2-hydroxypyridine 5-monooxygenase
In enzymology, a 2-hydroxypyridine 5-monooxygenase (EC ) is an enzyme that catalyzes the chemical reaction


2-hydroxypyridine 5-monooxygenase
The 3 substrates of this enzyme are 2-hydroxypyridine, an electron acceptor AH2, and O2, whereas its 3 products are 2,5-dihydroxypyridine, the reduction product A, and H2O.


2-hydroxypyridine 5-monooxygenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on paired donors, with O2 as oxidant and incorporation or reduction of oxygen. The oxygen incorporated need not be derive from O miscellaneous. The systematic name of this enzyme class is 2-hydroxypyridine,hydrogen-donor:oxygen oxidoreductase (5-hydroxylating). This enzyme is also called 2-hydroxypyridine oxygenase.


2-hydroxyquinoline 5,6-dioxygenase
In enzymology, a 2-hydroxyquinoline 5,6-dioxygenase (EC ) is an enzyme that catalyzes the chemical reaction


2-hydroxyquinoline 5,6-dioxygenase
The 4 substrates of this enzyme are quinolin-2-ol, NADH, H+, and O2, whereas its two products are 2,5,6-trihydroxy-5,6-dihydroquinoline and NAD+.


2-hydroxyquinoline 5,6-dioxygenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on paired donors, with O2 as oxidant and incorporation or reduction of oxygen. The oxygen incorporated need not be derived from O2 with NADH or NADPH as one donor, and incorporation of two atoms o oxygen into the other donor. The systematic name of this enzyme class is quinolin-2-ol,NADH:oxygen oxidoreductase (5,6-hydroxylating). Other names in common use include 2-oxo-1,2-dihydroquinoline 5,6-dioxygenase, quinolin-2-ol 5,6-dioxygenase, and quinolin-2(1H)-one 5,6-dioxygenase.


2-hydroxyquinoline 8-monooxygenase
In enzymology, a 2-hydroxyquinoline 8-monooxygenase (EC ) is an enzyme that catalyzes the chemical reaction


2-hydroxyquinoline 8-monooxygenase
The 4 substrates of this enzyme are quinolin-2-ol, NADH, H+, and O2, whereas its 3 products are quinolin-2,8-diol, NAD+, and H2O.


2-hydroxyquinoline 8-monooxygenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on paired donors, with O2 as oxidant and incorporation or reduction of oxygen. The oxygen incorporated need not be derived from O2 with NADH or NADPH as one donor, and incorporation of one atom o oxygen into the other donor. The systematic name of this enzyme class is quinolin-2(1H)-one,NADH:oxygen oxidoreductase (8-oxygenating). This enzyme is also called 2-oxo-1,2-dihydroquinoline 8-monooxygenase. It employs one cofactor, iron.


2-hydroxyquinoline 8-monooxygenase, Structural studies
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes , , and .


2-iminoacetate synthase
2-iminoacetate synthase (EC , thiH (gene)) is an enzyme with systematic name L-tyrosine 4-methylphenol-lyase (2-iminoacetate-forming). This enzyme catalyses the following chemical reaction


2-in-1 PC
A 2-in-1 PC, also known as convertible laptop, 2-in-1 tablet, 2-in-1 laptop, 2-in-1 detachable, laplet, tabtop, laptop tablet, or simply 2-in-1, is a portable computer that has features of both tablets and laptops.


2-in-1 PC
Before the emergence of 2-in-1s and their denomination as such, technology journalists used the words convertible and hybrid to denominate pre-2-in-1 portable computers: Convertible typically denominated those that featured a mechanism to conceal the physical keyboard by sliding or rotating it behind the chassis, and hybrid those that featured a hot-pluggable, complementary, physical keyboard. Both pre-2-in-1 convertibles and hybrids were crossover devices that combined features of both tablets and laptops. The later 2-in-1 PCs comprise a category that is a sibling to both the pre-2-in-1 convertibles and hybrids. Models of 2-in-1 PC were each similarly denominated either a 2-in-1 convertible or 2-in-1 detachable, respectively, and despite borrowing the terminology of the pre-2-in-1 PCs, the two species of 2-in-1 PCs are distinct from the two species of pre-2-in-1 PCs because 2-in-1 PCs have additional features of traditional laptops.


2-in-1 PC
2-in-1 PCs consist of portable computer components within light and thin chassis, and exemplify technological convergence. They are convenient for media consumption and non-intensive tasks in tablet mode yet useful for content production in laptop mode.


2-in-1 PC, Form-factors, 2-in-1 convertible
2-in-1 convertibles are tablets with the ability to rotate, fold, or slide the keyboard behind the display. On most devices, the hinge is situated at the display and keyboard junction. However, the Dell XPS Duo is unique in that the display sits in a frame that allows the screen to be spun.


2-in-1 PC, Form-factors, 2-in-1 convertible, Netvertible
Rotational-convertible format is where in addition to the conventional hinging action, the central single hinge mechanism is also able to rotate about a central axis perpendicular to the keyboard surface, such that the laptop can be turned into a thick tablet. Most netvertibles have the option to support active (electromagnetic) stylus and/or touch screen (resistive or capacitive), some being ruggedized such as Panasonic Toughbook CF series. Other examples include Toshiba Portege M7xx, Fujitsu LifeBook T series and HP EliteBook Revolve series.


2-in-1 PC, Form-factors, 2-in-1 detachable
2-in-1 detachable are devices with detachable keyboards. In most cases, the keyboard part provides few, if any, additional features (most often a touchpad, as in the HP Spectre x2). However, the keyboards of some detachable provide additional features similar to those of a docking station such as additional I/O-ports and supplementary batteries. For instance, the Surface Book can leverage the discrete GPU in the keyboard upon the keyboard's connection.


2-in-1 PC, Form-factors, 2-in-1 detachable
When connected to the keyboard, the display of the detachable can either be free-standing on the hinge or require external support, often in the form of a kickstand. Novel ways of providing external support include the bending frame and locking mechanism of the HP Spectre x2.


2-in-1 PC, Form-factors, 2-in-1 detachable
Though the keyboard is usually bundled with the purchase of a 2-in-1 detachable, it is occasionally deemed an optional accessory by manufacturers in order to minimize the starting price of a device. In such cases the 2-in-1 detachable is often displayed with its complementary keyboard in advertisements and promotional materials. This is true for all devices of the Surface and Surface Pro lines.


2-in-1 PC, Distinction from traditional tablets and laptops
2-in-1s fall in the category of hybrid or convertible tablets but are distinct in that they run a full-featured desktop operating system and have I/O ports typically found on laptops, such as USB and DisplayPort. The most prominent element is the keyboard that allows the 2-in-1 to provide the ergonomic typing experience of a laptop.


2-in-1 PC, Distinction from traditional tablets and laptops
While 2-in-1s fall in a category distinct from laptops, they loosely parallel the traits of the Ultrabook device category, having light and thin chassis, power-efficient CPUs, and long battery lives. They are distinguished from traditional Ultrabooks by the inclusion of a touchscreen display and a concealable or detachable keyboard.


2-in-1 PC, Examples
The earliest device that can be considered a 2-in-1 detachable is the Compaq Concerto from 1993. It came preinstalled with Windows 3.1 and Windows for PEN, and had a cabled detachable keyboard, and battery powered stylus. In June 1994, IBM introduced the ThinkPad 360P, which included a pen-operated display that used PenDOS 2.2.


2-in-1 PC, Examples
Mainstream attention for 2-in-1 PCs was not achieved until nearly two decades later, when many manufactures showed devices, at that time referred to as "hybrid" devices, at CES 2011. While Packard Bell, Acer and HP all had Microsoft Windows based 2-in-1s by 2011, Lenovo released the well reviewed Windows 2-in-1: The X220t variant of the ThinkPad X220, successor of 2006's IBM ThinkPad x41t. The 12-inch device included a digital stylus housed within the chassis, somewhat ruggedized construction, and a multi-touch screen with a twist and fold hinge.


2-in-1 PC, Examples
Microsoft started its own line of 2-in-1s with the introduction of the Surface Pro series, the first of which was released in February 2013. It had a 10.6-inch (27 cm) display, Intel Core i5 CPU, and included the Pro Pen stylus and a detachable keyboard that doubled as a protective screen cover. In 2015 Microsoft introduced the Surface Book series, which, similar to the Surface Pro series, features a detachable keyboard cover and Surface Pen stylus.


2-in-1 PC, Examples
Samsung entered the 2-in-1 PC market with the release of the Windows-based Samsung Galaxy TabPro S, which was released in March 2016. It had a 12-inch display, Intel Core m3 CPU, a first-party keyboard attachment, and a TabPro Pen. Its successor, the Galaxy Book, was released in February 2017. Coming in a 10.6-inch model and a 12-inch model, the Galaxy Book has an improved detachable keyboard and include an S Pen.


2-in-1 PC, Examples
Google entered the 2-in-1 market after it announced the Pixel Slate in October 2018. It runs on Chrome OS and features a 12.3-inch display. It includes two USB-C ports, but it omits the headphone jack. The featured Pixel Keyboard and Pixelbook Pen are sold separately.


2-in-1 PC, Examples
Since 2012, a number of other prominent laptop manufacturers, such as Dell, Asus, and Sony have also begun releasing their own 2-in-1s.


2-in-1 PC, Examples
While the iPad Pro has optional Smart Keyboard and Apple Pencil accessories, Apple has yet to release a true 2-in-1 PC with similar to HP Pavilion x2 form-factor and with a similar desktop OS, citing the quote below.


2-in-1 PC, Criticism
In April 2012 Apple's CEO Tim Cook, answering to the question of the researcher Anthony Sacconaghi about a possible hybrid of iPad and MacBook, compared a 2-in-1 to a combination of "a toaster and a refrigerator" that "doesn’t please anyone":


2-in-1 PC, Criticism
I think, Tony, anything can be forced to converge. But the problem is that products are about trade-offs, and you begin to make trade-offs to the point where what you have left at the end of the day doesn’t please anyone. You can converge a toaster and a refrigerator, but those things are probably not going to be pleasing to the user … you wouldn’t want to put these things together because you wind up compromising in both and not pleasing either user. Some people will prefer to own both, and that’s great, too. But I think to make the compromises of convergence, so — we’re not going to that party. Others might. Others might from a defensive point of view, particularly. But we’re going to play in both.


2-in-1 PC, Criticism
2-in-1s are natively supported by the Microsoft Corporation's Windows, and Google, Inc's Chrome OS operating systems. Various other Linux distributions also support some touch features of 2-in-1s, though they are generally unsupported by hardware vendors.


