Split (1)

train · 2.1M rows

title stringlengths 1 149 ⌀	section stringlengths 1 1.9k ⌀	text stringlengths 13 73.5k
Negative temperature	Examples	ln ⁡(1+2e−β+e−2β)+2e−β+2e−2β(1+2e−β+e−2β)T The resulting values for S, E, and Z all increase with T and never need to enter a negative temperature regime.
Negative temperature	Examples	Nuclear spins The previous example is approximately realized by a system of nuclear spins in an external magnetic field. This allows the experiment to be run as a variation of nuclear magnetic resonance spectroscopy. In the case of electronic and nuclear spin systems, there are only a finite number of modes available, often just two, corresponding to spin up and spin down. In the absence of a magnetic field, these spin states are degenerate, meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.
Negative temperature	Examples	In the absence of a magnetic field, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we will assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using radio frequency techniques. This causes atoms to flip from spin-down to spin-up.
Negative temperature	Examples	Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position. In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature. In NMR spectroscopy, this corresponds to pulses with a pulse width of over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures. It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the nuclear Overhauser effect with other spins).
Negative temperature	Examples	Lasers This phenomenon can also be observed in many lasing systems, wherein a large fraction of the system's atoms (for chemical and gas lasers) or electrons (in semiconductor lasers) are in excited states. This is referred to as a population inversion. The Hamiltonian for a single mode of a luminescent radiation field at frequency ν is H=(hν−μ)a†a. The density operator in the grand canonical ensemble is Tr ⁡(e−βH). For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful, βH must be positive semidefinite. So if hν < μ, and H is negative semidefinite, then β must itself be negative, implying a negative temperature.
Negative temperature	Examples	Motional degrees of freedom Negative temperatures have also been achieved in motional degrees of freedom. Using an optical lattice, upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold potassium-39 atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a Feshbach resonance and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the Bose-Hubbard Hamiltonian from Ĥ → −Ĥ. Performing this transformation adiabatically while keeping the atoms in the Mott insulator regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long lifetimes in an anti-trapping harmonic potential.
Negative temperature	Examples	Two-dimensional vortex motion The two-dimensional systems of vortices confined to a finite area can form thermal equilibrium states at negative temperature, and indeed negative temperature states were first predicted by Onsager in his analysis of classical point vortices. Onsager's prediction was confirmed experimentally for a system of quantum vortices in a Bose-Einstein condensate in 2019.
Polytopological space	Polytopological space	In general topology, a polytopological space consists of a set X together with a family {τi}i∈I of topologies on X that is linearly ordered by the inclusion relation ( I is an arbitrary index set). It is usually assumed that the topologies are in non-decreasing order, but some authors prefer to put the associated closure operators {ki}i∈I in non-decreasing order (operators ki and kj satisfy ki≤kj if and only if kiA⊆kjA for all A⊆X ), in which case the topologies have to be non-increasing.
Polytopological space	Polytopological space	Polytopological spaces were introduced in 2008 by the philosopher Thomas Icard for the purpose of defining a topological model of Japaridze's polymodal logic (GLP). They subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.
Polytopological space	Definition	An L -topological space (X,τ) is a set X together with a monotone map τ:L→ Top (X) where (L,≤) is a partially ordered set and Top (X) is the set of all possible topologies on X, ordered by inclusion. When the partial order ≤ is a linear order, then (X,τ) is called a polytopological space. Taking L to be the ordinal number n={0,1,…,n−1}, an n -topological space (X,τ0,…,τn−1) can be thought of as a set X together with n topologies τ0⊆⋯⊆τn−1 on it (or τ0⊇⋯⊇τn−1, depending on preference). More generally, a multitopological space (X,τ) is a set X together with an arbitrary family τ of topologies on X.
Liebermann–Burchard test	Liebermann–Burchard test	The Liebermann–Burchard or acetic anhydride test is used for the detection of cholesterol. The formation of a green or green-blue colour after a few minutes is positive.
Liebermann–Burchard test	Liebermann–Burchard test	Lieberman–Burchard is a reagent used in a colourimetric test to detect cholesterol, which gives a deep green colour. This colour begins as a purplish, pink colour and progresses through to a light green then very dark green colour. The colour is due to the hydroxyl group (-OH) of cholesterol reacting with the reagents and increasing the conjugation of the un-saturation in the adjacent fused ring. Since this test uses acetic anhydride and sulfuric acid as reagents, caution must be exercised so as not to receive severe burns.
Liebermann–Burchard test	Liebermann–Burchard test	Method: Dissolve one or two crystals of cholesterol in dry chloroform in a dry test tube. Add several drops of acetic anhydride and then 2 drops of concentrated H2SO4 and mix carefully. After the reaction is finished, the concentration of cholesterol can be measured using spectrophotometry.
Edupunk	Edupunk	Edupunk is a do it yourself (DIY) attitude to teaching and learning practices. Tom Kuntz described edupunk as "an approach to teaching that avoids mainstream tools like PowerPoint and Blackboard, and instead aims to bring the rebellious attitude and DIY ethos of ’70s bands like The Clash to the classroom." Many instructional applications can be described as DIY education or edupunk.
Edupunk	Edupunk	The term was first used on May 25, 2008, by Jim Groom in his blog, and covered less than a week later in the Chronicle of Higher Education. Stephen Downes, an online education theorist and an editor for the International Journal of Instructional Technology and Distance Learning, noted that "the concept of edupunk has totally caught wind, spreading through the blogosphere like wildfire".
Edupunk	Aspects	Edupunk has risen from an objection to the efforts of government and corporate interests in reframing and bundling emerging technologies into cookie-cutter products with pre-defined application—somewhat similar to traditional punk ideologies.The reaction to corporate influence on education is only one part of edupunk, though. Stephen Downes has identified three aspects to this approach: Reaction against commercialization of learning Do-it-yourself attitude Thinking and learning for yourself
Edupunk	Examples	An example of edupunk was the University of British Columbia's course "Wikipedia:WikiProject Murder Madness and Mayhem" experiment of creating articles on Wikipedia in spring 2008, "(having) one’s students as partners and peers." A video clip illustrating an edupunk approach, produced by Tony Hirst at the Open University in the UK, on 8 June 2008, illustrated how quickly the edupunk concept has been adopted outside North America.
Edupunk	Examples	A website set up by Australian educators illustrates how edupunk spread, and a presentation by Norm Friesen of Thompson Rivers University identifies a number of possible intellectual precursors for the movement.Hampshire College, Evergreen State College, Marlboro College, New College of Florida, and Warren Wilson College are collegiate institutions imbued with edupunk ideology.
Branding iron	Branding iron	A branding iron is used for branding, pressing a heated metal shape against an object or livestock with the intention of leaving an identifying mark.
