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The Tolman–Oppenheimer–Volkoff limit (or TOV limit ) is an upper bound to the mass of cold, non-rotating neutron stars , analogous to the Chandrasekhar limit for white dwarf stars. Stars more massive than the TOV limit collapse into a black hole . The original calculation in 1939, which neglected complications such as nuclear forces between neutrons, placed this limit at approximately 0.7 solar masses ( M ☉ ). Later, more refined analyses have resulted in larger values.
Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 M ☉ , [ 1 ] corresponding to an original stellar mass of 15 to 20 M ☉ ; additional work in the same year gave a more precise range of 2.2 to 2.9 M ☉ . [ 2 ]
Data from GW170817 , the first gravitational wave observation attributed to merging neutron stars (thought to have collapsed into a black hole [ 3 ] within a few seconds after merging [ 4 ] ) placed the limit in the range of 2.01 to 2.17 M ☉ . [ 5 ]
In the case of a rigidly spinning neutron star, meaning that different levels in the interior of the star all rotate at the same rate, the mass limit is thought to increase by up to 18–20%. [ 4 ] [ 5 ]
The idea that there should be an absolute upper limit for the mass of a cold (as distinct from thermal pressure supported) self-gravitating body dates back to the 1932 work of Lev Landau , based on the Pauli exclusion principle . Pauli's principle shows that the fermionic particles in sufficiently compressed matter would be forced into energy states so high that their rest mass contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevant quantum wavelength λ , which would be of the order of the mean interparticle separation. In terms of Planck units , with the reduced Planck constant ħ , the speed of light c , and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by
At the upper mass limit, that pressure will equal the pressure needed to resist gravity. The pressure to resist gravity for a body of mass M will be given according to the virial theorem roughly by
where ρ is the density. This will be given by ρ = m / λ 3 , where m is the relevant mass per particle. It can be seen that the wavelength cancels out so that one obtains an approximate mass limit formula of the very simple form
In this relationship, m can be taken to be given roughly by the proton mass . This even applies in the white dwarf case (that of the Chandrasekhar limit ) for which the fermionic particles providing the pressure are electrons. This is because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons. Likewise, the protons, for charge neutrality, must be exactly as numerous as the electrons outside.
In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman . Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas . They thereby obtained a limiting mass of approximately 0.7 solar masses , [ 6 ] [ 7 ] which was less than the Chandrasekhar limit for white dwarfs.
Oppenheimer and Volkoff's paper notes that "the effect of repulsive forces, i.e., of raising the pressure for a given density above the value given by the Fermi equation of state ... could tend to prevent the collapse." [ 7 ] And indeed, the most massive neutron star detected so far, PSR J0952–0607 , is estimated to be much heavier than Oppenheimer and Volkoff's TOV limit at 2.35 ± 0.17 M ☉ . [ 8 ] [ 9 ] More realistic models of neutron stars that include baryon strong force repulsion predict a neutron star mass limit of 2.2 to 2.9 M ☉ . [ 10 ] [ 11 ] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known.
In a star less massive than the limit, the gravitational compression is balanced by short-range repulsive neutron–neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons, preventing collapse. [ 12 ] : 74 If its mass is above the limit, the star will collapse to some denser form. It could form a black hole , or change composition and be supported in some other way (for example, by quark degeneracy pressure if it becomes a quark star ). Because the properties of hypothetical, more exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter, most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole.
A black hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of mass loss during stellar evolution , a black hole formed from an isolated star of solar metallicity can have a mass of no more than approximately 10 solar masses . [ 13 ] :Fig. 16 Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects in X-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20 solar masses . [ 14 ] [ 15 ] LIGO has detected black hole mergers involving black holes in the 7.5–50 solar mass range; it is possible – although unlikely – that these black holes were themselves the result of previous mergers.
Oppenheimer and Volkoff discounted the influence of heat, stating in reference to work by Landau (1932), 'even [at] 10 7 degrees... the pressure is determined essentially by the density only and not by the temperature' [ 7 ] – yet it has been estimated [ 16 ] that temperatures can reach up to approximately >10 9 K during formation of a neutron star, mergers and binary accretion. Another source of heat and therefore collapse-resisting pressure in neutron stars is 'viscous friction in the presence of differential rotation.' [ 16 ]
Oppenheimer and Volkoff's calculation of the mass limit of neutron stars also neglected to consider the rotation of neutron stars, however we now know that neutron stars are capable of spinning at much faster rates than were known in Oppenheimer and Volkoff's time. The fastest-spinning neutron star known is PSR J1748-2446ad, rotating at a rate of 716 times per second [ 17 ] [ 18 ] or 43,000 revolutions per minute, giving a linear (tangential) speed at the surface on the order of 0.24c (i.e., nearly a quarter the speed of light). Star rotation interferes with convective heat loss during supernova collapse, so rotating stars are more likely to collapse directly to form a black hole [ 19 ] : 1044
This list contains objects that may be neutron stars, black holes, quark stars, or other exotic objects. This list is distinct from the list of least massive black holes due to the undetermined nature of these objects, largely because of indeterminate mass, or other poor observation data. | https://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit |
Toluene ( / ˈ t ɒ l . j u iː n / ), also known as toluol ( / ˈ t ɒ l . j u . ɒ l , - ɔː l , - oʊ l / ), is a substituted aromatic hydrocarbon [ 15 ] with the chemical formula C 6 H 5 CH 3 , often abbreviated as PhCH 3 , where Ph stands for the phenyl group. It is a colorless, water -insoluble liquid with the odor associated with paint thinners . It is a mono-substituted benzene derivative, consisting of a methyl group (CH 3 ) attached to a phenyl group by a single bond . As such, its systematic IUPAC name is methylbenzene . Toluene is predominantly used as an industrial feedstock and a solvent .