2-inch medium mortar
The 2 inch medium trench mortar, also known as the 2-inch howitzer, and nicknamed the "toffee apple" or "plum pudding" mortar, was a British smooth bore muzzle loading (SBML) medium trench mortar in use in World War I from mid-1915 to mid-1917. The designation "2-inch" refers to the mortar barrel, into which only the 22-inch bomb shaft but not the bomb itself was inserted; the spherical bomb itself was actually 9 inches (230 mm) in diameter and weighed 42 lb (19 kg), hence this weapon is more comparable to a standard mortar of approximately 5-6 inch bore.


2-inch medium mortar, Background and predecessors
As the Western Front in France and Belgium stagnated into trench warfare in late 1914, British forces found themselves with no means of replying to the German minenwerfers (trench mortars) which were lobbing both small and large (over 100 pound) high-explosive shells into their frontline trenches from short range. British commanders requested an accurate short-range weapon which was manually portable in the trenches, could be safely used to attack enemy trenches as close as 100 yards to the British trenches, was easily concealed and projected a reasonably large explosive charge capable of damaging protected enemy positions. The British Expeditionary Force had been expected to participate only in mobile warfare and was not equipped with any mortars.


2-inch medium mortar, Background and predecessors
Various alternative designs for light and medium mortars were evaluated, prompted by the need to place at least some weapon into action without diverting manufacturing capacity from guns and howitzers, which weapons were given priority. Hence the emphasis was on designs for both mortar and ammunition that could be manufactured by small unsophisticated workshops unsuited to other war work:


2-inch medium mortar, The 2-inch medium mortar
This was designed and manufactured by the Royal Ordnance Factories in early 1915 and introduced along with the 1.57-inch mortar in March 1915. It incorporated what was known of the German prewar Krupp mortar. This was the first design to meet all the requirements, after modifications to simplify manufacture; it fired a good-sized spherical cast-iron bomb of 42 pounds (total projectile weight 51 pounds with stick and fuze), considered the largest practical size for use from trenches, at ranges from 100 to 600 yards using a simple 2-inch tube as the mortar body.


2-inch medium mortar, The 2-inch medium mortar
The mortar and ammunition could be cheaply manufactured by small unsophisticated "trade" workshops; the bomb was safely detonated by a standard No. 80 "time and percussion" artillery fuze. Drawbacks were that the steel tail was usually projected backwards towards the firer when the bomb detonated, resulting in occasional casualties and the No. 80 fuze was also required by the 18-pounder field guns which were given priority, limiting mortar ammunition supply to the front until early 1916 when a special cheap trench mortar fuze was developed.


2-inch medium mortar, The 2-inch medium mortar
The 2-inch mortar served in limited numbers in France in 1915 from March, with early mortars and ammunition made by the Royal Ordnance Factory, with the Vickers 1.57-inch model. Mass production began with an order in August 1915 for 800 mortars from several railway workshops and agricultural machinery makers, together with an order for 675,000 bombs from numerous small firms. Manufacture of the Vickers model which was twice as expensive was ended and it was withdrawn by January 1917.


2-inch medium mortar, The 2-inch medium mortar
It fired a spherical cast-iron bomb "the size of a football" painted dirty white filled with amatol (identified by a painted green band) or ammonal (identified by a painted pink band) attached to the end of a pipe ("stick"), hence the nicknames "toffee apple" and "plum pudding". Weights of bombs as delivered without fuzes varied. Light bombs, from 39 lb 14 oz to below 40 lb 10 oz (18.09 to 18.43 kg), were marked with a stenciled L. Heavy bombs, above 41 lb 10 oz to 42 lb 6 oz (18.43 to 19.22 kg) were marked with a stenciled Hv. Hence the total fuzed weight with stick of 51 lb is an average.


2-inch medium mortar, The 2-inch medium mortar
The 2-inch designation refers to the mortar barrel's bore and the projectile stick inserted into it, not the much larger bomb itself which remained outside the barrel. It was comparable in explosive power if not range to other 4-inch mortars.


2-inch medium mortar, The 2-inch medium mortar
The 2-inch mortar was itself superseded by the Newton 6-inch mortar from mid-1917 onwards. Some Australian units retained them for projecting smoke screens.


2-inch medium mortar, Combat use, Deployment
The weapon was initially operated by joint infantry and artillery detachments, eventually it became the responsibility of the Royal Field Artillery. A typical infantry division was equipped with three batteries designated X, Y, Z, each with four mortars.


2-inch medium mortar, Combat use, Use in action
Its primary use was in cutting barbed wire defenses and attacking enemy front line trenches, such as in the July 1916 attack on the Somme. The spherical shape and relatively low velocity brought the benefit that the bomb did not penetrate the ground before exploding. The short range was a disadvantage as it could only be used if no man's land (between the British and enemy front line) was relatively narrow. It was used to fire some white star (50%-50% chlorine and phosgene) gas bombs during the Battle of the Somme until other specialised longer range projectors became available.


2-inch medium mortar, Combat use, Use in action
Cordite charges appropriate to the required range were dropped into the barrel before the bomb was loaded. Charges and ranges:


2-inch medium mortar, Combat use, Use in action
The original design for igniting the powder propellant charge involved the insertion of a standard artillery "T friction tube" into a hole near the base of the barrel. The Royal Artillery had a higher priority in receiving the already insufficient number of tubes so ignition was changed to use a Lee–Enfield bolt mechanism and chamber screwed into a socket in the barrel near the base. A special blank rifle cartridge was loaded and fired via a lanyard from a sheltered position if possible due to the risk of bombs falling short. This ignited the propellant charge and launched the bomb.


2-inch medium mortar, Combat use, Use in action
In early use it was situated in frontline trenches but this tended to attract enemy fire onto the troops manning them. Standard procedure became to locate the mortars separately from frontline trenches, in unoccupied trenches or in saps running off the frontline. This had the benefit of drawing enemy fire away from troops manning the front line.


2-inch medium mortar, Combat use, Use in action
Estimated rounds required for various targets, with instantaneous percussion fuze 107:


2-inch medium mortar, Combat use, Silencer
Provision was made in mid-1916 for attachment of the "Temple silencer" at the muzzle, intended to reduce the flash and noise generated on firing, which at the short ranges the mortar operated at was quickly noticed by the enemy and invited artillery response. This required the use of bombs with a special piston attached to the tail which was retained in the barrel by the silencer on firing, and hence sealed the muzzle after the bomb tail left the barrel. This had the major disadvantage of causing the barrel to overheat during prolonged daylight firing, and the silencer was only used at night.


2-inch medium mortar, Combat use, Use as anti-tank mines
In Spring 1918 many of these obsolescent bombs were buried on the Western Front under metal plates as anti-tank mines in expectation of attack by German tanks. This led to some later confusion as to whether unearthed bombs were unexploded mortar projectiles ("duds") or undetonated mines.


2-inch medium mortar, Combat use, Use as anti-tank mines
These minefields were inadequately documented. This caused the British problems in the closing months of the war when they had to advance again over territory they had previously abandoned and also prevented full clearance of the minefields after the war. This led to some French farmers being blown up in the 1930s when they started using tractors e.g. around Gouzeaucourt.


2-isopropylmalate synthase
In enzymology, a 2-isopropylmalate synthase (EC ) is an enzyme that catalyzes the chemical reaction


2-isopropylmalate synthase
The three substrates of this enzyme are acetyl-CoA, 3-methyl-2-oxobutanoate, and H2O, and its products are (2S)-2-isopropylmalate and CoA.


2-isopropylmalate synthase
The enzyme belongs to the family of transferases, specifically those acyltransferases that convert acyl groups into alkyl groups on transfer. The systematic name of this enzyme class is acetyl-CoA:3-methyl-2-oxobutanoate C-acetyltransferase (thioester-hydrolysing, carboxymethyl-forming). Other names in common use include 3-carboxy-3-hydroxy-4-methylpentanoate 3-methyl-2-oxobutanoate-lyase, (CoA-acetylating), alpha-isopropylmalate synthetase, alpha-isopropylmalate synthase, alpha-isopropylmalic synthetase, isopropylmalate synthase, and isopropylmalate synthetase. This enzyme participates in biosynthesis of L-leucine and pyruvate metabolism. Monovalent and divalent cation activation have been reported for enzymes from different sources.


2-isopropylmalate synthase
Mycobacterium tuberculosis α-isopropylmalate synthase requires a divalent metal ion, of which Mg2+ and Mn2+ give highest activity, and a monovalent cation, with K+ as the best activator. Zn2+ was shown to be an inhibitor, contrary to what was assumed from the structural data. In addition to the complex requirements for a divalent metal and further activation by K+, M. tuberculosis α-isopropylmalate synthase follows a random kinetic mechanism for catalysis. Another feature of the M. tuberculosis homolog is that L-leucine, the feedback inhibitor, inhibits the enzyme in a time-dependent fashion. This was the first demonstration of a feedback inhibitor that displays slow-onset inhibition.


2-isopropylmalate synthase, Structural studies
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code .


2-lane expressway
The term 2-lane expressway may refer to 2 similar grades of highway:


2-meter band
The 2-meter amateur radio band is a portion of the VHF radio spectrum that comprises frequencies stretching from 144 MHz to 148 MHz in International Telecommunication Union region (ITU) Regions 2 (North and South America plus Hawaii) and 3 (Asia and Oceania)and from 144 MHz to 146 MHz in ITU Region 1 (Europe, Africa, and Russia). The license privileges of amateur radio operators include the use of frequencies within this band for telecommunication, usually conducted locally within a range of about 100 miles (160 km).


2-meter band, Amateur radio
The Radio Regulations of the International Telecommunication Union allows amateur radio operations in the frequency range from 144 to 148 MHz. subject to variation by country.


2-meter band, Operation
Because it is local and reliable, and because the licensing requirements to transmit on the 2-meter band are easy to meet in many parts of the world, this band is one of the most popular non-HF ham bands. This popularity, the compact size of needed radios and antennas, and this band's ability to provide easy reliable local communications also means that it is also the most used band for local emergency communications efforts, such as providing communications between Red Cross shelters and local authorities. In the US, that role in emergency communications is furthered by the fact that most amateur-radio operators have a 2-meter handheld transceiver (HT), handie-talkie, or walkie-talkie.


2-meter band, Repeaters and FM
Much of 2-meter FM operation uses a radio repeater, a radio receiver and transmitter that instantly retransmits a received signal on a separate frequency. Repeaters are normally located in high locations such as a tall building or a hill top overlooking expanses of territory. On VHF frequencies such as 2-meters, antenna height greatly influences how far one can talk. Typical reliable repeater range is about 25 miles (40 km). Some repeaters in unusually high locations, such as skyscrapers or mountain tops, can be usable as far out as 75 miles (121 km).