Branding iron	History	The history of branding is very much tied to the history of using animals as a commodity. The act of marking livestock with fire-heated marks to identify ownership begins in ancient times with the ancient Egyptians. The process continued throughout the ages, with both Romans and American colonists using the process to brand slaves as well.In the English lexicon, the Germanic word "brand" originally meant anything hot or burning, such as a fire-brand, a burning stick. By the European Middle Ages it commonly identified the process of burning a mark into a stock animals with thick hides, such as cattle, so as to identify ownership under animus revertendi. In England, the rights of common including the common pasture system meant that cattle could be grazed on certain land with commoner's rights and the cattle were branded to show ownership, often with the commoner's or Lord of the manor's mark. The practice was widespread in most European nations with large cattle grazing regions, including Spain. With colonialism, many cattle branding traditions and techniques were spread via the Spanish Empire to South America and to countries of the British Empire including the Americas, Australasia & South Africa where distinct sets of traditions and techniques developed respectively.
Branding iron	History	In the Americas these European systems continued with English tradition being used in the New England Colonies and spread outwards with the western expansion of the U.S. The Spanish system evolved from the south with the vaquero tradition in what today is the southwestern United States and northern Mexico. The branding iron consisted of an iron rod with a simple symbol or mark which was heated in a fire. After the branding iron turned red-hot, the cowhand pressed the branding iron against the hide of the cow. The unique brand meant that cattle owned by multiple owners could then graze freely together on the commons or open range. Drovers or cowboys could then separate the cattle at roundup time for driving to market.
Branding iron	Types of branding irons	Branding Irons come in a variety of styles, designed primarily by their method of heating. Fire-heated The traditional fire-heated method is still in use today. While they require longer lengths of time to heat, are inconsistent in temperature and all around inferior to more advanced forms of branding, they are inexpensive to produce and purchase. Fire-heated branding irons are used to brand wood, steak, leather, livestock and plastics.
Branding iron	Types of branding irons	Electric Electric branding irons utilize an electric heating element to heat a branding iron to the desired temperature. Electric branding irons come in many variations from irons designed to brand cattle, irons designed to mark wood and leather and models designed to be placed inside a drill press for the purposes of manufacturing. An electric branding iron’s temperature can be controlled by increasing or decreasing the flow of electricity.
Branding iron	Types of branding irons	Propane Propane Branding Irons use a continuous flow of propane to heat the iron head. They are commonly used where electricity is not available. Utilizing the flow of propane, the temperature can be adjusted for varying branding environments. A commercially built branding iron heater fired with L.P. gas is a common method of heating several branding irons at once.
Branding iron	Types of branding irons	Freeze-branding In stark contrast to traditional hot-iron branding, freeze branding uses an iron that has been chilled with a coolant such as dry ice or liquid nitrogen. Instead of burning a scar into the animal's skin, a freeze brand damages the pigment-producing hair cells, causing the animal's hair to grow back white within the branded area. This white-on-dark pattern is prized by cattle ranchers as its contrast allows some range work to be conducted with binoculars rather than individual visits to every animal. To apply a freeze brand the hair coat of the animal is first shaved very closely so that bare skin is exposed. Then the frozen iron is pressed to the animal's bare skin for a period of time that varies with both the species of animal and the color of its hair coat. Shorter times are used on dark-colored animals, as this causes follicle melanocyte death and hence permanent pigment loss to the hair when it regrows. Longer times, sometimes as little as five seconds more, are needed for animals with white hair coats. In these cases the brand is applied for long enough to kill the cells of the growth follicle, those that create the hair filaments themselves. This leaves the animal permanently bald in the branded area. The somewhat darker epidermis then contrasts well with a pale animal's coat.
Branding iron	Popular use	Livestock Livestock branding is perhaps the most prevalent use of a branding iron. Modern use includes gas heating, the traditional fire-heated method, an iron heated by electricity (electric cattle branding iron) or an iron super cooled by dry ice (freeze branding iron). Cattle, horses and other livestock are commonly branded today for the same reason they were in Ancient times, to prove ownership.
Branding iron	Popular use	Wood branding Woodworkers will often use Electric or Fire-Heated Branding Irons to leave their maker's mark or company logo. Timber pallets and other timber export packaging is often marked in this way in accordance with ISPM 15 to indicate that the timber has been treated to prevent it carrying pests. Steak Steak branding irons are used commonly by barbecue enthusiasts and professional chefs to leave a mark indicating how well done a steak is or to identify the chef or grill master. Leather Branding Irons are used often by makers of horse tack often in place of a steel leather stamp to indicate craftsmanship.
Johnson Bar (locomotive)	Johnson Bar (locomotive)	On a steam locomotive, the reversing gear is used to control the direction of travel of the locomotive. It also adjusts the cutoff of the steam locomotive.
Johnson Bar (locomotive)	Reversing lever	This is the most common form of reverser. It is also known as a Johnson bar in the United States. It consists of a long lever mounted parallel to the direction of travel, on the driver’s side of the cab. It has a handle and sprung trigger at the top and is pivoted at the bottom to pass between two notched sector plates. The reversing rod, which connects to the valve gear, is attached to this lever, either above or below the pivot, in such a position as to give good leverage. A square pin is arranged to engage with the notches in the plates and hold the lever in the desired position when the trigger is released.
Johnson Bar (locomotive)	Reversing lever	The advantages of this design are that change between forward and reverse gear can be made very quickly (as is needed in, for example, a shunting engine).
Johnson Bar (locomotive)	Reversing lever	Limitations and drawbacks The reversing lever has a catch mechanism which engages with a series of notches to hold the lever at the desired cut-off position. This means that the operator does not have a full choice of cut-off positions between maximum and mid-gear, but only those which correspond with the notches. The position of the notches is chosen by the locomotive designer or constructor with a view to the locomotive's intended purpose. In general engines designed for freight will have fewer notches with a 'longer' minimum cut-off (providing high tractive effort at low speeds but poor efficiency at high speeds) while a passenger locomotive will have more notches and a shorter minimum cut-off (allowing efficiency at high speeds at the expense of tractive effort). If the minimum cut-off provided for by the notches was too high, it would not be possible to run the locomotive in the efficient way described above (with a fully open regulator) without leading to steam wastage or 'choking' of the steam passages, so the regulator would have to be closed. That limits efficiency. The Johnson Bar is effectively part of the entire valve gear, being connected to the various linkages and arms in order to serve its function in adjusting them. This means that the forces in the valve gear can be transmitted to the lever. This is especially the case if the engine has unbalanced slide valves, which have a high operating friction and are subject to steam forces on both sides of the valve. This friction meant that if the Johnson Bar is unlatched while the engine is operating under high steam pressure (wide regulator openings and high cut-off) or at high speeds, the forces that are supposed to act on the slide valves can instead be transmitted back through the linkage to the now-free reversing lever. This will suddenly and violently throw the lever into the full cut-off position, carrying with it the real danger of injury to the driver, damage to the valve gear and triggering wheel slip in the locomotive. The only way to prevent this is to close the regulator and allow the steam pressure in the valve chest to drop. The reversing lever can then be unlatched and set to a new cut-off position and then the regulator could be opened again. During this process the locomotive is not under power. On ascending gradients it was a matter of great skill to reduce the regulator opening by enough to safely unlatch the Johnson Bar while maintaining sufficient steam pressure to the cylinders. Each time the regulator was re-opened was a chance to encounter wheel slip and in loose coupled trains each closure and opening of the regulator set up dynamic forces throughout the length of the train which risked broken couplings. The screw reverser overcame all these issues.