As the solvent in some types of paint thinner , permanent markers , contact cement and certain types of glue, toluene is sometimes used as a recreational inhalant [ 16 ] and has the potential of causing severe neurological harm. [ 17 ] [ 18 ]
The compound was first isolated in 1837 through a distillation of pine oil by Pierre Joseph Pelletier and Filip Neriusz Walter , who named it rétinnaphte . [ 19 ] [ 20 ] In 1841, Henri Étienne Sainte-Claire Deville isolated a hydrocarbon from balsam of Tolu (an aromatic extract from the tropical Colombian tree Myroxylon balsamum ), which Deville recognized as similar to Walter's rétinnaphte and to benzene; hence he called the new hydrocarbon benzoène . [ 21 ] [ 22 ] [ 23 ] In 1843, Jöns Jacob Berzelius recommended the name toluin . [ 24 ] In 1850, French chemist Auguste Cahours isolated from a distillate of wood a hydrocarbon which he recognized as similar to Deville's benzoène and which Cahours named toluène . [ 25 ] [ 26 ]
The distance between carbon atoms in the toluene ring is 0.1399 nm. The C-CH 3 bond is longer at 0.1524 nm, while the average C-H bond length is 0.111 nm. [ 27 ]
Toluene reacts as a normal aromatic hydrocarbon in electrophilic aromatic substitution . [ 28 ] [ 29 ] [ 30 ] Because the methyl group has greater electron-releasing properties than a hydrogen atom in the same position, toluene is more reactive than benzene toward electrophiles. It undergoes sulfonation to give p -toluenesulfonic acid , and chlorination by Cl 2 in the presence of FeCl 3 to give ortho and para isomers of chlorotoluene .
Nitration of toluene gives mono-, di-, and trinitrotoluene, all of which are widely used. Dinitrotoluene is the precursor to toluene diisocyanate , a precursor to polyurethane foam. Trinitrotoluene (TNT) is an explosive.
Complete hydrogenation of toluene gives methylcyclohexane . The reaction requires a high pressure of hydrogen and a catalyst .
The C-H bonds of the methyl group in toluene are benzylic , therefore they are weaker than C-H bonds in simpler alkanes. Reflecting this weakness, the methyl group in toluene undergoes a variety of free radical reactions. For example, when heated with N -bromosuccinimide (NBS) in the presence of AIBN , toluene converts to benzyl bromide . The same conversion can be effected with elemental bromine in the presence of UV light or even sunlight.
Toluene may also be brominated by treating it with HBr and H 2 O 2 in the presence of light. [ 31 ]
Benzoic acid and benzaldehyde are produced commercially by partial oxidation of toluene with oxygen . Typical catalysts include cobalt or manganese naphthenates . [ 32 ] Related but laboratory-scale oxidations involve the use of potassium permanganate to yield benzoic acid and chromyl chloride to yield benzaldehyde ( Étard reaction ).
The methyl group in toluene undergoes deprotonation only with very strong bases; its p K a is estimated using acidity trends to be approximately 43 in dimethyl sulfoxide (DMSO) [ 33 ] [ 34 ] and its ion pair acidity is extrapolated to be 41.2 in cyclohexylamine (Cesium Cyclohexylamide) using a Bronsted correlation. [ 35 ] [ 36 ]
Toluene is miscible (soluble in all proportions) with ethanol , benzene , diethyl ether , acetone , chloroform , glacial acetic acid and carbon disulfide , but immiscible with water. [ 37 ]
Toluene occurs naturally at low levels in crude oil and is a byproduct in the production of gasoline by a catalytic reformer or ethylene cracker . It is also a byproduct of the production of coke from coal . Final separation and purification is done by any of the distillation or solvent extraction processes used for BTX aromatics (benzene, toluene, and xylene isomers). [ 15 ]
Toluene can be prepared by a variety of methods. For example, benzene reacts with methanol in presence of a solid acid to give toluene and water: [ 15 ]
Toluene is one of the most abundantly produced chemicals. Its main uses are (1) as a precursor to benzene and xylenes , (2) as a solvent for thinners, paints, lacquers , adhesives, and (3) as an additive for gasoline. [ 15 ]
Toluene is converted to benzene via hydrodealkylation :
Its transalkylation gives a mixture of benzene and xylenes .
