bonding patterns of carbon, with its valence of four—formal single, double, and triple bonds, plus structures with delocalized electrons —make the array of organic compounds structurally diverse, and their range of applications enormous. They form the basis of, or are constituents of, many commercial products including pharmaceuticals ; petrochemicals and agrichemicals , and products made from them including lubricants , solvents ; plastics ; fuels and explosives . The study of organic chemistry overlaps organometallic chemistry and biochemistry , but also with medicinal chemistry , polymer chemistry , and materials science . [ 1 ]
Organic chemistry is typically taught at the college or university level. [ 4 ] It is considered a very challenging course but has also been made accessible to students. [ 5 ]
Before the 18th century, chemists generally believed that compounds obtained from living organisms were endowed with a vital force that distinguished them from inorganic compounds . According to the concept of vitalism (vital force theory), organic matter was endowed with a "vital force". [ 6 ] During the first half of the nineteenth century, some of the first systematic studies of organic compounds were reported. Around 1816 Michel Chevreul started a study of soaps made from various fats and alkalis . He separated the acids that, in combination with the alkali, produced the soap. Since these were all individual compounds, he demonstrated that it was possible to make a chemical change in various fats (which traditionally come from organic sources), producing new compounds, without "vital force". In 1828 Friedrich Wöhler produced the organic chemical urea (carbamide), a constituent of urine , from inorganic starting materials (the salts potassium cyanate and ammonium sulfate ), in what is now called the Wöhler synthesis . Although Wöhler himself was cautious about claiming he had disproved vitalism, this was the first time a substance thought to be organic was synthesized in the laboratory without biological (organic) starting materials. The event is now generally accepted as indeed disproving the doctrine of vitalism. [ 7 ]
After Wöhler, Justus von Liebig worked on the organization of organic chemistry, being considered one of its principal founders. [ 8 ]
In 1856, William Henry Perkin , while trying to manufacture quinine , accidentally produced the organic dye now known as Perkin's mauve . His discovery, made widely known through its financial success, greatly increased interest in organic chemistry. [ 9 ]
A crucial breakthrough for organic chemistry was the concept of chemical structure, developed independently in 1858 by both Friedrich August Kekulé and Archibald Scott Couper . [ 10 ] Both researchers suggested that tetravalent carbon atoms could link to each other to form a carbon lattice, and that the detailed patterns of atomic bonding could be discerned by skillful interpretations of appropriate chemical reactions. [ 11 ]
The era of the pharmaceutical industry began in the last decade of the 19th century when the German company, Bayer , first manufactured acetylsalicylic acid—more commonly known as aspirin . [ 12 ] By 1910 Paul Ehrlich and his laboratory group began developing arsenic-based arsphenamine (Salvarsan) as the first effective medicinal treatment of syphilis , and thereby initiated the medical practice of chemotherapy . Ehrlich popularized the concepts of "magic bullet" drugs and of systematically improving drug therapies. [ 13 ] [ 14 ] His laboratory made decisive contributions to developing antiserum for diphtheria and standardizing therapeutic serums. [ 15 ]
Early examples of organic reactions and applications were often found because of a combination of luck and preparation for unexpected observations. The latter half of the 19th century however witnessed systematic studies of organic compounds. The development of synthetic indigo is illustrative. The production of indigo from plant sources dropped from 19,000 tons in 1897 to 1,000 tons by 1914 thanks to the synthetic methods developed by Adolf von Baeyer . In 2002, 17,000 tons of synthetic indigo were produced from petrochemicals . [ 17 ]
In the early part of the 20th century, polymers and enzymes were shown to be large organic molecules, and petroleum was shown to be of biological origin.
The multiple-step synthesis of complex organic compounds is called total synthesis. Total synthesis of complex natural compounds increased in complexity to glucose and terpineol . For example, cholesterol -related compounds have opened ways to synthesize complex human hormones and their modified derivatives. Since the start of the 20th century, complexity of total syntheses has been increased to include molecules of high complexity such as lysergic acid and vitamin B 12 . [ 18 ]
The discovery of petroleum and the development of the petrochemical industry spurred the development of organic chemistry. Converting individual petroleum compounds into types of compounds by various chemical processes led to organic reactions enabling a broad range of industrial and commercial products including, among (many) others: plastics , synthetic rubber , organic adhesives , and various property-modifying petroleum additives and catalysts .
The majority of chemical compounds occurring in biological organisms are carbon compounds, so the association between organic chemistry and biochemistry is so close that biochemistry might be regarded as in essence a branch of organic chemistry. Although the history of biochemistry might be taken to span some four centuries, fundamental understanding of the field only began to develop in the late 19th century and the actual term biochemistry was coined around the start of 20th century. Research in the field increased throughout the twentieth century, without any indication of slackening in the rate of increase, as may be verified by inspection of abstraction and indexing services such as BIOSIS Previews and Biological Abstracts , which began in the 1920s as a single annual volume, but has grown so drastically that by the end of the 20th century it was only available to the everyday user as an online electronic database . [ 19 ]
Since organic compounds often exist as mixtures , a variety of techniques have also been developed to assess purity; chromatography techniques are especially important for this application, and include HPLC and gas chromatography . Traditional methods of separation include distillation , crystallization , evaporation , magnetic separation and solvent extraction .
Organic compounds were traditionally characterized by a variety of chemical tests, called "wet methods", but such tests have been largely displaced by spectroscopic or other computer-intensive methods of analysis. [ 20 ] Listed in approximate order of utility, the chief analytical methods are:
Traditional spectroscopic methods such as infrared spectroscopy , optical rotation , and UV/VIS spectroscopy provide relatively nonspecific structural information but remain in use for specific applications. Refractive index and density can also be important for substance identification.
The physical properties of organic compounds typically of interest include both quantitative and qualitative features. Quantitative information includes a melting point, boiling point, solubility, and index of refraction. Qualitative properties include odor, consistency, and color.
Organic compounds typically melt and many boil. In contrast, while inorganic materials generally can be melted, many do not boil, and instead tend to degrade. In earlier times, the melting point (m.p.) and boiling point (b.p.) provided crucial information on the purity and identity of organic compounds. The melting and boiling points correlate with the polarity of the molecules and their molecular weight. Some organic compounds, especially symmetrical ones, sublime . A well-known example of a sublimable organic compound is para -dichlorobenzene , the odiferous constituent of modern mothballs. Organic compounds are usually not very stable at temperatures above 300 °C, although some exceptions exist.
Neutral organic compounds tend to be hydrophobic ; that is, they are less soluble in water than in organic solvents. Exceptions include organic compounds that contain ionizable groups as well as low molecular weight alcohols , amines , and carboxylic acids where hydrogen bonding occurs. Otherwise, organic compounds tend to dissolve in organic solvents . Solubility varies widely with the organic solute and with the organic solvent.
Various specialized properties of molecular crystals and organic polymers with conjugated systems are of interest depending on applications, e.g. thermo-mechanical and electro-mechanical such as piezoelectricity , electrical conductivity (see conductive polymers and organic semiconductors ), and electro-optical (e.g. non-linear optics ) properties. For historical reasons, such properties are mainly the subjects of the areas of polymer science and materials science .
The names of organic compounds are either systematic, following logically from a set of rules, or nonsystematic, following various traditions. Systematic nomenclature is stipulated by specifications from IUPAC (International Union of Pure and Applied Chemistry). Systematic nomenclature starts with the name for a parent structure within the molecule of interest. This parent name is then modified by prefixes, suffixes, and numbers to unambiguously convey the structure. Given that millions of organic compounds are known, rigorous use of systematic names can be cumbersome. Thus, IUPAC recommendations are more closely followed for simple compounds, but not complex molecules. To use the systematic naming, one must know the structures and names of the parent structures. Parent structures include unsubstituted hydrocarbons, heterocycles, and mono functionalized derivatives thereof.
Nonsystematic nomenclature is simpler and unambiguous, at least to organic chemists. Nonsystematic names do not indicate the structure of the compound. They are common for complex molecules, which include most natural products. Thus, the informally named lysergic acid diethylamide is systematically named
(6a R ,9 R )- N , N -diethyl-7-methyl-4,6,6a,7,8,9-hexahydroindolo-[4,3- fg ] quinoline-9-carboxamide.
With the increased use of computing, other naming methods have evolved that are intended to be interpreted by machines. Two popular formats are SMILES and InChI .
Organic molecules are described more commonly by drawings or structural formulas , combinations of drawings and chemical symbols. The line-angle formula is simple and unambiguous. In this system, the endpoints and intersections of each line represent one carbon, and hydrogen atoms can either be notated explicitly or assumed to be present as implied by tetravalent carbon.
By 1880 an explosion in the number of chemical compounds being discovered occurred assisted by new synthetic and analytical techniques. Grignard described the situation as "chaos le plus complet" (complete chaos) due to the lack of convention it was possible to have multiple names for the same compound. This led to the creation of the Geneva rules in 1892. [ 21 ]
The concept of functional groups is central in organic chemistry, both as a means to classify structures and for predicting properties. A functional group is a molecular module, and the reactivity of that functional group is assumed, within limits, to be the same in a variety of molecules. Functional groups can have a decisive influence on the chemical and physical properties of organic compounds. Molecules are classified based on their functional groups. Alcohols, for example, all have the subunit C-O-H. All alcohols tend to be somewhat hydrophilic , usually form esters , and usually can be converted to the corresponding halides . Most functional groups feature heteroatoms (atoms other than C and H). Organic compounds are classified according to functional groups, alcohols, carboxylic acids, amines, etc. [ 22 ] Functional groups make the molecule more acidic or basic due to their electronic influence on surrounding parts of the molecule.
As the p K a (aka basicity ) of the molecular addition/functional group increases, there is a corresponding dipole , when measured, increases in strength. A dipole directed towards the functional group (higher p K a therefore basic nature of group) points towards it and decreases in strength with increasing distance. Dipole distance (measured in Angstroms ) and steric hindrance towards the functional group have an intermolecular and intramolecular effect on the surrounding environment and pH level.
Different functional groups have different p K a values and bond strengths (single, double, triple) leading to increased electrophilicity with lower p K a and increased nucleophile strength with higher p K a . More basic/nucleophilic functional groups desire to attack an electrophilic functional group with a lower p K a on another molecule (intermolecular) or within the same molecule (intramolecular). Any group with a net acidic p K a that gets within range, such as an acyl or carbonyl group is fair game. Since the likelihood of being attacked decreases with an increase in p K a , acyl chloride components with the lowest measured p K a values are most likely to be attacked, followed by carboxylic acids (p K a = 4), thiols (13), malonates (13), alcohols (17), aldehydes (20), nitriles (25), esters (25), then amines (35). [ 23 ] Amines are very basic, and are great nucleophiles/attackers.
The aliphatic hydrocarbons are subdivided into three groups of homologous series according to their state of saturation :
The rest of the group is classified according to the functional groups present. Such compounds can be "straight-chain", branched-chain or cyclic. The degree of branching affects characteristics, such as the octane number or cetane number in petroleum chemistry.
Both saturated ( alicyclic ) compounds and unsaturated compounds exist as cyclic derivatives. The most stable rings contain five or six carbon atoms, but large rings (macrocycles) and smaller rings are common. The smallest cycloalkane family is the three-membered cyclopropane ((CH 2 ) 3 ). Saturated cyclic compounds contain single bonds only, whereas aromatic rings have an alternating (or conjugated) double bond. Cycloalkanes do not contain multiple bonds, whereas the cycloalkenes and the cycloalkynes do.
Aromatic hydrocarbons contain conjugated double bonds. This means that every carbon atom in the ring is sp 2 hybridized, allowing for added stability. The most important example is benzene , the structure of which was formulated by Kekulé who first proposed the delocalization or resonance principle for explaining its structure. For "conventional" cyclic compounds, aromaticity is conferred by the presence of 4n + 2 delocalized pi electrons, where n is an integer. Particular instability ( antiaromaticity ) is conferred by the presence of 4n conjugated pi electrons.
The characteristics of the cyclic hydrocarbons are again altered if heteroatoms are present, which can exist as either substituents attached externally to the ring (exocyclic) or as a member of the ring itself (endocyclic). In the case of the latter, the ring is termed a heterocycle . Pyridine and furan are examples of aromatic heterocycles while piperidine and tetrahydrofuran are the corresponding alicyclic heterocycles. The heteroatom of heterocyclic molecules is generally oxygen, sulfur, or nitrogen, with the latter being particularly common in biochemical systems.
Heterocycles are commonly found in a wide range of products including aniline dyes and medicines. Additionally, they are prevalent in a wide range of biochemical compounds such as alkaloids , vitamins, steroids, and nucleic acids (e.g. DNA, RNA).
Rings can fuse with other rings on an edge to give polycyclic compounds . The purine nucleoside bases are notable polycyclic aromatic heterocycles. Rings can also fuse on a "corner" such that one atom (almost always carbon) has two bonds going to one ring and two to another. Such compounds are termed spiro and are important in several natural products .
One important property of carbon is that it readily forms chains, or networks, that are linked by carbon-carbon (carbon-to-carbon) bonds. The linking process is called polymerization , while the chains, or networks, are called polymers . The source compound is called a monomer .
Two main groups of polymers exist: synthetic polymers and biopolymers . Synthetic polymers are artificially manufactured, and are commonly referred to as industrial polymers . [ 24 ] Biopolymers occur within a respectfully natural environment, or without human intervention.
Biomolecular chemistry is a major category within organic chemistry which is frequently studied by biochemists . Many complex multi-functional group molecules are important in living organisms. Some are long-chain biopolymers , and these include peptides , DNA , RNA and the polysaccharides such as starches in animals and celluloses in plants. The other main classes are amino acids (monomer building blocks of peptides and proteins), carbohydrates (which includes the polysaccharides), the nucleic acids (which include DNA and RNA as polymers), and the lipids . Besides, animal biochemistry contains many small molecule intermediates which assist in energy production through the Krebs cycle , and produces isoprene , the most common hydrocarbon in animals. Isoprenes in animals form the important steroid structural ( cholesterol ) and steroid hormone compounds; and in plants form terpenes , terpenoids , some alkaloids , and a class of hydrocarbons called biopolymer polyisoprenoids present in the latex of various species of plants, which is the basis for making rubber . Biologists usually classify the above-mentioned biomolecules into four main groups, i.e., proteins, lipids, carbohydrates, and nucleic acids. Petroleum and its derivatives are considered organic molecules, which is consistent with the fact that this oil comes from the fossilization of living beings, i.e., biomolecules. [ 25 ] See also: peptide synthesis , oligonucleotide synthesis and carbohydrate synthesis .
In pharmacology, an important group of organic compounds is small molecules , also referred to as 'small organic compounds'. In this context, a small molecule is a small organic compound that is biologically active but is not a polymer . In practice, small molecules have a molar mass less than approximately 1000 g/mol.
Fullerenes and carbon nanotubes , carbon compounds with spheroidal and tubular structures, have stimulated much research into the related field of materials science . The first fullerene was discovered in 1985 by Sir Harold W. Kroto of the United Kingdom and by Richard E. Smalley and Robert F. Curl Jr., of the United States. Using a laser to vaporize graphite rods in an atmosphere of helium gas, these chemists and their assistants obtained cagelike molecules composed of 60 carbon atoms (C60) joined by single and double bonds to form a hollow sphere with 12 pentagonal and 20 hexagonal faces—a design that resembles a football, or soccer ball. In 1996 the trio was awarded the Nobel Prize for their pioneering efforts. The C60 molecule was named buckminsterfullerene (or, more simply, the buckyball) after the American architect R. Buckminster Fuller, whose geodesic dome is constructed on the same structural principles.
Organic compounds containing bonds of carbon to nitrogen, oxygen and the halogens are not normally grouped separately. Others are sometimes put into major groups within organic chemistry and discussed under titles such as organosulfur chemistry , organometallic chemistry , organophosphorus chemistry and organosilicon chemistry .
Organic reactions are chemical reactions involving organic compounds . [ 22 ] Many of these reactions are associated with functional groups. The general theory of these reactions involves careful analysis of such properties as the electron affinity of key atoms, bond strengths and steric hindrance . These factors can determine the relative stability of short-lived reactive intermediates , which usually directly determine the path of the reaction.
The basic reaction types are: addition reactions , elimination reactions , substitution reactions , pericyclic reactions , rearrangement reactions and redox reactions . [ 22 ] An example of a common reaction is a substitution reaction written as:
where X is some functional group and Nu is a nucleophile .
The number of possible organic reactions is infinite. However, certain general patterns are observed that can be used to describe many common or useful reactions. Each reaction has a stepwise reaction mechanism that explains how it happens in sequence—although the detailed description of steps is not always clear from a list of reactants alone.
The stepwise course of any given reaction mechanism can be represented using arrow pushing techniques in which curved arrows are used to track the movement of electrons as starting materials transition through intermediates to final products.
Synthetic organic chemistry is an applied science as it borders engineering , the "design, analysis, and/or construction of works for practical purposes". [ 26 ] Organic synthesis of a novel compound is a problem-solving task, where a synthesis is designed for a target molecule by selecting optimal reactions from optimal starting materials. Complex compounds can have tens of reaction steps that sequentially build the desired molecule. The synthesis proceeds by utilizing the reactivity of the functional groups in the molecule. For example, a carbonyl compound can be used as a nucleophile by converting it into an enolate , or as an electrophile ; the combination of the two is called the aldol reaction . Designing practically useful syntheses always requires conducting the actual synthesis in the laboratory. The scientific practice of creating novel synthetic routes for complex molecules is called total synthesis . [ 22 ]
Strategies to design a synthesis include retrosynthesis , popularized by E.J. Corey , which starts with the target molecule and splices it to pieces according to known reactions. The pieces, or the proposed precursors, receive the same treatment, until available and ideally inexpensive starting materials are reached. Then, the retrosynthesis is written in the opposite direction to give the synthesis. A "synthetic tree" can be constructed because each compound and also each precursor has multiple syntheses. | https://en.wikipedia.org/wiki/Organic_chemistry |
Some chemical authorities define an organic compound as a chemical compound that contains a carbon–hydrogen or carbon–carbon bond ; others consider an organic compound to be any chemical compound that contains carbon. For example, carbon-containing compounds such as alkanes (e.g. methane CH 4 ) and its derivatives are universally considered organic, but many others are sometimes considered inorganic , such as certain compounds of carbon with nitrogen and oxygen (e.g. cyanide ion CN − , hydrogen cyanide HCN , chloroformic acid ClCO 2 H , carbon dioxide CO 2 , and carbonate ion CO 2− 3 ). [ citation needed ]
Due to carbon's ability to catenate (form chains with other carbon atoms ), millions of organic compounds are known. The study of the properties, reactions, and syntheses of organic compounds comprise the discipline known as organic chemistry . For historical reasons, a few classes of carbon-containing compounds (e.g., carbonate salts and cyanide salts), along with a few other exceptions (e.g., carbon dioxide , and even hydrogen cyanide despite the fact it contains a carbon-hydrogen bond), are generally considered inorganic . Other than those just named, little consensus exists among chemists on precisely which carbon-containing compounds are excluded, making any rigorous definition of an organic compound elusive. [ 1 ]
Although organic compounds make up only a small percentage of Earth's crust , they are of central importance because all known life is based on organic compounds. Living things incorporate inorganic carbon compounds into organic compounds through a network of processes (the carbon cycle ) that begins with the conversion of carbon dioxide and a hydrogen source like water into simple sugars and other organic molecules by autotrophic organisms using light ( photosynthesis ) or other sources of energy. Most synthetically-produced organic compounds are ultimately derived from petrochemicals consisting mainly of hydrocarbons , which are themselves formed from the high pressure and temperature degradation of organic matter underground over geological timescales. [ 2 ] This ultimate derivation notwithstanding, organic compounds are no longer defined as compounds originating in living things, as they were historically.
In chemical nomenclature, an organyl group , frequently represented by the letter R, refers to any monovalent substituent whose open valence is on a carbon atom. [ 3 ]
For historical reasons discussed below, a few types of carbon-containing compounds, such as carbides , carbonates (excluding carbonate esters ), simple oxides of carbon (for example, CO and CO 2 ) and cyanides are generally considered inorganic compounds . Different forms ( allotropes ) of pure carbon, such as diamond , graphite , fullerenes and carbon nanotubes [ 4 ] are also excluded because they are simple substances composed of a single element and so not generally considered chemical compounds . The word "organic" in this context does not mean "natural". [ 5 ]
Vitalism was a widespread conception that substances found in organic nature are formed from the chemical elements by the action of a "vital force" [ 6 ] or "life-force" ( vis vitalis ) that only living organisms possess. [ 7 ]
In the 1810s, Jöns Jacob Berzelius argued that a regulative force must exist within living bodies. Berzelius also contended that compounds could be distinguished by whether they required any organisms in their synthesis (organic compounds) or whether they did not ( inorganic compounds ). [ 8 ] Vitalism taught that formation of these "organic" compounds were fundamentally different from the "inorganic" compounds that could be obtained from the elements by chemical manipulations in laboratories. [ 9 ] [ 10 ]
Vitalism survived for a short period after the formulation of modern ideas about the atomic theory and chemical elements . It first came under question in 1824, when Friedrich Wöhler synthesized oxalic acid , a compound known to occur only in living organisms, from cyanogen . A further experiment was Wöhler's 1828 synthesis of urea from the inorganic salts potassium cyanate and ammonium sulfate . Urea had long been considered an "organic" compound, as it was known to occur only in the urine of living organisms. Wöhler's experiments were followed by many others, in which increasingly complex "organic" substances were produced from "inorganic" ones without the involvement of any living organism, thus disproving vitalism. [ 11 ]
Although vitalism has been discredited, scientific nomenclature retains the distinction between organic and inorganic compounds. The modern meaning of organic compound is any compound that contains a significant amount of carbon—even though many of the organic compounds known today have no connection to any substance found in living organisms. The term carbogenic has been proposed by E. J. Corey as a modern alternative to organic , but this neologism remains relatively obscure. [ citation needed ]
The organic compound L -isoleucine molecule presents some features typical of organic compounds: carbon–carbon bonds , carbon–hydrogen bonds , as well as covalent bonds from carbon to oxygen and to nitrogen. [ citation needed ]
As described in detail below, any definition of organic compound that uses simple, broadly-applicable criteria turns out to be unsatisfactory, to varying degrees. The modern, commonly accepted definition of organic compound essentially amounts to any carbon-containing compound, excluding several classes of substances traditionally considered "inorganic". The list of substances so excluded varies from author to author. Still, it is generally agreed upon that there are (at least) a few carbon-containing compounds that should not be considered organic. For instance, almost all authorities would require the exclusion of alloys that contain carbon, including steel (which contains cementite , Fe 3 C ), as well as other metal and semimetal carbides (including "ionic" carbides, e.g, Al 4 C 3 and CaC 2 and "covalent" carbides, e.g. B 4 C and SiC , and graphite intercalation compounds, e.g. KC 8 ). Other compounds and materials that are considered 'inorganic' by most authorities include: metal carbonates , simple oxides of carbon ( CO , CO 2 , and arguably, C 3 O 2 ), the allotropes of carbon, cyanide derivatives not containing an organic residue (e.g., KCN , (CN) 2 , BrCN , cyanate anion OCN − , etc.), and heavier analogs thereof (e.g., cyaphide anion CP − , CSe 2 , COS ; although carbon disulfide CS 2 is often classed as an organic solvent). Halides of carbon without hydrogen (e.g., CF 4 and CClF 3 ), phosgene ( COCl 2 ), carboranes , metal carbonyls (e.g., nickel tetracarbonyl ), mellitic anhydride ( C 12 O 9 ), and other exotic oxocarbons are also considered inorganic by some authorities. [ citation needed ]
Nickel tetracarbonyl ( Ni(CO) 4 ) and other metal carbonyls are often volatile liquids, like many organic compounds, yet they contain only carbon bonded to a transition metal and to oxygen, and are often prepared directly from metal and carbon monoxide . Nickel tetracarbonyl is typically classified as an organometallic compound as it satisfies the broad definition that organometallic chemistry covers all compounds that contain at least one carbon to metal covalent bond; it is unknown whether organometallic compounds form a subset of organic compounds. For example, the evidence of covalent Fe-C bonding in cementite , [ 12 ] a major component of steel, places it within this broad definition of organometallic, yet steel and other carbon-containing alloys are seldom regarded as organic compounds. Thus, it is unclear whether the definition of organometallic should be narrowed, whether these considerations imply that organometallic compounds are not necessarily organic, or both. [ citation needed ]
Metal complexes with organic ligands but no carbon-metal bonds (e.g., (CH 3 CO 2 ) 2 Cu ) are not considered organometallic; instead, they are called metal-organic compounds (and might be considered organic).