Johnson Bar (locomotive)	Reversing lever	Ban in the US The dangers of the traditional Johnson Bar (which grew as locomotive power, weight and operating steam pressures increased through the first half of the 20th century) led to it being banned in the USA by the Interstate Commerce Commission. From 1939 all new-build steam locomotives had to be fitted with power reversers and from 1942 Johnson Bar-fitted engines undergoing heavy overhaul or rebuilding had to be retro-fitted with power reverse. Exceptions existed for light, low-powered locomotives and switchers. For switching, which required frequent changes of direction from full-ahead to full-reverse gear, the Johnson Bar was favored because the change could be made quickly in a single motion instead of the multiple turns of the handle of a low-geared screw reverser.
Johnson Bar (locomotive)	Screw reverser	In the screw reverser mechanism (sometimes called a bacon slicer in the UK), the reversing rod is controlled by a screw and nut, worked by a wheel in the cab. The nut either operates on the reversing rod directly or through a lever, as above. The screw and nut may be cut with a double thread and a coarse pitch to move the mechanism as quickly as possible. The wheel is fitted with a locking lever to prevent creep and there is an indicator to show the percentage of cutoff in use. This method of altering the cutoff offers finer control than the sector lever, but it has the disadvantage of slow operation. It is most suitable for long-distance passenger engines where frequent changes of cutoff are not required and where fine adjustments offer the most benefit. On locomotives fitted with Westinghouse air brake equipment and Stephenson valve gear, it was common to use the screw housing as an air cylinder, with the nut extended to form a piston. Compressed air from the brake reservoirs was applied to one side of the piston to reduce the effort required to lift the heavy expansion link, with gravity assisting in the opposite direction.
Johnson Bar (locomotive)	Screw reverser	Power reverse gear With larger engines, the linkages involved in controlling cutoff and direction grew progressively heavier and there was a need for power assistance in adjusting them. Steam (later, compressed air) powered reversing gears were developed in the late 19th and early 20th centuries. Typically, the operator worked a valve that admitted steam to one side or the other of a cylinder connected to the reversing mechanism until the indicator showed the intended position. A second mechanism—usually a piston in an oil-filled cylinder held in position by closing a control cock—was required to keep the linkages in place. Stirling gearThe first locomotive engineer to fit such a device was James Stirling of the Glasgow and South Western Railway in 1873. Several engineers then tried them, including William Dean of the GWR and Vincent Raven of the North Eastern Railway, but they found them little to their liking, mainly because of maintenance difficulties: any oil leakage from the locking cylinder, either through the piston gland or the cock, allowed the mechanism to creep, or worse “nose-dive”, into full forward gear while running. Stirling moved to the South Eastern Railway and Harry Smith Wainwright, his successor at that company, incorporated them into most of his designs, which were in production about thirty years after Stirling’s innovation. Later still the forward-looking Southern Railway engineer Oliver Bulleid fitted them to his famous Merchant Navy Class of locomotives, but they were mostly removed at rebuild.
Johnson Bar (locomotive)	Screw reverser	Henszey's reversing gearPatented in 1882, the Henszey's reversing gear illustrates a typical early solution. Henszey's device consists of two pistons mounted on a single piston rod. Both pistons are double-ended. One is a steam piston to move the rod as required. The other, containing oil, holds the rod in a fixed position when the steam is turned off. Control is by a small three-way steam valve (“forward”, “stop”, “back”) and a separate indicator showing the position of the rod and thus the percentage of cutoff in use. When the steam valve is at “stop”, an oil cock connecting the two ends of the locking piston is also closed, thus holding the mechanism in position. The piston rod connects by levers to the reversing gear, which operates in the usual way, according to the type of valve gear in use.
Johnson Bar (locomotive)	Screw reverser	The Ragonnet power reverserThe Ragonnet power reverse, patented in 1909, was a true feedback controlled servomechanism. The power reverse amplified small motions of the reversing lever made in the locomotive cab with modest force into much larger and more forceful motions of the reach rod that controlled the engine cutoff and direction. It was usually air powered, but could also be steam powered. The term servomotor was explicitly used by the developers of some later power reverse mechanisms. The use of feedback control in these later power reverse mechanisms eliminated the need for a second cylinder for a hydraulic locking mechanism, and it restored the simplicity of a single operating lever that both controlled the reversing linkage and indicated its position.
Johnson Bar (locomotive)	Screw reverser	Power reverse impetus The development of articulated locomotives was a major impetus to the development of power reverse systems, because these typically had two or even three sets of reverse gear, instead of just one on a simple locomotive. The Baldwin Locomotive Works used the Ragonnet reversing gear, and other US builders generally abandoned positive locking features sooner than later. Many American locomotives were built, or retro-fitted, with power reversers, including the PRR K4s, PRR N1s, PRR B6, and PRR L1s, but in Britain locking cylinders remained in use. The Hadfield reversing gear, patented in 1950, was in most particulars a Ragonnet reversing gear with added locking cylinder. Most Beyer Garratt locomotives used the Hadfield system.
Johnson Bar (locomotive)	Sources	Allen, Cecil J; (1949); Locomotive Practice and Performance in the Twentieth Century; W. Heffer and Sons Ltd.; Cambridge Bell, A. Morton; (1950); Locomotives : Volume one; Seventh edition; London, Virtue and Company Ltd.
Noncommutative logic	Noncommutative logic	Noncommutative logic is an extension of linear logic that combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders that may be viewed as a species of structure), and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.
Noncommutative logic	Noncommutativity in logic	By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible. The remainder of this article is devoted to a presentation of this acceptance of the term. The oldest noncommutative logic is the Lambek calculus, which gave rise to the class of logics known as categorial grammars. Since the publication of Jean-Yves Girard's linear logic there have been several new noncommutative logics proposed, namely the cyclic linear logic of David Yetter, the pomset logic of Christian Retoré, and the noncommutative logics BV and NEL.
Noncommutative logic	Noncommutativity in logic	Noncommutative logic is sometimes called ordered logic, since it is possible with most proposed noncommutative logics to impose a total or partial order on the formulae in sequents. However this is not fully general since some noncommutative logics do not support such an order, such as Yetter's cyclic linear logic. Although most noncommutative logics do not allow weakening or contraction together with noncommutativity, this restriction is not necessary.
Noncommutative logic	Noncommutativity in logic	The Lambek calculus Joachim Lambek proposed the first noncommutative logic in his 1958 paper Mathematics of Sentence Structure to model the combinatory possibilities of the syntax of natural languages. His calculus has thus become one of the fundamental formalisms of computational linguistics.
Noncommutative logic	Noncommutativity in logic	Cyclic linear logic David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic. Sequents of cyclic linear logic form a ring, and so are invariant under rotation, where multipremise rules glue their rings together at the formulae described in the rules. The calculus supports three structural modalities, a self-dual modality allowing exchange, but still linear, and the usual exponentials (? and !) of linear logic, allowing nonlinear structural rules to be used together with exchange.