Toluene is widely used in the paint, dye , rubber, chemical, glue, printing, and pharmaceutical industries as a solvent. [ 38 ] Nail polish, paintbrush cleaners, and stain removers may contain toluene. Manufacturing of explosives (TNT) uses it as well. Toluene is also found in cigarette smoke and car exhaust. If not in contact with air, toluene can remain unchanged in soil or water for a long time. [ 39 ]
Toluene is a common solvent , e.g. for paints , paint thinners, strippers, silicone sealants, [ 40 ] many chemical reactants , rubber , printing ink, adhesives (glues), lacquers , leather tanners , and disinfectants . [ 15 ]
Toluene is an octane booster in gasoline fuels for internal combustion engines as well as jet fuel and turbocharged engines in Formula One . [ 41 ]
In Australia in 2003, toluene was found to have been illegally combined with petrol in fuel outlets for sale as standard vehicular fuel. Toluene incurs no fuel excise tax, while other fuels are taxed at more than 40%, providing a greater profit margin for fuel suppliers. The extent of toluene substitution is claimed to be 60%. [ 42 ] [ 43 ]
In the laboratory, toluene is used as a solvent for carbon nanomaterials, including nanotubes and fullerenes, and it can also be used as a fullerene indicator. The color of the toluene solution of C 60 is bright purple. Toluene is used as a cement for fine polystyrene kits (by dissolving and then fusing surfaces) as it can be applied very precisely by brush and contains none of the bulk of an adhesive. Toluene can be used to break open red blood cells in order to extract hemoglobin in biochemistry experiments. Toluene has also been used as a coolant for its good heat transfer capabilities in sodium cold traps used in nuclear reactor system loops. Toluene had also been used in the process of removing the cocaine from coca leaves in the production of Coca-Cola syrup. [ 44 ]
The environmental and toxicological effects of toluene have been extensively studied. [ 45 ]
Toluene is irritating to the eyes, skin, and respiratory tract. It is absorbed slowly through the skin. It can cause systemic toxicity by inhalation or ingestion. Inhalation is the most common route of exposure. Symptoms of toluene poisoning include central nervous system effects (headache, dizziness, drowsiness, ataxia , euphoria , tremors , hallucinations , seizures , and coma), chemical pneumonitis , respiratory depression, ventricular arrhythmias , nausea, vomiting, and electrolyte imbalances . [ 38 ]
Inhalation of toluene in low to moderate levels can cause tiredness, confusion, weakness, drunken-type actions, memory loss, nausea, loss of appetite, hearing loss , [ 46 ] [ 47 ] [ 48 ] and colour vision loss. [ 49 ] Some of these symptoms usually disappear when exposure is stopped. Inhaling high levels of toluene in a short time may cause light-headedness, nausea, or sleepiness, unconsciousness, and even death . [ 50 ] [ 51 ] Toluene is, however, much less toxic than benzene , and as a consequence, largely replaced it as an aromatic solvent in chemical preparation. The US Environmental Protection Agency (EPA) states that the carcinogenic potential of toluene cannot be evaluated due to insufficient information. [ 52 ] In 2013, worldwide sales of toluene amounted to about 24.5 billion US dollars. [ 53 ]
Toluene occurs as an indoor air pollutant in a number of processes including electrosurgery, and can be removed from the air with an activated carbon filter. [ 54 ]
Similarly to many other solvents such as 1,1,1-trichloroethane and some alkylbenzenes , toluene has been shown to act as a non-competitive NMDA receptor antagonist and GABA A receptor positive allosteric modulator . [ 55 ] Additionally, toluene has been shown to display antidepressant -like effects in rodents in the forced swim test (FST) and the tail suspension test (TST), [ 55 ] likely due to its NMDA antagonist properties.
Toluene is sometimes used as a recreational inhalant ("glue sniffing"), likely on account of its euphoric and dissociative effects. [ 55 ]
Toluene inhibits excitatory ion channels such as the NMDA receptor , nicotinic acetylcholine receptor , and the serotonin 5-HT 3 receptor . It also potentiates the function of inhibitory ion channels, such as the GABA A and glycine receptors . In addition, toluene disrupts voltage-gated calcium channels and ATP-gated ion channels. [ 56 ]
Toluene is used as an intoxicative inhalant in a manner unintended by manufacturers. People inhale toluene-containing products (e.g., paint thinner , contact cement , correction pens, model glue, etc.) for its intoxicating effect . The possession and use of toluene and products containing it are regulated in many jurisdictions, for the supposed reason of preventing minors from obtaining these products for recreational drug purposes. As of 2007, 24 US states had laws penalizing use, possession with intent to use, and/or distribution of such inhalants. [ 57 ] In 2005 the European Union banned the general sale of products consisting of greater than 0.5% toluene. [ 58 ]
Several types of fungi including Cladophialophora , Exophiala , Leptodontidium ( syn. Leptodontium ), Pseudeurotium zonatum , and Cladosporium sphaerospermum , and certain species of bacteria can degrade toluene using it as a source of carbon and energy. [ 59 ] | https://en.wikipedia.org/wiki/Toluene |
This page provides supplementary chemical data on toluene . | https://en.wikipedia.org/wiki/Toluene_(data_page) |
Toluidine blue , also known as TBO or tolonium chloride ( INN ) is a blue cationic (basic) dye used in histology (as the toluidine blue stain ) and sometimes clinically.
Toluidine blue solution is used in testing for lignin , a complex organic molecule that bonds to cellulose fibres and strengthens and hardens the cell walls in plants . A positive toluidine blue test causes the solution to turn from blue to blue-green. [ 1 ] A similar test can be performed with phloroglucinol - HCl solution, which turns red.
Toluidine blue is a basic thiazine metachromatic dye with high affinity for acidic tissue components. [ 2 ] It stains nucleic acids blue and polysaccharides purple and also increases the sharpness of histology slide images. It is especially useful today for staining chromosomes in plant or animal tissues, as a replacement for Aceto-orcein stain .
Toluidine blue is often used to identify mast cells , by virtue of the heparin in their cytoplasmic granules . [ 3 ] It is also used to stain proteoglycans and glycosaminoglycans in tissues such as cartilage. The strongly acidic macromolecular carbohydrates of mast cells and cartilage are coloured red by the blue dye, a phenomenon called metachromasia .
Alkaline solutions of toluidine blue are commonly used for staining semi-thin (0.5 to 1 μm) sections of resin-embedded tissue. At high pH (about 10) the dye binds to nucleic acids and all proteins. Although everything in the tissue is stained, structural details are clearly visible because of the thinness of the sections. Semi-thin sections are used in conjunction with ultra-thin sections examined by electron microscopy .
Toluidine blue is also commonly used to stain frozen sections (rapid microscopic analysis of a specimen). Because time is of the essence for a frozen section, toluidine blue allows for the frozen section to be stained and reviewed in 10 to 20 seconds. [ 4 ] The other staining method for frozen sections (rapid H&E) takes approximately 60 to 90 seconds.
The results depend on the studied organs: [ 5 ]
It is used in forensic examination, [ 6 ] renal pathology [ 7 ] and neuropathology .