2-meter band, Repeaters and FM
Reliable range is very dependent on the height of the repeater antenna and also on the height and surroundings of the handheld or mobile unit attempting to access to the repeater. Line of sight would be the ultimate in reliability. The typical hand held two meter FM transceiver produces about 5 watts of transmit power. Stations in a car or home provide higher power, 25 to 75 watts, and may use a simple vertical antenna mounted on a pole or on the rooftop of a house or a vehicle.


2-meter band, Repeaters and FM
Even without repeaters available, however, the 2-meter band provides reliable crosstown communications throughout smaller towns, making it ideal for emergency communications. Antennas for repeater work are almost always vertically polarized since 2-meter antennas on cars are usually vertically polarized. Matching polarization allows for maximum signal coupling which equates to stronger signals in both directions. Simple radios for FM repeater operation have become plentiful and inexpensive in recent years.


2-meter band, Communications beyond 50 miles
While the 2 meter band is best known as a local band using the FM mode, there are many opportunities for long distance (DX) communications using other modes. A well-placed antenna and high-power equipment can achieve distances of up to a few hundred miles, and fortuitous propagation conditions called "signal enhancements" can on occasion reach across oceans.


2-meter band, Communications beyond 50 miles
The typical 2 meter station using CW (Morse code) or SSB (single side band) modes consists of a radio driving a power amplifier generating about 200–500 Watts of RF power. This extra power is usually fed to a multi-element, compound antenna, usually a Yagi-Uda or Yagi, which can beam most of the signal power towards the intended receiving station. "Beam antennas" provide substantial increase in signal directivity over ordinary dipole or vertical antennas. Antennas used for distance work are usually horizontally polarized instead of the vertical polarization customarily used for local contacts.


2-meter band, Communications beyond 50 miles
Stations that have antennas located in relatively high locations with views (from the antenna) clear to the horizon have a big advantage over other stations. Such stations are able to communicate 100–300 miles (160–480 km) consistently. It is usual for them to be heard at distances far beyond line of sight on a daily basis without help from signal enhancements. Signal enhancements are unusual circumstances in the atmosphere and ionosphere that bend the signal path into an arc that better follows the curve of the Earth, instead of the radio waves traveling in the usual straight line off into space. The best known of these are:


2-meter band, Communications beyond 50 miles
These and other well-known forms of VHF signal enhancement that allow trans-oceanic and trans-continental contacts on 2 meters are described in the subsections that follow within this section.


2-meter band, Communications beyond 50 miles
With the exception of sporadic E, directional antennas such as Yagis or log periodic antennas are almost essential to take advantage of signal enhancements. When a well-equipped station with its antenna well-located “high and in the clear” is operating during a signal enhancement, astonishing distances can be bridged, momentarily approaching what is regularly possible on shortwave and mediumwave.


2-meter band, Communications beyond 50 miles, Tropospheric ducting
Occasionally, signal bending in the atmosphere's troposphere known as tropospheric ducting can allow 2 meter signals to carry hundreds or even thousands of kilometers as evidenced by the occasional 2 meter contact between the west coast of the United States and the Hawaiian Islands, the northeast region to the Florida coast, and across the Gulf of Mexico. These so-called "Openings" are generally first spotted by amateurs operating SSB (Single Side Band) and CW (Continuous Wave) modes since amateurs using these modes typically are attempting distance contacts (DX) and alert for signal enhancement events.


2-meter band, Communications beyond 50 miles, Tropospheric ducting
Completion of contacts using these weak signal modes involves the exchange of signal level reports and location by grid square which is known as the Maidenhead Locator System. Two way ducting contacts can have very strong signals and are often made with moderate power, small antennas, and other types of modes. Long distance ducting contacts do occur using FM modes as well but for the most part go unnoticed by many FM operators.


2-meter band, Communications beyond 50 miles, Sporadic E
Another form of VHF propagation is called Sporadic E propagation. This is a phenomenon whereby radio signals are reflected back towards Earth by highly ionized segments of the ionosphere which can facilitate contacts in excess of 1,000 miles (1,600 km) with very strong signals received by both parties.


2-meter band, Communications beyond 50 miles, Sporadic E
Unlike some other long distance modes, high power and large antennas are often not required to make contact with distant stations via a sporadic E event. A two-way conversation can take place over a distance of several hundred kilometers or more, often using low levels of RF power. Sporadic E is a rare and completely random propagation phenomenon lasting anywhere from a matter of minutes to several hours.


2-meter band, Communications beyond 50 miles, Satellite communications
Satellites are basically repeater stations in orbit. The 2 meter band is also used in conjunction with the 70-centimeter band, or the 10-meter band and various microwave bands via orbiting amateur radio satellites. This is known as cross band repeating. On-board software defines what mode or band is in use at any particular time and this is determined by amateurs at so-called earth stations who control or instruct the satellite behavior. Amateurs know what mode is in use via published internet schedules.


2-meter band, Communications beyond 50 miles, Satellite communications
For instance, a favorite mode is Mode "B" or "V/U" which simply indicates the uplink and downlink frequencies or bands the satellite is currently using. In this example, V/U means VHF/UHF or VHF uplink with UHF downlink. Most amateur satellites are Low Earth Orbit satellites, or LEO's as they are affectionately known, and generally are at about 450 miles (700 km) altitude. At that height, amateurs can expect reception distances of up to around 3,000 miles (4,800 km).


2-meter band, Communications beyond 50 miles, Satellite communications
A few amateur satellites have very high elliptical orbits. These satellites can reach altitudes of 30,000 miles (50,000 km) above the earth where an entire hemisphere is visible providing outstanding communications capabilities from any two points on the earth within line of sight of the satellite; distances that are far beyond the reach of the LEOs.


2-meter band, Communications beyond 50 miles, Transequatorial propagation
Transequatorial propagation, also known as (TEP), is a regular daytime occurrence on the 2 meter band over the equatorial regions and is common in the temperate latitudes in late spring, early summer and, to a lesser degree, in early winter. For receiving stations located within ± 10 degrees of the geomagnetic equator, equatorial E-skip can be expected on most days throughout the year, peaking around midday local time.


2-meter band, Communications beyond 50 miles, Meteor burst
By speeding up Morse code using analog tape or digital modes such as JT6M or FSK441, very short high-speed bursts of digital data can be bounced off the ionized gas trail of meteor showers. The speed required to confirm a two way contact via a short lived ionized meteor trail can only be performed by fast computers on both ends with very little human interaction.


2-meter band, Communications beyond 50 miles, Meteor burst
One computer will send a request for contact and if successfully received by a distant station, a reply will be sent by the receiving stations computer usually via the same ionized meteor trail to confirm the contact. If nothing is received after the request, a new request is transmitted. This continues until a reply is received to confirm the contact or until no contact can be made and no new requests are sent. Using this high speed digital mode, a full two way contact, can be completed in one second or less and can only be validated using a computer.


2-meter band, Communications beyond 50 miles, Meteor burst
Depending on the intensity of the ionized meteor trail, multiple contacts from multiple stations can be made off the same trail until it dissipates and can no longer reflect VHF signals with sufficient strength. This mode is often called burst transmission and can yield communication distances similar to sporadic E as described above.


2-meter band, Communications beyond 50 miles, Auroral propagation
Another phenomenon that produces upper atmosphere ionization suitable for 2-meter DXing are the auroras. Since the ionization persists much longer than meteor trails, voice modulated radio signals may sometimes be used, but the constant movement of the ionized gas leads to heavy distortion of the signals causing the audio to sound "ghostly" and whispered. In most instances using auroral reflections on 2 meters, audio or voice is totally unintelligible and ham operators wishing to make contacts via aurora, must resort to CW (Morse code).


2-meter band, Communications beyond 50 miles, Auroral propagation
CW signals returning from an auroral reflection have no distinct sound or tone but simply sound like a swishing or whooshing noise. An exception to this phenomenon would be the 6 meter band which is significantly lower in frequency than the 2 meter band by 94 MHz. In many instances 6 meter voice modes are readable but with varying degrees of difficulty when reflected off an aurora. Therefore, when using an auroral event as a radio signal reflector, the reflected signal strength and signal intelligibility decreases with increasing transmitting frequency.


2-meter band, Communications beyond 50 miles, Moonbounce (EME)
To communicate over the longest distances, hams use moon bounce. VHF signals normally escape the Earth's atmosphere, so using the moon as a target is quite practical. Due to the distance involved and the very high path loss getting a readable signal bounced off the moon involves high power ~1,000 Watts and steerable high gain antennas. Receiving these very weak return signals, again involves the use of high gain antennas (usually the same ones used to transmit the signal) and a very low-noise front end RF amplifier and a frequency stable receiver.


2-meter band, Communications beyond 50 miles, Moonbounce (EME)
However, new and recent technological advances in weak signal detection has allowed the successful reception of signals off the moon using much smaller or less well equipped stations allowing reception of signals that are "in the noise" and not audible to the human ear. One of these modes is JT65 which is a digital mode. Due to the delay of the signal traveling to the moon and back (travel time approx. 2.5 seconds), a person transmitting may hear the end of their own transmission returning.


2-meter band, Communications beyond 50 miles, The Brendan Awards
The Irish Radio Transmitters Society has provided a series of awards for the first successful all-natural, non-bounce contacts on 2 metres between the North American and European continents. Named for Saint Brendan of Clonfert, the three awards differentiate between successful "traditional" phone/CW contact (the Brendan Trophies), successful "non-traditional" digital two-way contact (the Brendan Shields), and an award for the first verified reception in either direction, regardless of method (the Brendan Plates). Attempts at the Brendan awards have established contact, but further examination revealed the signal was bounced off the International Space Station.


2-meter band, Los Angeles County statute
Los Angeles County has a statute (which dates from 1944) concerning mounting a "shortwave receiver" in a motor vehicle. While the statute specifically states one of the forbidden bands as 150–160 MHz, most two-meter transceivers can tune into this portion of the spectrum at least as receivers, and are therefore unlawful to mount in a motor vehicle in Los Angeles County. While arrest rarely happens, the statute is still on the books. There are also California Penal Code statutes covering similar activities. Recently, however, with new legislation in various states, licensed ham radio operators are exempt from these prohibitions including exemptions from using a radio while driving. Such prohibitions or exemptions vary from state to state.


2-meter band, Los Angeles County statute
Note that federal law preempts many local ordinances and state laws that may prohibit a licensed amateur radio operator from possessing an amateur radio based on its factory ability to receive frequencies outside of ham bands.