Noncommutative logic	Noncommutativity in logic	Pomset logic Pomset logic was proposed by Christian Retoré in a semantic formalism with two dual sequential operators existing together with the usual tensor product and par operators of linear logic, the first logic proposed to have both commutative and noncommutative operators. A sequent calculus for the logic was given, but it lacked a cut-elimination theorem; instead the sense of the calculus was established through a denotational semantics.
Noncommutative logic	Noncommutativity in logic	BV and NEL Alessio Guglielmi proposed a variation of Retoré's calculus, BV, in which the two noncommutative operations are collapsed onto a single, self-dual, operator, and proposed a novel proof calculus, the calculus of structures to accommodate the calculus. The principal novelty of the calculus of structures was its pervasive use of deep inference, which it was argued is necessary for calculi combining commutative and noncommutative operators; this explanation concurs with the difficulty of designing sequent systems for pomset logic that have cut-elimination.
Noncommutative logic	Noncommutativity in logic	Lutz Straßburger devised a related system, NEL, also in the calculus of structures in which linear logic with the mix rule appears as a subsystem. Structads Structads are an approach to the semantics of logic that are based upon generalising the notion of sequent along the lines of Joyal's combinatorial species, allowing the treatment of more drastically nonstandard logics than those described above, where, for example, the ',' of the sequent calculus is not associative.
P21 holin family	P21 holin family	The Phage 21 S (P21 Holin) Family (TC# 1.E.1) is a member of the Holin Superfamily II.The Bacteriophage P21 Lysis protein S holin (TC# 1.E.1.1.1) is the prototype for class II holins. Lysis S proteins have two transmembrane segments (TMSs), with both the N- and C-termini on the cytoplasmic side of the inner membrane. TMS1 may be dispensable for function.A homologue of the P21 holin is the holin of bacteriophage H-19B (TC# 1.E.1.1.3). The gene encoding it has been associated with the Shiga-like Toxin I gene in E. coli. It may function in toxin export as has been proposed for the X. nematophila holin-1 (TC #1.E.2.1.4).A representative list of proteins belonging to the P21 holin family can be found in the Transporter Classification Database.
A3 coupling reaction	A3 coupling reaction	The A3 coupling (also known as A3 coupling reaction or the aldehyde-alkyne-amine reaction), coined by Prof. Chao-Jun Li of McGill University, is a type of multicomponent reaction involving an aldehyde, an alkyne and an amine which react to give a propargylamine.
A3 coupling reaction	A3 coupling reaction	The reaction proceeds via direct dehydrative condensation and requires a metal catalyst, typically based on ruthenium/copper, gold or silver. Chiral catalyst can be used to give an enantioselective reaction, yielding a chiral amine. The solvent can be water. In the catalytic cycle the metal activates the alkyne to a metal acetylide, the amine and aldehyde combine to form an imine which then reacts with the acetylide in a nucleophilic addition. The reaction type was independently reported by three research groups in 2001 -2002; one report on a similar reaction dates back to 1953.If the amine substituents have an alpha hydrogen present and provided a suitable zinc or copper catalyst is used, the A3 coupling product may undergo a further internal hydride transfer and fragmentation to give an allene in a Crabbé reaction.
A3 coupling reaction	Decarboxylative A3 reaction	One variation is called the decarboxylative A3 coupling. In this reaction the amine is replaced by an amino acid. The imine can isomerise and the alkyne group is placed at the other available nitrogen alpha position. This reaction requires a copper catalyst. The redox A3 coupling has the same product outcome but the reactants are again an aldehyde, an amine and an alkyne as in the regular A3 coupling.
Fei–Ranis model of economic growth	Fei–Ranis model of economic growth	The Fei–Ranis model of economic growth is a dualism model in developmental economics or welfare economics that has been developed by John C. H. Fei and Gustav Ranis and can be understood as an extension of the Lewis model. It is also known as the Surplus Labor model. It recognizes the presence of a dual economy comprising both the modern and the primitive sector and takes the economic situation of unemployment and underemployment of resources into account, unlike many other growth models that consider underdeveloped countries to be homogenous in nature. According to this theory, the primitive sector consists of the existing agricultural sector in the economy, and the modern sector is the rapidly emerging but small industrial sector. Both the sectors co-exist in the economy, wherein lies the crux of the development problem. Development can be brought about only by a complete shift in the focal point of progress from the agricultural to the industrial economy, such that there is augmentation of industrial output. This is done by transfer of labor from the agricultural sector to the industrial one, showing that underdeveloped countries do not suffer from constraints of labor supply. At the same time, growth in the agricultural sector must not be negligible and its output should be sufficient to support the whole economy with food and raw materials. Like in the Harrod–Domar model, saving and investment become the driving forces when it comes to economic development of underdeveloped countries.
Fei–Ranis model of economic growth	Basics of the model	One of the biggest drawbacks of the Lewis model was the undermining of the role of agriculture in boosting the growth of the industrial sector. In addition to that, he did not acknowledge that the increase in productivity of labor should take place prior to the labor shift between the two sectors. However, these two ideas were taken into account in the Fei–Ranis dual economy model of three growth stages. They further argue that the model lacks in the proper application of concentrated analysis to the change that takes place with agricultural development In Phase 1 of the Fei–Ranis model, the elasticity of the agricultural labor work-force is infinite and as a result, suffers from disguised unemployment. Also, the marginal product of labor is zero. This phase is similar to the Lewis model. In Phase 2 of the model, the agricultural sector sees a rise in productivity and this leads to increased industrial growth such that a base for the next phase is prepared. In Phase 2, agricultural surplus may exist as the increasing average product (AP), higher than the marginal product (MP) and not equal to the subsistence level of wages.Using the help of the figure on the left, we see that Phase 1 from figure and from figure )=AP According to Fei and Ranis, AD amount of labor (see figure) can be shifted from the agricultural sector without any fall in output. Hence, it represents surplus labor.
Fei–Ranis model of economic growth	Basics of the model	Phase2 :AP>MP After AD, MP begins to rise, and industrial labor rises from zero to a value equal to AD. AP of agricultural labor is shown by BYZ and we see that this curve falls downward after AD. This fall in AP can be attributed to the fact that as agricultural laborers shift to the industrial sector, the real wage of industrial laborers decreases due to shortage of food supply, since less laborers are now working in the food sector. The decrease in the real wage level decreases the level of profits, and the size of surplus that could have been re-invested for more industrialization. However, as long as surplus exists, growth rate can still be increased without a fall in the rate of industrialization. This re-investment of surplus can be graphically visualized as the shifting of MP curve outwards. In Phase2 the level of disguised unemployment is given by AK. This allows the agricultural sector to give up a part of its labor-force until Real wages Constant institutional wages (CIW) Phase 3 begins from the point of commercialization which is at K in the Figure. This is the point where the economy becomes completely commercialized in the absence of disguised unemployment. The supply curve of labor in Phase 3 is steeper and both the sectors start bidding equally for labor.