The dye is sometimes used by surgeons to help highlight areas of mucosal dysplasia (which preferentially take up the dye compared to normal tissue) in premalignant lesions (e.g. leukoplakia ). [ 8 ] This can be used to choose the best site of the lesion to biopsy , or during surgery to remove the lesion to decide whether to remove more tissue from the margins of the excision defect or leave it behind. | https://en.wikipedia.org/wiki/Toluidine_blue_stain |
In organic chemistry , tolyl groups are functional groups related to toluene . [ 1 ] They have the general formula CH 3 C 6 H 4 −R . The change of the relative position of the methyl and the R substituent on the aromatic ring can generate three possible structural isomers : 1,2 ( ortho ), 1,3 ( meta ), and 1,4 ( para ). Tolyl groups are aryl groups which are commonly found in the structure of diverse chemical compounds . They are considered nonpolar and hydrophobic groups.
The functionalization to include tolyl groups into compounds is often done by Williamson etherification , using tolyl alcohols as reagents, or by C-C coupling reactions. Tolyl sulfonates are excellent leaving groups in nucleophilic substitutions , for this reason, they are commonly generated as intermediaries to activate alcohols. To this end, 4-toluenesulfonyl chloride is reacted in the presence of a base with the corresponding alcohol.
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tolyl_group |
Tom's Hardware is an online publication owned by Future plc and focused on technology. It was founded in 1996 by Thomas Pabst. [ 1 ] It provides articles, news, price comparisons, videos and reviews on computer hardware and high technology. The site features coverage on CPUs , motherboards , RAM , PC cases, graphic cards , display technology, power supplies and displays , storage , smartphones, tablets, gaming, consoles, and computer peripherals .
Tom's Hardware has a forum and featured blogs .
Tom's Hardware was founded in 1996 as Tom's Hardware Guide in Canada by Thomas Pabst. [ 1 ] It started using the domain tomshardware.com in September 1997 and was followed by several foreign language versions, including Italian, French, Finnish and Russian based on franchise agreements. [ 2 ] [ 3 ] [ 4 ] [ 5 ]
While the initial testing labs were in Germany and California, [ 6 ] much of Tom's Hardware 's testing now occurs in New York and a facility in Ogden, Utah owned by its parent company. In April 2007, the site was acquired by the French company Bestofmedia Group. [ 7 ] In July 2013, that company was acquired by TechMediaNetwork, Inc. , [ 8 ] which changed its name to Purch in April 2014. [ 9 ] Purch's consumer brands, including Tom's Hardware , were acquired by Future in 2018. [ 10 ] [ 11 ]
The site celebrated its 20th anniversary in May 2016. [ 12 ] Beyond continuous publication of the website, it is known for its overclocking championships and other contests. [ 13 ] [ 14 ]
Avram Piltch is the current editor-in-chief of Tom's Hardware . [ 15 ] Prior to starting the position in 2018, he worked for sister sites Tom's Guide and Laptop Mag. Prior to that, John A. Burek, formerly of Computer Shopper , briefly held the role. [ 16 ]
Burek succeeded Fritz Nelson, who served from August 2014 through 2017. Other former editors-in-chief include Chris Angelini (July 2008 – July 2014), Patrick Schmid (2005–2006), David Strom (2005), Omid Rahmat (1999–2003) and founder Thomas Pabst (1996–2001). [ 12 ]
Tom's Hardware is owned by Future plc , which also owns a number of other websites. In technology, those include Tom's Guide (formerly Gear Digest ), [ 17 ] Laptop Mag and AnandTech , [ 18 ] as well as science sites like LiveScience and Space.com .
In March 2018 the German spin-off was to be closed because of the new data/privacy laws, but continued as an independent site (tomshw.de), with an exclusive license for the local usage of the brand name. [ 19 ]
In July 2019 the license was returned. After that the German CEO and editor-in-chief of the gotIT! Tech Media GmbH started a new website Igor´sLAB [ 20 ] and his own Youtube channel. [ 21 ]
Tom's Guide (formerly known as GearDigest [ 22 ] ) is an online publication owned by Future that focuses on technology, with editorial teams in the US, UK and Australia. Tom's Guide was launched in 2007 by Bestofmedia, which was subsequently acquired by TechMediaNetwork in 2013; in 2014, TechMediaNetwork changed its name to Purch, which was acquired by Future in 2018. [ 23 ] Primarily focused on news, reviews, price comparisons, how-tos and guides, Tom's Guide also features opinion articles and deals content.
The site features coverage on CPUs , motherboards , RAM , PC cases, graphic cards , display technology, displays , storage , smartphones, tablets, gaming, consoles, fitness and health, home, smart home, streaming, security and computer peripherals .
It is the second-largest consumer technology, news and review site from the US with 68.4 million visits in September 2022. [ 24 ]
Tom’s Guide was originally launched as GearDigest by Bestofmedia before being re-named to Tom's Guide . The publication was subsequently acquired by TechMediaNetwork in 2013; in 2014, TechMediaNetwork changed its name to Purch, which was then acquired by Future in 2018.
While the initial testing labs were in Germany and California, much of Tom’s Guide 's testing now occurs in New York and a facility in Ogden, Utah owned by its parent company, Purch .
In April 2007, the site was acquired by the French company Bestofmedia Group. In July 2013, that company was acquired by TechMediaNetwork, Inc. , which changed its name to Purch in April 2014.
The site celebrated its 15th anniversary in 2022. Beyond continuous publication of the website, it is known for its annual CES awards [ 25 ] and Tom's Guide Awards [ 26 ] that are held in June and July each year.