The relatively narrow definition of organic compounds as those containing C-H bonds excludes compounds that are (historically and practically) considered organic. Neither urea CO(NH 2 ) 2 nor oxalic acid (COOH) 2 are organic by this definition, yet they were two key compounds in the vitalism debate. However, the IUPAC Blue Book on organic nomenclature specifically mentions urea [ 13 ] and oxalic acid [ 14 ] as organic compounds. Other compounds lacking C-H bonds but traditionally considered organic include benzenehexol , mesoxalic acid , and carbon tetrachloride . Mellitic acid , which contains no C-H bonds, is considered a possible organic compound in Martian soil. [ 15 ] Terrestrially, it, and its anhydride, mellitic anhydride , are associated with the mineral mellite ( Al 2 C 6 (COO) 6 ·16H 2 O ).
A slightly broader definition of the organic compound includes all compounds bearing C-H or C-C bonds. This would still exclude urea. Moreover, this definition still leads to somewhat arbitrary divisions in sets of carbon-halogen compounds. For example, CF 4 and CCl 4 would be considered by this rule to be "inorganic", whereas CHF 3 , CHCl 3 , and C 2 Cl 6 would be organic, though these compounds share many physical and chemical properties. [ citation needed ]
Organic compounds may be classified in a variety of ways. One major distinction is between natural and synthetic compounds. Organic compounds can also be classified or subdivided by the presence of heteroatoms , e.g., organometallic compounds , which feature bonds between carbon and a metal , and organophosphorus compounds , which feature bonds between carbon and a phosphorus . [ citation needed ]
Another distinction, based on the size of organic compounds, distinguishes between small molecules and polymers .
Natural compounds refer to those that are produced by plants or animals. Many of these are still extracted from natural sources because they would be more expensive to produce artificially. Examples include most sugars , some alkaloids and terpenoids , certain nutrients such as vitamin B 12 , and, in general, those natural products with large or stereoisometrically complicated molecules present in reasonable concentrations in living organisms.
Further compounds of prime importance in biochemistry are antigens , carbohydrates , enzymes , hormones , lipids and fatty acids , neurotransmitters , nucleic acids , proteins , peptides and amino acids , lectins , vitamins , and fats and oils .
Compounds that are prepared by reaction of other compounds are known as " synthetic ". They may be either compounds that are already found in plants/animals or those artificial compounds that do not occur naturally . [ citation needed ]
Most polymers (a category that includes all plastics and rubbers ) are organic synthetic or semi-synthetic compounds.
Many organic compounds—two examples are ethanol and insulin —are manufactured industrially using organisms such as bacteria and yeast. [ 16 ] Typically, the DNA of an organism is altered to express compounds not ordinarily produced by the organism. Many such biotechnology -engineered compounds did not previously exist in nature. [ 17 ]
A great number of more specialized databases exist for diverse branches of organic chemistry. [ 18 ]
The main tools are proton and carbon-13 NMR spectroscopy , IR Spectroscopy , Mass spectrometry , UV/Vis Spectroscopy and X-ray crystallography . [ 19 ] | https://en.wikipedia.org/wiki/Organic_compound |
Organic engineering systems ( OES ) are organic, hydroponic , aeroponic or aquaponic farming technologies that are designed as a building engineering system. The main purpose of these technologies is to grow food in buildings. OES are agricultural technologies and are considered a sub-section of ecological engineering , primarily because they mimic natural ecosystems to produce food and are defined as urban horticulture . [ 1 ] | https://en.wikipedia.org/wiki/Organic_engineering_systems |
The effect of organic farming has been a subject of interest for researchers. Theory suggests that organic farming practices, which exclude the use of most synthetic pesticides and fertilizers, may be beneficial for biodiversity. This is generally shown to be true for soils scaled to the area of cultivated land, where species abundance is, on average, 30% richer than that of conventional farms. However, for crop yield-scaled land the effect of organic farming on biodiversity is highly debated due to the significantly lower yields compared to conventional farms. [ 1 ]
In ancient farming practices, farmers did not possess the technology or manpower to have a significant impact on the destruction of biodiversity even as mass-production agriculture was rising. Nowadays, common farming methods generally rely on pesticides to maintain high yields. With such, most agricultural landscapes favor mono-culture crops with very little flora or fauna co-existence (van Elsen 2000). Modern organic farm practices such as the removal of pesticides and the inclusion of animal manure, crop rotation , and multi-cultural crops provides the chance for biodiversity to thrive. [ 2 ]
Nearly all non-crop, naturally occurring species observed in comparative farm land practice studies show a preference in organic farming both by population and richness. [ 3 ] [ 4 ] Spanning all associated species, there is an average of 30% more on organic farms versus conventional farming methods, however this does not account for possible loss of biodiversity due to decreased yields. [ 1 ] Birds, butterflies, soil microbes, beetles, earthworms, spiders, vegetation, and mammals are particularly affected. Some organic farms may use less pesticides and thus biodiversity fitness and population density may benefit. [ 4 ] Larger farms however tend to use pesticides more liberally and in some cases to larger extent than conventional farms. [ 5 ] Many weed species attract beneficial insects that improve soil qualities and forage on weed pests. [ 6 ] Soil-bound organisms often benefit because of increased bacteria populations due to natural fertilizer spread such as manure, while experiencing reduced intake of herbicides and pesticides commonly associated with conventional farming methods. [ 3 ] Increased biodiversity, especially from soil microbes such as mycorhizae, have been proposed as an explanation for the high yields experienced by some organic plots, especially in light of the differences seen in a 21-year comparison of organic and control fields. [ 7 ]
The level of biodiversity that can be yielded from organic farming provides a natural capital to humans. Species found in most organic farms provides a means of agricultural sustainability by reducing amount of human input (e.g. fertilizers, pesticides). [ 8 ] Farmers that produce with organic methods reduce risk of poor yields by promoting biodiversity. Common game birds such as the ring-necked pheasant and the northern bobwhite often reside in agriculture landscapes, and are a natural capital yielded from high demands of recreational hunting. Because bird species richness and population are typically higher on organic farm systems, promoting biodiversity can be seen as logical and economical.
Earthworm population and diversity appears to have the most significant data out of all studies. Out of six studies comparing earthworm biodiversity to organic and conventional farming methods, all six suggested a preference for organic practices including a study at the pioneering Haughley farm in 1980/1981 that compared earthworm populations and soil properties after 40 years. [ 9 ] Hole et al. (2005) summarized a study conducted by Brown (1999) and found nearly double the population and diversity when comparing farming methods.
Organic farms are said to be beneficial to birds while remaining economical. Bird species are one of the most prominent animal groups that benefit from organic farming methods. Many species rely on farmland for foraging, feeding, and migration phases. With such, bird populations often relate directly to the natural quality of farmland. The more natural diversity of organic farms provides better habitats to bird species, and is especially beneficial when the farmland is located within a migration zone. [ 10 ] In 5 recent studies almost all bird species including locally declining species, both population and variation increased on organic farmland,. [ 3 ] [ 11 ] Making a switch from conventional farming methods to organic practices also seems to directly improve bird species in the area. [ 12 ] While organic farming improves bird populations and diversity, species populations receive the largest boost when organic groups are varied within a landscape. Bird populations are increased further with optimal habitat for biodiversity, rather than organic alone, with systems such as Conservation Grade. [ 13 ]
A specific study done in the UK in 2006 found substantially more butterflies on organic farms versus standard farming methods except for two pest species. The study also observed higher populations in uncropped field margins compared with cropland edges regardless of farm practice. [ 2 ] Conversely, Weibull et al. (2000) found no significant differences in species diversity or population.
Ten studies have been conducted involving spider species and abundance on farm systems. All but three of the studies indicated that there was a higher diversity of spider species on organic farms, in addition to populations of species. Two of the studies indicated higher species diversity, but statistically insignificant populations between organic and standard farming methods.
Out of 13 studies comparing bacteria and fungus communities between organic and standard farming, 8 of the studies showed heightened level of growth on organic farm systems. One study concluded that the use of “green” fertilizers and manures was the primary cause of higher bacterial levels on organic farms. On the other hand, nematode population/diversity depended on what their primary food intake was. Bacteria-feeding nematodes showed preference towards organic systems whereas fungus-feeding nematodes showed preference for standard farm systems. The heightened level of bacteria-feeding nematodes makes sense due to higher levels of bacteria in organic soils, but the fungus-feeding populations being higher on standard farms seems to contradict the data since more fungi are generally found on organic farms. [ 3 ]
According to Hole et al. (2005), beetle species are among the most commonly studied animal species on farming systems. Twelve studies have found a higher population and species richness of carabids on organic systems. The overall conclusion of significantly higher carabid population species and diversity is that organic farms have a higher level of weed species where they can thrive. Staphylinid populations and diversity have seemed to show no specific preference with some studies showing higher population and diversity, some with lower population and diversity, and one study showed no statistical significance between the organic and conventional farming systems.
Two comparative studies have been conducted involving mammal populations and diversity among farm practices. A study done by Brown (1999) found that small mammal population density and diversity did not depend on farming practices, however overall activity was higher on organic farms. It was concluded that more food resources were available to small mammals on organic farms because of the reduction or lack of herbicides and pesticides. Another study conducted by Wickramasinghe et al. (2003) compared bat species and activity. Species activity and foraging were both more than double on organic farms compared to conventional farms. Species richness was also higher on organic farms, and 2 of the sixteen species sighted were found only on organic farms. [ 14 ]
Approximately ten studies have been conducted to compare non-crop vegetation between organic and conventional farming practices. Hedgerow, inner-crop and grassland observations were made within these studies and all but one showed a higher weed preference and diversity in or around organic farms. Most of these studies showed significant overall preference for organic farming preferences especially for broad-leafed species, but many grass species showed far less on conventional farms likely because pesticide interaction was low or non-existent. Organic farm weed population and richness was believed to be lower in mid-crop land because of weed-removal methods such as under sowing. [ 3 ] Switching from conventional to organic farming often results in a “boom” of weed speciation due to intense chemical change of soil composition from the lack of herbicides and pesticides. Natural plant species can also vary on organic farms from year-to-year because crop rotation creates new competition based on the chemical needs of each crop. [ 15 ]
Biological research on soil and soil organisms has proven beneficial to the system of organic farming. Varieties of bacteria and fungi break down chemicals, plant matter and animal waste into productive soil nutrients. In turn, the producer benefits by healthier yields and more arable soil for future crops. [ 16 ] Furthermore, a 21-year study was conducted testing the effects of organic soil matter and its relationship to soil quality and yield. Controls included actively managed soil with varying levels of manure, compared to a plot with no manure input. After the study commenced, there was significantly lower yields on the control plot when compared to the fields with manure. The concluded reason was an increased soil microbe community in the manure fields, providing a healthier, more arable soil system. [ 7 ]
Organic farming practices still require active participation from the farmer to effectively boost biodiversity. Making a switch to organic farming methods does not automatically or guarantee improved biodiversity. Pro-conservation ethics are required to create arable farm land that generates biodiversity. Conservationist ideals are commonly overlooked because they require additional physical and economical efforts from the producer. [ 6 ] Common weed-removal processes like undercutting and controlled burning provides little opportunity for species survival, and often leads to comparable populations and richness to conventionally managed landscapes when performed in excess. Another common process is the addition of biotopes in the form of hedgerows and ponds to further improve species richness. Farmers commonly make the mistake of over-using these resources for more intense crop production because organic yields are typically lower. Another error comes from the over-stratification of biotopes. A series of small clusters does not provide adequate land area for high biodiversity potential. [ 6 ] | https://en.wikipedia.org/wiki/Organic_farming_and_biodiversity |
An organic field-effect transistor ( OFET ) is a field-effect transistor using an organic semiconductor in its channel. OFETs can be prepared either by vacuum evaporation of small molecules, by solution-casting of polymers or small molecules, or by mechanical transfer of a peeled single-crystalline organic layer onto a substrate. These devices have been developed to realize low-cost, large-area electronic products and biodegradable electronics . OFETs have been fabricated with various device geometries. The most commonly used device geometry is bottom gate with top drain and source electrodes , because this geometry is similar to the thin-film silicon transistor (TFT) using thermally grown SiO 2 as gate dielectric . Organic polymers, such as poly(methyl-methacrylate) ( PMMA ), can also be used as dielectric. [ 1 ] One of the benefits of OFETs, especially compared with inorganic TFTs, is their unprecedented physical flexibility, [ 2 ] which leads to biocompatible applications, for instance in the future health care industry of personalized biomedicines and bioelectronics. [ 3 ]
In May 2007, Sony reported the first full-color, video-rate, flexible, all plastic display, [ 4 ] [ 5 ] in which both the thin-film transistors and the light-emitting pixels were made of organic materials.
The concept of a field-effect transistor (FET) was first proposed by Julius Edgar Lilienfeld , who received a patent for his idea in 1930. [ 6 ] He proposed that a field-effect transistor behaves as a capacitor with a conducting channel between a source and a drain electrode. Applied voltage on the gate electrode controls the amount of charge carriers flowing through the system.
The first insulated-gate field-effect transistor was designed and prepared by Frosch and Derrick in 1957, using masking and predeposition, were able to manufacture silicon dioxide transistors and showed that silicon dioxide insulated, protected silicon wafers and prevented dopants from diffusing into the wafer. [ 7 ] [ 8 ] Later, following this research, Mohamed Atalla and Dawon Kahng proposed a silicon MOS transistor in 1959 [ 9 ] and successfully demonstrated a working MOS device with their Bell Labs team in 1960. [ 10 ] [ 11 ] Their team included E. E. LaBate and E. I. Povilonis who fabricated the device; M. O. Thurston, L. A. D’Asaro, and J. R. Ligenza who developed the diffusion processes, and H. K. Gummel and R. Lindner who characterized the device. [ 12 ] [ 13 ] Also known as the MOS transistor, the MOSFET is the most widely manufactured device in the world. [ 14 ] [ 15 ]
The concept of a thin-film transistor (TFT) was first proposed by John Wallmark who in 1957 filed a patent for a thin film MOSFET in which germanium monoxide was used as a gate dielectric. Thin-film transistor was developed in 1962 by Paul K. Weimer who implemented Wallmark's ideas. [ 16 ] The TFT is a special type of MOSFET. [ 17 ]
Rising costs of materials and manufacturing, [ citation needed ] as well as public interest in more environmentally friendly electronics materials, have supported development of organic based electronics in more recent years. In 1986, Mitsubishi Electric researchers H. Koezuka, A. Tsumura and Tsuneya Ando reported the first organic field-effect transistor, [ 18 ] [ 19 ] based on a polymer of thiophene molecules. [ 20 ] The thiophene polymer is a type of conjugated polymer that is able to conduct charge, eliminating the need to use expensive metal oxide semiconductors. Additionally, other conjugated polymers have been shown to have semiconducting properties. OFET design has also improved in the past few decades. Many OFETs are now designed based on the thin-film transistor (TFT) model, which allows the devices to use less conductive materials in their design. Improvement on these models in the past few years have been made to field-effect mobility and on–off current ratios.
One common feature of OFET materials is the inclusion of an aromatic or otherwise conjugated π-electron system, facilitating the delocalization of orbital wavefunctions. Electron withdrawing groups or donating groups can be attached that facilitate hole or electron transport.
OFETs employing many aromatic and conjugated materials as the active semiconducting layer have been reported, including small molecules such as rubrene , tetracene , pentacene , diindenoperylene , perylenediimides , tetracyanoquinodimethane (TCNQ), and polymers such as polythiophenes (especially poly(3-hexylthiophene) (P3HT)), polyfluorene , polydiacetylene , poly(2,5-thienylene vinylene) , poly(p-phenylene vinylene) (PPV).
The field is very active, with newly synthesized and tested compounds reported weekly in prominent research journals. Many review articles exist documenting the development of these materials. [ 21 ] [ 22 ] [ 23 ] [ 24 ] [ 25 ]
Rubrene-based OFETs show the highest carrier mobility 20–40 cm 2 /(V·s). Another popular OFET material is pentacene, which has been used since the 1980s, but with mobilities 10 to 100 times lower (depending on the substrate) than rubrene. [ 25 ] The major problem with pentacene, as well as many other organic conductors, is its rapid oxidation in air to form pentacene-quinone. However if the pentacene is preoxidized, and the thus formed pentacene-quinone is used as the gate insulator, then the mobility can approach the rubrene values. This pentacene oxidation technique is akin to the silicon oxidation used in the silicon electronics. [ 21 ]
Polycrystalline tetrathiafulvalene and its analogues result in mobilities in the range 0.1–1.4 cm 2 /(V·s). However, the mobility exceeds 10 cm 2 /(V·s) in solution-grown or vapor-transport-grown single crystalline hexamethylene-tetrathiafulvalene (HMTTF). The ON/OFF voltage is different for devices grown by those two techniques, presumably due to the higher processing temperatures using in the vapor transport grows. [ 21 ]
All the above-mentioned devices are based on p-type conductivity. N-type OFETs are yet poorly developed. They are usually based on perylenediimides or fullerenes or their derivatives, and show electron mobilities below 2 cm 2 /(V·s). [ 22 ]
Three essential components of field-effect transistors are the source, the drain and the gate. Field-effect transistors usually operate as a capacitor . They are composed of two plates. One plate works as a conducting channel between two ohmic contacts , which are called the source and the drain contacts. The other plate works to control the charge induced into the channel, and it is called the gate. The direction of the movement of the carriers in the channel is from the source to the drain. Hence the relationship between these three components is that the gate controls the carrier movement from the source to the drain. [ 26 ]
When this capacitor concept is applied to the device design, various devices can be built up based on the difference in the controller – i.e. the gate. This can be the gate material, the location of the gate with respect to the channel, how the gate is isolated from the channel, and what type of carrier is induced by the gate voltage into channel (such as electrons in an n-channel device, holes in a p-channel device, and both electrons and holes in a double injection device).
Classified by the properties of the carrier, three types of FETs are shown schematically in Figure 1. [ 27 ] They are MOSFET (metal–oxide–semiconductor field-effect transistor), MESFET (metal–semiconductor field-effect transistor) and TFT (thin-film transistor).
The most prominent and widely used FET in modern microelectronics is the MOSFET (metal–oxide–semiconductor FET). There are different kinds in this category, such as MISFET (metal–insulator–semiconductor field-effect transistor), and IGFET (insulated-gate FET). A schematic of a MISFET is shown in Figure 1a. The source and the drain are connected by a semiconductor and the gate is separated from the channel by a layer of insulator. If there is no bias (potential difference) applied on the gate, the Band bending is induced due to the energy difference of metal conducting band and the semiconductor Fermi level . Therefore, a higher concentration of holes is formed on the interface of the semiconductor and the insulator. When an enough positive bias is applied on the gate contact, the bended band becomes flat. If a larger positive bias is applied, the band bending in the opposite direction occurs and the region close to the insulator-semiconductor interface becomes depleted of holes. Then the depleted region is formed. At an even larger positive bias, the band bending becomes so large that the Fermi level at the interface of the semiconductor and the insulator becomes closer to the bottom of the conduction band than to the top of the valence band, therefore, it forms an inversion layer of electrons, providing the conducting channel. Finally, it turns the device on. [ 28 ]
The second type of device is described in Fig.1b. The only difference of this one from the MISFET is that the n-type source and drain are connected by an n-type region. In this case, the depletion region extends all over the n-type channel at zero gate voltage in a normally “off” device (it is similar to the larger positive bias in MISFET case). In the normally “on” device, a portion of the channel is not depleted, and thus leads to passage of a current at zero gate voltage.
A thin-film transistor (TFT) is illustrated in Figure 1c. Here the source and drain electrodes are directly deposited onto the conducting channel (a thin layer of semiconductor) then a thin film of insulator is deposited between the semiconductor and the metal gate contact. This structure suggests that there is no depletion region to separate the device from the substrate. If there is zero bias, the electrons are expelled from the surface due to the Fermi-level energy difference of the semiconductor and the metal. This leads to band bending of semiconductor. In this case, there is no carrier movement between the source and drain. When the positive charge is applied, the accumulation of electrons on the interface leads to the bending of the semiconductor in an opposite way and leads to the lowering of the conduction band with regards to the Fermi-level of the semiconductor. Then a highly conductive channel forms at the interface (shown in Figure 2).