Fei–Ranis model of economic growth	Basics of the model	Phase3 MP CIW The amount of labor that is shifted and the time that this shifting takes depends upon: The growth of surplus generated within the agricultural sector, and the growth of industrial capital stock dependent on the growth of industrial profits; The nature of the industry's technical progress and its associated bias; Growth rate of population.So, the three fundamental ideas used in this model are: Agricultural growth and industrial growth are both equally important; Agricultural growth and industrial growth are balanced; Only if the rate at which labor is shifted from the agricultural to the industrial sector is greater than the rate of growth of population will the economy be able to lift itself up from the Malthusian population trap.This shifting of labor can take place by the landlords' investment activities and by the government's fiscal measures. However, the cost of shifting labor in terms of both private and social cost may be high, for example transportation cost or the cost of carrying out construction of buildings. In addition to that, per capita agricultural consumption can increase, or there can exist a wide gap between the wages of the urban and the rural people. These three occurrences- high cost, high consumption and high gap in wages, are called as leakages, and leakages prevent the creation of agricultural surplus. In fact, surplus generation might be prevented due to a backward-sloping supply curve of labor as well, which happens when high income-levels are not consumed. This would mean that the productivity of laborers with rise in income will not rise. However, the case of backward-sloping curves is mostly unpractical.
Fei–Ranis model of economic growth	Connectivity between sectors	Fei and Ranis emphasized strongly on the industry-agriculture interdependency and said that a robust connectivity between the two would encourage and speedup development. If agricultural laborers look for industrial employment, and industrialists employ more workers by use of larger capital good stock and labor-intensive technology, this connectivity can work between the industrial and agricultural sector. Also, if the surplus owner invests in that section of industrial sector that is close to soil and is in known surroundings, he will most probably choose that productivity out of which future savings can be channelized. They took the example of Japan's dualistic economy in the 19th century and said that connectivity between the two sectors of Japan was heightened due to the presence of a decentralized rural industry which was often linked to urban production. According to them, economic progress is achieved in dualistic economies of underdeveloped countries through the work of a small number of entrepreneurs who have access to land and decision-making powers and use industrial capital and consumer goods for agricultural practices.
Fei–Ranis model of economic growth	Connectivity between sectors	Agricultural sector In (A), land is measured on the vertical axis, and labor on the horizontal axis. Ou and Ov represent two ridge lines, and the production contour lines are depicted by M, M1 and M2. The area enclosed by the ridge lines defines the region of factor substitutability, or the region where factors can easily be substituted. Let us understand the repercussions of this. If te amount of labor is the total labor in the agricultural sector, the intersection of the ridge line Ov with the production curve M1 at point s renders M1 perfectly horizontal below Ov. The horizontal behavior of the production line implies that outside the region of factor substitutability, output stops and labor becomes redundant once land is fixed and labor is increased.If Ot is the total land in the agricultural sector, ts amount of labor can be employed without it becoming redundant, and es represents the redundant agricultural labor force. This led Fei and Ranis to develop the concept of Labor Utilization Ratio, which they define as the units of labor that can be productively employed (without redundancy) per unit of land. In the left-side figure, labor utilization ratio R=tsOt which is graphically equal to the inverted slope of the ridge line Ov.
Fei–Ranis model of economic growth	Connectivity between sectors	Fei and Ranis also built the concept of endowment ratio, which is a measure of the relative availability of the two factors of production. In the figure, if Ot represents agricultural land and tE represents agricultural labor, then the endowment ratio is given by S=tEOt which is equal to the inverted slope of OE. The actual point of endowment is given by E. Finally, Fei and Ranis developed the concept of non-redundancy coefficient T which is measured by T=tste These three concepts helped them in formulating a relationship between T, R and S. If :: T=tste, then or T=RS This mathematical relation proves that the non-redundancy coefficient is directly proportional to labor utilization ratio and is inversely proportional to the endowment ratio.
Fei–Ranis model of economic growth	Connectivity between sectors	(B) displays the total physical productivity of labor (TPPL) curve. The curve increases at a decreasing rate, as more units of labor are added to a fixed amount of land. At point N, the curve shapes horizontally and this point N conforms to the point G in (C, which shows the marginal productivity of labor (MPPL) curve, and with point s on the ridge line Ov in (A).
Fei–Ranis model of economic growth	Connectivity between sectors	Industrial sector Like in the agricultural sector, Fei and Ranis assume constant returns to scale in the industrial sector. However, the main factors of production are capital and labor. In the graph (A) right hand side, the production functions have been plotted taking labor on the horizontal axis and capital on the vertical axis. The expansion path of the industrial sector is given by the line OAoA1A2. As capital increases from Ko to K1 to K2 and labor increases from Lo to L1 and L2, the industrial output represented by the production contour Ao, A1 and A3 increases accordingly.
Fei–Ranis model of economic growth	Connectivity between sectors	According to this model, the prime labor supply source of the industrial sector is the agricultural sector, due to redundancy in the agricultural labor force. (B) shows the labor supply curve for the industrial sector S. PP2 represents the straight line part of the curve and is a measure of the redundant agricultural labor force on a graph with industrial labor force on the horizontal axis and output/real wage on the vertical axis. Due to the redundant agricultural labor force, the real wages remain constant but once the curve starts sloping upwards from point P2, the upward sloping indicates that additional labor would be supplied only with a corresponding rise in the real wages level.
Fei–Ranis model of economic growth	Connectivity between sectors	MPPL curves corresponding to their respective capital and labor levels have been drawn as Mo, M1, M2 and M3. When capital stock rises from Ko to K1, the marginal physical productivity of labor rises from Mo to M1. When capital stock is Ko, the MPPL curve cuts the labor supply curve at equilibrium point Po. At this point, the total real wage income is Wo and is represented by the shaded area POLoPo. λ is the equilibrium profit and is represented by the shaded area qPPo. Since the laborers have extremely low income-levels, they barely save from that income and hence industrial profits (πo) become the prime source of investment funds in the industrial sector.
Fei–Ranis model of economic growth	Connectivity between sectors	Kt=Ko+So+Πo Here, Kt gives the total supply of investment funds (given that rural savings are represented by So) Total industrial activity rises due to increase in the total supply of investment funds, leading to increased industrial employment.
Fei–Ranis model of economic growth	Agricultural surplus	Agricultural surplus in general terms can be understood as the produce from agriculture which exceeds the needs of the society for which it is being produced, and may be exported or stored for future use. Generation of agricultural surplus To understand the formation of agricultural surplus, we must refer to graph (B) of the agricultural sector. The figure on the left is a reproduced version of a section of the previous graph, with certain additions to better explain the concept of agricultural surplus.
Fei–Ranis model of economic growth	Agricultural surplus	We first derive the average physical productivity of the total agricultural labor force (APPL). Fei and Ranis hypothesize that it is equal to the real wage and this hypothesis is known as the constant institutional wage hypothesis. It is also equal in value to the ratio of total agricultural output to the total agricultural population. Using this relation, we can obtain APPL = MP/OP. This is graphically equal to the slope of line OM, and is represented by the line WW in (C).