Mark Spoonauer is the current Global Editor-in-Chief and has been since 2013. [ 27 ] Before that, he worked as the Editor-in-Chief of Laptop Mag since 2003. [ 28 ]
Mike Prospero is the current US Editor-in-Chief alongside Managing Editors Philip Michaels, Jason England, Nick Pino and Senior Deals Editor Louis Ramirez. [ 23 ] | https://en.wikipedia.org/wiki/Tom's_Hardware |
James Thomas Brenna (born October 15, 1959) is an American scientist specializing in analytical chemistry , mass spectrometry , and in human nutrition and foods, specifically fats . He is a professor of Pediatrics and chemistry at Dell Medical School , having previously been a professor of human nutrition, chemistry, chemical biology and food science at Cornell University . [ 1 ]
Brenna was the key expert witness in the action of the United States Anti-Doping Agency (USADA) against 2006 Tour de France first place finisher Floyd Landis , adjudicated in a hearing of the Tribunal Arbitral du Sport/Court for Arbitration in Sport held in Malibu, California. He testified to the accuracy of Carbon Isotope Ratio (CIR) tests conducted by the French Antidoping laboratory at Châtenay-Malabry. [ 2 ] [ 3 ] Landis conducted a high profile, multimillion dollar defense but lost the 2007 original hearing [ 4 ] with the decision relying for technical opinion on Brenna’s testimony. [ 5 ] Landis later lost a 2008 de novo case before the American Arbitration Association also relying on Brenna’s testimony [ 6 ] and was stripped of his title. Years later Landis revealed he was doping and filed a whistle-blower lawsuit under the federal False Claims Act against Lance Armstrong that was settled with multimillion dollar payments by Armstrong. [ 7 ]
Brenna visited the FDA infant formula team in late 2001 to encourage omega-3 DHA to be included in infant formula. A few weeks later the FDA issued its “no questions” letter accepting this suggestion and citing his work. [ 8 ]
Brenna was a member of the Dietary Guidelines Advisory Committee advising on the 2015 U.S. Dietary Guidelines for Americans. [ 9 ] He was one of four members of the Food Sustainability and Safety subcommittee whose work on sustainability was excluded from consideration by an act of Congress. [ 10 ]
Comments in the New York Times about the healthfulness of coconut oil in late 2015 [ 11 ] [ 12 ] were covered in 200+ newspapers globally. Brenna opined that 21st-century virgin coconut oil does not cause heart disease but that earlier coconut oils may cause heart disease due to process contaminants. [ 13 ]
In 2016 he was co-lead of a team that discovered an insertion-deletion polymorphism, rs66698963, is under positive selective pressure depending on whether ancestral diets were primarily animal/seafood-based or plant-based. [ 14 ] [ 15 ] [ 16 ] Global news widely reported that it would lead to potentially greater risk of disease, [ 17 ] though this was corrected later. [ 18 ] | https://en.wikipedia.org/wiki/Tom_Brenna |
Thomas Linn Dinwoodie (born November 15, 1954) is a cleantech entrepreneur, inventor, and founder of SunPower Corporation Systems (formerly PowerLight Corporation). He holds a long-standing interest in accelerating the transition to clean energy and other climate-sustaining practices. Dinwoodie is also an architect.
From 1978 to 1983, Dinwoodie was a research assistant at the MIT Energy Laboratory, where he authored numerous papers on the economics and policy of distributed solar and wind generation, as well as flywheel energy storage. [ citation needed ]
In 1981, Dinwoodie was awarded a contract from the U.S. DOE for development of an ultra-low-cost, polymer solar thermal collector. [ citation needed ]
He was founder, president and CEO of TDEnergy, a wind power developer in the U.S. Northeast, from 1982 to 1988. TDEnergy constructed one of the earliest New England wind power facilities in Canaan, NH. [ citation needed ]
Dinwoodie then founded PowerLight Corporation in 1994, and served as its CEO and chairman of the board from 1995 to 2007. PowerLight was a global manufacturer, supplier, and systems integrator of solar products and services for the residential, commercial, and utility sectors. [ 1 ] In 2004, PowerLight was inducted into the INC 500 Hall of Fame. [ 2 ] PowerLight merged with SunPower Corporation in 2007, [ 3 ] where Dinwoodie served as CEO and later CTO for the subsidiary, SunPower Corporation, Systems. [ 4 ] SunPower produces high-efficiency solar cells and modules. [ 5 ]
Dinwoodie holds over 30 patents on PV-related products. [ 6 ]
Dinwoodie holds a BS in Civil and Environmental Engineering from Cornell University , [ 7 ] an MS from the department of mechanical engineering from MIT , and an MArch in architecture from the University of California at Berkeley .
Dinwoodie is executive producer of Time to Choose , a film by Charles Ferguson that seeks to raise awareness of the causes of, and solutions to, climate change. The film released in June, 2016.
Dinwoodie is or has been an advisor to, board member, seed investor or founder of a range of clean-tech startups, spanning wind and solar technologies, electric transportation, off-grid micropower, financial services, and clean utilities. These companies include AllPower Labs, Ethical Electric, Etrion Corporation, Fenix Int’l, Keystone Tower Systems, Mosaic, MyDomino, NEW GmbH, NEXTracker, PowerLight, SCOOT, Sistine Solar, Sungevity, SunPower Systems, Solar Grid Storage, and TDEnergy. [ 8 ] [ 9 ]
Dinwoodie serves as lead independent trustee of the Rocky Mountain Institute , a nonprofit think-and-do tank focused on solutions for the transition to a clean energy economy. He also serves on the Sierra Club's Climate Cabinet and Scientific Advisory Panel, the MIT Mechanical Engineering Visiting Committee, and the advisory board to The Solutions Project, and is an advisor to the MIT Energy Club. | https://en.wikipedia.org/wiki/Tom_Dinwoodie |
Thomas C. Hull is an associate professor of applied mathematics at Franklin & Marshall College [ 1 ] and is known for his expertise in the mathematics of paper folding . [ 2 ] [ 3 ] [ 4 ]
Hull was an undergraduate at Hampshire College . He earned a master's degree and Ph.D. in mathematics at the University of Rhode Island . [ 5 ] His 1997 dissertation, Some Problems in List Coloring Bipartite Graphs , involved graph coloring , and was supervised by Nancy Eaton. [ 6 ]
Prior to his appointment at Franklin & Marshall College, Hull taught at Merrimack College (1997–2008) and Western New England University (2008–2023). [ 1 ] He has also taught at the Hampshire College Summer Studies in Mathematics for many years: as junior staff from 1991 to 1995, and as senior staff in 1998 to 2007. Since 2013, he has taught at MathILy, an intensive residential summer program for mathematically excellent high school students. [ 5 ]
Hull was a member of the board of directors of origami association OrigamiUSA from 1995 to 2008. [ 7 ]
Hull is the author or co-author of several books on origami, including:
He is also featured in the 2010 origami documentary Between the Folds . [ 2 ]
With Tomohiro Tachi of the University of Tokyo , Hull was the recipient of the 2016 A. T. Yang Memorial Award in Theoretical Kinematics of the American Society of Mechanical Engineers , for their joint work on predicting the motion of rigid origami patterns when forces are applied to them in their flat state. [ 10 ] | https://en.wikipedia.org/wiki/Tom_Hull_(mathematician) |
Tom Meier is a sculptor , a founding partner of Ral Partha Enterprises , and the winner of numerous awards for the design and sculpture of historical, fantasy and science fiction gaming miniatures.