OFETs adopt the architecture of TFT. With the development of the conducting polymer, the semiconducting properties of small conjugated molecules have been recognized. The interest in OFETs has grown enormously in the past ten years. The reasons for this surge of interest are manifold. The performance of OFETs, which can compete with that of amorphous silicon (a-Si) TFTs with field-effect mobilities of 0.5–1 cm 2 V −1 s −1 and ON/OFF current ratios (which indicate the ability of the device to shut down) of 10 6 –10 8 , has improved significantly. Currently, thin-film OFET mobility values of 5 cm 2 V −1 s −1 in the case of vacuum-deposited small molecules [ 29 ] and 0.6 cm 2 V −1 s −1 for solution-processed polymers [ 30 ] have been reported. As a result, there is now a greater industrial interest in using OFETs for applications that are currently incompatible with the use of a-Si or other inorganic transistor technologies. One of their main technological attractions is that all the layers of an OFET can be deposited and patterned at room temperature by a combination of low-cost solution-processing and direct-write printing, which makes them ideally suited for realization of low-cost, large-area electronic functions on flexible substrates. [ 31 ]
Thermally oxidized silicon is a traditional substrate for OFETs where the silicon dioxide serves as the gate insulator. The active FET layer is usually deposited onto this substrate using either (i) thermal evaporation, (ii) coating from organic solution, or (iii) electrostatic lamination. The first two techniques result in polycrystalline active layers; they are much easier to produce, but result in relatively poor transistor performance. Numerous variations of the solution coating technique (ii) are known, including dip-coating , spin-coating , inkjet printing and screen printing . The electrostatic lamination technique is based on manual peeling of a thin layer off a single organic crystal; it results in a superior single-crystalline active layer, yet it is more tedious. The thickness of the gate oxide and the active layer is below one micrometer. [ 21 ]
The carrier transport in OFET is specific for two-dimensional (2D) carrier propagation through the device. Various experimental techniques were used for this study, such as Haynes - Shockley experiment on the transit times of injected carriers, time-of-flight (TOF) experiment [ 32 ] for the determination of carrier mobility, pressure-wave propagation experiment for probing electric-field distribution in insulators, organic monolayer experiment for probing orientational dipolar changes, optical time-resolved second harmonic generation (TRM-SHG), etc. Whereas carriers propagate through polycrystalline OFETs in a diffusion-like (trap-limited) manner, [ 33 ] they move through the conduction band in the best single-crystalline OFETs. [ 21 ]
The most important parameter of OFET carrier transport is carrier mobility. Its evolution over the years of OFET research is shown in the graph for polycrystalline and single crystalline OFETs. The horizontal lines indicate the comparison guides to the main OFET competitors – amorphous (a-Si) and polycrystalline silicon. The graph reveals that the mobility in polycrystalline OFETs is comparable to that of a-Si whereas mobility in rubrene -based OFETs (20–40 cm 2 /(V·s)) approaches that of best poly-silicon devices. [ 21 ]
Development of accurate models of charge carrier mobility in OFETs is an active field of research. Fishchuk et al. have developed an analytical model of carrier mobility in OFETs that accounts for carrier density and the polaron effect . [ 34 ]
While average carrier density is typically calculated as function of gate voltage when used as an input for carrier mobility models, [ 35 ] modulated amplitude reflectance spectroscopy (MARS) has been shown to provide a spatial map of carrier density across an OFET channel. [ 36 ]
Because an electric current flows through such a transistor, it can be used as a light-emitting device, thus integrating current modulation and light emission. In 2003, a German group reported the first organic light-emitting field-effect transistor (OLET). [ 37 ] The device structure comprises interdigitated gold source- and drain electrodes and a polycrystalline tetracene thin film. Both positive charges ( holes ) as well as negative charges ( electrons ) are injected from the gold contacts into this layer leading to electroluminescence from the tetracene. | https://en.wikipedia.org/wiki/Organic_field-effect_transistor |
Organic geochemistry is the study of the impacts and processes that organisms have had on the Earth. It is mainly concerned with the composition and mode of origin of organic matter in rocks and in bodies of water. [ 1 ] The study of organic geochemistry is traced to the work of Alfred E. Treibs , "the father of organic geochemistry." [ 2 ] Treibs first isolated metalloporphyrins from petroleum. This discovery established the biological origin of petroleum, which was previously poorly understood. [ 3 ] Metalloporphyrins in general are highly stable organic compounds, and the detailed structures of the extracted derivatives made clear that they originated from chlorophyll.
The relationship between the occurrence of organic compounds in sedimentary deposits and petroleum deposits has long been of interest. [ 4 ] Studies of ancient sediments and rock provide insights into the origins and sources of oil and petroleum, as well as the biochemical antecedents of life. Oil spills in particular have been of interest to geochemists in regards to the impact of petroleum and oil on the current geological environment. Following the Exxon Valdez Oil Spill , organic geochemistry knowledge on oil-spill chemistry bloomed with the analyses of samples from the spill. [ 5 ]
Geochemists study petroleum-inclusions in geological samples to compare present-day fluid-inclusions to dated samples. This analysis provides insight into the age of the petroleum samples and the surrounding rock. Spectrographic, optical, destructive, and nondestructive methods are used to analyze samples via mass spectrometry or Raman spectroscopy. The discovered differences in samples, such as oil-to-gas ratio or viscosity are typically attributed to the rock source of the sample. Other characteristics typically noted are pressure/volume/temperature properties, sample texture, and sample composition. Complications in analysis arise when the source rock is near or in a water source. [ 6 ]
Petroleum is also studied via carbon isotope analysis. Carbon isotopes provide insight into the Earth's carbon cycle and geological processes. Geochemists are able to discern the composition of petroleum deposits by examining the ratio of carbon isotopes and comparing this ratio to known values for carbon based structures of which the petroleum could be composed. [ 7 ]
Vast knowledge about coal has been attained since the inception of its use as an energy source. However, modern geochemists are still studying how plant material changes into coal. They have determined coalification results from a selective degradation of plant materials, while other plant material is preserved. Coal macromolecules are usually made up of these degradation-resistant biopolymers contained in algae, spores, and wood. Geochemists have unraveled the mysteries behind coal formation by comparing properties of the biopolymers to properties found in existing coal macromolecules. The analytical methods of Carbon NMR and gas chromatography-mass spectrometry (GC-MS) combined with flash pyrolysis has greatly enhanced the ability of organic geochemists to analyse the minute structural units of coal. [ 8 ]
Further knowledge into the age of coal sediments has been attained via isochron dating of uranium in the coalified samples. Examination of the parent to daughter ratio of uranium isotopes has led to the dating of select samples to the Late Cretaceous Period . [ 9 ]
Modern organic geochemistry includes studies of recent sediments to understand the carbon cycle, climate change, and ocean processes. In connection with petroleum studies, petroleum-focused geochemists also examine the impact of petroleum on the geological environment. [ 10 ] Geochemistry also examines other pollutants in geological systems, such as metabolites formed from the degradation of hydrocarbons . Organic geochemistry analytical techniques, such as GC-MS, allow chemists to determine the intricate effects of organic metabolites and human-derived waste products on the geological environment. [ 11 ] Of specific concern are the human-derived pollutants stemming from agricultural work. The use of animal manure, in combination with general municipal and sewage waste management, has changed many physical properties of the agricultural soil involved and the surrounding soils. [ 12 ]
Organic geochemistry is also relevant to aqueous environments. Pollutants, their metabolites, and how both enter bodies of water are of particular importance in the field. This organic matter can also be derived from geological processes in or near bodies of water, similarly influencing nearby lifeforms and protein production. Fluorescence spectroscopy has been introduced as a technique to examine organic matter in bodies of water, as dissolved organic matter is typically fluorescent. [ 13 ]
The study of organic geochemistry also extends to the atmosphere. Particularly, geochemists in this field study the makeup of insoluble material in the lower atmosphere. They have defined certain consequences of organic aerosols including physiological toxicity, direct and indirect climate forcing , smog, rain acidification, and incorporation into the natural carbon cycle. [ 14 ] | https://en.wikipedia.org/wiki/Organic_geochemistry |
An organic light-emitting transistor ( OLET ) is a form of transistor that emits light. These transistors have potential for digital displays and on-chip optical interconnects . [ 1 ] OLET is a new light-emission concept, providing planar light sources that can be easily integrated in substrates like silicon, glass, and paper using standard microelectronic techniques. [ 2 ]
OLETs differ from OLEDs in that an active matrix can be made entirely of OLETs, whereas OLEDs must be combined with switching elements such as TFTs .
This electronics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Organic_light-emitting_transistor |
Organic matter , organic material or natural organic matter is the large source of carbon-based compounds found within natural and engineered, terrestrial, and aquatic environments. It is matter composed of organic compounds that have come from the feces and remains of organisms such as plants and animals . [ 1 ] Organic molecules can also be made by chemical reactions that do not involve life. [ 2 ] Basic structures are created from cellulose , tannin , cutin , and lignin , along with other various proteins , lipids , and carbohydrates . Organic matter is very important in the movement of nutrients in the environment and plays a role in water retention on the surface of the planet. [ 3 ]
Living organisms are composed of organic compounds. In life, they secrete or excrete organic material into their environment, shed body parts such as leaves and roots and after organisms die, their bodies are broken down by bacterial and fungal action. Larger molecules of organic matter can be formed from the polymerization of different parts of already broken down matter. [ citation needed ] The composition of natural organic matter depends on its origin, transformation mode, age, and existing environment, thus its bio-physicochemical functions and properties vary with different environments. [ 4 ]
Organic matter is common throughout the ecosystem and is cycled through decomposition processes by soil microbial communities that are crucial for nutrient availability. [ 5 ] After degrading and reacting, it can move into soil and mainstream water via waterflow. Organic matter provides nutrition to living organisms. Organic matter acts as a buffer in aqueous solutions to maintain a neutral pH in the environment. The buffer acting component has been proposed to be relevant for neutralizing acid rain . [ 6 ]
Some organic matter not already in the soil comes from groundwater . When the groundwater saturates the soil or sediment around it, organic matter can freely move between the phases. Groundwater has its own sources of natural organic matter including:
Organisms decompose into organic matter, which is then transported and recycled. Not all biomass migrates, some is rather stationary, turning only over the course of millions of years. [ 8 ]
The organic matter in soil derives from plants, animals and microorganisms. In a forest, for example, leaf litter and woody materials fall to the forest floor. This is sometimes referred to as organic material. [ 9 ] When it decays to the point in which it is no longer recognizable, it is called soil organic matter. When the organic matter has broken down into a stable substance that resists further decomposition it is called humus . Thus soil organic matter comprises all of the organic matter in the soil exclusive of the material that has not decayed. [ 10 ]
An important property of soil organic matter is that it improves the capacity of a soil to hold water and nutrients, and allows their slow release, thereby improving the conditions for plant growth. Another advantage of humus is that it helps the soil to stick together which allows nematodes , or microscopic bacteria, to easily decay the nutrients in the soil. [ 11 ]
There are several ways to quickly increase the amount of humus. Combining compost, plant or animal materials/waste, or green manure with soil will increase the amount of humus in the soil.
These three materials supply nematodes and bacteria with nutrients for them to thrive and produce more humus, which will give plants enough nutrients to survive and grow. [ 11 ]
Soil organic matter is crucial to all ecology and to all agriculture , but it is especially emphasized in organic farming , where it is relied upon especially heavily.
The priming effect is characterized by intense changes in the natural process of soil organic matter (SOM) turnover, resulting from relatively moderate intervention with the soil. [ 12 ] The phenomenon is generally caused by either pulsed or continuous changes to inputs of fresh organic matter (FOM). [ 13 ] Priming effects usually result in an acceleration of mineralization due to a trigger such as the FOM inputs. The cause of this increase in decomposition has often been attributed to an increase in microbial activity resulting from higher energy and nutrient availability released from the FOM. After the input of FOM, specialized microorganisms are believed to grow quickly and only decompose this newly added organic matter. [ 14 ] The turnover rate of SOM in these areas is at least one order of magnitude higher than the bulk soil. [ 13 ]
Other soil treatments, besides organic matter inputs, which lead to this short-term change in turnover rates, include "input of mineral fertilizer, exudation of organic substances by roots, mere mechanical treatment of soil or its drying and rewetting." [ 12 ]
Priming effects can be either positive or negative depending on the reaction of the soil with the added substance. A positive priming effect results in the acceleration of mineralization while a negative priming effect results in immobilization, leading to N unavailability. Although most changes have been documented in C and N pools, the priming effect can also be found in phosphorus and sulfur, as well as other nutrients. [ 12 ]
Löhnis was the first to discover the priming effect phenomenon in 1926 through his studies of green manure decomposition and its effects on legume plants in soil. He noticed that when adding fresh organic residues to the soil, it resulted in intensified mineralization by the humus N. It was not until 1953, though, that the term priming effect was given by Bingeman in his paper titled, The effect of the addition of organic material on the decomposition of an organic soil . Several other terms had been used before priming effect was coined, including priming action, added nitrogen interaction (ANI), extra N and additional N. [ 12 ] Despite these early contributions, the concept of the priming effect was widely disregarded until about the 1980s-1990s. [ 13 ]
The priming effect has been found in many different studies and is regarded as a common occurrence, appearing in most plant soil systems. [ 15 ] However, the mechanisms which lead to the priming effect are more complex than originally thought, and still remain generally misunderstood. [ 14 ]
Although there is a lot of uncertainty surrounding the reason for the priming effect, a few undisputed facts have emerged from the collection of recent research:
Recent findings suggest that the same priming effect mechanisms acting in soil systems may also be present in aquatic environments, which suggests a need for broader considerations of this phenomenon in the future. [ 13 ] [ 16 ]
One suitable definition of organic matter is biological material in the process of decaying or decomposing , such as humus . A closer look at the biological material in the process of decaying reveals so-called organic compounds ( biological molecules ) in the process of breaking up (disintegrating).
The main processes by which soil molecules disintegrate are by bacterial or fungal enzymatic catalysis . If bacteria or fungi were not present on Earth, the process of decomposition would have proceeded much slower.
Various factors impact the decomposition of organic matter including its chemical properties and other environmental parameters. Metabolic capabilities of the microbial communities play a crucial role on decomposition since they are highly connected with the energy availability and processing. [ 17 ] In terrestrial ecosystems the energy status of soil organic matter has been shown to affect microbial substrate preferences. [ 18 ] Some organic matter pools may be energetically favorable for the microbial communities resulting in their fast oxidation and decomposition, in comparison with other pools where microbial degraders get less return from the energy they invest. By extension, soil microorganisms preferentially mineralize high-energy organic matter, avoiding decomposing less energetically dense organic matter. [ 19 ]
Measurements of organic matter generally measure only organic compounds or carbon , and so are only an approximation of the level of once living or decomposed matter. Some definitions of organic matter likewise only consider "organic matter" to refer to only the carbon content or organic compounds and do not consider the origins or decomposition of the matter. In this sense, not all organic compounds are created by living organisms, and living organisms do not only leave behind organic material. A clam's shell, for example, while biotic , does not contain much organic carbon , so it may not be considered organic matter in this sense. Conversely, urea is one of many organic compounds that can be synthesized without any biological activity.
Organic matter is heterogeneous and very complex. Generally, organic matter, in terms of weight, is: [ 6 ]
The molecular weights of these compounds can vary drastically, depending on if they repolymerize or not, from 200 to 20,000 amu. Up to one-third of the carbon present is in aromatic compounds in which the carbon atoms form usually six-membered rings. These rings are very stable due to resonance stabilization , so they are challenging to break down. The aromatic rings are also susceptible to electrophilic and nucleophilic attacks from other electron-donating or electron-accepting material, which explains the possible polymerization to create larger molecules of organic matter.
Some reactions occur with organic matter and other materials in the soil to create compounds never seen before. Unfortunately, it is challenging to characterize these because so little is known about natural organic matter in the first place. Research is currently being done to determine more about these new compounds and how many are being formed. [ 20 ]
Aquatic organic matter can be further divided into two components: (1) dissolved organic matter (DOM), measured as colored dissolved organic matter (CDOM) or dissolved organic carbon (DOC), and (2) particulate organic matter (POM). They are typically differentiated by that which can pass through a 0.45 micrometre filter (DOM), and that which cannot (POM).
Organic matter is important in water and wastewater treatment and recycling, natural aquatic ecosystems, aquaculture, and environmental rehabilitation. It is, therefore, important to have reliable methods of detection and characterisation, for both short- and long-term monitoring. Various analytical detection methods for organic matter have existed for up to decades to describe and characterise organic matter. These include, but are not limited to: total and dissolved organic carbon, mass spectrometry , nuclear magnetic resonance (NMR) spectroscopy , infrared (IR) spectroscopy , UV-Visible spectroscopy , and fluorescence spectroscopy . Each of these methods has its advantages and limitations.
The same capability of natural organic matter that helps with water retention in the soil creates problems for current water purification methods. In water, organic matter can still bind to metal ions and minerals. The purification process does not necessarily stop these bound molecules but does not cause harm to any humans, animals, or plants. However, because of the high reactivity of organic matter, by-products that do not contain nutrients can be made. These by-products can induce biofouling , which essentially clogs water filtration systems in water purification facilities, as the by-products are larger than membrane pore sizes. This clogging problem can be treated by chlorine disinfection ( chlorination ), which can break down residual material that clogs systems. However, chlorination can form disinfection by-products . [ 20 ]
Water with organic matter can be disinfected with ozone -initiated radical reactions. The ozone (three oxygens) has powerful oxidation characteristics. It can form hydroxyl radicals (OH) when it decomposes, which will react with the organic matter to shut down the problem of biofouling. [ 21 ]
The equation of "organic" with living organisms comes from the now-abandoned idea of vitalism , which attributed a special force to life that alone could create organic substances. This idea was first questioned after Friedrich Wöhler artificially synthesized urea in 1828.
Compare with: | https://en.wikipedia.org/wiki/Organic_matter |
Organic molecular cages represent a unique class of porous materials characterized by their discrete molecular nature and well-defined internal cavities, formed through covalent bonds between precisely designed organic building blocks. [ 1 ] [ 2 ] These molecular structures contain organized frameworks surrounding a central cavity, where organic components are precisely arranged to create functional internal spaces. Unlike extended networks such as metal-organic frameworks (MOFs) and covalent organic frameworks (COFs), [ 3 ] these cage compounds exist as distinct molecular entities, offering advantages in solution processability and structural precision. [ 4 ] [ 5 ]
The field of organic molecular cages emerged in the early 2000s, pioneered by the work of Cram , Lehn , and Pedersen , whose foundational research on host-guest chemistry and molecular recognition earned them the 1987 Nobel Prize. [ 6 ] The first discrete organic cages were reported by Tozawa and Cooper in 2009, introducing permanently porous organic cages with intrinsic cavities. [ 7 ] Since then, the field has grown significantly, driven by advances in synthetic chemistry and characterization techniques. [ 4 ] Early examples demonstrated basic molecular containment, but modern designs achieve sophisticated functions, including selective molecular recognition , catalysis , and stimuli-responsive behavior. The ability to control cavity size and chemical environment at the molecular level distinguishes these materials from traditional porous systems. [ 5 ] [ 8 ]
The architecture of organic molecular cages follows precise geometric principles that dictate their three-dimensional structure and functional properties. [ 9 ] Understanding these structural components is crucial for the design of cages with desired characteristics. The assembly process relies on the careful selection and combination of two primary components: nodes and linkers, whose geometric relationship determines the final cage structure. [ 9 ] [ 10 ]
Nodes are the cornerstones of cage architecture and are typically composed of rigid organic molecules with specific geometric arrangements. The geometry of these nodes directly influences the final cage structure and its internal cavity dimensions, making their selection crucial for targeted applications. [ 10 ] [ 11 ] Common node geometries include trigonal (three-directional), tetrahedral (four-directional), and octahedral (six-directional).
Complementing the nodes, linkers act as edges that connect these vertices to form the complete cage structure. [ 11 ] These linkers are typically linear or slightly bent organic molecules that contain reactive end groups essential for cage formation. The careful selection of linker length and flexibility helps control the size of the internal cavity and the overall cage stability. [ 12 ] The most employed linkers are dialdehyde units, diamine compounds, and diboronic acids.
This modular approach of combining specific nodes and linkers enables the rational design of cages with predetermined geometries and functionalities. The interplay between node geometry and linker characteristics determines not only the structural features but also the chemical environment of the resulting cavity, which is crucial for applications in molecular recognition and catalysis . [ 5 ]
The most extensively studied category of organic molecular cages are imine -based cages, formed through Schiff base condensation reactions. [ 13 ] [ 14 ] This reaction involves the condensation of aldehyde and amine groups to form imine bonds (C=N). The reversible nature of imine bond formation enables error correction during synthesis, leading to highly ordered structures, where multiple imine bonds connect the organic linkers and nodes to form a well-defined cage structure. This self-correcting mechanism makes imine-based cages particularly attractive for developing new cage architectures and has contributed to their widespread study in the field. [ 13 ] [ 15 ]
Boronic ester cages are another important class, characterized by their reversible boronic ester bonds and remarkable stability in non-aqueous conditions. [ 16 ] Their unique chemical nature allows for post-synthetic modification, enabling the fine-tuning of cage properties after initial synthesis. This adaptability makes them valuable for applications requiring specific chemical functionalities, particularly in conditions where imine bonds are unstable. [ 16 ] [ 17 ]
A third major category includes alkyne -based cages, which feature irreversible acetylene linkages that provide enhanced structural rigidity. [ 3 ] The strong covalent bonds in these structures result in high thermal stability , making them suitable for applications under demanding conditions. Their rigid framework ensures consistent cavity size and shape, making them particularly valuable for selective molecular recognition applications where structural integrity is crucial. [ 18 ]
Beyond chemical composition, cages are also classified based on their structural characteristics. [ 7 ] Shape-persistent cages maintain fixed conformations due to their rigid building blocks and strong covalent bonds , providing stable and predictable cavity environments. [ 19 ] In contrast, flexible cages exhibit dynamic structures that can adapt to guest molecules through conformational changes, [ 20 ] allowing for responsive host-guest interactions . This flexibility can be advantageous in applications requiring adaptive binding, such as selective molecular capture under varying conditions. Some systems even form hierarchical assemblies, creating cage-of-cage structures with complex internal architectures that can provide multiple distinct environments for guest molecules or cascade reactions. [ 21 ]
The classification of cages by cavity size provides practical guidance for applications and directly relates to their synthetic components and geometry. [ 4 ] Small cages (< 1 nm internal diameter) are typically constructed from compact building blocks and feature tight binding pockets suitable for gas molecule separation and storage, particularly for gases like CO 2 and CH 4 . [ 22 ] Medium cages (1-2 nm) represent the most versatile category, finding applications in selective molecular recognition and catalysis due to their ability to accommodate a wide range of organic molecules and maintain specific chemical environments. Large cages (> 2 nm), often synthesized using extended linear components or through hierarchical assembly, can accommodate bigger guest molecules such as proteins or large organic compounds, making them valuable for applications in drug delivery and enzyme encapsulation. The relationship between cage size and function has been extensively studied, revealing optimal size ranges for specific applications and guiding the design of new cage systems. [ 23 ]
Organic molecular cages exhibit diverse chemical properties determined by their structural components and functional groups. Imine-based cages show pH-responsive behavior, with the imine bonds being stable at neutral to basic conditions but susceptible to hydrolysis in acidic environments. This pH sensitivity can be exploited for controlled release applications. [ 24 ] Boronic ester cages demonstrate selective binding with diols and sugars, enabling specific molecular recognition . [ 16 ] Post-synthetic modification of cage structures through chemical reactions on peripheral groups allows tuning of cavity properties and introduction of new functionalities. [ 25 ]
Organic molecular cages exhibit permanent porosity in both solution and solid state, a unique characteristic that distinguishes them from conventional porous materials. [ 7 ] Typical surface areas range from 500 to 3000 m²/g, with pore volumes varying based on cage geometry. [ 21 ] [ 23 ] Most organic cages demonstrate high thermal stability up to 300 °C, though this varies with chemical composition [19,28]. [ 18 ]
Solubility represents another key physical property, with most cages showing good solubility in common organic solvents. [ 15 ] This solution processability enables their incorporation into membranes and composite materials. [ 22 ] The mechanical properties of cage crystals depend on packing arrangements and intermolecular interactions. [ 21 ] [ 7 ] While individual cage molecules are robust due to their covalent nature, crystal mechanical properties can range from brittle to flexible depending on intermolecular forces. Shape persistence varies with cage structure, affecting their stability and guest binding properties. [ 19 ] [ 20 ]
Dynamic covalent chemistry enables the formation of thermodynamically stable cage structures through reversible bond formation. [ 13 ] [ 14 ] The formation of imine bonds through the reaction between aldehyde and amine groups represents a fundamental example of this chemistry. [ 13 ] This reversible nature allows for continuous bond breaking and reforming during synthesis, enabling error correction and driving the system toward the most thermodynamically stable products. [ 16 ] [ 17 ] The dynamic nature of these reactions is particularly crucial in cage synthesis for several reasons. The ability to self-correct defects during formation ensures the production of highly ordered structures with minimal imperfections. [ 16 ] The formation of thermodynamically favored products leads to stable and well-defined cage architectures that can maintain their structural integrity under various conditions. [ 21 ] Furthermore, the reaction conditions can be carefully controlled for desired product distribution, allowing for selective synthesis of desired cage structures. [ 17 ] This dynamic character also enables template-directed synthesis, where specific molecular templates can guide the assembly process toward predetermined architectures. [ 16 ]
Both experimentally and computationally, various synthetic approaches have been developed to control cage formation and optimize yields. [ 23 ] The choice of synthetic strategy significantly influences the final cage structure, purity, and scalability of the synthesis. [ 21 ] Strategic synthetic approaches range from simple one-pot reactions to sophisticated template-directed methods, each offering distinct advantages.