Fei–Ranis model of economic growth	Agricultural surplus	Observe point Y, somewhere to the left of P on the graph. If a section of the redundant agricultural labor force (PQ) is removed from the total agricultural labor force (OP) and absorbed into the industrial sector, then the labor force remaining in the industrial sector is represented by the point Y. Now, the output produced by the remaining labor force is represented by YZ and the real income of this labor force is given by XY. The difference of the two terms yields the total agricultural surplus of the economy. It is important to understand that this surplus is produced by the reallocation of labor such that it is absorbed by the industrial sector. This can be seen as deployment of hidden rural savings for the expansion of the industrial sector. Hence, we can understand the contribution of the agricultural sector to the expansion of industrial sector by this allocation of redundant labor force and the agricultural surplus that results from it.
Fei–Ranis model of economic growth	Agricultural surplus	Agricultural surplus as wage fund Agricultural surplus plays a major role as a wage fund. Its importance can be better explained with the help of the graph on the right, which is an integration of the industrial sector graph with an inverted agricultural sector graph, such that the origin of the agricultural sector falls on the upper-right corner. This inversion of the origin changes the way the graph is now perceived. While the labor force values are read from the left of 0, the output values are read vertically downwards from O. The sole reason for this inversion is for the sake of convenience. The point of commercialization as explained before (See Section on Basics of the model) is observed at point R, where the tangent to the line ORX runs parallel to OX.
Fei–Ranis model of economic growth	Agricultural surplus	Before a section of the redundant labor force is absorbed into the industrial sector, the entire labor OA is present in the agricultural sector. Once AG amount of labor force (say) is absorbed, it represented by OG' in the industrial sector, and the labor remaining in the agricultural sector is then OG. But how is the quantity of labor absorbed into the industrial sector determined? (A) shows the supply curve of labor SS' and several demand curves for labor df, d'f' and d"f". When the demand for labor is df, the intersection of the demand-supply curves gives the equilibrium employment point G'. Hence OG represents the amount of labor absorbed into the industrial sector. In that case, the labor remaining in the agricultural sector is OG. This OG amount of labor produces an output of GF, out of which GJ amount of labor is consumed by the agricultural sector and JF is the agricultural surplus for that level of employment. Simultaneously, the unproductive labor force from the agricultural sector turns productive once it is absorbed by the industrial sector, and produces an output of OG'Pd as shown in the graph, earning a total wage income of OG'PS.
Fei–Ranis model of economic growth	Agricultural surplus	The agricultural surplus JF created is needed for consumption by the same workers who left for the industrial sector. Hence, agriculture successfully provides not only the manpower for production activities elsewhere, but also the wage fund required for the process.
Fei–Ranis model of economic growth	Agricultural surplus	Significance of agriculture in the Fei–Ranis model The Lewis model is criticised on the grounds that it neglects agriculture. Fei–Ranis model goes a step beyond and states that agriculture has a very major role to play in the expansion of the industrial sector. In fact, it says that the rate of growth of the industrial sector depends on the amount of total agricultural surplus and on the amount of profits that are earned in the industrial sector. So, larger the amount of surplus and the amount of surplus put into productive investment and larger the amount of industrial profits earned, the larger will be the rate of growth of the industrial economy. As the model focuses on the shifting of the focal point of progress from the agricultural to the industrial sector, Fei and Ranis believe that the ideal shifting takes place when the investment funds from surplus and industrial profits are sufficiently large so as to purchase industrial capital goods like plants and machinery. These capital goods are needed for the creation of employment opportunities. Hence, the condition put by Fei and Ranis for a successful transformation is that Rate of increase of capital stock & rate of employment opportunities > Rate of population growth
Fei–Ranis model of economic growth	The indispensability of labor reallocation	As an underdeveloped country goes through its development process, labor is reallocated from the agricultural to the industrial sector. More the rate of reallocation, faster is the growth of that economy. The economic rationale behind this idea of labor reallocation is that of faster economic development. The essence of labor reallocation lies in Engel's Law, which states that the proportion of income being spent on food decreases with increase in the income-level of an individual, even if there is a rise in the actual expenditure on food. For example, if 90 per cent of the entire population of the concerned economy is involved in agriculture, that leaves just 10 per cent of the population in the industrial sector. As the productivity of agriculture increases, it becomes possible for just 35 per cent of population to maintain a satisfactory food supply for the rest of the population. As a result, the industrial sector now has 65 per cent of the population under it. This is extremely desirable for the economy, as the growth of industrial goods is subject to the rate of per capita income, while the growth of agricultural goods is subject only to the rate of population growth, and so a bigger labor supply to the industrial sector would be welcome under the given conditions. In fact, this labor reallocation becomes necessary with time since consumers begin to want more of industrial goods than agricultural goods in relative terms.
Fei–Ranis model of economic growth	The indispensability of labor reallocation	However, Fei and Ranis were quick to mention that the necessity of labor reallocation must be linked more to the need to produce more capital investment goods as opposed to the thought of industrial consumer goods following the discourse of Engel's Law. This is because the assumption that the demand for industrial goods is high seems unrealistic, since the real wage in the agricultural sector is extremely low and that hinders the demand for industrial goods. In addition to that, low and mostly constant wage rates will render the wage rates in the industrial sector low and constant. This implies that demand for industrial goods will not rise at a rate as suggested by the use of Engel's Law.
Fei–Ranis model of economic growth	The indispensability of labor reallocation	Since the growth process will observes a slow-paced increase in the consumer purchasing power, the dualistic economies follow the path of natural austerity, which is characterized by more demand and hence importance of capital good industries as compared to consumer good ones. However, investment in capital goods comes with a long gestation period, which drives the private entrepreneurs away. This suggests that in order to enable growth, the government must step in and play a major role, especially in the initial few stages of growth. Additionally, the government also works on the social and economic overheads by the construction of roads, railways, bridges, educational institutions, health care facilities and so on.
Fei–Ranis model of economic growth	Growth without development	In the Fei-Ranis model, it is possible that as technological progress takes place and there is a shift to labor-saving production techniques, growth of the economy takes place with increase in profits but no economic development takes place. This can be explained well with the help of graph in this section.
Fei–Ranis model of economic growth	Growth without development	The graph displays two MPL lines plotted with real wage and MPL on the vertical axis and employment of labor on the horizontal axis. OW denotes the subsistence wage level, which is the minimum wage level at which a worker (and his family) would survive. The line WW' running parallel to the X-axis is considered to be infinitely elastics since supply of labor is assumed to be unlimited at the subsistence-wage level. The square area OWEN represents the wage bill and DWE represents the surplus or the profits collected. This surplus or profit can increase if the MPL curve changes.If the MPL curve changes from MPL1 to MPL2 due to a change in production technique, such that it becomes labor-saving or capital-intensive, then the surplus or profit collected would increase. This increase can be seen by comparing DWE with D1WE since D1WE since is greater in area compared to DWE. However, there is no new point of equilibrium and as E continues to be the point of equilibrium, there is no increase in the level of labor employment, or in wages for that matter. Hence, labor employment continues as ON and wages as OW. The only change that accompanies the change in production technique is the one in surplus or profits.This makes for a good example of a process of growth without development, since growth takes place with increase in profits but development is at a standstill since employment and wages of laborers remain the same.