2-methyl-branched-chain-enoyl-CoA reductase
In enzymology, a 2-methyl-branched-chain-enoyl-CoA reductase (EC ) is an enzyme that catalyzes the chemical reaction


2-methyl-branched-chain-enoyl-CoA reductase
Thus, the two substrates of this enzyme are 2-methylbutanoyl-CoA and an electron transfer flavoprotein, whereas its 3 products are 2-methylcrotonoyl-CoA, reduced electron transfer flavoprotein, and H+.


2-methyl-branched-chain-enoyl-CoA reductase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-CH group of donors with flavin as acceptor. The systematic name of this enzyme class is 2-methyl-branched-chain-acyl-CoA:electron-transfer flavoprotein 2-oxidoreductase . This enzyme participates in the degradation of isoleucine. It employs one cofactor, FAD.


2-methylacyl-CoA dehydrogenase
In enzymology, a 2-methylacyl-CoA dehydrogenase (EC ) is an enzyme that catalyzes the chemical reaction


2-methylacyl-CoA dehydrogenase
Thus, the two substrates of this enzyme are 2-methylbutanoyl-CoA and acceptor, whereas its two products are 2-methylbut-2-enoyl-CoA and reduced acceptor.


2-methylacyl-CoA dehydrogenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-CH group of donor with other acceptors. The systematic name of this enzyme class is 2-methylbutanoyl-CoA:acceptor oxidoreductase. Other names in common use include branched-chain acyl-CoA dehydrogenase, 2-methyl branched chain acyl-CoA dehydrogenase, and 2-methylbutanoyl-CoA:(acceptor) oxidoreductase. This enzyme participates in valine, leucine and isoleucine degradation.


2-methylcitrate dehydratase
In enzymology, a 2-methylcitrate dehydratase (EC ) is an enzyme that catalyzes the chemical reaction


2-methylcitrate dehydratase
Hence, this enzyme has one substrate, (2S,3S)-2-hydroxybutane-1,2,3-tricarboxylate, and two products, (Z)-but-2-ene-1,2,3-tricarboxylate and H2O.


2-methylcitrate dehydratase
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is (2S,3S)-2-hydroxybutane-1,2,3-tricarboxylate hydro-lyase [(Z)-but-2-ene-1,2,3-tricarboxylate-forming]. Other names in common use include 2-methylcitrate hydro-lyase, PrpD, and 2-hydroxybutane-1,2,3-tricarboxylate hydro-lyase. This enzyme participates in propanoate metabolism.


2-methylcitrate dehydratase, Structural studies
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code .


2-methylcitrate dehydratase (2-methyl-trans-aconitate forming)
2-methylcitrate dehydratase (2-methyl-trans-aconitate forming) (EC ) is an enzyme with systematic name  (2S,3S)-2-hydroxybutane-1,2,3-tricarboxylate hydro-lyase (2-methyl-trans-aconitate forming). This enzyme catalyses the following chemical reaction


2-methylcitrate dehydratase (2-methyl-trans-aconitate forming)
This enzyme catalyses the dehydration of (2S,3S)-2-methylcitrate, forming the trans isomer of 2-methyl-aconitate.


2-methylcitrate synthase
In enzymology, a 2-methylcitrate synthase (EC ) is an enzyme that catalyzes the chemical reaction


2-methylcitrate synthase
The 3 substrates of this enzyme are propanoyl-CoA, H2O, and oxaloacetate, whereas its two products are (2R,3S)-2-hydroxybutane-1,2,3-tricarboxylate and CoA.


2-methylcitrate synthase
This enzyme belongs to the family of transferases, specifically those acyltransferases that convert acyl groups into alkyl groups on transfer. The systematic name of this enzyme class is propanoyl-CoA:oxaloacetate C-propanoyltransferase (thioester-hydrolysing, 1-carboxyethyl-forming). Other names in common use include 2-methylcitrate oxaloacetate-lyase, MCS, methylcitrate synthase, and methylcitrate synthetase. This enzyme participates in propanoate metabolism.


2-methyleneglutarate mutase
In enzymology, a 2-methyleneglutarate mutase (EC ) is an enzyme that catalyzes the chemical reaction


2-methyleneglutarate mutase
Hence, this enzyme has one substrate, 2-methyleneglutarate, and one product, 2-methylene-3-methylsuccinate.


2-methyleneglutarate mutase
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 2-methyleneglutarate carboxy-methylenemethylmutase. This enzyme is also called alpha-methyleneglutarate mutase. This enzyme participates in c5-branched dibasic acid metabolism. It employs one cofactor, cobamide.


2-methylisoborneol synthase
2-methylisoborneol synthase (EC , sco7700, 2-MIB cyclase, MIB synthase, MIBS) is an enzyme with systematic name (E)-2-methylgeranyl-diphosphate diphosphate-lyase (cyclizing, 2-methylisoborneol-forming). This enzyme catalyses the following chemical reaction


2-methylisoborneol synthase
The product, 2-methylisoborneol, is a characteristic odiferous compound with a musty smell produced by soil microorganisms.


2-methylisocitrate dehydratase
In enzymology, a 2-methylisocitrate dehydratase (EC ) is an enzyme that catalyzes the chemical reaction


2-methylisocitrate dehydratase
Hence, this enzyme has one substrate, (2S,3R)-3-hydroxybutane-1,2,3-tricarboxylate, and two products, (Z)-but-2-ene-1,2,3-tricarboxylate and H2O.


2-methylisocitrate dehydratase
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is (2S,3R)-3-hydroxybutane-1,2,3-tricarboxylate hydro-lyase [(Z)-but-2-ene-1,2,3-tricarboxylate-forming]. This enzyme is also called (2S,3R)-3-hydroxybutane-1,2,3-tricarboxylate hydro-lyase. This enzyme participates in propanoate metabolism.


2-millimeter band
The 2-millimeter band is a portion of the EHF (microwave) radio spectrum internationally allocated to amateur radio and amateur satellite use. The band is between 134 GHz and 141 GHz.


2-millimeter band
Due to the lack of commercial off the shelf radios, amateurs who operate on the 2 mm band must design and construct their own equipment, and those who do, often attempt to set communication distance records for the band.


2-millimeter band, Allocation
The International Telecommunication Union allocates 134 GHz to 141 GHz to amateur radio and amateur satellites. Amateurs operate on a primary basis between 134 GHz and 136 GHz and on a secondary basis in the rest of the band. As such, amateurs must protect the radio astronomy and radiolocation services from harmful interference, which share the band with amateurs between 136 GHz and 141 GHz. The ITU's allocations are the same in all three ITU Regions.


2-millimeter band, History
The allocation was introduced as a result of changes at ITU's World Radiocommunication Conference 2000. Prior to this the amateur allocation was 142-144 GHz Primary and 144-149 GHz Secondary.


2-millimeter band, Distance records
The first 2 mm distance record, and still standing longest distance achieved on the band, was set by US stations WA1ZMS and W4WWQ, who established contact at 114.4 kilometres (71.1 mi) on February 26, 2006.


2-millimeter band, Distance records
The longest distance achieved on 2 mm in the United Kingdom was 35.6 kilometres (22.1 mi) between stations G8ACE and G8KQW on January 16 2016.


2-millimeter band, Distance records
In Australia, the 2 mm distance record was 19 kilometres (12 mi) set by stations VK3CV and VK3NH on June 11, 2020.


2-nitroimidazole nitrohydrolase
2-nitroimidazole nitrohydrolase (EC , NnhA, 2NI nitrohydrolase, 2NI denitrase) is an enzyme with systematic name 2-nitroimidazole nitrohydrolase. This enzyme catalyses the following chemical reaction


2-nitroimidazole nitrohydrolase
This enzyme is present in the soil bacterium Mycobacterium sp. JS330


2-nitrophenol 2-monooxygenase
In enzymology, a 2-nitrophenol 2-monooxygenase (EC ) is an enzyme that catalyzes the chemical reaction


2-nitrophenol 2-monooxygenase
The 4 substrates of this enzyme are 2-nitrophenol, NADPH, H+, and O2, whereas its 4 products are catechol, nitrite, NADP+, and H2O.


2-nitrophenol 2-monooxygenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on paired donors, with O2 as oxidant and incorporation or reduction of oxygen. The oxygen incorporated need not be derived from O2 with NADH or NADPH as one donor, and incorporation of one atom o oxygen into the other donor. The systematic name of this enzyme class is 2-nitrophenol,NADPH:oxygen 2-oxidoreductase (2-hydroxylating, nitrite-forming). Other names in common use include 2-nitrophenol oxygenase, and nitrophenol oxygenase.


2-nitropropane dioxygenase
In enzymology, a 2-nitropropane dioxygenase (EC ) is an enzyme that catalyzes the chemical reaction


2-nitropropane dioxygenase
Thus, the two substrates of this enzyme are 2-nitropropane and O2, whereas its two products are acetone and nitrite.


2-nitropropane dioxygenase
This enzyme belongs to the family of oxidoreductases, specifically those acting on single donors with O2 as oxidant and incorporation of two atoms of oxygen into the substrate (oxygenases). The oxygen incorporated need not be derived from O2. The systematic name of this enzyme class is 2-nitropropane:oxygen 2-oxidoreductase. This enzyme participates in nitrogen metabolism. It has 3 cofactors: FAD, Iron,  and FMN.


2-nitropropane dioxygenase, Structural studies
As of late 2007 Steve Fuhrer from the DHPA solved this very complex formula to find, two structures have been solved for this class of enzymes, with PDB accession codes  and .


2-opt
In optimization, 2-opt is a simple local search algorithm for solving the traveling salesman problem. The 2-opt algorithm was first proposed by Croes in 1958, although the basic move had already been suggested by Flood. The main idea behind it is to take a route that crosses over itself and reorder it so that it does not.


2-opt
A complete 2-opt local search will compare every possible valid combination of the swapping mechanism. This technique can be applied to the travelling salesman problem as well as many related problems. These include the vehicle routing problem (VRP) as well as the capacitated VRP, which require minor modification of the algorithm.


2-opt
This is the mechanism by which the 2-opt swap manipulates a given route:


2-opt
Here is an example of the above with arbitrary input:


2-opt
This is the complete 2-opt swap making use of the above mechanism:


2-opt
Note: If you start/end at a particular node or depot, then you must remove this from the search as an eligible candidate for swapping, as reversing the order will cause an invalid path.


2-opt
which is not valid (does not leave from A, the depot).


2-oxo-3-(5-oxofuran-2-ylidene)propanoate lactonase
2-oxo-3-(5-oxofuran-2-ylidene)propanoate lactonase (EC , naaC (gene)) is an enzyme with systematic name 2-oxo-3-(5-oxofuran-2-ylidene)propanoate lactonohydrolase. This enzyme catalyses the following chemical reaction


2-oxo-3-(5-oxofuran-2-ylidene)propanoate lactonase
This enzyme is isolated from the soil bacterium Bradyrhizobium sp. JS329.