One-pot reactions involve combining all reactants simultaneously under appropriate conditions. [ 13 ] This straightforward approach has proven successful for many simpler cage structures, particularly those formed through imine condensation. [ 13 ] [ 15 ] The success of one-pot synthesis often depends on the reversible nature of bond formation, allowing the system to self-correct and converge on the thermodynamically favored product. [ 16 ] [ 17 ] The reaction conditions, such as temperature, solvent choice, and concentration, play crucial roles in determining the outcome. For example, in imine cage synthesis, polar aprotic solvents like dichloromethane or chloroform are often preferred as they facilitate imine formation while allowing the removal of water byproduct. [ 17 ] Additionally, techniques such as slow addition of components or temperature control can be employed to enhance selectivity towards the desired cage product.
More complex cage architectures often require stepwise assembly strategies. [ 25 ] This approach involves the systematic construction of cage fragments followed by their controlled combination into the final structure. [ 21 ] While more time-consuming, stepwise assembly offers greater control over the final product and is particularly useful for asymmetric cage structures. The key advantage of this method lies in its ability to isolate and characterize intermediate products, ensuring the quality of each synthetic step. [ 25 ] For instance, in the synthesis of large cages, building blocks can be first combined into smaller sub-cages or fragments, which are then purified before final assembly. This strategy is particularly valuable when working with expensive or sophisticated building blocks, as it minimizes material waste and allows for optimization of each step.
Template-directed methods employ molecular templates to guide cage formation around a specific guest molecule. [ 16 ] This approach can enhance selectivity and yield while providing control over cage size and shape. [ 26 ] The template can either be removed post-synthesis or remain as a functional component of the final structure. [ 27 ] The choice of template is crucial and depends on several factors including size compatibility, chemical affinity, and reversible binding capability. Common templates include metal ions, organic molecules, and solvent molecules. The template effect can operate through various mechanisms, such as geometric pre-organization of building blocks, electronic effects , or hydrogen bonding interactions. [ 21 ] [ 26 ] After cage formation, template removal strategies must be carefully considered to maintain the integrity of the cage structure.
The structural analysis of organic molecular cages requires a comprehensive suite of analytical techniques. Both 2D chemical structures and 3D physical arrangements are crucial for understanding cage architecture. [ 28 ] [ 29 ] These complementary structural representations serve different purposes: the 2D chemical structure provides connectivity information while the 3D ball-and-stick model reveals the spatial arrangement and actual cavity formation. This multi-faceted visualization is essential for understanding the relationship between molecular design and functional properties. The characterization process usually involves multiple complementary methods to fully understand the cage structure and properties.
Single-crystal X-ray diffraction serves as the definitive method for determining atomic-level structure of organic cages. [ 28 ] This technique provides precise information about spatial arrangements of atoms, revealing exact bond lengths, angles, and the three-dimensional architecture of the cage framework. Critical structural features such as cavity dimensions, shape, and packing arrangements in the solid state can be determined with high accuracy. [ 28 ] [ 30 ]
NMR spectroscopy plays a crucial role in characterizing cage formation and dynamics in solution. [ 31 ] Solution-state NMR allows researchers to monitor reaction progress and confirm structural features through chemical shift analysis and coupling patterns. Two-dimensional NMR techniques help establish connectivity patterns and verify the successful formation of key structural features. [ 31 ] This solution-phase characterization complements solid-state analysis by providing insight into the cage's behavior under practical application conditions. [ 15 ]
Mass spectrometry confirms molecular mass and provides insights into cage assembly processes. [ 25 ] High-resolution mass spectrometry precisely determines the molecular weight of completed cages and can identify reaction intermediates during synthesis. The technique is particularly valuable for large cage structures where multiple charge states may be observed. Tandem mass spectrometry reveals fragmentation patterns that confirm structural assignments and can provide information about the strength of various bonds within the cage framework. Modern ionization techniques enable the study of host-guest complexes, offering insights into molecular recognition properties. [ 15 ]
Gas sorption analysis provides crucial information about the porosity and surface properties of organic molecular cages. [ 22 ] [ 30 ] Nitrogen adsorption-desorption isotherms determine surface area, pore volume, and pore size distribution. BET surface area measurements typically reveal values ranging from 500 to 3000 m²/g, depending on cage structure. [ 12 ] The analysis of adsorption isotherms also provides insights into pore accessibility and connectivity. Carbon dioxide and hydrogen adsorption measurements evaluate potential applications in gas storage and separation. [ 22 ] [ 7 ]
Thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC) assess the thermal stability and phase behavior of cage compounds. [ 30 ] [ 32 ] TGA reveals decomposition temperatures and solvent loss patterns, while DSC identifies phase transitions and structural changes. These techniques are particularly important for evaluating cage stability under application conditions. Many organic cages show remarkable thermal stability up to 300 °C, though this varies significantly with chemical composition. [ 4 ] [ 18 ]
The well-defined pore structure of organic cages enables selective molecular separations. [ 22 ] [ 33 ] Molecular cages can discriminate between molecules based on size, shape, and chemical affinity . [ 22 ] [ 34 ] Gas separation represents a major application, where cages are incorporated into mixed-matrix membranes for selective gas transport. [ 22 ] Studies have demonstrated effective separation of CO 2 /N 2 , CO 2 /CH 4 , and other industrially relevant gas mixtures. [ 33 ] [ 35 ] The uniform pore size and chemical environment ensure consistent separation performance. [ 22 ] In liquid-phase separations, organic cages show promise for challenging molecular separations. [ 20 ] [ 34 ] Their solution processability enables incorporation into chromatographic stationary phases. [ 20 ] The intrinsic chirality of some cage structures allows for enantioselective separations of racemic mixtures , achieving high separation factors for pharmaceutical and fine chemical applications. [ 34 ]
Organic cages function as molecular containers through specific host-guest interactions . [ 15 ] [ 30 ] The defined cavity can encapsulate guest molecules of appropriate size and chemical compatibility. [ 35 ] The binding process often induces measurable changes in cage properties, enabling their use as molecular sensors. [ 15 ] Selective binding of specific analytes can trigger optical, electronic, or structural responses that provide detection signals. [ 36 ] These features enable applications in environmental monitoring and chemical detection. The host-guest chemistry extends to selective capture of environmental pollutants and valuable chemicals. [ 22 ] [ 37 ] The tunable cavity size and surface chemistry allow targeting specific molecules, while some cages demonstrate stimuli-responsive guest release for controlled delivery applications. [ 22 ]
Organic cages act as nano reactors for catalytic transformations. [ 38 ] The confined environment enhances reaction rates and selectivity through concentration effects in the cavity, specific orientation of reactants, stabilization of transition states , and control over product distribution. [ 38 ] [ 39 ] The well-defined cavity creates a unique microenvironment that can accelerate reactions and influence product selectivity. Asymmetric catalysis benefits from chiral cage environments, enabling stereoselective transformations. [ 38 ] The catalytic activity can be tuned through modification of cage structure and functionalization of the cavity interior. Integration of catalytic sites within the cage framework allows for size-selective catalysis, where only appropriately sized substrates can access the active sites. [ 39 ]
Recent developments have expanded the applications of organic molecular cages into new areas. Energy storage and conversion applications utilize cages as components in battery electrolytes and fuel cells . [ 40 ] In environmental applications, cages demonstrate the potential for carbon capture and water purification through selective molecular binding . [ 22 ]
Biological applications represent another growing field. [ 41 ] The biocompatibility of certain cage structures enables their use in drug delivery systems. [ 41 ] Some cages can encapsulate and protect therapeutic molecules, releasing them under specific physiological conditions. [ 7 ] Additionally, enzyme-mimetic cages catalyze biological transformations in artificial systems. [ 41 ]
Smart materials incorporating organic cages show stimuli-responsive behavior. [ 32 ] These materials change properties in response to external stimuli such as light, temperature, or chemical signals. Applications include switchable membranes and responsive sensing systems. [ 32 ] [ 36 ]
Organic molecular cages represent a versatile class of porous materials that bridge the gap between molecular chemistry and materials science. Their discrete molecular nature combined with well-defined cavities enables applications ranging from molecular separations to catalysis and sensing. The field has progressed significantly since the first report of permanently porous organic cages, with advances in synthetic methods, characterization techniques, and applications. [ 4 ] [ 5 ] [ 7 ]
Future directions in this field include the development of more complex cage architectures with multiple functional sites, expansion of stimuli-responsive systems, and integration of cages into devices and practical applications. [ 40 ] [ 42 ] Computational approaches are increasingly important for predicting structure-property relationships and designing cages with targeted functions. [ 9 ] The combination of experimental and theoretical methods promises to accelerate the discovery of new cage systems with enhanced properties. [ 23 ]
Challenges remain in scaling up the synthesis of organic cages for industrial applications, improving thermal and chemical stability for demanding environments, and developing predictive models for guest selectivity. [ 5 ] [ 42 ] | https://en.wikipedia.org/wiki/Organic_molecular_cages |
The Chinese Chemical Society [ a ] (CCS; simplified Chinese : 中国化学会 ; traditional Chinese : 中國化學會 ) lays out a set of rules based on those given by the International Union of Pure and Applied Chemistry (IUPAC) for the purpose of systematic organic nomenclature in Chinese. The chemical names derived from these rules are meant to correspond with the English IUPAC name in a manner that is close to one-to-one, while being adapted to and taking advantage of the logographic nature of the Chinese written language. A standard set of characters invented during the 20th century, along with characters for the chemical elements and characters corresponding to standard chemical prefixes and suffixes, are used for this purpose.
The majority of the Chinese characters used for this purpose are phonosemantic compounds, with part of the character giving a general semantic category and the other part providing a pronunciation, usually based on the international (European) pronunciation. There are four common radicals (the part of the character that gives the semantic category) for these characters:
Additionally, the mouth radical (口, kǒu ) is affixed to characters that are used for their sound only. This occurs often in the transliteration of the names of heterocyclic compounds , (e.g., 吡啶, "bǐdìng", pyridine). These characters are also used for the transliteration of non-chemical terms from foreign languages.
Below is a table, in pinyin order, of the Chinese names of major organic compounds, radicals, and functional groups. Characters given are in traditional Chinese , followed by simplified Chinese where possible. Since the characters are modern creations, the traditional Chinese characters are analogous (with traditional components in place of simplified components). The Mandarin pronunciation of each character, as said in mainland China, is in pronunciation column. Any Taiwanese pronunciations that differ from the mainland Chinese pronunciations are put in the notes. Other usages of characters are etymologically unrelated to the character's meaning as names for organic compounds, radicals, and functional groups unless otherwise stated. This list is not exhaustive, although many of the other characters used for this purpose can only be found in specialist dictionaries.
Warning: Verbally, 酮 ('ketone') and 铜 ('copper') are both pronounced tóng and are indistinguishable. In modern times, new chemical terminology is chosen to avoid conflicts like this.
In the CCS system, carbon chain lengths are denoted by celestial stems (甲 jiǎ , 乙 yǐ , 丙 bǐng , 丁 dīng , 戊 wù , 己 jǐ , 庚 gēng , 辛 xīn , 壬 rén , 癸 guǐ ), characters used since the Shang dynasty (16th–11th centuries BCE) for naming days (and later, to name years). For example, hexane is 己烷 jǐwán , since 己 jǐ is the sixth celestial stem. Longer carbon chains are specified by number followed by '碳' tàn 'carbon'. For example, 1-hexadecene is 1-十六碳烯 (read as [1, yī ] [-, wèi ] [十六, shíliù , '16'] [碳, tàn ] [烯, xī ]), where the hyphen is read as 位 ( wèi , 'position').
For a more complex example, consider 3-buten-1-ol. Its Chinese name is 3-丁烯-1-醇 (read as [3, sān ] [-, wèi ] [丁, dīng ] [烯, xī ] [1, yī ] [-, wèi ] [醇, chún ]).
The descriptors for degree of substitution, primary , secondary , tertiary , and quaternary , are translated as 伯 ( bó ), 仲 ( zhòng ), 叔 ( shū ), 季 ( jì ), which refer to the first, second, third, and fourth male siblings in a family. For instance, tert- butyllithium is translated as 叔丁基锂 ([叔, shū , ' tert '], [丁, dīng , 'but-'], [基, jī , 'yl'], [锂, lǐ , 'lithium']). Other commonly used isomeric descriptors normal- , iso-, and neo- are translated as 正 ( zhèng , 'proper'), 异 ( yì , 'different'), and 新 ( xīn , 'new'), respectively.
The numerical prefix bis- is translated as 双 ( shuāng , 'double'), while larger multiplicities are simply given by the Chinese word for the number (e.g., 四 ( sì , 'four') for tetrakis-). For example, tetrakis(triphenylphosphine)palladium is rendered 四(三苯基膦)钯, in which 三苯基膦 is triphenylphosphine and 钯 is palladium. The prefix bi- (for joining of ring systems) is translated as 联 (lián, 'join', 'couple'), as in 联苯 for biphenyl.
The stereochemical descriptors cis- and trans- are translated as 顺 ( shùn , 'along') and 反 ( fǎn , 'against'). The relational prefixes ortho- , meta- , and para- are translated as 邻 ( lín , 'neighboring'), 间 ( jiàn , 'between'), and 对 ( duì , 'opposing'), respectively.
The structural modification descriptors cyclo- , nor- and homo- are translated as 环 ( huán , 'ring'), 降 ( jiàng , 'lowered'), and 高 ( gāo , 'high'). For example, norbornene is translated as 降冰片烯, in which the trivial name (冰片) for bornyl [literally, camphor] is used. When substitutive nomenclature is used for naming heterocycles, the suffix 杂 (zá, 'mixed') is used in the same way as '-a' in English (as in aza , thia , oxa , etc.). As an example, DABCO (1,4-diazabicylo[2.2.2]octane) is named 1,4-二氮杂二环［2.2.2］辛烷.
The common unsaturated groups allyl and propargyl are translated as 烯丙(基) ( xībǐng(jī) , 'alkene-prop-(yl)') and 炔丙(基) ( qūebǐng(jī) , 'alkyne-prop-(yl)'). Thus, using 高 for homo- and 烯丙 for allyl, 3-buten-1-ol is also called 高烯丙醇 (i.e., homoallyl alcohol) in Chinese via semisystematic nomenclature.
The Chinese Wikipedia page may be consulted for further details.
Although not substances, the thermochemical concepts entropy and enthalpy were assigned Chinese characters based on similar considerations. The 'fire' radical, 火, is used as the semantic category. The character for entropy, 熵 (pinyin: shāng), is derived from 商 (pinyin: shāng), which means 'quotient' in this context. This recognizes the Clausius equation for the differential change in entropy as the differential heat absorbed divided by the temperature: dS = dQ / T . The character for enthalpy, 焓 (pinyin: hán), is derived from 含 (pinyin: hán), which means 'to contain.' This character phonetically approximates the first syllable of ' en thalpy', and recognizes the definition of enthalpy as heat content. | https://en.wikipedia.org/wiki/Organic_nomenclature_in_Chinese |
In organic chemistry , organic peroxides are organic compounds containing the peroxide functional group ( R−O−O−R′ ). If the R′ is hydrogen , the compounds are called hydroperoxides , which are discussed in that article. The O−O bond of peroxides easily breaks, producing free radicals of the form RO • (the dot represents an unpaired electron ). Thus, organic peroxides are useful as initiators for some types of polymerization , such as the acrylic , unsaturated polyester, and vinyl ester resins used in glass-reinforced plastics . MEKP and benzoyl peroxide are commonly used for this purpose. However, the same property also means that organic peroxides can explosively combust. Organic peroxides, like their inorganic counterparts, are often powerful bleaching agents. [ 1 ]
Organic peroxides are classified (i) by the presence or absence of a hydroxyl ( −OH ) terminus and (ii) by the presence of alkyl vs acyl substituents. [ 2 ]
One gap in the classes of organic peroxides is diphenyl peroxide. Quantum chemical calculations predict that it undergoes a nearly barrierless reaction akin to the benzidine rearrangement . [ 3 ]
The O−O bond length in peroxides is about 1.45 Å , and the R−O−O angles (R = H, C) are about 110° (water-like). Characteristically, the C−O−O−R (R = H, C) dihedral angles are about 120°. The O−O bond is relatively weak, with a bond dissociation energy of 45–50 kcal/mol (190–210 kJ/mol ), less than half the strengths of C−C, C−H, and C−O bonds. [ 4 ] [ 5 ]
Peroxides play important roles in biology. Hundreds of peroxides and hydroperoxides are known, being derived from fatty acids, steroids, and terpenes. [ 6 ] The prostaglandins are biosynthesized by initial formation of a bicyclic peroxide ("endoperoxide") derived from arachidonic acid . [ 7 ]
Many aspects of biodegradation or aging are attributed to the formation and decay of peroxides formed from oxygen in air. Countering these effects, an array of biological and artificial antioxidants destroy peroxides.
In fireflies , oxidation of luciferins , which is catalyzed by luciferases , yields a peroxy compound 1,2-dioxetane . The dioxetane is unstable and decays spontaneously to carbon dioxide and excited ketones , which release excess energy by emitting light ( bioluminescence ). [ 8 ]
Many peroxides are used as a radical initiators , e.g., to enable polymerization of acrylates. Industrial resins based on acrylic and/or methacrylic acid esters are invariably produced by radical polymerization with organic peroxides at elevated temperatures. [ 9 ] The polymerization rate is adjusted by suitable choice of temperature and type of peroxide. [ 10 ]
Methyl ethyl ketone peroxide , benzoyl peroxide and to a smaller degree acetone peroxide are used as initiators for radical polymerization of some thermosets , e.g. unsaturated polyester and vinyl ester resins, often encountered when making fiberglass or carbon fiber composites (CFRP), with examples including boats, RV units, bath tubs, pools, sporting equipment, wind turbine blades, and a variety of industrial applications.
Benzoyl peroxide , peroxyesters / peroxyketals , and alkylperoxy monocarbonates are used in production of polystyrene , expanded polystyrene , and High Impact Polystyrene , and benzoyl peroxide is utilized for many acrylate based adhesive applications.
Thermoplastic production techniques for many industrial polymerization applications include processes which are carried out in bulk, solution, or suspension type batches. Relevant polymers include: polyvinyl chloride (PVC), low-density polyethylene (LDPE), high-density polyethylene (HDPE), polymethyl methacrylate (PMMA), Polystyrene , and Polycarbonates .
Benzoyl peroxide and hydrogen peroxide are used as bleaching and "maturing" agents for treating flour to make its grain release gluten more easily; the alternative is letting the flour slowly oxidize by air, which is too slow for the industrialized era. Benzoyl peroxide is an effective topical medication for treating most forms of acne .
Dialkyl peroxides, e.g., dicumyl peroxide , are synthesized by addition of hydrogen peroxide to alkenes or by O-alkylation of hydroperoxides.
Diacyl peroxides are typically prepared by treating hydrogen peroxide with acid chlorides or acid anhydrides in the presence of base: [ 1 ]
The reaction competes with hydrolysis of the acylating agent but the hydroperoxide anion is a superior nucleophile relative to hydroxide. Unsymmetrical diacyl peroxides can be produced by treating acyl chlorides with the peroxy acid.
Peresters , an example being tert -Butyl peroxybenzoate , are produced by treating acid anhydrides or acid chlorides with hydroperoxides.
Cyclic peroxides can be obtained by cycloaddition of singlet oxygen (generated by UV radiation) to dienes. An important example is rubrene . Six-membered cyclic peroxides are called endo peroxides. [ 11 ] The four-membered dioxetanes can be obtained by 2+2 cycloaddition of oxygen to alkenes . [ 12 ] [ 13 ]
The hazards associated with storage of ethers in air is attributed to the formation of hydroperoxides via the direct albeit slow reaction of triplet oxygen with C-H bonds .
Organic peroxides are widely used to initiate polymerization of olefins , e.g. the formation of polyethylene . A key step is homolysis :
The tendency to homolyze is also exploited to modify polymers by grafting or visbreaking , or cross-link polymers to create a thermoset . When used for these purposes, the peroxide is highly diluted, so the heat generated by the exothermic decomposition is safely absorbed by the surrounding medium (e.g. polymer compound or emulsion ).
Especially when in concentrated form, organic peroxides can decompose by self-oxidation, since organic peroxides contain both an oxidizer (the O-O bond) and fuel (C-H and C-C bonds). A "self-accelerating decomposition" occurs when the rate of peroxide decomposition generates heat at a faster rate than it can be dissipated to the environment. Temperature is the main factor in the rate of decomposition. The lowest temperature at which a packaged organic peroxide will undergo a self-accelerating decomposition within a week is defined as the self-accelerating decomposition temperature (SADT). A large fire at the Arkema Chemical Plant in Crosby, Texas (USA) in 2017 was caused by the decomposition of various organic peroxides following power failure and subsequent loss of cooling systems. [ 14 ] This occurred due to extreme flooding from Hurricane Harvey , which destroyed main and back-up power generators at the site. [ 14 ]
Hydroperoxides are intermediates or reagents in major commercial processes. In the cumene process , acetone and phenol are produced by decomposition of cumene hydroperoxide (Me = methyl):
Anthrahydroquinone reacts spontaneously with oxygen to form anthraquinone and hydrogen peroxide, possibly through some organic peroxide intermediate. After extracting the hydrogen peroxide the anthraquinone is catalytically reduced to anthrahydroquinone and reused in the process. There are other hydroquinones reacting in a similar fashion.
Organoperoxides can be reduced to alcohols with lithium aluminium hydride , as described in this idealized equation:
The phosphite esters and tertiary phosphines also effect reduction:
Cleavage to ketones and alcohols occurs in the base-catalyzed Kornblum–DeLaMare rearrangement , which involves the breaking of bonds within peroxides to form these products.
Some peroxides are drugs , whose action is based on the formation of radicals at desired locations in the organism. For example, artemisinin and its derivatives, such as artesunate , possess the most rapid action of all current drugs against falciparum malaria . [ 15 ] Artesunate is also efficient in reducing egg production in Schistosoma haematobium infection. [ 16 ]
tert-Butyl hydroperoxide is used for epoxidation and hydroxylation reagents in conjunction with metal catalysts. [ 17 ]
Several analytical methods are used for qualitative and quantitative determination of peroxides. [ 18 ] A simple qualitative detection of peroxides is carried out with the iodine-starch reaction . [ 19 ] Here peroxides, hydroperoxides or peracids oxidize the added potassium iodide into iodine , which reacts with starch producing a deep-blue color. Commercial paper indicators using this reaction are available. This method is also suitable for quantitative evaluation, but it can not distinguish between different types of peroxide compounds. Discoloration of various indigo dyes in presence of peroxides is used instead for this purpose. [ 20 ] For example, the loss of blue color in leuco- methylene blue is selective for hydrogen peroxide. [ 21 ]
Quantitative analysis of hydroperoxides can be performed using potentiometric titration with lithium aluminium hydride . [ 22 ] Another way to evaluate the content of peracids and peroxides is the volumetric titration with alkoxides such as sodium ethoxide . [ 23 ]
Each peroxy group is considered to contain one active oxygen atom. The concept of active oxygen content is useful for comparing the relative concentration of peroxy groups in formulations, which is related to the energy content. In general, energy content increases with active oxygen content, and thus the higher the molecular weight of the organic groups, the lower the energy content and, usually, the lower the hazard.