Fei–Ranis model of economic growth	Reactions to the model	Fei–Ranis model of economic growth has been criticized on multiple grounds, although if the model is accepted, then it will have a significant theoretical and policy implications on the underdeveloped countries' efforts towards development and on the persisting controversial statements regarding the balanced vs. unbalanced growth debate. It has been asserted that Fei and Ranis did not have a clear understanding of the sluggish economic situation prevailing in the developing countries. If they had thoroughly scrutinized the existing nature and causes of it, they would have found that the existing agricultural backwardness was due to the institutional structure, primarily the system of feudalism that prevailed.
Fei–Ranis model of economic growth	Reactions to the model	Fei and Ranis say, "It has been argued that money is not a simple substitute for physical capital in an aggregate production function. There are reasons to believe that the relationship between money and physical capital could be complementary to one another at some stage of economic development, to the extent that credit policies could play an important part in easing bottlenecks on the growth of agriculture and industry." This indicates that in the process of development they neglect the role of money and prices. They fail to differ between wage labor and household labor, which is a significant distinction for evaluating prices of dualistic development in an underdeveloped economy.
Fei–Ranis model of economic growth	Reactions to the model	Fei and Ranis assume that MPPL is zero during the early phases of economic development, which has been criticized by Harry T.Oshima and some others on the grounds that MPPL of labor is zero only if the agricultural population is very large, and if it is very large, some of that labor will shift to cities in search of jobs. In the short run, this section of labor that has shifted to the cities remains unemployed, but over the long run it is either absorbed by the informal sector, or it returns to the villages and attempts to bring more marginal land into cultivation. They have also neglected seasonal unemployment, which occurs due to seasonal change in labor demand and is not permanent.To understand this better, we refer to the graph in this section, which shows Food on the vertical axis and Leisure on the horizontal axis. OS represents the subsistence level of food consumption, or the minimum level of food consumed by agricultural labor that is necessary for their survival. I0 and I1 between the two commodities of food and leisure (of the agriculturists). The origin falls on G, such that OG represents maximum labor and labor input would be measured from the right to the left.
Fei–Ranis model of economic growth	Reactions to the model	The transformation curve SAG falls from A, which indicates that more leisure is being used to same units of land. At A, the marginal transformation between food and leisure and MPL = 0 and the indifference curve I0 is also tangent to the transformation curve at this point. This is the point of leisure satiation.
Fei–Ranis model of economic growth	Reactions to the model	Consider a case where a laborer shifts from the agricultural to the industrial sector. In that case, the land left behind would be divided between the remaining laborers and as a result, the transformation curve would shift from SAG to RTG. Like at point A, MPL at point T would be 0 and APL would continue to be the same as that at A (assuming constant returns to scale). If we consider MPL = 0 as the point where agriculturalists live on the subsistence level, then the curve RTG must be flat at point T in order to maintain the same level of output. However, that would imply leisure satiation or leisure as an inferior good, which are two extreme cases. It can be surmised then that under normal cases, the output would decline with shift of labor to industrial sector, although the per capita output would remain the same. This is because, a fall in the per capita output would mean fall in consumption in a way that it would be lesser than the subsistence level, and the level of labor input per head would either rise or fall.
Fei–Ranis model of economic growth	Reactions to the model	Berry and Soligo in their 1968 paper have criticized this model for its MPL=0 assumption, and for the assumption that the transfer of labor from the agricultural sector leaves the output in that sector unchanged in Phase 1. They show that the output changes, and may fall under various land tenure systems, unless the following situations arise:1. Leisure falls under the inferior good category 2. Leisure satiation is present.
Fei–Ranis model of economic growth	Reactions to the model	3. There is perfect substitutability between food and leisure, and the marginal rate of substitution is constant for all real income levels. Now if MPL>0 then leisure satiation option becomes invalid, and if MPL=0 then the option of food and leisure as perfect substitutes becomes invalid. Therefore, the only remaining viable option is leisure as an inferior good.
Fei–Ranis model of economic growth	Reactions to the model	While mentioning the important role of high agricultural productivity and the creation of surplus for economic development, they have failed to mention the need for capital as well. Although it is important to create surplus, it is equally important to maintain it through technical progress, which is possible through capital accumulation, but the Fei-Ranis model considers only labor and output as factors of production.
Fei–Ranis model of economic growth	Reactions to the model	The question of whether MPL = 0 is that of an empirical one. The underdeveloped countries mostly exhibit seasonality in food production, which suggests that especially during favorable climatic conditions, say that of harvesting or sowing, MPL would definitely be greater than zero.
Fei–Ranis model of economic growth	Reactions to the model	Fei and Ranis assume a close model and hence there is no presence of foreign trade in the economy, which is very unrealistic as food or raw materials can not be imported. If we take the example of Japan again, the country imported cheap farm products from other countries and this made better the country's terms of trade. Later they relaxed the assumption and said that the presence of a foreign sector was allowed as long as it was a "facilitator" and not the main driving force.
Fei–Ranis model of economic growth	Reactions to the model	The reluctant expansionary growth in the industrial sector of underdeveloped countries can be attributed to the lagging growth in the productivity of subsistence agriculture. This suggests that increase in surplus becomes more important a determinant as compared to re-investment of surplus, an idea that was utilized by Jorgenson in his 1961 model that centered around the necessity of surplus generation and surplus persistence.
Fei–Ranis model of economic growth	Reactions to the model	Stagnation has not been taken into consideration, and no distinction is made between labor through family and labor through wages. There is also no explanation of the process of self-sustained growth, or of the investment function. There is complete negligence of terms of trade between agriculture and industry, foreign exchange, money and price.
Canonical basis	Canonical basis	In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given by the monomials, (Xi)i For finite extension fields, it means the polynomial basis. In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix A , if the set is composed entirely of Jordan chains. In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.
Canonical basis	Representation theory	The canonical basis for the irreducible representations of a quantized enveloping algebra of type ADE and also for the plus part of that algebra was introduced by Lusztig by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter q to q=1 yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter q to q=0 yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara; it is sometimes called the crystal basis.
Canonical basis	Representation theory	The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara (by an algebraic method) and by Lusztig (by a topological method).