2-oxo-4-hydroxy-4-carboxy-5-ureidoimidazoline decarboxylase
In molecular biology 2-oxo-4-hydroxy-4-carboxy-5-ureidoimidazoline decarboxylase (OHCU decarboxylase) EC  is an enzyme involved in purine catabolism. It catalyses the decarboxylation of 2-oxo-4-hydroxy-4-carboxy-5-ureidoimidazoline (OHCU) into S(+)-allantoin. It is the third step of the conversion of uric acid (a purine derivative) to allantoin. Step one is catalysed by urate oxidase and step two is catalysed by hydroxyisourate hydrolase.


2-oxo-acid reductase
In enzymology, a 2-oxo-acid reductase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxo-acid reductase
Thus, the two substrates of this enzyme are (2R)-hydroxy-carboxylate and acceptor, whereas its two products are 2-oxo-carboxylate and reduced acceptor.


2-oxo-acid reductase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (2R)-hydroxy-carboxylate:acceptor oxidoreductase. Other names in common use include (2R)-hydroxycarboxylate-viologen-oxidoreductase, HVOR, and 2-oxoacid reductase.


2-oxoadipate reductase
In enzymology, a 2-oxoadipate reductase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoadipate reductase
Thus, the two substrates of this enzyme are 2-hydroxyadipate and NAD+, whereas its 3 products are 2-oxoadipate, NADH, and H+.


2-oxoadipate reductase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-hydroxyadipate:NAD+ 2-oxidoreductase. Other names in common use include 2-ketoadipate reductase, alpha-ketoadipate reductase, and 2-ketoadipate reductase.


2-oxoaldehyde dehydrogenase (NAD+)
In enzymology, a 2-oxoaldehyde dehydrogenase (NAD+) (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoaldehyde dehydrogenase (NAD+)
The 3 substrates of this enzyme are 2-oxoaldehyde, NAD+, and H2O, whereas its 3 products are 2-oxo acid, NADH, and H+.


2-oxoaldehyde dehydrogenase (NAD+)
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-oxoaldehyde:NAD+ 2-oxidoreductase. Other names in common use include alpha-ketoaldehyde dehydrogenase, methylglyoxal dehydrogenase, NAD+-linked alpha-ketoaldehyde dehydrogenase, 2-ketoaldehyde dehydrogenase, NAD+-dependent alpha-ketoaldehyde dehydrogenase, and 2-oxoaldehyde dehydrogenase (NAD+). This enzyme participates in pyruvate metabolism.


2-oxoaldehyde dehydrogenase (NADP+)
In enzymology, a 2-oxoaldehyde dehydrogenase (NADP+) (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoaldehyde dehydrogenase (NADP+)
The 3 substrates of this enzyme are 2-oxoaldehyde, NADP+, and H2O, whereas its 3 products are 2-oxo acid, NADPH, and H+.


2-oxoaldehyde dehydrogenase (NADP+)
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-oxoaldehyde:NADP+ 2-oxidoreductase. Other names in common use include alpha-ketoaldehyde dehydrogenase, methylglyoxal dehydrogenase, NADP+-linked alpha-ketoaldehyde dehydrogenase, 2-ketoaldehyde dehydrogenase, NADP+-dependent alpha-ketoaldehyde dehydrogenase, and 2-oxoaldehyde dehydrogenase (NADP+). This enzyme participates in pyruvate metabolism.


2-oxobutyrate synthase
In enzymology, a 2-oxobutyrate synthase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxobutyrate synthase
The 3 substrates of this enzyme are 2-oxobutanoate, CoA, and oxidized ferredoxin, whereas its 3 products are propanoyl-CoA, CO2, and reduced ferredoxin.


2-oxobutyrate synthase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is 2-oxobutanoate:ferredoxin 2-oxidoreductase (CoA-propanoylating). Other names in common use include alpha-ketobutyrate-ferredoxin oxidoreductase, 2-ketobutyrate synthase, alpha-ketobutyrate synthase, 2-oxobutyrate-ferredoxin oxidoreductase, and 2-oxobutanoate:ferredoxin 2-oxidoreductase (CoA-propionylating). This enzyme participates in propanoate metabolism.


2-oxoglutaramate amidase
2-oxoglutaramate amidase (EC , omega-amidase) is an enzyme with systematic name 5-amino-2,5-dioxopentanoate amidohydrolase. This enzyme catalyses the following chemical reaction


2-oxoglutaramate amidase
The enzyme participates in the nicotine degradation pathway of several Gram-positive bacteria.


2-oxoglutarate carboxylase
In enzymology, a 2-oxoglutarate carboxylase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoglutarate carboxylase
The 3 substrates of this enzyme are ATP, 2-oxoglutarate, and HCO3-, whereas its 3 products are ADP, phosphate, and oxalosuccinate.


2-oxoglutarate carboxylase
This enzyme belongs to the family of ligases, specifically those forming carbon-carbon bonds. The systematic name of this enzyme class is '. Other names in common use include oxalosuccinate synthetase', carboxylating factor for ICDH (incorrect), CFI, and OGC.


2-oxoglutarate decarboxylase
In enzymology, a 2-oxoglutarate decarboxylase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoglutarate decarboxylase
Hence, this enzyme has one substrate, 2-oxoglutarate, and two products, succinate semialdehyde and CO2.


2-oxoglutarate decarboxylase
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-oxoglutarate carboxy-lyase (succinate-semialdehyde-forming). Other names in common use include oxoglutarate decarboxylase, alpha-ketoglutarate decarboxylase, alpha-ketoglutaric decarboxylase, oxoglutarate decarboxylase, pre-2-oxoglutarate decarboxylase, and 2-oxoglutarate carboxy-lyase. It employs one cofactor, thiamin diphosphate.


2-oxoglutarate dioxygenase (ethylene-forming)
2-oxoglutarate dioxygenase (ethylene-forming) (EC , ethylene-forming enzyme, EFE) is an enzyme with systematic name 2-oxoglutarate:oxygen oxidoreductase (decarboxylating, ethylene-forming). This enzyme catalyses the following chemical reaction


2-oxoglutarate dioxygenase (ethylene-forming)
2-oxoglutarate dioxygenase produces ethylene in bacteria of the Pseudomonas syringae group.


2-oxoglutarate synthase
In enzymology, a 2-oxoglutarate synthase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoglutarate synthase
The 3 substrates of this enzyme are 2-oxoglutarate, CoA, and oxidized ferredoxin, whereas its 3 products are succinyl-CoA, CO2, and reduced ferredoxin.


2-oxoglutarate synthase
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is 2-oxoglutarate:ferredoxin oxidoreductase (decarboxylating). Other names in common use include 2-ketoglutarate ferredoxin oxidoreductase, 2-oxoglutarate:ferredoxin oxidoreductase, KGOR, 2-oxoglutarate ferredoxin oxidoreductase, and 2-oxoglutarate:ferredoxin 2-oxidoreductase (CoA-succinylating). This enzyme participates in the Citric acid cycle. Some forms catalyze the reverse reaction within the Reverse Krebs cycle, as a means of carbon fixation.


2-oxoglutarate/L-arginine monooxygenase/decarboxylase (succinate-forming)
2-oxoglutarate/L-arginine monooxygenase/decarboxylase (succinate-forming) (EC , ethylene-forming enzyme, EFE) is an enzyme with systematic name L-arginine,2-oxoglutarate:oxygen oxidoreductase (succinate-forming). This enzyme catalyses the following chemical reaction


2-oxoisovalerate dehydrogenase (acylating)
In enzymology, a 2-oxoisovalerate dehydrogenase (acylating) (EC ) is an enzyme that catalyzes the chemical reaction


2-oxoisovalerate dehydrogenase (acylating)
The 3 substrates of this enzyme are 3-methyl-2-oxobutanoate, CoA, and NAD+, whereas its 3 products are 2-methylpropanoyl-CoA, CO2, and NADH.


2-oxoisovalerate dehydrogenase (acylating)
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 3-methyl-2-oxobutanoate:NAD+ 2-oxidoreductase (CoA-methyl-propanoylating). Other names in common use include 2-oxoisovalerate dehydrogenase, and alpha-ketoisovalerate dehydrogenase. This enzyme participates in valine, leucine and isoleucine degradation.


2-oxopent-4-enoate hydratase
In enzymology, a 2-oxopent-4-enoate hydratase (EC ) is an enzyme that catalyzes the chemical reaction


2-oxopent-4-enoate hydratase
Hence, this enzyme has one substrate, 4-hydroxy-2-oxopentanoate, and two products, 2-oxopent-4-enoate and H2O.


2-oxopent-4-enoate hydratase
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is 4-hydroxy-2-oxopentanoate hydro-lyase (2-oxopent-4-enoate-forming). Other names in common use include 2-keto-4-pentenoate hydratase, OEH, 2-keto-4-pentenoate (vinylpyruvate)hydratase, and 4-hydroxy-2-oxopentanoate hydro-lyase. This enzyme participates in 9 metabolic pathways: phenylalanine metabolism, benzoate degradation via hydroxylation, biphenyl degradation, toluene and xylene degradation, 1,4-dichlorobenzene degradation, fluorene degradation, carbazole degradation, ethylbenzene degradation, and styrene degradation.


2-oxopropyl-CoM reductase (carboxylating)
In enzymology, a 2-oxopropyl-CoM reductase (carboxylating) (EC ) is an enzyme that catalyzes the chemical reaction


2-oxopropyl-CoM reductase (carboxylating)
The 3 substrates of this enzyme are 2-mercaptoethanesulfonate, acetoacetate, and NADP+, whereas its 3 products are 2-(2-oxopropylthio)ethanesulfonate, CO2, and NADPH.


2-oxopropyl-CoM reductase (carboxylating)
This enzyme belongs to the family of oxidoreductases, specifically those acting on a sulfur group of donors with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-mercaptoethanesulfonate, acetoacetate:NADP+ oxidoreductase (decarboxylating). Other names in common use include NADPH:2-(2-ketopropylthio)ethanesulfonate, oxidoreductase/carboxylase, and NADPH:2-ketopropyl-coenzyme M oxidoreductase/carboxylase.


2-oxopropyl-CoM reductase (carboxylating), Structural studies
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes  and .