Meier began sculpting professionally at the age of 15 and won his first H.G. Wells award just two years later. His earliest work was in the bulky style of Heritage Miniatures for whom he briefly worked. After the founding of Ral Partha in 1975, he was inspired by the art in fables and fairy tales and developed a style which emphasized beauty and natural proportion. A second lasting contribution was the popularization of sculpting in two-part ribbon epoxy putty designed for automotive repair. Commonly known as "green stuff," the epoxy held detail better than traditional media. Between 1977 and 1992 Tom's work won more than two dozen awards and he was inducted into the Origins Hall of Fame in 1991. In 1988, he left Ral Partha to start up his own company, Thunderbolt Mountain and does commission work for large and small game manufacturers. [ 1 ]
Meier's signature lines include:
Meier also contributed to:
Origins Awards / H.G. Wells Awards [ 14 ]
Strategist Club "Creativity in Wargaming" Award [ 7 ]
The Courier Award [ 7 ]
Games Day Awards | https://en.wikipedia.org/wiki/Tom_Meier |
The Tom W. Bonner Prize in Nuclear Physics is an annual prize awarded by the American Physical Society's Division of Nuclear Physics . Established in 1964, [ 1 ] and currently consisting of $10,000 and a certificate, the Bonner Prize was founded in memory of physicist Tom W. Bonner . The aim of the prize, as stated by the American Physical Society is:
The Bonner Prize is generally awarded for individual achievement in experimental research, but can be awarded for exceptional theoretical work and to groups who have contributed to a single accomplishment. | https://en.wikipedia.org/wiki/Tom_W._Bonner_Prize |
The tomahawk is a tool in geometry for angle trisection , the problem of splitting an angle into three equal parts. The boundaries of its shape include a semicircle and two line segments , arranged in a way that resembles a tomahawk , a Native American axe. [ 1 ] [ 2 ] The same tool has also been called the shoemaker's knife , [ 3 ] but that name is more commonly used in geometry to refer to a different shape, the arbelos (a curvilinear triangle bounded by three mutually tangent semicircles). [ 4 ]
The basic shape of a tomahawk consists of a semicircle (the "blade" of the tomahawk), with a line segment the length of the radius extending along the same line as the diameter of the semicircle (the tip of which is the "spike" of the tomahawk), and with another line segment of arbitrary length (the "handle" of the tomahawk) perpendicular to the diameter. In order to make it into a physical tool, its handle and spike may be thickened, as long as the line segment along the handle continues to be part of the boundary of the shape. Unlike a related trisection using a carpenter's square , the other side of the thickened handle does not need to be made parallel to this line segment. [ 1 ]
In some sources a full circle rather than a semicircle is used, [ 5 ] or the tomahawk is also thickened along the diameter of its semicircle, [ 6 ] but these modifications make no difference to the action of the tomahawk as a trisector.
To use the tomahawk to trisect an angle , it is placed with its handle line touching the apex of the angle, with the blade inside the angle, tangent to one of the two rays forming the angle, and with the spike touching the other ray of the angle. One of the two trisecting lines then lies on the handle segment, and the other passes through the center point of the semicircle. [ 1 ] [ 6 ] If the angle to be trisected is too sharp relative to the length of the tomahawk's handle, it may not be possible to fit the tomahawk into the angle in this way, but this difficulty may be worked around by repeatedly doubling the angle until it is large enough for the tomahawk to trisect it, and then repeatedly bisecting the trisected angle the same number of times as the original angle was doubled. [ 2 ]
If the apex of the angle is labeled A , the point of tangency of the blade is B , the center of the semicircle is C , the top of the handle is D , and the spike is E , then triangles △ ACD and △ ADE are both right triangles with a shared base and equal height, so they are congruent triangles . Because the sides AB and BC of triangle △ ABC are respectively a tangent and a radius of the semicircle, they are at right angles to each other and △ ABC is also a right triangle; it has the same hypotenuse as △ ACD and the same side lengths BC = CD , so again it is congruent to the other two triangles, showing that the three angles formed at the apex are equal. [ 5 ] [ 6 ]
Although the tomahawk may itself be constructed using a compass and straightedge , [ 7 ] and may be used to trisect an angle, it does not contradict Pierre Wantzel 's 1837 theorem that arbitrary angles cannot be trisected by compass and unmarked straightedge alone. [ 8 ] The reason for this is that placing the constructed tomahawk into the required position is a form of neusis that is not allowed in compass and straightedge constructions. [ 9 ]
The inventor of the tomahawk is unknown, [ 1 ] [ 10 ] but the earliest references to it come from 19th-century France. It dates back at least as far as 1835, when it appeared in a book by Claude Lucien Bergery , Géométrie appliquée à l'industrie, à l'usage des artistes et des ouvriers (3rd edition). [ 1 ] Another early publication of the same trisection was made by Henri Brocard in 1877; [ 11 ] Brocard in turn attributes its invention to an 1863 memoir by French naval officer Pierre-Joseph Glotin [ d ] . [ 12 ] [ 13 ] [ 14 ] | https://en.wikipedia.org/wiki/Tomahawk_(geometry) |
Tomato grafting is a horticulture technique that has been utilized in Asia and Europe for greenhouse and high tunnel production and is gaining popularity in the United States. [ 1 ] Typically, stock or rootstock are selected for their ability to resist infection by certain soilborne pathogens or their ability to increase vigor and fruit yield. The scion of the grafted tomato represents the upper portion of the plant and is selected for its fruit quality characteristics. There are several methods for grafting tomatoes and they have certain advantages and disadvantages. Once the grafts are made, the plants are moved into a chamber or environment with high relative humidity (>90%) and low light levels to reduce water stress in the scion while the graft union forms.