The term active oxygen is used to specify the amount of peroxide present in any organic peroxide formulation. One of the oxygen atoms in each peroxide group is considered "active". The theoretical amount of active oxygen can be described by the following equation: [ 24 ]
where p is the number of peroxide groups in the molecule, and m is the molecular mass of the pure peroxide.
Organic peroxides are often sold as formulations that include one or more phlegmatizing agents . That is, for safety sake or performance benefits the properties of an organic peroxide formulation are commonly modified by the use of additives to phlegmatize (desensitize), stabilize, or otherwise enhance the organic peroxide for commercial use. Commercial formulations occasionally consist of mixtures of organic peroxides, which may or may not be phlegmatized.
Peroxides are also strong oxidizers and easily react with skin, cotton and wood pulp. [ 25 ] For safety reasons, peroxidic compounds are stored in a cool, opaque container, as heating and illumination accelerate their chemical reactions . Small amounts of peroxides, which emerge from storage or reaction vessels are neutralized using reducing agents such as iron(II) sulfate . Safety measures in industrial plants producing large amounts of peroxides include the following:
1) The equipment is located within reinforced concrete structures with foil windows, which would relieve pressure and not shatter in case of explosion.
2) The products are bottled in small containers and are moved to a cold place promptly after the synthesis.
3) The containers are made of non-reactive materials such as stainless steel, some aluminium alloys or dark glass. [ 26 ]
For safe handling of concentrated organic peroxides, an important parameter is temperature of the sample, which should be maintained below the self accelerating decomposition temperature of the compound. [ 27 ]
The shipping of organic peroxides is restricted. The US Department of Transportation lists organic peroxide shipping restrictions and forbidden materials in 49 CFR 172.101 Hazardous Materials Table based on the concentration and physical state of the material: | https://en.wikipedia.org/wiki/Organic_peroxides |
Organic photochemistry encompasses organic reactions that are induced by the action of light. [ 1 ] [ 2 ] The absorption of ultraviolet light by organic molecules often leads to reactions. In the earliest days, sunlight was employed, while in more modern times ultraviolet lamps are employed. Organic photochemistry has proven to be a very useful synthetic tool. Complex organic products can be obtained simply.
Early examples were often uncovered by the observation of precipitates or color changes from samples that were exposed to sunlights. The first reported case was by Ciamician that sunlight converted santonin to a yellow photoproduct: [ 3 ]
An early example of a precipitate was the photodimerization of anthracene , characterized by Yulii Fedorovich Fritzsche and confirmed by Elbs. [ 4 ] Similar observations focused on the dimerization of cinnamic acid to truxillic acid . Many photodimers are now recognized, e.g. pyrimidine dimer , thiophosgene , diamantane .
Another example was uncovered by Egbert Havinga in 1956. [ 5 ] The curious result was activation on photolysis by a meta nitro group in contrast to the usual activation by ortho and para groups.
Organic photochemistry advanced with the development of the Woodward-Hoffmann rules . [ 6 ] [ 7 ] Illustrative, these rules help rationalize the photochemically driven electrocyclic ring-closure of hexa-2,4-diene, which proceeds in a disrotatory fashion.
Organic reactions that obey these rules are said to be symmetry allowed. Reactions that take the opposite course are symmetry forbidden and require substantially more energy to take place if they take place at all.
Organic photochemical reactions are explained in the context of the relevant excited states . [ 8 ] [ 9 ]
Parallel to the structural studies described above, the role of spin multiplicity – singlet vs triplet – on reactivity was evaluated. The importance of triplet excited species was emphasized. Triplets tend to be longer-lived than singlets and of lower energy than the singlet of the same configuration. Triplets may arise from (A) conversion of the initially formed singlets or by (B) interaction with a higher energy triplet (sensitization).
It is possible to quench triplet reactions. [ 10 ]
Common organic photochemical reactions include: Norrish Type I , the Norrish Type II , the racemization of optically active biphenyls, the type A cyclohexadienone rearrangement, the type B cyclohexenone rearrangement, the di- π -methane rearrangement , the type B bicyclo[3.1.0]hexanone rearrangement to phenols, photochemical electrocyclic processes, the rearrangement of epoxyketones to beta-diketones, ring opening of cyclopropyl ketones, heterolysis of 3,5-dimethoxylbenzylic derivatives, and photochemical cyclizations of dienes.
Reactants of the photoreactions can be both gaseous and liquids. [ 11 ] In general, it is necessary to bring the reactants close to the light source in order to obtain the highest possible luminous efficacy . For this purpose, the reaction mixture can be irradiated either directly or in a flow-through side arm of a reactor with a suitable light source. [ 12 ]
A disadvantage of photochemical processes is the low efficiency of the conversion of electrical energy in the radiation energy of the required wavelength . In addition to the radiation, light sources generate plenty of heat, which in turn requires cooling energy. In addition, most light sources emit polychromatic light, even though only monochromatic light is needed. [ 13 ] A high quantum yield , however, compensates for these disadvantages.
Working at low temperatures is advantageous since side reactions are avoided (as the selectivity is increased) and the yield is increased (since gaseous reactants are driven out less from the solvent).
The starting materials can sometimes be cooled before the reaction to such an extent that the reaction heat is absorbed without further cooling of the mixture. In the case of gaseous or low-boiling starting materials, work under overpressure is necessary. Due to the large number of possible raw materials, a large number of processes have been described. [ 14 ] [ 15 ] Large scale reactions are usually carried out in a stirred tank reactor , a bubble column reactor or a tube reactor, followed by further processing depending on the target product. [ 16 ] In case of a stirred tank reactor, the lamp (generally shaped as an elongated cylinder) is provided with a cooling jacket and placed in the reaction solution. Tube reactors are made from quartz or glass tubes, which are irradiated from the outside. Using a stirred tank reactor has the advantage that no light is lost to the environment. However, the intensity of light drops rapidly with the distance to the light source due to adsorption by the reactants. [ 12 ]
The influence of the radiation on the reaction rate can often be represented by a power law based on the quantum flow density, i.e. the mole light quantum (previously measured in the unit einstein ) per area and time. One objective in the design of reactors is therefore to determine the economically most favorable dimensioning with regard to an optimization of the quantum current density. [ 17 ]
Olefins dimerize upon UV-irradiation. [ 18 ]
Quite parallel to the santonin to lumisantonin example is the rearrangement of 4,4-diphenylcyclohexadienone [ 9 ] Here the n-pi* triplet excited state undergoes the same beta-beta bonding. This is followed by intersystem crossing (i.e. ISC) to form the singlet ground state which is seen to be a zwitterion . The final step is the rearrangement to the bicyclic photoproduct. The reaction is termed the type A cyclohexadienone rearrangement.
To provide further evidence on the mechanism of the dienone in which there is bonding between the two double bonds,
the case of 4,4-diphenylcyclohexenone is presented here. It is seen that the rearrangement is quite different; thus two double bonds are required for a type A rearrangement. With one double bond one of the phenyl groups, originally at C-4, has migrated to C-3 (i.e. the beta carbon). [ 19 ]
When one of the aryl groups has a para-cyano or para-methoxy group, that substituted aryl group migrates in preference. [ 20 ] Inspection of the alternative phenonium-type species, in which an aryl group has begun to migrate to the beta-carbon, reveals the greater electron delocalization with a substituent para on the migrating aryl group and thus a more stabilized pathway.
Still another type of photochemical reaction is the di- π -methane rearrangement . [ 21 ] Two further early examples were the rearrangement of 1,1,5,5-tetraphenyl-3,3-dimethyl-1,4-pentadiene (the "Mariano" molecule) [ 22 ] and the rearrangement of barrelene to semibullvalene . [ 23 ] We note that, in contrast to the cyclohexadienone reactions which used n- π * excited states, the di- π -methane rearrangements utilize π - π * excited states.
In photoredox catalysis , the photon is absorbed by a sensitizer (antenna molecule or ion) which then effects redox reactions on the organic substrate. A common sensitizer is ruthenium(II) tris(bipyridine) . Illustrative of photoredox catalysis are some aminotrifluoromethylation reactions. [ 24 ]
Photochlorination is one of the largest implementations of photochemistry to organic synthesis. The photon is however not absorbed by the organic compound, but by chlorine . Photolysis of Cl 2 gives chlorine atoms, which abstract H atoms from hydrocarbons, leading to chlorination. | https://en.wikipedia.org/wiki/Organic_photochemistry |
Organic reactions are chemical reactions involving organic compounds . [ 1 ] [ 2 ] [ 3 ] The basic organic chemistry reaction types are addition reactions , elimination reactions , substitution reactions , pericyclic reactions , rearrangement reactions , photochemical reactions and redox reactions . In organic synthesis , organic reactions are used in the construction of new organic molecules. The production of many man-made chemicals such as drugs, plastics , food additives , fabrics depend on organic reactions.
The oldest organic reactions are combustion of organic fuels and saponification of fats to make soap. Modern organic chemistry starts with the Wöhler synthesis in 1828. In the history of the Nobel Prize in Chemistry awards have been given for the invention of specific organic reactions such as the Grignard reaction in 1912, the Diels–Alder reaction in 1950, the Wittig reaction in 1979 and olefin metathesis in 2005.
Organic chemistry has a strong tradition of naming a specific reaction to its inventor or inventors and a long list of so-called named reactions exists, conservatively estimated at 1000. A very old named reaction is the Claisen rearrangement (1912) and a recent named reaction is the Bingel reaction (1993). When the named reaction is difficult to pronounce or very long as in the Corey–House–Posner–Whitesides reaction it helps to use the abbreviation as in the CBS reduction . The number of reactions hinting at the actual process taking place is much smaller, for example the ene reaction or aldol reaction .
Another approach to organic reactions is by type of organic reagent , many of them inorganic , required in a specific transformation. The major types are oxidizing agents such as osmium tetroxide , reducing agents such as lithium aluminium hydride , bases such as lithium diisopropylamide and acids such as sulfuric acid .
Finally, reactions are also classified by mechanistic class. Commonly these classes are (1) polar, (2) radical, and (3) pericyclic. Polar reactions are characterized by the movement of electron pairs from a well-defined source (a nucleophilic bond or lone pair) to a well-defined sink (an electrophilic center with a low-lying antibonding orbital). Participating atoms undergo changes in charge, both in the formal sense as well as in terms of the actual electron density. The vast majority of organic reactions fall under this category. Radical reactions are characterized by species with unpaired electrons ( radicals ) and the movement of single electrons. Radical reactions are further divided into chain and nonchain processes. Finally, pericyclic reactions involve the redistribution of chemical bonds along a cyclic transition state . Although electron pairs are formally involved, they move around in a cycle without a true source or sink. These reactions require the continuous overlap of participating orbitals and are governed by orbital symmetry considerations . Of course, some chemical processes may involve steps from two (or even all three) of these categories, so this classification scheme is not necessarily straightforward or clear in all cases. Beyond these classes, transition-metal mediated reactions are often considered to form a fourth category of reactions, although this category encompasses a broad range of elementary organometallic processes, many of which have little in common and very specific.
Factors governing organic reactions are essentially the same as that of any chemical reaction . Factors specific to organic reactions are those that determine the stability of reactants and products such as conjugation , hyperconjugation and aromaticity and the presence and stability of reactive intermediates such as free radicals , carbocations and carbanions .
An organic compound may consist of many isomers . Selectivity in terms of regioselectivity , diastereoselectivity and enantioselectivity is therefore an important criterion for many organic reactions. The stereochemistry of pericyclic reactions is governed by the Woodward–Hoffmann rules and that of many elimination reactions by Zaitsev's rule .
Organic reactions are important in the production of pharmaceuticals . In a 2006 review, [ 4 ] it was estimated that 20% of chemical conversions involved alkylations on nitrogen and oxygen atoms, another 20% involved placement and removal of protective groups , 11% involved formation of new carbon–carbon bond and 10% involved functional group interconversions .
There is no limit to the number of possible organic reactions and mechanisms. [ 5 ] [ 6 ] However, certain general patterns are observed that can be used to describe many common or useful reactions. Each reaction has a stepwise reaction mechanism that explains how it happens, although this detailed description of steps is not always clear from a list of reactants alone. Organic reactions can be organized into several basic types. Some reactions fit into more than one category. For example, some substitution reactions follow an addition-elimination pathway. This overview isn't intended to include every single organic reaction. Rather, it is intended to cover the basic reactions.
In condensation reactions a small molecule, usually water, is split off when two reactants combine in a chemical reaction. The opposite reaction, when water is consumed in a reaction, is called hydrolysis . Many polymerization reactions are derived from organic reactions. They are divided into addition polymerizations and step-growth polymerizations .
In general the stepwise progression of reaction mechanisms can be represented using arrow pushing techniques in which curved arrows are used to track the movement of electrons as starting materials transition to intermediates and products.
Organic reactions can be categorized based on the type of functional group involved in the reaction as a reactant and the functional group that is formed as a result of this reaction. For example, in the Fries rearrangement the reactant is an ester and the reaction product an alcohol .
An overview of functional groups with their preparation and reactivity is presented below:
In heterocyclic chemistry , organic reactions are classified by the type of heterocycle formed with respect to ring-size and type of heteroatom. See for instance the chemistry of indoles . Reactions are also categorized by the change in the carbon framework. Examples are ring expansion and ring contraction , homologation reactions , polymerization reactions , insertion reactions , ring-opening reactions and ring-closing reactions .
Organic reactions can also be classified by the type of bond to carbon with respect to the element involved. More reactions are found in organosilicon chemistry , organosulfur chemistry , organophosphorus chemistry and organofluorine chemistry . With the introduction of carbon-metal bonds the field crosses over to organometallic chemistry . | https://en.wikipedia.org/wiki/Organic_reaction |
Organic reductions or organic oxidations or organic redox reactions are redox reactions that take place with organic compounds . In organic chemistry oxidations and reductions are different from ordinary redox reactions, because many reactions carry the name but do not actually involve electron transfer . [ 1 ] Instead the relevant criterion for organic oxidation is gain of oxygen and/or loss of hydrogen. [ 2 ] Simple functional groups can be arranged in order of increasing oxidation state . The oxidation numbers are only an approximation: [ 1 ]
When methane is oxidized to carbon dioxide its oxidation number changes from −4 to +4. Classical reductions include alkene reduction to alkanes and classical oxidations include oxidation of alcohols to aldehydes . In oxidations electrons are removed and the electron density of a molecule is reduced. In reductions electron density increases when electrons are added to the molecule. This terminology is always centered on the organic compound. For example, it is usual to refer to the reduction of a ketone by lithium aluminium hydride , but not to the oxidation of lithium aluminium hydride by a ketone. Many oxidations involve removal of hydrogen atoms from the organic molecule, and reduction adds hydrogens to an organic molecule.
Many reactions classified as reductions also appear in other classes. For instance, conversion of the ketone to an alcohol by lithium aluminium hydride can be considered a reduction but the hydride is also a good nucleophile in nucleophilic substitution . Many redox reactions in organic chemistry have coupling reaction reaction mechanism involving free radical intermediates. True organic redox chemistry can be found in electrochemical organic synthesis or electrosynthesis . Examples of organic reactions that can take place in an electrochemical cell are the Kolbe electrolysis . [ 3 ]
In disproportionation reactions the reactant is both oxidised and reduced in the same chemical reaction forming two separate compounds.
Asymmetric catalytic reductions and asymmetric catalytic oxidations are important in asymmetric synthesis .
Most oxidations are conducted with air or oxygen , especially in industry. These oxidation include routes to chemical compounds, remediation of pollutants, and combustion . Some commercially important oxidations are listed:
Many reagents have been invented for organic oxidations. Organic oxidations reagents are usually classified according to the functional group attacked by the oxidant:
Often the substrate to be oxidized features more than one functional group. In such cases, selective oxidations become important.
In organic chemistry, reduction is equivalent to the addition of hydrogen atoms, usually in pairs. The reaction of unsaturated organic compounds with hydrogen gas is called hydrogenation . The reaction of saturated organic compounds with hydrogen gas is called hydrogenolysis . Hydrogenolyses necessarily cleaves C-X bonds (X = C, O, N, etc.). Reductions can also be effected by adding hydride and proton sources, the so-called heterolytic pathway . Such reactions are often effected using stoichiometric hydride reagents such as sodium borohydride or lithium aluminium hydride . [ 5 ] | https://en.wikipedia.org/wiki/Organic_redox_reaction |
Organic selenocyanates are organoselenium compounds with the general formula RSeCN. They are generally colorless, air-stable solids or liquids with repulsive odors. In terms of structure, synthesis, and reactivity, selenocyanates and thiocyanates behave similarly.
Alkyl selenocyanates are generally prepared by treatment of potassium selenocyanate with alkyl halides in alcohol or acetone solution. Aryl selenocyanates are generally prepared by treatment of potassium selenocyanate with aryl diazonium salts . [ 1 ]
Organic selenocyanates can be reduced to the selenol, which readily oxidize to the diselenide:
Oxidation of selenocyanates gives the seleninic acids . [ 1 ] | https://en.wikipedia.org/wiki/Organic_selenocyanates |
Organic semiconductors are solids whose building blocks are pi-bonded molecules or polymers made up by carbon and hydrogen atoms and – at times – heteroatoms such as nitrogen , sulfur and oxygen . They exist in the form of molecular crystals or amorphous thin films . In general, they are electrical insulators , but become semiconducting when charges are injected from appropriate electrodes or are introduced by doping or photoexcitation .
In molecular crystals the energetic separation between the top of the valence band and the bottom conduction band , i.e. the band gap , is typically 2.5–4 eV, while in inorganic semiconductors the band gaps are typically 1–2 eV. This implies that molecular crystals are, in fact, insulators rather than semiconductors in the conventional sense. They become semiconducting only when charge carriers are either injected from the electrodes or generated by intentional or unintentional doping.
Charge carriers can also be generated in the course of optical excitation. It is important to realize, however, that the primary optical excitations are neutral excitons with a Coulomb -binding energy of typically 0.5–1.0 eV. The reason is that in organic semiconductors their dielectric constants are as low as 3–4. This impedes efficient photogeneration of charge carriers in neat systems in the bulk. Efficient photogeneration can only occur in binary systems due to charge transfer between donor and acceptor moieties. Otherwise neutral excitons decay radiatively to the ground state – thereby emitting photoluminescence – or non-radiatively. The optical absorption edge of organic semiconductors is typically 1.7–3 eV, equivalent to a spectral range from 700 to 400 nm (which corresponds to the visible spectrum).
In 1862, Henry Letheby obtained a partly conductive material by anodic oxidation of aniline in sulfuric acid . The material was probably polyaniline . [ 2 ] In the 1950s, researchers discovered that polycyclic aromatic compounds formed semi-conducting charge-transfer complex salts with halogens. In particular, high conductivity of 0.12 S/cm was reported in perylene – iodine complex in 1954. [ 3 ] This finding indicated that organic compounds could carry current.
The fact that organic semiconductors are, in principle, insulators but become semiconducting when charge carriers are injected from the electrode(s) was discovered by Kallmann and Pope. [ 4 ] [ 5 ] They found that a hole current can flow through an anthracene crystal contacted with a positively biased electrolyte containing iodine that can act as a hole injector. This work was stimulated by the earlier discovery by Akamatu et al. [ 6 ] that aromatic hydrocarbons become conductive when blended with molecular iodine because a charge-transfer complex is formed. Since it was readily realized that the crucial parameter that controls injection is the work function of the electrode, it was straightforward to replace the electrolyte by a solid metallic or semiconducting contact with an appropriate work function. When both electrons and holes are injected from opposite contacts, they can recombine radiatively and emit light ( electroluminescence ). It was observed in organic crystals in 1965 by Sano et al. [ 7 ]
In 1972, researchers found metallic conductivity in the charge-transfer complex TTF-TCNQ. Superconductivity in charge-transfer complexes was first reported in the Bechgaard salt (TMTSF) 2 PF 6 in 1980. [ 8 ]
In 1973 Dr. John McGinness produced the first device incorporating an organic semiconductor. This occurred roughly eight years before the next such device was created. The " melanin ( polyacetylenes ) bistable switch" currently is part of the chips collection of the Smithsonian Institution . [ 9 ]
In 1977, Shirakawa et al. reported high conductivity in oxidized and iodine-doped polyacetylene. [ 10 ] They received the 2000 Nobel prize in Chemistry for "The discovery and development of conductive polymers ". [ 11 ] Similarly, highly conductive polypyrrole was rediscovered in 1979. [ 12 ]
Rigid-backbone organic semiconductors are now used as active elements in optoelectronic devices such as organic light-emitting diodes (OLED), organic solar cells , organic field-effect transistors (OFET), electrochemical transistors and recently in biosensing applications. Organic semiconductors have many advantages, such as easy fabrication, mechanical flexibility, and low cost.
The discovery by Kallman and Pope paved the way for applying organic solids as active elements in semiconducting electronic devices, such as organic light-emitting diodes (OLEDs) that rely on the recombination of electrons and holes injected from "ohmic" electrodes, i.e. electrodes with unlimited supply of charge carriers. [ 13 ] The next major step towards the technological exploitation of the phenomenon of electron and hole injection into a non-crystalline organic semiconductor was the work by Tang and Van Slyke. [ 14 ] They showed that efficient electroluminescence can be generated in a vapor-deposited thin amorphous bilayer of an aromatic diamine (TAPC) and Alq3 sandwiched between an indium-tin-oxide (ITO) anode and an Mg:Ag cathode. Another milestone towards the development of organic light-emitting diodes (OLEDs) was the recognition that also conjugated polymers can be used as active materials. [ 15 ] The efficiency of OLEDs was greatly improved when realizing that phosphorescent states ( triplet excitons) may be used for emission when doping an organic semiconductor matrix with a phosphorescent dye, such as complexes of iridium with strong spin–orbit coupling . [ 16 ]
Work on conductivity of anthracene crystals contacted with an electrolyte showed that optically excited dye molecules adsorbed at the surface of the crystal inject charge carriers. [ 17 ] The underlying phenomenon is called sensitized photoconductivity. It occurs when photo-exciting a dye molecule with appropriate oxidation/reduction potential adsorbed at the surface or incorporated in the bulk. This effect revolutionized electrophotography, which is the technological basis of today's office copying machines. [ 18 ] It is also the basis of organic solar cells (OSCs), in which the active element is an electron donor, and an electron acceptor material is combined in a bilayer or a bulk heterojunction .