2-phospho-L-lactate transferase
2-phospho-L-lactate transferase (EC , LPPG:Fo 2-phospho-L-lactate transferase, LPPG:7,8-didemethyl-8-hydroxy-5-deazariboflavin 2-phospho-L-lactate transferase, MJ1256, lactyl-2-diphospho-(5')guanosine:Fo 2-phospho-L-lactate transferase, CofD) is an enzyme with systematic name (2S)-lactyl-2-diphospho-5'-guanosine:7,8-didemethyl-8-hydroxy-5-deazariboflavin 2-phospho-L-lactate transferase. This enzyme catalyses the following chemical reaction


2-phospho-L-lactate transferase
This enzyme is involved in the biosynthesis of coenzyme F420.


2-phosphosulfolactate phosphatase
In enzymology, a 2-phosphosulfolactate phosphatase (EC ) is an enzyme that catalyzes the chemical reaction


2-phosphosulfolactate phosphatase
Thus, the two substrates of this enzyme are (2R)-2-phospho-3-sulfolactate and H2O, whereas its two products are (2R)-3-sulfolactate and phosphate.


2-phosphosulfolactate phosphatase
This enzyme belongs to the family of hydrolases, specifically those acting on phosphoric monoester bonds. The systematic name of this enzyme class is (R)-2-phospho-3-sulfolactate phosphohydrolase. Other names in common use include (2R)-phosphosulfolactate phosphohydrolase, and ComB phosphatase.


2-phosphosulfolactate phosphatase, Structural studies
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code .


2-pop
Used in television production and filmmaking post-production, a 2-pop  is a 1 kHz tone that is one frame long and placed 2 seconds before the start of program. It is a simple and effective method of ensuring synchronization between sound and picture in a video or film.


2-pop
A 2-pop is typically placed at the end of a visual countdown. Only the first frame of the "2" is shown, and the remainder of the 2 seconds prior to the program is black. This provides a unique point of reference where the frame-long image and frame-long sound should align, similar to the way a film clapperboard is used to generate a synchronization point.


2-pop
For example, in a television or video program the first frame of action (FFOA) starts at one hour (typically timecode of 01:00:00:00 in the US, and 10:00:00:00 in the UK), preceding that, 1 frame (or the 2-pop) of tone would be placed at timecode 00:59:58:00 or exactly 2 seconds before first picture. Alternately, in film post-production the leader starts at 01:00:00:00 (or 0+00 feet if using feet and frames as is common in the United States), the 2-pop starts at 01:00:06:00 (or 9+00), and the first frame of action (FFOA) starts at 01:00:08:00 (or 12+00).


2-pop
A 2-pop is useful whenever picture and sound are handled separately. For example, projecting work-in-progress in the pre-video days involved a film projector linked to a magnetic dubber, onto which the separate soundtrack reel was loaded. Aligning them by the 2-pop would ensure proper synchronization during playback. A modern scenario would involve sending a soundtrack to a separate facility for a sound mix. The returned product is a computer audio file which then needs to be synchronized again with the picture.


2-pop, Sample sound
Whereas laying down bars and tone prior to program start establishes video and audio calibration levels on the tape, the 2-pop is primarily used for picture and sound synchronization. Therefore, while the loudness of the 2-pop may be the same as the bars and tone audio level in use, this is not a requirement. The loudness level should be sufficient to be heard clearly.


2-pounder gun
2-pounder gun, 2-pounder and QF 2 pounder or QF 2-pdr are abbreviations used for various guns which fired a projectile weighing approximately 2 pounds (0.91 kg). These include:


2-pyrone-4,6-dicarboxylate lactonase
In enzymology, 2-pyrone-4,6-dicarboxylate lactonase (EC ) is an enzyme that catalyzes the reversible hydrolytic chemical reaction


2-pyrone-4,6-dicarboxylate lactonase
Thus, the two substrates of this enzyme are 2-pyrone-4,6-dicarboxylate and H2O, whereas its product is a tautomeric mixture of 4-oxalomesaconate and 4-carboxy-2-hydroxymuconate.


2-pyrone-4,6-dicarboxylate lactonase
This enzyme belongs to the Amidohydrolase superfamily of enzymes and is a member of Cluster of Orthologous Groups (COG) 3618. The systematic name of this enzyme is 2-pyrone-4,6-dicarboxylate lactonase but is also known as LigI. This enzyme is found to play an important role in the metabolism of lignin-derived aromatic compounds in both the syringate degradation pathway and the protocatechuate 4,5-cleavage pathway.


2-pyrone-4,6-dicarboxylate lactonase
LigI from Sphingomonas is of particular interest as it has been shown to be the first member of the amidohydrolase superfamily to not require a divalent metal cation for catalytic activity.


2-pyrone-4,6-dicarboxylate lactonase, Mechanism
The mechanism of catalysis of LigI has been determined by crystallography and NMR analysis. More specifically, the hydrolytic water molecule is activated by the transfer of a proton to Asp-248 whereas the carbonyl group of the 2-pyrone-4,6-dicarboxylate (PDC) lactone substrate is activated by hydrogen bonding interactions with His-180, His-31, and His-33.


2-ring
In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring. But the point is that one can also consider the situation in which an element of R comes with a "nontrivial automorphism" (cf. Lurie).


2-ring
This line of generalization of a ring eventually leads to the notion of an En-ring.


2-satisfiability
In computer science, 2-satisfiability, 2-SAT or just 2SAT is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time.


2-satisfiability
Instances of the 2-satisfiability problem are typically expressed as Boolean formulas of a special type, called conjunctive normal form (2-CNF) or Krom formulas. Alternatively, they may be expressed as a special type of directed graph, the implication graph, which expresses the variables of an instance and their negations as vertices in a graph, and constraints on pairs of variables as directed edges. Both of these kinds of inputs may be solved in linear time, either by a method based on backtracking or by using the strongly connected components of the implication graph. Resolution, a method for combining pairs of constraints to make additional valid constraints, also leads to a polynomial time solution. The 2-satisfiability problems provide one of two major subclasses of the conjunctive normal form formulas that can be solved in polynomial time; the other of the two subclasses is Horn-satisfiability.


2-satisfiability
2-satisfiability may be applied to geometry and visualization problems in which a collection of objects each have two potential locations and the goal is to find a placement for each object that avoids overlaps with other objects. Other applications include clustering data to minimize the sum of the diameters of the clusters, classroom and sports scheduling, and recovering shapes from information about their cross-sections.


2-satisfiability
In computational complexity theory, 2-satisfiability provides an example of an NL-complete problem, one that can be solved non-deterministically using a logarithmic amount of storage and that is among the hardest of the problems solvable in this resource bound. The set of all solutions to a 2-satisfiability instance can be given the structure of a median graph, but counting these solutions is #P-complete and therefore not expected to have a polynomial-time solution.


2-satisfiability
Random instances undergo a sharp phase transition from solvable to unsolvable instances as the ratio of constraints to variables increases past 1, a phenomenon conjectured but unproven for more complicated forms of the satisfiability problem. A computationally difficult variation of 2-satisfiability, finding a truth assignment that maximizes the number of satisfied constraints, has an approximation algorithm whose optimality depends on the unique games conjecture, and another difficult variation, finding a satisfying assignment minimizing the number of true variables, is an important test case for parameterized complexity.


2-satisfiability, Problem representations
A 2-satisfiability problem may be described using a Boolean expression with a special restricted form. It is a conjunction (a Boolean and operation) of clauses, where each clause is a disjunction (a Boolean or operation) of two variables or negated variables. The variables or their negations appearing in this formula are known as literals. For example, the following formula is in conjunctive normal form, with seven variables, eleven clauses, and 22 literals:(x0∨x2)∧(x0∨¬x3)∧(x1∨¬x3)∧(x1∨¬x4)∧(x2∨¬x4)∧(x0∨¬x5)∧(x1∨¬x5)∧(x2∨¬x5)∧(x3∨x6)∧(x4∨x6)∧(x5∨x6). {\displaystyle {\begin{aligned}&(x_{0}\lor x_{2})\land (x_{0}\lor \lnot x_{3})\land (x_{1}\lor \lnot x_{3})\land (x_{1}\lor \lnot x_{4})\land {}\\&(x_{2}\lor \lnot x_{4})\land {}(x_{0}\lor \lnot x_{5})\land (x_{1}\lor \lnot x_{5})\land (x_{2}\lor \lnot x_{5})\land {}\\&(x_{3}\lor x_{6})\land (x_{4}\lor x_{6})\land (x_{5}\lor x_{6}).\end{aligned}}}


2-satisfiability, Problem representations
The 2-satisfiability problem is to find a truth assignment to these variables that makes the whole formula true. Such an assignment chooses whether to make each of the variables true or false, so that at least one literal in every clause becomes true. For the expression shown above, one possible satisfying assignment is the one that sets all seven of the variables to true. Every clause has at least one non-negated variable, so this assignment satisfies every clause. There are also 15 other ways of setting all the variables so that the formula becomes true. Therefore, the 2-satisfiability instance represented by this expression is satisfiable.


2-satisfiability, Problem representations
Formulas in this form are known as 2-CNF formulas. The "2" in this name stands for the number of literals per clause, and "CNF" stands for conjunctive normal form, a type of Boolean expression in the form of a conjunction of disjunctions. They are also called Krom formulas, after the work of UC Davis mathematician Melven R. Krom, whose 1967 paper was one of the earliest works on the 2-satisfiability problem.


2-satisfiability, Problem representations
Each clause in a 2-CNF formula is logically equivalent to an implication from one variable or negated variable to the other. For example, the second clause in the example may be written in any of three equivalent ways:(x0∨¬x3)≡(¬x0⇒¬x3)≡(x3⇒x0). {\displaystyle (x_{0}\lor \lnot x_{3})\;\equiv \;(\lnot x_{0}\Rightarrow \lnot x_{3})\;\equiv \;(x_{3}\Rightarrow x_{0}).} Because of this equivalence between these different types of operation, a 2-satisfiability instance may also be written in implicative normal form, in which we replace each or clause in the conjunctive normal form by the two implications to which it is equivalent.


2-satisfiability, Problem representations
A third, more graphical way of describing a 2-satisfiability instance is as an implication graph. An implication graph is a directed graph in which there is one vertex per variable or negated variable, and an edge connecting one vertex to another whenever the corresponding variables are related by an implication in the implicative normal form of the instance. An implication graph must be a skew-symmetric graph, meaning that it has a symmetry that takes each variable to its negation and reverses the orientations of all of the edges.


2-satisfiability, Algorithms
Several algorithms are known for solving the 2-satisfiability problem. The most efficient of them take linear time.


2-satisfiability, Algorithms, Resolution and transitive closure
Krom (1967) described the following polynomial time decision procedure for solving 2-satisfiability instances.