Grafting of woody plants has been common for centuries, but herbaceous grafting has only become popular recently [ when? ] in agricultural systems. The cultivation of grafted vegetable plants began in Korea and Japan at the end of the 1920s when watermelon plants were grafted onto squash rootstock [1] . Since this time, this technique has spread throughout Asia and Europe. Currently, 81% of Korean and 54% of Japanese vegetable cultivation uses grafting. [ 2 ] The use of this cultural technique is mainly carried out for intensive cropping systems like greenhouse and tunnel production. This method is especially popular for vegetable production in the orient, and the number of vegetables in 1998 was estimated to be 540 million transplants in Korea and 750 million in Japan. [ 3 ] This technique has moved to the Mediterranean region as well, where the use of grafting has been proposed as a major component of an integrated management strategy for managing soilborne disease and increasing crop productivity. Grafted tomato transplant production has increased in Spain from less than one million plants in 1999–2000 to over 45 million plants in 2003–2004. Grafted tomato is also cultivated in France and Italy , and over 20 million tomato plants were grafted in Morocco in 2004 as a way to reduce soilborne disease and increase crop production. [ 4 ]
Grafting can take place on a number of crops. However, because of the added expense, it is typically associated with melons, cucurbits , and members of the family Solanaceae such as eggplant and tomato. Tomato grafting became popular in the 1960s as a way to reduce certain diseases caused by soilborne plant pathogens such as Raletonia solanacearum [1] . Currently, however, grafting is used to offer not only protection from certain diseases, but also tolerance to abiotic stress like flooding, drought, and salinity [2] .
The first grafts in the early 20th century were made in order to diminish attacks by infectious organisms, such as Fusarium oxysporum on watermelons. [2] However, research has shown that this technique can be effective against a variety of fungal , bacterial , viral , and nematode diseases. [ 5 ] Furthermore, many researchers are looking to utilize specific rootstocks as an alternative to methyl bromide -a soil fumigant that has been widely used until recently. [ when? ] [4] Grafting has been highly effective at overcoming abiotic sources of stress, such as soil salinity , temperature extremes, and excessive soil moisture. [2] Grafting has also been utilized to reduce the effects of flooding in areas where a wet season may occur. [ 6 ]
Grafting tomatoes with tolerant rootstocks has been highly effective at producing a saline-tolerant plants. Research indicates that several rootstocks prevent the translocation of sodium and chloride into the shoot. [ 7 ] Many of the most economically important vegetable crops like tomato, squash, cucumber, and watermelon are highly sensitive to thermal stress in the roots throughout vegetative development and reproduction. Whether using rootstock tolerant of hot or cold temperatures, the use of temperature tolerant rootstocks often leads to the extension of the growing season in either direction, resulting in better yield and economic stability through the year. [2] Although the vegetable grafting is typically associated with reduction of disease or abiotic stress, yield is often increased without the presence of these identified sources of stress.
In tomatoes, increases in fruit yield are typically the results of increased fruit size. [ 8 ] Research has shown that possible mechanisms for increased yield are likely due to increased water and nutrient uptake among vigorous rootstock genotypes. Conductance through the stoma was improved in tomato plants when grafted onto vigorous rootstock. [7] Nutrient uptake for the macronutrients , such as phosphorus and nitrogen , were enhanced by grafting. [ 9 ] [ 10 ]
There are a variety of methods for grafting vegetable crops. Cleft grafting occurs when a V-shape is cut into the rootstock and a complementing wedge-shaped scion is inserted. The graft is then held with a small clip until healing occurs. [ 11 ] Approach grafting involves notching opposing sides of the stems of the rootstock and scion, and then using a clip to hold the stems together while they fuse. Once the graft has healed, the original scion is then cut off of the desired rootstock and the unused rootstock is detached from the scion. [ 12 ] Micrografting is a new technique that has been recently [ when? ] integrated into micropropagation production for hybrid tomato. This method uses micropropagated scion shoots that grafted onto 3 week-old rootstock seedlings. [ 13 ] The most common commercial technique for grafting tomato is tube grafting. Tube grafting takes place when the scion and rootstock are severed as seedlings and reattached with a small, silicone tube or clip. [ 14 ] [ 15 ] This technique has been highly effective as it can be carried out when plants are very small, thereby eliminating the need for large healing chambers while increasing the output. Tube grafting has been adopted as the primary method for vegetable grafting on the farm as it can be easily carried out with small healing chambers with typical success rates ranging from 85 to 90 percent [14] . | https://en.wikipedia.org/wiki/Tomato_grafting |
Tomaž (Tomo) Pisanski (born 24 May 1949 in Ljubljana , Yugoslavia , which is now in Slovenia ) is a Slovenian mathematician working mainly in discrete mathematics and graph theory . He is considered by many Slovenian mathematicians to be the "father of Slovenian discrete mathematics." [ 1 ]
As a high school student, Pisanski competed in the 1966 and 1967 International Mathematical Olympiads as a member of the Yugoslav team, winning a bronze medal in 1967. [ 2 ] He studied at the University of Ljubljana where he obtained a B.Sc, M.Sc and PhD in mathematics. His 1981 PhD thesis in topological graph theory was written under the guidance of Torrence Parsons . He also obtained an M.Sc. in computer science from Pennsylvania State University in 1979. [ 3 ]
Currently, Pisanski is a professor of discrete and computational mathematics and Head of the Department of Information Sciences and Technology at University of Primorska in Koper . [ 4 ] In addition, he is a professor at the University of Ljubljana Faculty of Mathematics and Physics (FMF). He has been a member of the Institute of Mathematics, Physics and Mechanics (IMFM) in Ljubljana since 1980, and the leader of several IMFM research projects. [ 5 ] In 1991 he established the Department of Theoretical Computer Science at IMFM, of which he has served as both head and deputy head.