Doping with strong electron donors or acceptors can render organic solids conductive even in the absence of light. Examples are doped polyacetylene [ 19 ] and doped light-emitting diodes. [ 20 ]
Amorphous molecular films are produced by evaporation or spin-coating. They have been investigated for device applications such as OLEDs, OFETs, and OSCs. Illustrative materials are tris(8-hydroxyquinolinato)aluminium , C 60 , phenyl-C61-butyric acid methyl ester (PCBM), pentacene , carbazoles , and phthalocyanine .
Molecularly doped polymers are prepared by spreading a film of an electrically inert polymer, e.g. polycarbonate, doped with typically 30% of charge transporting molecules, on a base electrode. Typical materials are the triphenylenes . They have been investigated for use as photoreceptors in electrophotography. [ 18 ] This requires films to have a thickness of several micrometers, which can be prepared using the doctor-blade technique.
In the early days of fundamental research into organic semiconductors the prototypical materials were free-standing single crystals of the acene family, e.g. anthracene and tetracene . [ 21 ] The advantage of employing molecular crystals instead of amorphous film is that their charge carrier mobilities are much larger. This is of particular advantage for OFET applications. Examples are thin films of crystalline rubrene prepared by hot wall epitaxy. [ 22 ] [ 23 ]
They are usually processed from solution employing variable deposition techniques including simple spin-coating, ink-jet deposition or industrial reel-to-reel coating which allows preparing thin films on a flexible substrate. The materials of choice are conjugated polymers such as poly-thiophene, poly-phenylenevinylene, and copolymers of alternating donor and acceptor units such as members of the poly(carbazole-dithiophene-benzothiadiazole (PCDTBT) family. [ 24 ] For solar cell applications they can be blended with C60 or PCBM as electron acceptors.
Aromatic short peptides self-assemblies are a kind of promising candidate for bioinspired and durable nanoscale semiconductors. [ 25 ] The highly ordered and directional intermolecular π-π interactions and hydrogen-bonding network allow the formation of quantum confined structures within the peptide self-assemblies, thus decreasing the band gaps of the superstructures into semiconductor regions. [ 26 ] As a result of the diverse architectures and ease of modification of peptide self-assemblies, their semiconductivity can be readily tuned, doped, and functionalized. Therefore, this family of electroactive supramolecular materials may bridge the gap between the inorganic semiconductor world and biological systems.
Organic semiconductors can be characterized by UV-photoemission spectroscopy. The equivalent technique for electron states is inverse photoemission. [ 27 ]
To measure the mobility of charge carriers, the traditional technique is the so-called time of flight (TOF) method. This technique requires relatively thick samples; it is not applicable to thin films. Alternatively, one can extract the charge carrier mobility from the current in a field effect transistor as a function of both the source-drain and the gate voltage. Other ways to determine the charge carrier mobility involve measuring space charge limited current (SCLC) flow and "carrier extraction by linearly increasing voltage (CELIV). [ 28 ]
In order to characterize the morphology of semiconductor films, one can apply atomic force microscopy (AFM), scanning electron microscopy (SEM), and grazing-incidence small-angle scattering (GISAS).
In contrast to organic crystals investigated in the 1960-70s, organic semiconductors that are nowadays used as active media in optoelectronic devices are usually more or less disordered. Combined with the fact that the structural building blocks are held together by comparatively weak van der Waals forces this precludes charge transport in delocalized valence and conduction bands. Instead, charge carriers are localized at molecular entities, e.g. oligomers or segments of a conjugated polymer chain, and move by incoherent hopping among adjacent sites with statistically variable energies. Quite often the site energies feature a Gaussian distribution. Also the hopping distances can vary statistically (positional disorder).
A consequence of the energetic broadening of the density of states (DOS) distribution is that charge motion is both temperature and field dependent and the charge carrier mobility can be several orders of magnitude lower than in an equivalent crystalline system. This disorder effect on charge carrier motion is diminished in organic field-effect transistors because current flow is confined in a thin layer. Therefore, the tail states of the DOS distribution are already filled so that the activation energy for charge carrier hopping is diminished. For this reason the charge carrier mobility inferred from FET experiments is always higher than that determined from TOF experiments. [ 28 ]
In organic semiconductors, charge carriers couple to vibrational modes and are referred to as polarons. Therefore, the activation energy for hopping motion contains an additional term due to structural site relaxation upon charging a molecular entity. It turns out, however, that usually the disorder contribution to the temperature dependence of the mobility dominates over the polaronic contribution. [ 29 ]
Source: [ 30 ]
The elastic modulus can be measured through tensile testing, which captures the material's stress-strain response. Additionally, the buckling method, employing buckling equations and measured wavelengths, can be used to determine the mechanical modulus of film materials. [ 31 ] The elastic modulus significantly impacts the applications of organic semiconductors; lower moduli are preferable for wearable and flexible electronics to ensure flexibility, [ 32 ] while higher moduli are required for devices needing greater resistance to mechanical stresses and enhanced structural integrity. [ 33 ]
The yield point of organic semiconductors is the stress or strain level at which the material starts to deform permanently. After this point, the material loses its elasticity and undergoes permanent deformation. Yield strength is usually measured by conducting tensile testing. Understanding and regulating the yield point of organic semiconductors is essential to designing devices that can endure operational stress without permanent deformation. [ 34 ] This helps maintain the device's functionality and prolong its lifetime.
As polymers, organic semiconductors exhibit viscoelasticity, meaning they exhibit both viscous and elastic characteristics during deformation. [ 35 ] Viscoelasticity allows materials to return to their original shape after being deformed and to exhibit strain that varies over time. Viscoelasticity is typically measured using dynamic mechanical analysis (DMA). Viscoelasticity is crucial for wearable devices, which are subjected to stretching and bending during use. The viscoelastic properties help the materials absorb energy during these processes, enhancing durability and ensuring long-term functionality under continuous physical stress. [ 36 ] [ 37 ] | https://en.wikipedia.org/wiki/Organic_semiconductor |
In organic chemistry , a sulfide ( British English sulphide ) or thioether is an organosulfur functional group with the connectivity R−S−R' as shown on right. Like many other sulfur-containing compounds, volatile sulfides have foul odors. [ 1 ] A sulfide is similar to an ether except that it contains a sulfur atom in place of the oxygen. The grouping of oxygen and sulfur in the periodic table suggests that the chemical properties of ethers and sulfides are somewhat similar, though the extent to which this is true in practice varies depending on the application.
Sulfides are sometimes called thioethers, especially in the old literature. The two organic substituents are indicated by the prefixes. (CH 3 ) 2 S is called dimethylsulfide . Some sulfides are named by modifying the common name for the corresponding ether. For example, C 6 H 5 SCH 3 is methyl phenyl sulfide, but is more commonly called thioanisole , since its structure is related to that for anisole , C 6 H 5 OCH 3 .
The modern systematic nomenclature in chemistry for the trival name thioether is sulfane . [ 2 ]
Sulfide is an angular functional group, the C–S–C angle approaching 90° The C–S bonds are about 180 pm .
For the prototype, dimethylsulfide, the C-S-C angles is 99°, which is smaller than the C-O-C angle in ether (~110°). The C-S distance in dimethylsulfide is 1.81 Å. [ 3 ]
Sulfides are characterized by their strong odors, which are similar to thiol odor. This odor limits the applications of volatile sulfides. In terms of their physical properties they resemble ethers, but are less volatile, higher melting, and less hydrophilic. These properties follow from the polarizability of the divalent sulfur center, which is greater than that for oxygen in ethers.
Thiophenes are a special class of sulfide-containing heterocyclic compounds . Because of their aromatic character, they are non-nucleophilic. The nonbonding electrons on sulfur are delocalized into the π-system. As a consequence, thiophene exhibits few properties expected for a sulfide – thiophene is non-nucleophilic at sulfur and, in fact, is sweet-smelling. Upon hydrogenation , thiophene gives tetrahydrothiophene , C 4 H 8 S, which indeed does behave as a typical sulfide.
Sulfides are important in biology, notably in the amino acid methionine and the cofactor biotin . Petroleum contains many organosulfur compounds, including sulfides. Polyphenylene sulfide is a useful high temperature plastic. Coenzyme M , CH 3 SCH 2 CH 2 SO − 3 , is the precursor to methane (i.e. natural gas) via the process of methanogenesis .
Sulfides are typically prepared by alkylation of thiols . Alkylating agents include not only alkyl halides, but also epoxides , aziridines , and Michael acceptors . [ 4 ]
Such reactions are usually conducted in the presence of a base, which converts the thiol into the more nucleophilic thiolate. [ 5 ] Analogously, the reaction of disulfides with organolithium reagents produces thioethers:
Analogous reactions are known starting with Grignard reagents .
Alternatively, sulfides can be synthesized by the addition of a thiol to an alkene in the thiol-ene reaction :
This reaction is often catalysed by free radicals produced from a photoinitiator . [ 6 ]
Sulfides can also be prepared by many other methods, such as the Pummerer rearrangement . Trialkysulfonium salts react with nucleophiles with a dialkyl sulfide as a leaving group:
This reaction is exploited in biological systems as a means of transferring an alkyl group . For example, S -adenosylmethionine acts as a methylating agent in biological S N 2 reactions .
An unusual but well tested method for the synthesis of thioethers involves addition of alkenes, especially ethylene across the S-Cl bond of sulfur dichloride . This method has been used in the production of bis(2-chloroethyl)sulfide , a mustard gas : [ 7 ]
The Lewis basic lone pairs on sulfur dominate the sulfides' reactivity. Sulfides readily alkylate to stable sulfonium salts, such as trimethylsulfonium iodide : [ 8 ]
Sulfides also oxidize easily to sulfoxides ( R−S(=O)−R ), which can themselves be further oxidized to sulfones ( R−S(=O) 2 −R ). Hydrogen peroxide is a typical oxidant—for example, with dimethyl sulfide ( S(CH 3 ) 2 ): [ 9 ]
In analogy to their easy alkylation, sulfides bind to metals to form thioether complexes . Consequently, Lewis acids do not decompose thioethers as they do ethers. [ 10 ] Sulfides are soft ligands, but their affinity for metals is lower than typical phosphines . Chelating thioethers are known, such as 1,4,7-trithiacyclononane .
Sulfides undergo hydrogenolysis in the presence of certain metals:
Raney nickel is useful for stoichiometric reactions in organic synthesis [ 11 ] whereas molybdenum-based catalysts are used to "sweeten" petroleum fractions, in the process called hydrodesulfurization . [ citation needed ] Similarly dissolving metal reductions can induce dealkylation or dearylation . [ 12 ]
Unlike ethers , thioethers are stable in the presence of Grignard reagents . [ 13 ] The protons adjacent to the sulfur atom are labile, and can be deprotonated with strong bases. [ 14 ] | https://en.wikipedia.org/wiki/Organic_sulfide |
An organic superconductor is a synthetic organic compound that exhibits superconductivity at low temperatures.
As of 2007 the highest achieved critical temperature for an organic superconductor at standard pressure is 33 K (−240 °C; −400 °F), observed in the alkali-doped fullerene RbCs 2 C 60 . [ 1 ] [ 2 ]
In 1979 Klaus Bechgaard synthesized the first organic superconductor (TMTSF) 2 PF 6 (the corresponding material class was named after him later) with a transition temperature of T C = 0.9 K, at an external pressure of 12 kbar. [ 3 ]
Many materials may be characterized as organic superconductors. These include the Bechgaard salts and Fabre salts which are both quasi-one-dimensional, and quasi-two-dimensional materials such as k -BEDT-TTF 2 X charge-transfer complex , λ -BETS 2 X compounds, graphite intercalation compounds and three-dimensional materials such as the alkali - doped fullerenes .
Organic superconductors are of special interest not only for scientists, looking for room-temperature superconductivity and for model systems explaining the origin of superconductivity but also for daily life issues as organic compounds are mainly built of carbon and hydrogen which belong to the most common elements on earth in contrast to copper or osmium .
Fabre-salts are composed of tetramethyltetrathiafulvalene (TMTTF) and Bechgaard salts of tetramethyltetraselenafulvalene (TMTSF). These two organic molecules are similar except for the sulfur -atoms of TMTTF being replaced by selenium -atoms in TMTSF. The molecules are stacked in columns (with a tendency to dimerization ) which are separated by anions . Typical anions are, for example, octahedral PF 6 , AsF 6 or tetrahedral ClO 4 or ReO 4 .
Both material classes are quasi-one-dimensional at room-temperature, only conducting along the molecule stacks, and share a very rich phase diagram containing antiferromagnetic ordering , charge order , spin-density wave state , dimensional crossover and superconductivity.
Only one Bechgaard salt was found to be superconducting at ambient pressure which is (TMTTF) 2 ClO 4 with a transition temperature of T C = 1.4 K. Several other salts become superconducting only under external pressure. The external pressure required to drive most Fabre-salts to superconductivity is so high, that under lab conditions superconductivity was observed only in one compound. A selection of the transition temperature and corresponding external pressure of several one-dimensional organic superconductors is shown in the table below.
BEDT-TTF is the short form of bisethylenedithio-tetrathiafulvalene commonly abbreviated with ET. These molecules form planes which are separated by anions. The pattern of the molecules in the planes is not unique but there are several different phases growing, depending on the anion and the growth conditions. Important phases concerning superconductivity are the α- and θ- phase with the molecules ordering in a fishbone structure and the β- and especially κ-phase which order in a checkerboard structure with molecules being dimerized in the κ-phase. This dimerization makes the κ-phases special as they are not quarter- but half-filled systems, driving them into superconductivity at higher temperatures compared to the other phases.
The amount of possible anions separating two sheets of ET-molecules is nearly infinite. There are simple anions such as triiodide ( I − 3 ), polymeric ones such as the very famous Cu[N(CN) 2 ]Br and anions containing solvents for example Ag(CF 3 ) 4 ·112DCBE. The electronic properties of the ET-based crystals are determined by its growing phase, its anion and by the external pressure applied. The external pressure needed to drive an ET-salt with insulating ground state to a superconducting one is much less than those needed for Bechgaard salts . For example, κ-(ET) 2 Cu[N(CN) 2 ]Cl needs only a pressure of about 300 bar to become superconducting, which can be achieved by placing a crystal in grease frozen below 0 °C (32 °F) and then providing sufficient stress to induce the superconducting transition. The crystals are very sensitive, which can be observed impressively in α-(ET) 2 I 3 lying several hours in the sun (or more controlled in an oven at 40 °C, 104 °F). After this treatment one gets α Tempered -(ET) 2 I 3 which is superconducting.
In contrast to the Fabre or Bechgaard salts universal phase diagrams for all the ET-based salts have only been proposed yet. Such a phase diagram would depend not only on temperature and pressure (i.e. bandwidth), but also on electronic correlations . In addition to the superconducting ground state these materials show charge-order , antiferromagnetism or remain metallic down to lowest temperatures. One compound is even predicted to be a spin liquid .
The highest transition temperatures at ambient pressure and with external pressure are both found in κ-phases with very similar anions. κ-(ET) 2 Cu[N(CN) 2 ]Br becomes superconducting at T C = 11.8 K at ambient pressure, and a pressure of 300 bar drives deuterated κ-(ET) 2 Cu[N(CN) 2 ]Cl from an antiferromagnetic to a superconducting ground state with a transition temperature of T C = 13.1 K. The following table shows only a few exemplary superconductors of this class. For more superconductors, see Lebed (2008) in the references.
Even more superconductors can be found by changing the ET-molecules slightly either by replacing the sulfur atoms by selenium (BEDT-TSF, BETS) or by oxygen (BEDO-TTF, BEDO).
Some two-dimensional organic superconductors of the κ-(ET) 2 X and λ(BETS) 2 X families are candidates for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase when superconductivity is suppressed by an external magnetic field. [ 5 ]
Superconducting fullerenes based on C 60 are fairly different from other organic superconductors. The building molecules are no longer manipulated hydrocarbons but pure carbon molecules. In addition these molecules are no longer flat but bulky which gives rise to a three-dimensional, isotropic superconductor. The pure C 60 grows in an fcc-lattice and is an insulator . By placing alkali atoms in the interstitials the crystal becomes metallic and eventually superconducting at low temperatures.
Unfortunately C 60 crystals are not stable at ambient atmosphere. They are grown and investigated in closed capsules, limiting the measurement techniques possible. The highest transition temperature measured so far was T C = 33 K for Cs 2 RbC 60 .The highest measured transition temperature of an organic superconductor was found in 1995 in Cs 3 C 60 pressurized with 15 kbar to be T C = 40 K. Under pressure this compound shows a unique behavior. Usually the highest T C is achieved with the lowest pressure necessary to drive the transition. Further increase of the pressure usually reduces the transition temperature. However, in Cs 3 C 60 superconductivity sets in at very low pressures of several 100 bar, and the transition temperature keeps increasing with increasing pressure. This indicates a completely different mechanism than just broadening of the bandwidth.
Next to the three major classes of organic superconductors (SCs) there are more organic systems becoming superconducting at low temperatures or under pressure. A few examples follow.
TMTTF as well as BEDT-TTF are based on the molecule TTF ( tetrathiafulvalene ). Using tetrathiapentalene (TTP) as basic molecules one receives a variety of new organic molecules serving as cations in organic crystals. Some of them are superconducting. This class of superconductors was only reported recently and investigations are still under process.
Instead of using sulfated molecules or the fairly big Buckminster fullerenes recently it became possible to synthesize crystals from the hydrocarbon picene and phenanthrene . Doping the crystal picene and phenanthrene with alkali metals such as potassium or rubidium and annealing for several days leads to superconductivity with transition temperatures up to 18 K (−255 °C; −427 °F). For AxPhenanthrene, the superconductivity is possible unconventional. Both phenanthrene and picene are called phenanthrene-edge-type polycyclic aromatic hydrocarbon . The increasing number of benzene rings results in higher T c .
Putting foreign molecules or atoms between hexagon graphite sheets leads to ordered structures and to superconductivity even if neither the foreign molecule or atom nor the graphite layers are metallic. Several stoichiometries have been synthesized using mainly alkali atoms as anions. | https://en.wikipedia.org/wiki/Organic_superconductor |
Organic synthesis is a branch of chemical synthesis concerned with the construction of organic compounds . Organic compounds are molecules consisting of combinations of covalently-linked hydrogen , carbon , oxygen , and nitrogen atoms. Within the general subject of organic synthesis, there are many different types of synthetic routes that can be completed including total synthesis , [ 1 ] stereoselective synthesis , [ 2 ] automated synthesis , [ 3 ] and many more. Additionally, in understanding organic synthesis it is necessary to be familiar with the methodology, techniques, and applications of the subject.
A total synthesis refers to the complete chemical synthesis of molecules from simple, natural precursors . [ 1 ] Total synthesis is accomplished either via a linear or convergent approach. In a linear synthesis —often adequate for simple structures—several steps are performed sequentially until the molecule is complete; the chemical compounds made in each step are called synthetic intermediates . [ 1 ] Most often, each step in a synthesis is a separate reaction taking place to modify the starting materials. For more complex molecules, a convergent synthetic approach may be better suited. This type of reaction scheme involves the individual preparations of several key intermediates, which are then combined to form the desired product. [ 4 ]
Robert Burns Woodward , who received the 1965 Nobel Prize for Chemistry for several total syntheses [ 5 ] including his synthesis of strychnine , [ 6 ] is regarded as the grandfather of modern organic synthesis. [ 7 ] Some latter-day examples of syntheses include Wender's , [ 8 ] Holton's , [ 9 ] Nicolaou's , [ 10 ] and Danishefsky's [ 11 ] total syntheses of the anti-cancer drug paclitaxel (trade name Taxol). [ 12 ]
Before beginning any organic synthesis, it is important to understand the chemical reactions , reagents , and conditions required in each step to guarantee successful product formation. When determining optimal reaction conditions for a given synthesis, the goal is to produce an adequate yield of pure product with as few steps as possible. [ 13 ] When deciding conditions for a reaction, the literature can offer examples of previous reaction conditions that can be repeated, or a new synthetic route can be developed and tested. For practical, industrial applications additional reaction conditions must be considered to include the safety of both the researchers and the environment, as well as product purity. [ 14 ]
Organic Synthesis requires many steps to separate and purify products. Depending on the chemical state of the product to be isolated, different techniques are required. For liquid products, a very common separation technique is liquid–liquid extraction and for solid products, filtration (gravity or vacuum) can be used. [ 15 ] [ 16 ]
Liquid–liquid extraction uses the density and polarity of the product and solvents to perform a separation. [ 16 ] Based on the concept of "like-dissolves-like", non-polar compounds are more soluble in non-polar solvents, and polar compounds are more soluble in polar solvents. [ 17 ] By using this concept, the relative solubility of compounds can be exploited by adding immiscible solvents into the same flask and separating the product into the solvent with the most similar polarity. Solvent miscibility is of major importance as it allows for the formation of two layers in the flask, one layer containing the side reaction material and one containing the product. As a result of the differing densities of the layers, the product-containing layer can be isolated and the other layer can be removed.
Many reactions require heat to increase reaction speed. [ 18 ] However, in many situations increased heat can cause the solvent to boil uncontrollably which negatively affects the reaction, and can potentially reduce product yield. To address this issue, reflux condensers can be fitted to reaction glassware. Reflux condensers are specially designed pieces of glassware that possess two inlets for water to run in and out through the glass against gravity. This flow of water cools any escaping substrate and condenses it back into the reaction flask to continue reacting [ 19 ] and ensure that all product is contained. The use of reflux condensers is an important technique within organic syntheses and is utilized in reflux steps, as well as recrystallization steps.
When being used for refluxing a solution, reflux condensers are fitted and closely observed. Reflux occurs when condensation can be seen dripping back into the reaction flask from the reflux condenser; 1 drop every second or few seconds. [ 19 ]
For recrystallization , the product-containing solution is equipped with a condenser and brought to reflux again. Reflux is complete when the product-containing solution is clear. Once clear, the reaction is taken off heat and allowed to cool which will cause the product to re-precipitate, yielding a purer product. [ 20 ]
Solid products can be separated from a reaction mixture using filtration techniques. To obtain solid products a vacuum filtration apparatus can be used.
Vacuum filtration uses suction to pull liquid through a Büchner funnel equipped with filter paper, which catches the desired solid product. [ 15 ] This process removes any unwanted solution in the reaction mixture by pulling it into the filtration flask and leaving the desired product to collect on the filter paper.
Liquid products can also be separated from solids by using gravity filtration . [ 15 ] In this separatory method, filter paper is folded into a funnel and placed on top of a reaction flask. The reaction mixture is then poured through the filter paper , at a rate such that the total volume of liquid in the funnel does not exceed the volume of the funnel. [ 15 ] This method allows for the product to be separated from other reaction components by the force of gravity, instead of a vacuum.
Most complex natural products are chiral, [ 2 ] [ 21 ] and the bioactivity of chiral molecules varies with the enantiomer . [ 22 ] Some total syntheses target racemic mixtures, which are mixtures of both possible enantiomers . A single enantiomer can then be selected via enantiomeric resolution .
As chemistry has developed methods of stereoselective catalysis and kinetic resolution have been introduced whereby reactions can be directed, producing only one enantiomer rather than a racemic mixture. [ 23 ] Early examples include stereoselective hydrogenations (e.g., as reported by William Knowles [ 24 ] and Ryōji Noyori [ 25 ] ) and functional group modifications such as the asymmetric epoxidation by Barry Sharpless ; [ 26 ] for these advancements in stereochemical preference, these chemists were awarded the Nobel Prize in Chemistry in 2001. [ 27 ] Such preferential stereochemical reactions give chemists a much more diverse choice of enantiomerically pure materials.