Canonical basis	Representation theory	There is a general concept underlying these bases: Consider the ring of integral Laurent polynomials := Z[v,v−1] with its two subrings := Z[v±1] and the automorphism ⋅¯ defined by := v−1 A precanonical structure on a free Z -module F consists of A standard basis (ti)i∈I of F An interval finite partial order on I , that is, := {j∈I∣j≤i} is finite for all i∈I A dualization operation, that is, a bijection F→F of order two that is ⋅¯ -semilinear and will be denoted by ⋅¯ as well.If a precanonical structure is given, then one can define the Z± submodule := {\textstyle F^{\pm }:=\sum {\mathcal {Z}}^{\pm }t_{j}} of F A canonical basis of the precanonical structure is then a Z -basis (ci)i∈I of F that satisfies: ci¯=ci and and mod vF+ for all i∈I . One can show that there exists at most one canonical basis for each precanonical structure. A sufficient condition for existence is that the polynomials rij∈Z defined by {\textstyle {\overline {t_{j}}}=\sum _{i}r_{ij}t_{i}} satisfy rii=1 and rij≠0⟹i≤j A canonical basis induces an isomorphism from F+∩F+¯=∑iZci to F+/vF+ Hecke algebras Let (W,S) be a Coxeter group. The corresponding Iwahori-Hecke algebra H has the standard basis (Tw)w∈W , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by := Tw−1−1 . This is a precanonical structure on H that satisfies the sufficient condition above and the corresponding canonical basis of H is the Kazhdan–Lusztig basis Cw′=∑y≤wPy,w(v2)Tw with Py,w being the Kazhdan–Lusztig polynomials.
Canonical basis	Linear algebra	If we are given an n × n matrix A and wish to find a matrix J in Jordan normal form, similar to A , we are interested only in sets of linearly independent generalized eigenvectors. A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible. A diagonal matrix D is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector.
Canonical basis	Linear algebra	Every n × n matrix A possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If λ is an eigenvalue of A of algebraic multiplicity μ , then A will have μ linearly independent generalized eigenvectors corresponding to λ For any given n × n matrix A , there are infinitely many ways to pick the n linearly independent generalized eigenvectors. If they are chosen in a particularly judicious manner, we can use these vectors to show that A is similar to a matrix in Jordan normal form. In particular, Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Canonical basis	Linear algebra	Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors xm−1,xm−2,…,x1 that are in the Jordan chain generated by xm are also in the canonical basis. Computation Let λi be an eigenvalue of A of algebraic multiplicity μi . First, find the ranks (matrix ranks) of the matrices (A−λiI),(A−λiI)2,…,(A−λiI)mi . The integer mi is determined to be the first integer for which (A−λiI)mi has rank n−μi (n being the number of rows or columns of A , that is, A is n × n). Now define rank rank ⁡(A−λiI)k(k=1,2,…,mi). The variable ρk designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue λi that will appear in a canonical basis for A . Note that rank rank ⁡(I)=n. Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector). Example This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order. The matrix A=(41100−1042001004100000510000052000004) has eigenvalues λ1=4 and λ2=5 with algebraic multiplicities μ1=4 and μ2=2 , but geometric multiplicities γ1=1 and γ2=1 For λ1=4, we have n−μ1=6−4=2, (A−4I) has rank 5, (A−4I)2 has rank 4, (A−4I)3 has rank 3, (A−4I)4 has rank 2.Therefore 4. rank rank ⁡(A−4I)4=3−2=1, rank rank ⁡(A−4I)3=4−3=1, rank rank ⁡(A−4I)2=5−4=1, rank rank 1. Thus, a canonical basis for A will have, corresponding to λ1=4, one generalized eigenvector each of ranks 4, 3, 2 and 1. For λ2=5, we have n−μ2=6−2=4, (A−5I) has rank 5, (A−5I)2 has rank 4.Therefore 2. rank rank ⁡(A−5I)2=5−4=1, rank rank 1. Thus, a canonical basis for A will have, corresponding to λ2=5, one generalized eigenvector each of ranks 2 and 1. A canonical basis for A is 27 25 25 36 12 −22−1),(321100),(−8−4−1010)}.
Canonical basis	Linear algebra	x1 is the ordinary eigenvector associated with λ1 . x2,x3 and x4 are generalized eigenvectors associated with λ1 . y1 is the ordinary eigenvector associated with λ2 . y2 is a generalized eigenvector associated with λ2 A matrix J in Jordan normal form, similar to A is obtained as follows: 27 25 25 36 12 1−1000−210000201000−100), J=(410000041000004100000400000051000005), where the matrix M is a generalized modal matrix for A and AM=MJ
Metiamide	Metiamide	Metiamide is a histamine H2 receptor antagonist developed from another H2 antagonist, burimamide. It was an intermediate compound in the development of the successful anti-ulcer drug cimetidine (Tagamet).
Metiamide	Development of metiamide from burimamide	After discovering that burimamide is largely inactive at physiological pH, due to the presence of its electron-donating side chain, the following steps were undertaken to stabilize burimamide: addition of a sulfide group close to the imidazole ring, giving thiaburimamide addition of methyl group to the 4-position on the imidazole ring to favor the tautomer of thiaburimamide which binds better to the H2 receptorThese changes increased the bioavailability metiamide so that it is ten times more potent than burimamide in inhibiting histamine-stimulated release of gastric acid. The clinical trials that began in 1973 demonstrated the ability of metiamide to provide symptomatic relief for ulcerous patients by increasing healing rate of peptic ulcers. However, during these trials, an unacceptable number of patients dosed with metiamide developed agranulocytosis (decreased white blood cell count).
Metiamide	Modification of metiamide to cimetidine	It was determined that the thiourea group was the cause of the agranulocytosis. Therefore, replacement of the thiocarbonyl in the thiourea group was suggested: with urea or guanidine resulted in a compound with much less activity (only 5% of the potency of metiamide) however, the NH form (the guanidine analog of metiamide) did not show agonistic effects to prevent the guanidine group being protonated at physiological pH, electron-withdrawing groups were added adding a nitrile or nitro group prevented the guanidine group from being protonated and did not cause agranulocytosisThe nitro and cyano groups are sufficiently electronegative to reduce the pKa of the neighboring nitrogens to the same acidity of the thiourea group, hence preserving the activity of the drug in a physiological environment.
Metiamide	Synthesis	Reacting ethyl 2-chloroacetoacetate (1) with 2 molar equivalents of formamide (2) gives 4-carboethoxy-5-methylimidazole (3). Reduction of the carboxylic ester (3) with sodium in liquid ammonia via Birch reduction gives the corresponding alcohol (4). Reaction of that with cysteamine (mercaptoethylamine), as its hydrochloride, leads to intermediate 5. In the strongly acid medium, the amine is completely protonated; this allows the thiol to express its nucleophilicity without competition and the acid also activates the alcoholic function toward displacement. Finally, condensation of the amine with methyl isothiocyanate gives metiamide (6).
Firework Code	Firework Code	In the United Kingdom, the Firework Code (sometimes Firework safety code) is the name given to a number of similar sets of guidelines for the safe use of fireworks by the general public. These include a thirteen-point guideline issued by the British government, a ten-point guide issued by the Royal Society for the Prevention of Accidents, a twelve-point guide from Cheshire Fire and Rescue Service, and a nine-point "firework safety code" from the London Fire Brigade.
VeNom Coding Group	VeNom Coding Group	The VeNom Coding Group is a group of veterinary academics and practitioners from across Britain who have devised a standardized terminology for use in veterinary medicine. These codes are available to academic or research institutes, software manufacturers or veterinary practices after a request for a permission.

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.