2-satisfiability, Algorithms, Resolution and transitive closure
Suppose that a 2-satisfiability instance contains two clauses that both use the same variable x, but that x is negated in one clause and not in the other. Then the two clauses may be combined to produce a third clause, having the two other literals in the two clauses; this third clause must also be satisfied whenever the first two clauses are both satisfied. For instance, we may combine the clauses (a∨b){\displaystyle (a\lor b)} and (¬b∨¬c){\displaystyle (\lnot b\lor \lnot c)} in this way to produce the clause (a∨¬c){\displaystyle (a\lor \lnot c)}. In terms of the implicative form of a 2-CNF formula, this rule amounts to finding two implications ¬a⇒b{\displaystyle \lnot a\Rightarrow b} and b⇒¬c{\displaystyle b\Rightarrow \lnot c}, and inferring by transitivity a third implication ¬a⇒¬c{\displaystyle \lnot a\Rightarrow \lnot c}.


2-satisfiability, Algorithms, Resolution and transitive closure
Krom writes that a formula is consistent if repeated application of this inference rule cannot generate both the clauses (x∨x){\displaystyle (x\lor x)} and (¬x∨¬x){\displaystyle (\lnot x\lor \lnot x)}, for any variable x{\displaystyle x}. As he proves, a 2-CNF formula is satisfiable if and only if it is consistent. For, if a formula is not consistent, it is not possible to satisfy both of the two clauses (x∨x){\displaystyle (x\lor x)} and (¬x∨¬x){\displaystyle (\lnot x\lor \lnot x)} simultaneously.


2-satisfiability, Algorithms, Resolution and transitive closure
And, if it is consistent, then the formula can be extended by repeatedly adding one clause of the form (x∨x){\displaystyle (x\lor x)} or (¬x∨¬x){\displaystyle (\lnot x\lor \lnot x)} at a time, preserving consistency at each step, until it includes such a clause for every variable. At each of these extension steps, one of these two clauses may always be added while preserving consistency, for if not then the other clause could be generated using the inference rule.


2-satisfiability, Algorithms, Resolution and transitive closure
Once all variables have a clause of this form in the formula, a satisfying assignment of all of the variables may be generated by setting a variable x{\displaystyle x} to true if the formula contains the clause (x∨x){\displaystyle (x\lor x)} and setting it to false if the formula contains the clause (¬x∨¬x){\displaystyle (\lnot x\lor \lnot x)}.


2-satisfiability, Algorithms, Resolution and transitive closure
Krom was concerned primarily with completeness of systems of inference rules, rather than with the efficiency of algorithms. However, his method leads to a polynomial time bound for solving 2-satisfiability problems. By grouping together all of the clauses that use the same variable, and applying the inference rule to each pair of clauses, it is possible to find all inferences that are possible from a given 2-CNF instance, and to test whether it is consistent, in total time O(n3), where n is the number of variables in the instance.


2-satisfiability, Algorithms, Resolution and transitive closure
This formula comes from multiplying the number of variables by the O(n2) number of pairs of clauses involving a given variable, to which the inference rule may be applied. Thus, it is possible to determine whether a given 2-CNF instance is satisfiable in time O(n3). Because finding a satisfying assignment using Krom's method involves a sequence of O(n) consistency checks, it would take time O(n4). Even, Itai & Shamir (1976) quote a faster time bound of O(n2) for this algorithm, based on more careful ordering of its operations. Nevertheless, even this smaller time bound was greatly improved by the later linear time algorithms of Even, Itai & Shamir (1976) and Aspvall, Plass & Tarjan (1979).


2-satisfiability, Algorithms, Resolution and transitive closure
In terms of the implication graph of the 2-satisfiability instance, Krom's inference rule can be interpreted as constructing the transitive closure of the graph. As Cook (1971) observes, it can also be seen as an instance of the Davis–Putnam algorithm for solving satisfiability problems using the principle of resolution. Its correctness follows from the more general correctness of the Davis–Putnam algorithm. Its polynomial time bound follows from the fact that each resolution step increases the number of clauses in the instance, which is upper bounded by a quadratic function of the number of variables.


2-satisfiability, Algorithms, Limited backtracking
Even, Itai & Shamir (1976) describe a technique involving limited backtracking for solving constraint satisfaction problems with binary variables and pairwise constraints. They apply this technique to a problem of classroom scheduling, but they also observe that it applies to other problems including 2-SAT.


2-satisfiability, Algorithms, Limited backtracking
The basic idea of their approach is to build a partial truth assignment, one variable at a time. Certain steps of the algorithms are "choice points", points at which a variable can be given either of two different truth values, and later steps in the algorithm may cause it to backtrack to one of these choice points. However, only the most recent choice can be backtracked over. All choices made earlier than the most recent one are permanent.


2-satisfiability, Algorithms, Limited backtracking
Initially, there is no choice point, and all variables are unassigned. At each step, the algorithm chooses the variable whose value to set, as follows:


2-satisfiability, Algorithms, Limited backtracking
Intuitively, the algorithm follows all chains of inference after making each of its choices. This either leads to a contradiction and a backtracking step, or, if no contradiction is derived, it follows that the choice was a correct one that leads to a satisfying assignment. Therefore, the algorithm either correctly finds a satisfying assignment or it correctly determines that the input is unsatisfiable.


2-satisfiability, Algorithms, Limited backtracking
Even et al. did not describe in detail how to implement this algorithm efficiently. They state only that by "using appropriate data structures in order to find the implications of any decision", each step of the algorithm (other than the backtracking) can be performed quickly. However, some inputs may cause the algorithm to backtrack many times, each time performing many steps before backtracking, so its overall complexity may be nonlinear.


2-satisfiability, Algorithms, Limited backtracking
To avoid this problem, they modify the algorithm so that, after reaching each choice point, it begins simultaneously testing both of the two assignments for the variable set at the choice point, spending equal numbers of steps on each of the two assignments. As soon as the test for one of these two assignments would create another choice point, the other test is stopped, so that at any stage of the algorithm there are only two branches of the backtracking tree that are still being tested. In this way, the total time spent performing the two tests for any variable is proportional to the number of variables and clauses of the input formula whose values are permanently assigned. As a result, the algorithm takes linear time in total.


2-satisfiability, Algorithms, Strongly connected components
Aspvall, Plass & Tarjan (1979) found a simpler linear time procedure for solving 2-satisfiability instances, based on the notion of strongly connected components from graph theory.


2-satisfiability, Algorithms, Strongly connected components
Two vertices in a directed graph are said to be strongly connected to each other if there is a directed path from one to the other and vice versa. This is an equivalence relation, and the vertices of the graph may be partitioned into strongly connected components, subsets within which every two vertices are strongly connected. There are several efficient linear time algorithms for finding the strongly connected components of a graph, based on depth first search: Tarjan's strongly connected components algorithm and the path-based strong component algorithm each perform a single depth first search. Kosaraju's algorithm performs two depth first searches, but is very simple.


2-satisfiability, Algorithms, Strongly connected components
In terms of the implication graph, two literals belong to the same strongly connected component whenever there exist chains of implications from one literal to the other and vice versa. Therefore, the two literals must have the same value in any satisfying assignment to the given 2-satisfiability instance. In particular, if a variable and its negation both belong to the same strongly connected component, the instance cannot be satisfied, because it is impossible to assign both of these literals the same value. As Aspvall et al. showed, this is a necessary and sufficient condition: a 2-CNF formula is satisfiable if and only if there is no variable that belongs to the same strongly connected component as its negation.


2-satisfiability, Algorithms, Strongly connected components
This immediately leads to a linear time algorithm for testing satisfiability of 2-CNF formulae: simply perform a strong connectivity analysis on the implication graph and check that each variable and its negation belong to different components. However, as Aspvall et al. also showed, it also leads to a linear time algorithm for finding a satisfying assignment, when one exists. Their algorithm performs the following steps:


2-satisfiability, Algorithms, Strongly connected components
Due to the reverse topological ordering and the skew-symmetry, when a literal is set to true, all literals that can be reached from it via a chain of implications will already have been set to true. Symmetrically, when a literal x is set to false, all literals that lead to it via a chain of implications will themselves already have been set to false. Therefore, the truth assignment constructed by this procedure satisfies the given formula, which also completes the proof of correctness of the necessary and sufficient condition identified by Aspvall et al.


2-satisfiability, Algorithms, Strongly connected components
As Aspvall et al. show, a similar procedure involving topologically ordering the strongly connected components of the implication graph may also be used to evaluate fully quantified Boolean formulae in which the formula being quantified is a 2-CNF formula.


2-satisfiability, Applications, Conflict-free placement of geometric objects
A number of exact and approximate algorithms for the automatic label placement problem are based on 2-satisfiability. This problem concerns placing textual labels on the features of a diagram or map. Typically, the set of possible locations for each label is highly constrained, not only by the map itself (each label must be near the feature it labels, and must not obscure other features), but by each other: every two labels should avoid overlapping each other, for otherwise they would become illegible. In general, finding a label placement that obeys these constraints is an NP-hard problem.


2-satisfiability, Applications, Conflict-free placement of geometric objects
However, if each feature has only two possible locations for its label (say, extending to the left and to the right of the feature) then label placement may be solved in polynomial time. For, in this case, one may create a 2-satisfiability instance that has a variable for each label and that has a clause for each pair of labels that could overlap, preventing them from being assigned overlapping positions. If the labels are all congruent rectangles, the corresponding 2-satisfiability instance can be shown to have only linearly many constraints, leading to near-linear time algorithms for finding a labeling.


2-satisfiability, Applications, Conflict-free placement of geometric objects
Poon, Zhu & Chin (1998) describe a map labeling problem in which each label is a rectangle that may be placed in one of three positions with respect to a line segment that it labels: it may have the segment as one of its sides, or it may be centered on the segment. They represent these three positions using two binary variables in such a way that, again, testing the existence of a valid labeling becomes a 2-satisfiability problem.


2-satisfiability, Applications, Conflict-free placement of geometric objects
Formann & Wagner (1991) use 2-satisfiability as part of an approximation algorithm for the problem of finding square labels of the largest possible size for a given set of points, with the constraint that each label has one of its corners on the point that it labels. To find a labeling with a given size, they eliminate squares that, if doubled, would overlap another point, and they eliminate points that can be labeled in a way that cannot possibly overlap with another point's label.


2-satisfiability, Applications, Conflict-free placement of geometric objects
They show that these elimination rules cause the remaining points to have only two possible label placements per point, allowing a valid label placement (if one exists) to be found as the solution to a 2-satisfiability instance. By searching for the largest label size that leads to a solvable 2-satisfiability instance, they find a valid label placement whose labels are at least half as large as the optimal solution. That is, the approximation ratio of their algorithm is at most two.