Using techniques developed by Robert B. Woodward paired with advancements in synthetic methodology, chemists have been able synthesize stereochemically selective complex molecules without racemization. Stereocontrol provides the target molecules to be synthesized as pure enantiomers (i.e., without need for resolution). Such techniques are referred to as stereoselective synthesis .
Many synthetic procedures are developed from a retrosynthetic framework , a type of synthetic design developed by Elias James Corey , for which he won the Nobel Prize in Chemistry in 1990. [ 28 ] In this approach, the synthesis is planned backwards from the product, obliging to standard chemical rules. [ 1 ] Each step breaks down the parent structure into achievable components, which are shown via the use of graphical schemes with retrosynthetic arrows (drawn as ⇒, which in effect, means "is made from"). Retrosynthesis allows for the visualization of desired synthetic designs.
A recent development within organic synthesis is automated synthesis . To conduct organic synthesis without human involvement, researchers are adapting existing synthetic methods and techniques to create entirely automated synthetic processes using organic synthesis software . This type of synthesis is advantageous as synthetic automation can increase yield with continual "flowing" reactions. In flow chemistry , substrates are continually fed into the reaction to produce a higher yield . Previously, this type of reaction was reserved for large-scale industrial chemistry but has recently transitioned to bench-scale chemistry to improve the efficiency of reactions on a smaller scale. [ 3 ]
Currently integrating automated synthesis into their work is SRI International , a nonprofit research institute. Recently SRI International has developed Autosyn, an automated multi-step chemical synthesizer that can synthesize many FDA -approved small molecule drugs. This synthesizer demonstrates the versatility of substrates and the capacity to potentially expand the type of research conducted on novel drug molecules without human intervention. [ 29 ]
Automated chemistry and the automated synthesizers used demonstrate a potential direction for synthetic chemistry in the future.
Necessary to organic synthesis is characterization . Characterization refers to the measurement of chemical and physical properties of a given compound, and comes in many forms. Examples of common characterization methods include: nuclear magnetic resonance (NMR), [ 30 ] mass spectrometry , [ 31 ] Fourier-transform infrared spectroscopy (FTIR), [ 32 ] and melting point analysis. [ 33 ] Each of these techniques allow for a chemist to obtain structural information about a newly synthesized organic compound. Depending on the nature of the product, the characterization method used can vary.
Organic synthesis is an important chemical process that is integral to many scientific fields. Examples of fields beyond chemistry that require organic synthesis include the medical industry, pharmaceutical industry, and many more. Organic processes allow for the industrial-scale creation of pharmaceutical products. An example of such a synthesis is Ibuprofen . Ibuprofen can be synthesized from a series of reactions including: reduction , acidification , formation of a Grignard reagent , and carboxylation . [ 34 ]
In the synthesis of Ibuprofen proposed by Kjonass et al ., p -isobutylacetophenone, the starting material, is reduced with sodium borohydride (NaBH 4 ) to form an alcohol functional group . The resulting intermediate is acidified with HCl to create a chlorine group. The chlorine group is then reacted with magnesium turnings to form a Grignard reagent. [ 34 ] This Grignard is carboxylated and the resulting product is worked up to synthesize ibuprofen.
This synthetic route is just one of many medically and industrially relevant reactions that have been created, and continued to be used. | https://en.wikipedia.org/wiki/Organic_synthesis |
Organic thiocyanates are organic compounds containing the functional group RSCN. the organic group is attached to sulfur: R−S−C≡N has a S–C single bond and a C≡N triple bond. [ 1 ]
Organic thiocyanates are valued building blocks. They allow to access efficiently various sulfur containing functional groups and scaffolds. [ 2 ]
Several synthesis routes exist, [ 3 ] the most common being the reaction between alkyl halides and alkali thiocyanate in aqueous media. [ 4 ] Illustrative is the preparation of isopropyl thiocyanate by treatment of isopropyl bromide with sodium thiocyanate in boiling ethanol. [ 5 ] The main complication with this route is the competing formation of alkyisothiocyanates. "SN1-type" substrates (e.g., benzyl halides) tend to give the isothiocyanate derivatives.
Some organic thiocyanates are generated by cyanation of some organosulfur compounds . Sulfenyl thiosulfates (RSSO 3 − ) react with alkali metal cyanides to give thiocyanates with displacement of sulfite. This approach has been applied to allyl thiocyanate: [ 6 ]
Sulfenyl chlorides (RSCl) also convert to thiocyanates.
Aryl thiocyanates are traditionally produced by the Sandmeyer reaction , which involves combining copper(I) thiocyanate and diazonium salts : [ 3 ]
Some arylthiocyanates can also often be obtained by thiocyanogenation , i.e. the reaction of thiocyanogen . This reaction is favored for electron-rich aromatic substrates. [ 1 ]
In methyl thiocyanate , N≡C and C−S distances are 116 and 176 pm. By contrast, N=C and C=S distances are 117 and 158 pm in isothiocyanates . [ 7 ]
Typical bond angles for C−S−C are 100°. [ 3 ] By contrast C−N=C in aryl isothiocyanates is 165°. Again, the thiocyanate isomers are quite different with C−S−C angle near 100°.
In both organic thiocyanate and isothiocyanate isomers the SCN angle approaches 180°.
Organic thiocyanates are hydrolyzed to thiocarbamates in the Riemschneider thiocarbamate synthesis .
Electrochemical reduction typically converts thiocyanates to thioates and cyanide , although sometimes it can replace the thiocyanate group as a whole with hydride. [ 8 ]
Some thiocyanates isomerize to the isothiocyanates. This reaction is especially rapid for the allyl isothiocyanate : [ 6 ]
Likewise, acyl thiocyanates are difficult to synthesize because any excess thiocyanate ion catalyzes isomerization to the acyl isothiocyanate. [ 9 ] | https://en.wikipedia.org/wiki/Organic_thiocyanates |
A light organ, or photophore , is a specialized anatomical structure found in a variety of organisms that emits light through the process of boluminescence . This light may be produced endogenously by the organism itself (symbiotic) or generated through a mutualistic relationship with bioluminescent bacteria (non-symbiotic), resulting in light production on a glandular organ of animals. Light organs are most commonly found in marine animals, including many species of fish and cephalopods . [ 1 ] The organ can be simple, or as complex as the human eye, equipped with lenses, shutters, color filters, and reflectors; unlike an eye, however, it is optimized to produce light, not absorb it.
In the context of developmental biology , light organs form through precise genetic regulation and, in some cases, microbial colonization during specific stages of an organism's life cycle. They play a crucial evolutionary role in enabling species to adapt to low-light or dark environments, particularly in the deep sea.
The development and use of light organs provide various biological advantages and have inspired numerous applications in science and technology. In ecosystems, these organs provide several benefits including camouflage, prey attraction, mating, and predator deterrence. [ 2 ] [ 3 ] Marine luminescence plays a role in predator-prey relationships in organisms like deep-sea siphonophores , which attract prey through the fluorescence in their tentacles. [ 4 ] Similarly, the bobtail squid maintains a symbiotic relationship with the bacterium Vibrio fischeri , which not only contributes to light production, but also plays a role in the proper development of the squid’s light organ. [ 5 ]
Organismal light organs are not only beneficial to animals, but are also useful in the field of medicine, inspiring the use of bioluminescence in research. Genes coding for bioluminescent proteins, such as luciferase , are used as reporters in molecular and cellular biology to monitor gene expression, track cellular processes, and study disease progression in real time. [ 6 ] [ 7 ] In medical diagnostics, bioluminescent imaging enables non-invasive visualization of infections, tumors, and other biological changes within living organisms. [ 8 ]
Bioluminescent organisms and engineered bacteria are also used in environmental monitoring . These systems can detect the presence of harmful pollutants such as heavy metals and organometallic compounds in soil and water by exhibiting a measurable change in light output. [ 9 ] In synthetic biology , efforts are ongoing to develop plants and microorganisms with customized bioluminescent capabilities for use in sustainable lighting and biosensing technologies.
Symbiotic light organs exist in many species of marine organisms. These light organs are used for various reasons, such as mating, warning predators, and communication. In a lot of marine animals, the photophores are usually along the sides of the body. [ 10 ] This is especially true for crustaceans and squids. This placement allows the light to bend around them and contrast with their bodies so they are harder to see by predators. [ 10 ]
There are several groups and families of fish that contain symbiotic light organs. Angler fish are a commonly known example of this, as they have a dangling light source that they use for mating and food. [ 11 ] Another example, though a predator, includes Dragonfish. This group has photophores on their cheeks that allows them to see red light underwater, helping in the capture of prey. [ 12 ]
Some of the fish included in the teleost fish groups fall under this category. [ 13 ] The flashlight fish, Anomalopidae , contain luminous bacteria that aid in navigation and predation. [ 14 ] Shrimp in the family Oplophoridae use bioluminescent secretions, which are used for mating and deterring predators. [ 15 ] The development of the light organ in Squids, specifically the Bobtail squid, differs from general fish light organ development.
Hawaiian Bobtail Squids, Euprymna scolopes , is a model organism in the study of marine light organ development. Its relationship with Vibrio fischeri has been studied by scientists for its long-term symbiosis and light organ colonization across generations of the organism. [ 16 ] The bacteria collects in pores on the sides of the squid light organ, caused by the movement of cilia during the juvenile stage. [ 16 ]
Lanternfish and Hatchetfish have photophores that create a pattern on the underside of the animals, causing predators to not see them from below. [ 17 ]
Compared to marine organisms, only a minimal amount of land-based organisms have developed photophores for bioluminescent capabilities. [ 18 ] This is likely due to the limited light availability in deeper waters. With the emission of light, aquatic animals have the ability to communicate movement, helping them in mating and distraction of predators.
However, though land animals have greater access to light sources, some species have evolved to benefit from light organs. Organisms such as fungi, insects, and several types of worms exhibit bioluminescent properties. [ 19 ] [ 20 ] These animals may use their bioluminescence to signal to mates, deter predators when in larval form, attract prey, and much more. [ 21 ] Bioluminescent light organs can be located on several parts of the organism including the head region in antenna or mouthpieces, the underside of the organism, or the backside of the organism. [ 21 ] Though it is not considered symbiotic, fireflies contain a light organ used primarily for mating. Their bioluminescence comes from the breaking down of luciferin , which is caused by a reaction with oxygen. [ 22 ]
Luminous Fungi - Basidiomycetes make up all known bioluminescent fungi, utilizing an enzyme system similar to that of fireflies to emit light. [ 21 ] These organisms inhabit subtropical forests and are best identified by the noticeable glow they emit from their wooden substrates. [ 23 ] A distinct function for fungi bioluminescence is unknown, however, the benefit of bioluminescence in these organisms is suggested to have significance in metabolism and dispersal. [ 23 ] Fungi in the genus Roridomyces have bioluminescent spores that are said to attract mobile terrestrial organisms to disperse the Fungi's reproductive structures.
Bioluminescent glow worm - Arachnocampa luminosa uses a similar luciferase mechanism as fireflies to produce a blue colored luminescence. [ 24 ] The worms, larvae of fungus gnats, inhabit caves of New Zealand and use web structures illuminated by their bioluminescence to attract prey. [ 25 ]
Certain light organs will begin development during embryogenesis, with a notable dependence on the homeobox factors AlABD-B and AlUNC-4 which both activate the gene AlLuc1, which is noted by instigating luciferase production. [ 26 ] During pupal development for fireflies, the aforementioned genes would be constantly upregulating, or increasing the response intensity to external signals when the organism is developing, whereas other genes like AlAbd-A, which regulates pigmentation, would be actively downregulating, or decreasing the response intensity for this organism, further increasing the activity for bioluminescence. [ 27 ]
The bioluminescence can be produced via organs designed to produce light called "photophores". The way that the light gets triggered for production varies among different animals, in animals like the Lampyridae genus, the production of light is triggered indirectly via the nervous system to produce a neurotransmitter called " octopamine ", simulating the production of nitric oxide, leading to the production of light. [ 28 ]
Photocytes are included in the process of light development, these are specialized cells with the sole purpose of producing light. It is important to note that, while some animals contain photocytes, others do not. The Lophiiformes genus, the light that gets produced is via the symbiotic relationship with bioluminescent bacteria, as the animal itself has no ability to produce light. [ 29 ]
As with several terrestrial animals, the effect of bioluminescence is a product of the enzyme Luciferase. [ 21 ] Luciferase is known to be located in the photophores of the organism. The type of Luciferase enzyme present in these light organs is directly related to the color representation that the photophore cells give off. This distinction can result in blue light, blue-green light, and even red light determined by their wavelengths. [ 30 ]
There are potential uses for bioluminescence in medical and research scenarios. In the past, ATP-driven bioluminescence has been used to detect the presence of plaque producing bacteria such as Streptococcus mutans . [ 31 ]
Bioluminescent bacteria also have the potential to disclose of environmental toxins, such as metals and organometallic compounds. Using bacteria that are sensitive to certain types of pollutants in the base parts of the environments, including the presence of metals in water and soil. [ 32 ] Detecting these contaminants within the environment has the potential to be beneficial, being informed on the health of the environment sooner allows for measures to be taken earlier than they would have otherwise. | https://en.wikipedia.org/wiki/Organismal_light_organs |
Organismal performance (or whole-organism performance) refers to the ability of an organism to conduct a task when maximally motivated . [ 1 ] Various aspects of performance are of primary concern in human athletics , horse racing , and dog racing . Performance in swimming tasks has been a subject of fisheries research since the 1960s. [ 2 ] In a broader biological context, the term first came to prominence with studies of locomotor abilities in lizards and snakes in the late 1970s and early 1980s. [ 3 ]
A seminal paper by Stevan J. Arnold in 1983 [ 4 ] focused on the importance of performance as an intermediary between lower-level traits and how natural selection acts. In particular, selection should act more directly on performance than on the subordinate traits (e.g., aspects of morphology , physiology , neurobiology ) that determine performance abilities. In other words, how fast a lizard can run is more important in escaping from predators than are the lengths of its legs , because they only party determine its ability to run fast. Since then, others have pointed out that behavior often acts as a "filter" between selection and performance [ 5 ] [ 6 ] [ 7 ] because animals do not always behave in ways that use their maximal performance abilities. [ 8 ] For example, if a lizard that saw a predator approaching did not choose to run, then its ability to sprint would be irrelevant. In any case, the original version of the conceptual model has stimulated much research in integrative organismal biology . [ 9 ] However, contrary to the hypothesis that selection should be stronger on whole-organism functional performance traits (such as sprinting ability) than on correlated morphological traits, a review of empirical studies did not find evidence that selection measured in the wild was stronger on performance. [ 10 ]
Although organismal performance is more commonly studied in animals (including human beings ) than in plants , various studies have focused on whole-plant performance in a similar vein. [ 11 ] For example, suction feeding abilities have been measured in carnivorous plants ( bladderworts ). [ 12 ] Although plants do not have either a nervous system or muscles, they can be said to have behavior. [ 13 ] [ 14 ] How such "behavior" may serve as a filter between performance and selection apparently has not been studied. | https://en.wikipedia.org/wiki/Organismal_performance |
Organisms can live at high altitude , either on land, in water, or while flying. Decreased oxygen availability and decreased temperature make life at such altitudes challenging, though many species have been successfully adapted via considerable physiological changes. As opposed to short-term acclimatisation (immediate physiological response to changing environment), high-altitude adaptation means irreversible, evolved physiological responses to high-altitude environments, associated with heritable behavioural and genetic changes . Among vertebrates, only few mammals (such as leaf-eared mice , yaks , ibexes , Tibetan gazelles , vicunas , llamas , mountain goats , etc.) and certain birds are known to have completely adapted to high-altitude environments. [ 1 ]
Human populations such as some Tibetans , South Americans and Ethiopians live in the otherwise uninhabitable high mountains of the Himalayas , Andes and Ethiopian Highlands respectively. The adaptation of humans to high altitude is an example of natural selection in action. [ 2 ]
High-altitude adaptations provide examples of convergent evolution , with adaptations occurring simultaneously on three continents. Tibetan humans and Tibetan domestic dogs share a genetic mutation in EPAS1 , but it has not been seen in Andean humans. [ 3 ]
Tardigrades live over the entire world, including the high Himalayas . [ 4 ] Tardigrades are also able to survive temperatures of close to absolute zero (−273 °C or −459 °F), [ 5 ] temperatures as high as 151 °C (304 °F), radiation that would kill other animals, [ 6 ] and almost a decade without water. [ 7 ] Since 2007, tardigrades have also returned alive from studies in which they have been exposed to the vacuum of outer space in low Earth orbit. [ 8 ] [ 9 ]
Other invertebrates with high-altitude habitats are Euophrys omnisuperstes , a spider that lives in the Himalaya range at altitudes of up to 6,700 m (22,000 ft); [ 10 ] it feeds on stray insects that are blown up the mountain by the wind. [ 11 ] The springtail Hypogastrura nivicola (one of several insects called snow fleas) also lives in the Himalayas. It is active in the dead of winter, its blood containing a compound similar to antifreeze . Some allow themselves to become dehydrated instead, preventing the formation of ice crystals within their body. [ 12 ]
Insects can fly and kite at very high altitude. Flies are common in the Himalayas up to 6,300 m (20,700 ft). [ 13 ] Bumble bees were discovered on Mount Everest at more than 5,600 m (18,400 ft) above sea level. [ 14 ] In subsequent tests, bumblebees were still able to fly in a flight chamber which recreated the thinner air of 9,000 m (30,000 ft). [ 15 ]
Ballooning is a term used for the mechanical kiting [ 16 ] [ 17 ] that many spiders , especially small species such as Erigone atra , [ 18 ] as well as certain mites and some caterpillars use to disperse through the air. Some spiders have been detected in atmospheric data balloons collecting air samples at slightly less than 5 km (16,000 ft) above sea level. [ 19 ] It is the most common way for spiders to pioneer isolated islands and mountaintops. [ 20 ] [ 21 ]
Fish at high altitudes have a lower metabolic rate, as has been shown in highland westslope cutthroat trout when compared to introduced lowland rainbow trout in the Oldman River basin. [ 22 ] There is also a general trend of smaller body sizes and lower species richness at high altitudes observed in aquatic invertebrates, likely due to lower oxygen partial pressures. [ 23 ] [ 24 ] [ 25 ] These factors may decrease productivity in high altitude habitats, meaning there will be less energy available for consumption, growth, and activity, which provides an advantage to fish with lower metabolic demands. [ 22 ]
The naked carp from Lake Qinghai , like other members of the carp family, can use gill remodelling to increase oxygen uptake in hypoxic environments . [ 26 ] The response of naked carp to cold and low-oxygen conditions seem to be at least partly mediated by hypoxia-inducible factor 1 (HIF-1) . [ 27 ] It is unclear whether this is a common characteristic in other high altitude dwelling fish or if gill remodelling and HIF-1 use for cold adaptation are limited to carp.
Mammals are also known to reside at high altitude and exhibit a striking number of adaptations in terms of morphology , physiology and behaviour . The Tibetan Plateau has very few mammalian species, ranging from wolf , kiang (Tibetan wild ass), goas , chiru (Tibetan antelope), wild yak , snow leopard , Tibetan sand fox , ibex , gazelle , Himalayan brown bear and water buffalo . [ 29 ] [ 30 ] [ 31 ] These mammals can be broadly categorised based on their adaptability in high altitude into two broad groups, namely eurybarc and stenobarc . Those that can survive a wide range of high-altitude regions are eurybarc and include yak, ibex, Tibetan gazelle of the Himalayas; and leaf-eared mice , vicuñas , llamas of the Andes. Stenobarc animals are those with lesser ability to endure a range of differences in altitude, such as rabbits , mountain goats , sheep , and cats . Among domesticated animals , yaks are perhaps the highest dwelling animals. The wild herbivores of the Himalayas such as the Himalayan tahr , markhor and chamois are of particular interest because of their ecological versatility and tolerance. [ 32 ]
A number of rodents live at high altitude, including deer mice , guinea pigs , and rats . The highest-elevation observation of a rodent is the yellow-rumped leaf-eared mouse ( Phyllotis xanthopygus rupestris ), trapped at the summit of Llullaillaco in the Andes at 6,739 metres (22,110 ft). This observation makes P. xanthopygus rupestris the highest-dwelling mammal. [ 33 ]
Several mechanisms help rodents survive these harsh conditions, including altered genetics of the hemoglobin gene in guinea pigs and deer mice. [ 34 ] [ 35 ] Deer mice use a high percentage of fats as metabolic fuel to retain carbohydrates for small bursts of energy. [ 36 ]
Other physiological changes that occur in rodents at high altitude include increased breathing rate [ 37 ] and altered morphology of the lungs and heart, allowing more efficient gas exchange and delivery. Lungs of high-altitude mice are larger, with more capillaries, [ 38 ] and their hearts have a heavier right ventricle (the latter applies to rats too), [ 39 ] [ 40 ] which pumps blood to the lungs.
At high altitudes, some rodents even shift their thermal neutral zone so they may maintain normal basal metabolic rate at colder temperatures. [ 41 ]
The deer mouse ( Peromyscus maniculatus ) is the best studied species, other than humans, in terms of high-altitude adaptation. [ 1 ] The deer mice native to Andes highlands (up to 3,000 m (9,800 ft)) are found to have relatively low hemoglobin content. [ 42 ] Measurement of food intake, gut mass, and cardiopulmonary organ mass indicated proportional increases in mice living at high altitudes, which in turn show that life at high altitudes demands higher levels of energy. [ 43 ] Variations in the globin genes ( α and β-globin ) seem to be the basis for increased oxygen-affinity of the hemoglobin and faster transport of oxygen. [ 44 ] [ 45 ] Structural comparisons show that in contrast to normal hemoglobin, the deer mouse hemoglobin lacks the hydrogen bond between α1Trp14 in the A helix and α1Thr67 in the E helix owing to the Thr 67 Ala substitution, and there is a unique hydrogen bond at the α1β1 interface between residues α1Cys34 and β1Ser128 . [ 46 ] The Peruvian native species of mice ( Phyllotis andium and Phyllotis xanthopygus ) have adapted to the high Andes by using proportionately more carbohydrates and have higher oxidative capacities of cardiac muscles compared to closely related native species residing at low-altitudes (100–300 m (330–980 ft)), ( Phyllotis amicus and Phyllotis limatus ). This shows that highland mice have evolved a metabolic process to economise oxygen usage for physical activities in the hypoxic conditions. [ 47 ]
Among domesticated animals , yaks ( Bos grunniens ) are the highest dwelling animals of the world, living at 3,000–5,000 m (9,800–16,400 ft). The yak is the most important domesticated animal for Tibet highlanders in Qinghai Province of China , as the primary source of milk , meat and fertilizer . Unlike other yak or cattle species, which suffer from hypoxia in the Tibetan Plateau, the Tibetan domestic yaks thrive only at high altitude, and not in lowlands. Their physiology is well-adapted to high altitudes, with proportionately larger lungs and heart than other cattle, as well as greater capacity for transporting oxygen through their blood. [ 48 ] In yaks, hypoxia-inducible factor 1 ( HIF-1 ) has high expression in the brain , lung and kidney , showing that it plays an important role in the adaptation to low oxygen environment. [ 49 ] On 1 July 2012 the complete genomic sequence and analyses of a female domestic yak was announced, providing important insights into understanding mammalian divergence and adaptation at high altitude. Distinct gene expansions related to sensory perception and energy metabolism were identified. [ 50 ] In addition, researchers also found an enrichment of protein domains related to the extracellular environment and hypoxic stress that had undergone positive selection and rapid evolution. For example, they found three genes that may play important roles in regulating the bodyʼs response to hypoxia, and five genes that were related to the optimisation of the energy from the food scarcity in the extreme plateau. One gene known to be involved in regulating response to low oxygen levels, ADAM17, is also found in human Tibetan highlanders. [ 51 ] [ 52 ]
Over 81 million people live permanently at high altitudes (>2,500 m or 8,200 ft) [ 53 ] in North , Central and South America , East Africa , and Asia , and have flourished for millennia in the exceptionally high mountains, without any apparent complications. [ 54 ] For average human populations, a brief stay at these places can risk mountain sickness . [ 55 ] For the native highlanders, there are no adverse effects to staying at high altitude.