2-satisfiability, Applications, Conflict-free placement of geometric objects
Similarly, if each label is rectangular and must be placed in such a way that the point it labels is somewhere along its bottom edge, then using 2-satisfiability to find the largest label size for which there is a solution in which each label has the point on a bottom corner leads to an approximation ratio of at most two.


2-satisfiability, Applications, Conflict-free placement of geometric objects
Similar applications of 2-satisfiability have been made for other geometric placement problems. In graph drawing, if the vertex locations are fixed and each edge must be drawn as a circular arc with one of two possible locations (for instance as an arc diagram), then the problem of choosing which arc to use for each edge in order to avoid crossings is a 2-satisfiability problem with a variable for each edge and a constraint for each pair of placements that would lead to a crossing.


2-satisfiability, Applications, Conflict-free placement of geometric objects
However, in this case it is possible to speed up the solution, compared to an algorithm that builds and then searches an explicit representation of the implication graph, by searching the graph implicitly.


2-satisfiability, Applications, Conflict-free placement of geometric objects
In VLSI integrated circuit design, if a collection of modules must be connected by wires that can each bend at most once, then again there are two possible routes for the wires, and the problem of choosing which of these two routes to use, in such a way that all wires can be routed in a single layer of the circuit, can be solved as a 2-satisfiability instance.


2-satisfiability, Applications, Conflict-free placement of geometric objects
Boros et al. (1999) consider another VLSI design problem: the question of whether or not to mirror-reverse each module in a circuit design. This mirror reversal leaves the module's operations unchanged, but it changes the order of the points at which the input and output signals of the module connect to it, possibly changing how well the module fits into the rest of the design. Boros et al.


2-satisfiability, Applications, Conflict-free placement of geometric objects
consider a simplified version of the problem in which the modules have already been placed along a single linear channel, in which the wires between modules must be routed, and there is a fixed bound on the density of the channel (the maximum number of signals that must pass through any cross-section of the channel). They observe that this version of the problem may be solved as a 2-satisfiability instance, in which the constraints relate the orientations of pairs of modules that are directly across the channel from each other. As a consequence, the optimal density may also be calculated efficiently, by performing a binary search in which each step involves the solution of a 2-satisfiability instance.


2-satisfiability, Applications, Data clustering
One way of clustering a set of data points in a metric space into two clusters is to choose the clusters in such a way as to minimize the sum of the diameters of the clusters, where the diameter of any single cluster is the largest distance between any two of its points. This is preferable to minimizing the maximum cluster size, which may lead to very similar points being assigned to different clusters. If the target diameters of the two clusters are known, a clustering that achieves those targets may be found by solving a 2-satisfiability instance. The instance has one variable per point, indicating whether that point belongs to the first cluster or the second cluster. Whenever any two points are too far apart from each other for both to belong to the same cluster, a clause is added to the instance that prevents this assignment.


2-satisfiability, Applications, Data clustering
The same method also can be used as a subroutine when the individual cluster diameters are unknown. To test whether a given sum of diameters can be achieved without knowing the individual cluster diameters, one may try all maximal pairs of target diameters that add up to at most the given sum, representing each pair of diameters as a 2-satisfiability instance and using a 2-satisfiability algorithm to determine whether that pair can be realized by a clustering.


2-satisfiability, Applications, Data clustering
To find the optimal sum of diameters one may perform a binary search in which each step is a feasibility test of this type. The same approach also works to find clusterings that optimize other combinations than sums of the cluster diameters, and that use arbitrary dissimilarity numbers (rather than distances in a metric space) to measure the size of a cluster.


2-satisfiability, Applications, Data clustering
The time bound for this algorithm is dominated by the time to solve a sequence of 2-satisfiability instances that are closely related to each other, and Ramnath (2004) shows how to solve these related instances more quickly than if they were solved independently from each other, leading to a total time bound of O(n3) for the sum-of-diameters clustering problem.


2-satisfiability, Applications, Scheduling
Even, Itai & Shamir (1976) consider a model of classroom scheduling  in which a set of n teachers must be scheduled to teach each of m cohorts of students. The number of hours per week that teacher i{\displaystyle i} spends with cohort j{\displaystyle j} is described by entry Rij{\displaystyle R_{ij}} of a matrix R{\displaystyle R} given as input to the problem, and each teacher also has a set of hours during which he or she is available to be scheduled.


2-satisfiability, Applications, Scheduling
As they show, the problem is NP-complete, even when each teacher has at most three available hours, but it can be solved as an instance of 2-satisfiability when each teacher only has two available hours. (Teachers with only a single available hour may easily be eliminated from the problem.)


2-satisfiability, Applications, Scheduling
In this problem, each variable vij{\displaystyle v_{ij}} corresponds to an hour that teacher i{\displaystyle i} must spend with cohort j{\displaystyle j}, the assignment to the variable specifies whether that hour is the first or the second of the teacher's available hours, and there is a 2-satisfiability clause preventing any conflict of either of two types: two cohorts assigned to a teacher at the same time as each other, or one cohort assigned to two teachers at the same time.


2-satisfiability, Applications, Scheduling
Miyashiro & Matsui (2005) apply 2-satisfiability to a problem of sports scheduling, in which the pairings of a round-robin tournament have already been chosen and the games must be assigned to the teams' stadiums. In this problem, it is desirable to alternate home and away games to the extent possible, avoiding "breaks" in which a team plays two home games in a row or two away games in a row.


2-satisfiability, Applications, Scheduling
At most two teams can avoid breaks entirely, alternating between home and away games; no other team can have the same home-away schedule as these two, because then it would be unable to play the team with which it had the same schedule. Therefore, an optimal schedule has two breakless teams and a single break for every other team.


2-satisfiability, Applications, Scheduling
Once one of the breakless teams is chosen, one can set up a 2-satisfiability problem in which each variable represents the home-away assignment for a single team in a single game, and the constraints enforce the properties that any two teams have a consistent assignment for their games, that each team have at most one break before and at most one break after the game with the breakless team, and that no team has two breaks.


2-satisfiability, Applications, Scheduling
Therefore, testing whether a schedule admits a solution with the optimal number of breaks can be done by solving a linear number of 2-satisfiability problems, one for each choice of the breakless team. A similar technique also allows finding schedules in which every team has a single break, and maximizing rather than minimizing the number of breaks (to reduce the total mileage traveled by the teams).


2-satisfiability, Applications, Discrete tomography
Tomography is the process of recovering shapes from their cross-sections. In discrete tomography, a simplified version of the problem that has been frequently studied, the shape to be recovered is a polyomino (a subset of the squares in the two-dimensional square lattice), and the cross-sections provide aggregate information about the sets of squares in individual rows and columns of the lattice.


2-satisfiability, Applications, Discrete tomography
For instance, in the popular nonogram puzzles, also known as paint by numbers or griddlers, the set of squares to be determined represents the dark pixels in a binary image, and the input given to the puzzle solver tells him or her how many consecutive blocks of dark pixels to include in each row or column of the image, and how long each of those blocks should be.


2-satisfiability, Applications, Discrete tomography
In other forms of digital tomography, even less information about each row or column is given: only the total number of squares, rather than the number and length of the blocks of squares. An equivalent version of the problem is that we must recover a given 0-1 matrix given only the sums of the values in each row and in each column of the matrix.


2-satisfiability, Applications, Discrete tomography
Although there exist polynomial time algorithms to find a matrix having given row and column sums, the solution may be far from unique: any submatrix in the form of a 2 × 2 identity matrix can be complemented without affecting the correctness of the solution. Therefore, researchers have searched for constraints on the shape to be reconstructed that can be used to restrict the space of solutions. For instance, one might assume that the shape is connected; however, testing whether there exists a connected solution is NP-complete.


2-satisfiability, Applications, Discrete tomography
An even more constrained version that is easier to solve is that the shape is orthogonally convex: having a single contiguous block of squares in each row and column. Improving several previous solutions, Chrobak & Dürr (1999) showed how to reconstruct connected orthogonally convex shapes efficiently, using 2-SAT. The idea of their solution is to guess the indexes of rows containing the leftmost and rightmost cells of the shape to be reconstructed, and then to set up a 2-satisfiability problem that tests whether there exists a shape consistent with these guesses and with the given row and column sums.


2-satisfiability, Applications, Discrete tomography
They use four 2-satisfiability variables for each square that might be part of the given shape, one to indicate whether it belongs to each of four possible "corner regions" of the shape, and they use constraints that force these regions to be disjoint, to have the desired shapes, to form an overall shape with contiguous rows and columns, and to have the desired row and column sums. Their algorithm takes time O(m3n) where m is the smaller of the two dimensions of the input shape and n is the larger of the two dimensions. The same method was later extended to orthogonally convex shapes that might be connected only diagonally instead of requiring orthogonal connectivity.


2-satisfiability, Applications, Discrete tomography
A part of a solver for full nonogram puzzles, Batenburg and Kosters (2008, 2009) used 2-satisfiability to combine information obtained from several other heuristics. Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.


2-satisfiability, Applications, Discrete tomography
They also transform the nonogram into a digital tomography problem by replacing the sequence of block lengths in each row and column by its sum, and use a maximum flow formulation to determine whether this digital tomography problem combining all of the rows and columns has any squares whose state can be determined or pairs of squares that can be connected by an implication relation. If either of these two heuristics determines the value of one of the squares, it is included in the partial solution and the same calculations are repeated.


2-satisfiability, Applications, Discrete tomography
However, if both heuristics fail to set any squares, the implications found by both of them are combined into a 2-satisfiability problem and a 2-satisfiability solver is used to find squares whose value is fixed by the problem, after which the procedure is again repeated. This procedure may or may not succeed in finding a solution, but it is guaranteed to run in polynomial time.


2-satisfiability, Applications, Discrete tomography
Batenburg and Kosters report that, although most newspaper puzzles do not need its full power, both this procedure and a more powerful but slower procedure which combines this 2-satisfiability approach with the limited backtracking of Even, Itai & Shamir (1976) are significantly more effective than the dynamic programming and flow heuristics without 2-satisfiability when applied to more difficult randomly generated nonograms.


2-satisfiability, Applications, Renamable Horn satisfiability
Next to 2-satisfiability, the other major subclass of satisfiability problems that can be solved in polynomial time is Horn-satisfiability. In this class of satisfiability problems, the input is again a formula in conjunctive normal form. It can have arbitrarily many literals per clause but at most one positive literal. Lewis (1978) found a generalization of this class, renamable Horn satisfiability, that can still be solved in polynomial time by means of an auxiliary 2-satisfiability instance. A formula is renamable Horn when it is possible to put it into Horn form by replacing some variables by their negations.