The physiological and genetic adaptations in native highlanders involve modification in the oxygen transport system of the blood , especially molecular changes in the structure and functions of hemoglobin , a protein for carrying oxygen in the body. [ 54 ] [ 56 ] This is to compensate for the low oxygen environment . This adaptation is associated with developmental patterns such as high birth weight , increased lung volumes , increased breathing , and higher resting metabolism . [ 57 ] [ 58 ]
The genome of Tibetans provided the first clue to the molecular evolution of high-altitude adaptation in 2010. [ 59 ] Genes such as EPAS1 , PPARA and EGLN1 are found to have significant molecular changes among the Tibetans, and the genes are involved in hemoglobin production . [ 60 ] These genes function in concert with transcription factors, hypoxia inducible factors ( HIF ), which in turn are central mediators of red blood cell production in response to oxygen metabolism. [ 61 ] Further, the Tibetans are enriched for genes in the disease class of human reproduction (such as genes from the DAZ , BPY2 , CDY , and HLA-DQ and HLA-DR gene clusters) and biological process categories of response to DNA damage stimulus and DNA repair (such as RAD51 , RAD52 , and MRE11A ), which are related to the adaptive traits of high infant birth weight and darker skin tone and, are most likely due to recent local adaptation. [ 62 ]
Among the Andeans, there are no significant associations between EPAS1 or EGLN1 and hemoglobin concentration, indicating variation in the pattern of molecular adaptation. [ 63 ] However, EGLN1 appears to be the principal signature of evolution, as it shows evidence of positive selection in both Tibetans and Andeans. [ 64 ] The adaptive mechanism is different among the Ethiopian highlanders. Genomic analysis of two ethnic groups, Amhara and Oromo , revealed that gene variations associated with hemoglobin differences among Tibetans or other variants at the same gene location do not influence the adaptation in Ethiopians. [ 65 ] Instead, several other genes appear to be involved in Ethiopians, including CBARA1 , VAV3 , ARNT2 and THRB , which are known to play a role in HIF genetic functions. [ 66 ]
The EPAS1 mutation in the Tibetan population has been linked to Denisovan -related populations. [ 67 ] The Tibetan haplotype is more similar to the Denisovan haplotype than any modern human haplotype. This mutation is seen at a high frequency in the Tibetan population, a low frequency in the Han population and is otherwise only seen in a sequenced Denisovan individual. This mutation must have been present before the Han and Tibetan populations diverged 2750 years ago. [ 67 ]
Birds have been especially successful at living at high altitudes. [ 68 ] In general, birds have physiological features that are advantageous for high-altitude flight. The respiratory system of birds moves oxygen across the pulmonary surface during both inhalation and exhalation, making it more efficient than that of mammals. [ 69 ] In addition, the air circulates in one direction through the parabronchioles in the lungs. Parabronchioles are oriented perpendicularly to the pulmonary arteries , forming a cross-current gas exchanger . This arrangement allows for more oxygen to be extracted compared to mammalian concurrent gas exchange ; as oxygen diffuses down its concentration gradient and the air gradually becomes more deoxygenated, the pulmonary arteries are still able to extract oxygen. [ 70 ] [ page needed ] Birds also have a high capacity for oxygen delivery to the tissues because they have larger hearts and cardiac stroke volume compared to mammals of similar body size. [ 71 ] Additionally, they have increased vascularization in their flight muscle due to increased branching of the capillaries and small muscle fibres (which increases surface-area-to-volume ratio). [ 72 ] These two features facilitate oxygen diffusion from the blood to muscle, allowing flight to be sustained during environmental hypoxia. Birds' hearts and brains, which are very sensitive to arterial hypoxia, are more vascularized compared to those of mammals. [ 73 ] The bar-headed goose ( Anser indicus ) is an iconic high-flyer that surmounts the Himalayas during migration, [ 74 ] and serves as a model system for derived physiological adaptations for high-altitude flight. Rüppell's vultures , whooper swans , alpine chough , and common cranes all have flown more than 8 km (26,000 ft) above sea level.
Adaptation to high altitude has fascinated ornithologists for decades, but only a small proportion of high-altitude species have been studied. In Tibet, few birds are found (28 endemic species ), including cranes , vultures , hawks , jays and geese . [ 29 ] [ 31 ] [ 75 ] The Andes is quite rich in bird diversity. The Andean condor , the largest bird of its kind in the Western Hemisphere , occurs throughout much of the Andes but generally in very low densities; species of tinamous (notably members of the genus Nothoprocta ), Andean goose , giant coot , Andean flicker , diademed sandpiper-plover , mountain parakeet , miners , sierra-finches and diuca-finches are also found in the highlands. [ 76 ] [ 77 ]
Evidence for adaptation is best investigated among the Andean birds. The water fowls and cinnamon teal ( Anas cyanoptera ) are found to have undergone significant molecular modifications . It is now known that the α-hemoglobin subunit gene is highly structured between elevations among cinnamon teal populations, which involves almost entirely a single non-synonymous amino acid substitution at position 9 of the protein , with asparagine present almost exclusively within the low-elevation species, and serine in the high-elevation species. This implies important functional consequences for oxygen affinity. [ 78 ] In addition, there is strong divergence in body size in the Andes and adjacent lowlands. These changes have shaped distinct morphological and genetic divergence within South American cinnamon teal populations. [ 79 ]
In 2013, the molecular mechanism of high-altitude adaptation was elucidated in the Tibetan ground tit ( Pseudopodoces humilis ) using a draft genome sequence. Gene family expansion and positively selected gene analysis revealed genes that were related to cardiac function in the ground tit. Some of the genes identified to have positive selection include ADRBK1 and HSD17B7 , which are involved in the adrenaline response and steroid hormone biosynthesis . Thus, the strengthened hormonal system is an adaptation strategy of this bird. [ 80 ]
Alpine Tibet hosts a limited diversity of animal species, among which snakes are common. There are only two endemic reptiles and ten endemic amphibians in the Tibetan highlands. [ 75 ] Gloydius himalayanus is perhaps the geographically highest living snake in the world, living at as high as 4,900 m (16,100 ft) in the Himalayas. [ 81 ] Another notable species is the Himalayan jumping spider , which can live at over 6,500 m (21,300 ft) of elevation. [ 29 ]
Many different plant species live in the high-altitude environment. These include perennial grasses , sedges , forbs , cushion plants , mosses , and lichens . [ 82 ] High-altitude plants must adapt to the harsh conditions of their environment, which include low temperatures, dryness, ultraviolet radiation, and a short growing season. Trees cannot grow at high altitude, because of cold temperature or lack of available moisture. [ 83 ] : 51 The lack of trees causes an ecotone , or boundary, that is obvious to observers. This boundary is known as the tree line .
The highest-altitude plant species is a moss that grows at 6,480 m (21,260 ft) on Mount Everest . [ 84 ] The sandwort Arenaria bryophylla is the highest flowering plant in the world, occurring as high as 6,180 m (20,280 ft). [ 85 ] | https://en.wikipedia.org/wiki/Organisms_at_high_altitude |
Most organisms involved in water purification originate from the waste, wastewater or water stream itself or arrive as resting spore of some form from the atmosphere. In a very few cases, mostly associated with constructed wetlands , specific organisms are planted to maximise the efficiency of the process.
Biota are an essential component of most sewage treatment processes and many water purification systems. Most of the organisms involved are derived from the waste, wastewater or water stream itself or from the atmosphere or soil water. However some processes, especially those involved in removing very low concentrations of contaminants, may use engineered eco-systems created by the introduction of specific plants and sometimes animals. Some full scale sewage treatment plants also use constructed wetlands to provide treatment.
Parasites , bacteria and viruses may be injurious to the health of people or livestock ingesting the polluted water . These pathogens may have originated from sewage or from domestic or wild bird or mammal feces . Pathogens may be killed by ingestion by larger organisms, oxidation, infection by phages or irradiation by ultraviolet sunlight unless that sunlight is blocked by plants or suspended solids. [ 1 ]
Particles of soil or organic matter may be suspended in the water. Such materials may give the water a cloudy or turbid appearance. The anoxic decomposition of some organic materials may give rise to obnoxious or unpleasant smells as sulphur containing compounds are released. [ 2 ]
Compounds containing nitrogen , potassium or phosphorus may encourage growth of aquatic plants and thus increase the available energy in the local food-web. this can lead to increased concentrations of suspended organic material. In some cases specific micro-nutrients may be required to allow the available nutrients to be fully utilised by living organisms. In other cases, the presence of specific chemical species may produce toxic effects limiting growth and abundance of living matter. [ 3 ]
Many dissolved or suspended metal salts exert harmful effects in the environment sometimes at very low concentrations. Some aquatic plants are able to remove very low metal concentrations, with the metals ending up bound to clay or other mineral particles.
Saprophytic bacteria and fungi can convert organic matter into living cell mass, carbon dioxide, water and a range of metabolic by-products. These saprophytic organisms may then be predated upon by protozoa , rotifers and, in cleaner waters, Bryozoa which consume suspended organic particles including viruses and pathogenic bacteria. Clarity of the water may begin to improve as the protozoa are subsequently consumed by rotifers and cladocera . [ 4 ] Purifying bacteria, protozoa, and rotifers must either be mixed throughout the water or have the water circulated past them to be effective. Sewage treatment plants mix these organisms as activated sludge or circulate water past organisms living on trickling filters or rotating biological contactors . [ 5 ]
Aquatic vegetation may provide similar surface habitat for purifying bacteria, protozoa, and rotifers in a pond or marsh setting; although water circulation is often less effective. Plants and algae have the additional advantage of removing nutrients from the water; but some of those nutrients will be returned to the water when the plants die unless the plants are removed from the water. Because of the complex chemistry of Phosphorus much of this element is in an unavailable form unless decomposition creates anoxic conditions which render the phosphorus available for re-uptake. Plants also provide shade , a refuge for fish , and oxygen for aerobic bacteria . In addition, fish can limit pests such as mosquitoes . Fish and waterfowl feces return waste to the water, and their feeding habits may increase turbidity . Cyanobacteria have the disadvantageous ability to add nutrients from the air to the water being purified and to generate toxins in some cases.
The choice of organism depends on the local climate different species and other factors. Indigenous species usually tend to be better adapted to the local environment.
The choice of plants in engineered wet-lands or managed lagoons is dependent on the purification requirements of the system and this may involve plantings of varying plant species at a range of depths to achieve the required goal.
Plants purify water by consuming excess nutrients and by providing surfaces upon which a wide range of other purifying organisms can live. They also are effective oxygenators in sunlight. They also have the ability to translocate chemicals between their submerged foliage and their root systems and this is of significance in engineered wet-lands designed to de-toxify waste waters. Plants that have been used in temperate climates include Nymphea alba , Phragmites australis , Sparganium erectum , Iris pseudacorus , Schoenoplectus lacustris and Carex acutiformis . [ citation needed ]
Where oxygenation is a critical requirement Stratiotes aloides , Hydrocharis morsus-ranae , Acorus calamus , Myriophyllum species and Elodea have been used. Hydrocharis morsus-ranae and Nuphar lutea have been used where shade and cover are required.
Fish are frequently the top level predators in a managed treatment eco-system and in some case may simply be a mono-culture of herbivorous species. Management of multi-species fisheries requires careful management and may involve a range of fish species including bottom-feeders and predatory species to limit population growth of the herbivorous fish.
Rotifers are microscopic complex organisms and are filter feeders removing fine particulate matter from water. They occur naturally in aerobic lagoons, activated sludge processes, in trickling filters and in final settlement tanks and are a significant factor in removing suspended bacterial cells and algae from the water column. [ 6 ] [ 7 ]
Annelid worms are essential to the effective operation of trickling filters [ 8 ] helping to remove excess bio-mass and enhancing natural sloughing of the bio-film. Supernumerary worms are very commonly found in the drainage troughs around trickling filters and in the final settlement sludge. Annelids also play a key role in lagoon treatment systems and in the effective working or engineered wet-lands. In this environment worms are a principal force in mixing in the upper few centimetres of the sediment layer exposing organic material to both oxidative and anoxic environments aiding the complete breakdown of most organics. They are also a key ingredient in the food-chain transferring energy upwards to fish and aquatic birds.
The range of protozoan species found is very wide but may include species of the following:
Bacteria are probably the most significant group of organisms involved in water purification and are ubiquitous in all biological purification environments. Some such as Sphaerotilus natans are typically associated with grossly polluted waters, but even in such environments the bacteria are degrading the organic material present. | https://en.wikipedia.org/wiki/Organisms_involved_in_water_purification |
Organix Inc is a US fine chemicals company specialising in chemical synthesis of analytical standards and custom synthesis of finished compounds and intermediates. [ citation needed ]
Organix carries out research and development of novel molecules used in a variety of pharmaceutical research applications. Some notable compounds include; | https://en.wikipedia.org/wiki/Organix_Inc |
The organization and expression of immunoglobulin genes are fundamental processes that enable the adaptive immune system to produce a vast repertoire of antibodies, essential for recognizing and neutralizing diverse antigens .
Antibody (or immunoglobulin) quaternary structure is made up of two heavy-chains and two light-chains . These chains are held together by disulfide bonds . [ 1 ] The arrangement of genes and processes that put together different parts of antibody molecules play important roles in antibody diversity and production of different classes or subclasses of antibodies. The organization of genes is relatively conserved in stem cell precursors, and processes take place during the development and differentiation of B cells that lead to many different arrangements of variable segments. [ 2 ] That is, the controlled gene expression during transcription and translation coupled with the rearrangements of immunoglobulin gene segments result in the generation of antibody repertoire during development and maturation of B cells.
During the development of B cells, the immunoglobulin gene undergoes sequences of rearrangements that lead to formation of the antibody repertoire. For example, in the early stages of transition from pro-B cell to pre-B cell, a partial rearrangement of the heavy-chain gene occurs which is followed by complete rearrangement of heavy-chain gene. [ 3 ] At this stage (Pre-B cell), the μ heavy chain and surrogate light chain are formed. [ 4 ] The final rearrangement of the light chain gene generates immature B cell and membrane-bound IgM (mIgM). [ 5 ] The process explained here occurs during development of naïve B cells, prior to exposure to exogenous antigens. The mature B cell formed as a result of these processing changes leaves the bone marrow and may then be stimulated by an antigen to develop into antibody-secreting plasma cells. [ 6 ] Also at first, the mature B cell expresses membrane-bound IgD and IgM. These two classes could switch to secretory IgD and IgM during the processing of mRNAs.
Finally, further class switching follows as the cell continues to divide and differentiate. For instance, a B cell expressing IgM can switch to IgG , IgA , or IgE depending on the stimulus provided (which may be dependent upon the antigenic source and the responding immune cells). [ 7 ]
From studies and predictions such as Dreyer and Bennett's, [ 8 ] it shows that the light chains and heavy chains are encoded by separate multigene families on different chromosomes . They are referred to as gene segments and are separated by non-coding regions. The rearrangement and organization of these gene segments during the maturation of B cells produce functional proteins. [ 9 ] The entire process of rearrangement and organization of these gene segments is the vital source where our body immune system gets its capabilities to recognize and respond to variety of antigens.
The light chain gene has three gene segments. These include: the light chain variable region (V), joining region (J), and constant region (C) gene segments. The variable region of light is therefore encoded by the rearrangement of VJ segments. The light chain can be either kappa,κ or lambda,λ. This process takes place at the level of mRNAs processing. Random rearrangements and recombinations of the gene segments at DNA level to form one kappa or lambda light chain occurs in an orderly fashion. As a result, "a functional variable region gene of a light chain contains two coding segments that are separated by a non-coding DNA sequence in unrearranged germ-line DNA". [ 10 ]
Heavy chain contains similar gene segments such as V H , J H , and C H , but also has another gene segment called D (diversity). Unlike the light chain multigene family, VDJ gene segments code for the variable region of the heavy chain. The rearrangement and reorganization of gene segments in this multigene family is more complex . The rearranging and joining of segments produced different end products because these are carried out by different RNA processes. The same reason is why the IgM and IgG are generates at the time. [ 11 ]
The variable region rearrangements happen in an orderly sequence in the bone marrow. [ 12 ] Usually, the assortment of these gene segments occurs at B cell maturation. [ 13 ]
The kappa and lambda light chains undergo rearrangements of the V and J gene segments. In this process, a functional Vlambda can combine with four functional Jλ –Cλ combinations. On the other hands, Vk gene segments can join with either one of the Jk functional gene segments. The overall rearrangements result in a gene segment order from 5 prime to 3 prime end. These are a short leader (L) exon, a noncoding sequence (intron), a joined VJ segment, a second intron, and the constant region. There is a promoter upstream from each leader gene segment. The leader exon is important in the transcription of light chain by the RNA polymerase. This sequence is translated to guide the polypeptide to the internal environment of the endoplasmic reticulum, but is cleaved and absent from the mature protein. [ 14 ] To remain with coding sequence only, the introns are removed during RNA- processing and repairing. [ 15 ]
The rearrangements of heavy-chains are different from the light chains because DNA undergoes rearrangements of V-D-J gene segments in the heavy chains. These reorganizations of gene segments produce gene sequence from 5 prime to 3 prime ends such as a short leader exon, an intron, a joined VDJ segment, a second intron and several gene segments. The final product of the rearrangement is transcribed when RNA polymerase
It is understood that rearrangement occurs between specific sites on the DNA called recombination signal sequences (RSSs). The signal sequences are composed of a conserved palindromic heptamer and a conserved AT- rich nonamer. These signal sequences are separated by non-conserved spacers of 12 or 23 base pairs called one-turn and two-turn respectively. They are within the lambda chain, k-chain and the processes of rearrangement in these regions are catalyzed by two recombination-activating genes: RAG-1 and RAG-2 and other enzymes and proteins. The segments joined due to signals generated RSSs that flank each V, D, and J segments. Only genes flank by 12 -bp that join to the genes flank by 23-bp spacer during the rearrangements and combinations to maintain VL-JL and VH-DH-JH joining.
Antibody diversity is produced by genetic rearrangement after shuffling and rejoining one of each of the various gene segments for the heavy and light chains. Due to mixing and random recombination of the gene segments errors can occur at the sites where gene segments join with each other. These errors are one of the sources of the antibody diversity that is commonly observed in both the light and heavy chains. Moreover, when B cells continue to proliferate, mutations accumulate at the variable regions through a process called somatic hypermutation. The high concentrations of these mutations at the variable region also produce high antibody diversity.
When the B cells get activated, class switching can occur. The class switching involves switch regions that made up of multiple copies of short repeats (GAGCT and TGGGG). These switches occur at the level of rearrangements of the DNA because there is a looping event that chops off the constant regions for IgM and IgD and form the IgG mRNAs . Any continuous looping occurrence will produce IgE or IgA mRNAs. In addition, cytokines are factors that have great effects on class switching of different classes of antibodies. Their interaction with B cells provides the appropriates signals needed for B cells differentiation and eventual class switching occurrence. For example, interleukin-4 induces the rearrangements of heavy chain immunoglobulin genes. That is IL- 4 induces the switching of Cμ to Cγ to Cκ | https://en.wikipedia.org/wiki/Organization_and_expression_of_immunoglobulin_genes |
Organization for Bat Conservation (OBC) was a national environmental education nonprofit based in Bloomfield Hills, Michigan , established to educate and inspire people to save bats. It was the largest grassroots bat conservation organization in the United States. [ 1 ] In February 2018, it was announced that the organization was ceasing operations due to unexpected financial problems and personnel changes. [ 2 ]
Founded in 1992, OBC was a leading environmental educator focused on bats. Its home base was at the Cranbrook Institute of Science in Bloomfield Hills, Michigan, where OBC operated the Bat Zone, a live animal center with approximately 200 animals including bats from around the world and other nocturnal animals. Each year, thousands of visitors came to the Bat Zone to attend tours and participate in live animal educational programs. OBC educators traveled throughout the country to present education programs to children and adults at schools, festivals, museums, science and nature centers each year. [ 1 ]
OBC attained non-profit status in August 1997. [ 3 ]
OBC also organized and participated in several special events. The Annual Great Lakes Bat Festival, started in 2002, was created to celebrate the role of bats in the Great Lakes ecosystem as insect eaters, while dispelling misconceptions that generate fears and threaten bats and their habitats around the world. The goal of the festival is to help people understand the impact to natural ecosystems and human economies should bat populations continue to decline.
In September 2014, OBC launched a new public action campaign called Save the Bats. Save the Bats is aimed at preventing the decline of bat populations. Save the Bats encourages people to take local action to conserve bats, including installing bat houses, planting wildlife gardens, and teaching others about the importance of bats. The campaign has many celebrity, government agency and corporate supporters.
In addition, OBC and Warner Brothers Entertainment worked together on the set of Batman v Superman: Dawn of Justice to re-purpose parts of the movie set into bat houses. Director Zack Snyder contacted OBC when he heard about bats dying off from white-nose syndrome and enlisted Rob Mies, OBC Executive Director, to assist in the bat house design and construction. More than 150 bat houses were made on the movie set in Pontiac, some of which were painted and signed by Snyder, Amy Adams, and Ben Affleck. [ 4 ] The bat houses were auctioned off to support the Save the Bats campaign. [ 4 ] Warner Brothers released a short PSA documenting the bat house build featuring Affleck encouraging people to join the campaign. [ 5 ] To date, more than 1,000,000 people have viewed the video.
In August 2017, Organization for Bat Conservation purchased the mineral rights for Magazine Mine, a silica mine in southern Illinois . [ 6 ] [ 7 ] Magazine Mine is one of the largest Indiana bat hibernacula in the world. [ 8 ]
After 15 years at the Cranbrook Institute of Science in Bloomfield Hills, OBC moved the Bat Zone to a larger location in Pontiac in June 2017. [ 5 ] [ 9 ] The 10,000 square foot building used to be a First Federal Bank, on west Huron Street near Woodward Avenue. [ 5 ]
Rob Mies was removed from his position as Executive Director of OBC in February 2018. Shortly after his dismissal, OBC announced that it would cease operation due to unexpected financial struggles. Mies disagreed with the decision to dissolve OBC and called for the resignation of its board members to save the organization. [ 2 ]
61 animals were transferred to the Detroit Zoo , and the organization worked with other institutions to find homes for over 200 animals. [ 10 ] | https://en.wikipedia.org/wiki/Organization_for_Bat_Conservation |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.