He has taught undergraduate and graduate courses in mathematics and computer science at the University of Ljubljana , University of Zagreb , University of Udine , University of Leoben , California State University, Chico , Simon Fraser University , University of Auckland and Colgate University . [ 6 ] Pisanski has been an adviser for M.Sc and PhD students in both mathematics and computer science. Notable students include John Shawe-Taylor (B.Sc in Ljubljana), Vladimir Batagelj , Bojan Mohar , Sandi Klavžar , and Sandra Sattolo (M.Sc in Udine).
Pisanski’s research interests span several areas of discrete and computational mathematics, including combinatorial configurations, abstract polytopes , maps on surfaces, chemical graph theory , and the history of mathematics and science. In 1980 he calculated the genus of the Cartesian product of any pair of connected, bipartite, d -valent graphs using a method that was later called the White–Pisanski method . [ 7 ] In 1982 Vladimir Batagelj and Pisanski proved that the Cartesian product of a tree and a cycle is Hamiltonian if and only if no degree of the tree exceeds the length of the cycle. They also proposed a conjecture concerning cyclic Hamiltonicity of graphs. Their conjecture was proved in 2005. [ 8 ] With Brigitte Servatius he is the co-author of the book Configurations from a Graphical Viewpoint (2013). [ 9 ]
From 1998-1999, Pisanski was chairman of the Society of Mathematicians, Physicists and Astronomers of Slovenia (DMFA Slovenije); he was appointed an honorary member in 2015. [ 10 ] He is a founding member of the International Academy of Mathematical Chemistry , serving as its vice president from 2007 to 2011. [ 11 ] In 2008, together with Dragan Marušič , he founded Ars Mathematica Contemporanea , the first international mathematical journal to be published in Slovenia. [ 12 ] In 2012 he was elected to the Academia Europaea . [ 13 ] He was the president of the Slovenian Discrete and Applied Mathematics Society (SDAMS) from its conception until December 12, 2024. SDAMS is the first Eastern European mathematical society not wholly devoted to theoretical mathematics to be accepted as a full member of the European Mathematical Society (EMS). [ 14 ]
In 2005, Pisanski was decorated with the Order of Merit (Slovenia) , [ 15 ] and in 2015 he received the Zois award for exceptional contributions to discrete mathematics and its applications. [ 16 ] In 2016, he received the Donald Michie and Alan Turing Prize for lifetime achievements in Information Science in Slovenia. [ 17 ] | https://en.wikipedia.org/wiki/Tomaž_Pisanski |
In mathematics, the tombstone , halmos , end-of-proof , or Q.E.D. symbol "∎" (or "□") is a symbol used to denote the end of a proof , in place of the traditional abbreviation "Q.E.D." for the Latin phrase " quod erat demonstrandum ". It is inspired by the typographic practice of end marks , an element that marks the end of an article. [ 1 ] [ 2 ]
In Unicode , it is represented as character U+220E ∎ END OF PROOF . Its graphic form varies, as it may be a hollow or filled rectangle or square.
In AMS-LaTeX , the symbol is automatically appended at the end of a proof environment \begin{proof} ... \end{proof} . It can also be obtained from the commands \qedsymbol , \qedhere or \qed (the latter causes the symbol to be right aligned). [ 3 ]
It is sometimes called a "Halmos finality symbol" or "halmos" after the mathematician Paul Halmos , who first used it in a mathematical context in 1950. [ 4 ] He got the idea of using it from seeing end marks in magazines , that is, typographic signs that indicate the end of an article. In his memoir I Want to Be a Mathematician , he wrote the following: [ 1 ]
The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.
This typography -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tombstone_(typography) |
Tomka gas test site ( German : Gas-Testgelände Tomka ) was a secret chemical weapons testing facility near a place codenamed Volsk-18 ( Wolsk , in German literature), 20 km off Volsk , now Shikhany , [ 1 ] Saratov Oblast , Russia created within the framework of German-Soviet military cooperation to circumvent the demilitarization provisions of the post- World War I Treaty of Versailles . It was co-directed by Yakov Moiseevich Fishman (начальник военно-химического управления Красной Армии), and German chemists Alexander von Grundherr and Ludwig von Sicherer. [ 2 ] [ 3 ] [ 4 ] It operated (according to an agreement undersigned by fictitious joint stock companies) during 1926-1933. [ 5 ]
After 1933 the area was used by the Red Army and expanded under the name "Volsk-18" or " Schichany-2 " to Russia's most important center for the development of chemical warfare agents and protective measures against NBC weapons.
Another chemical site was established by the settlement of Ukhtomsky, Moscow Region . [ 6 ] [ 2 ] | https://en.wikipedia.org/wiki/Tomka_gas_test_site |
The Tomlinson model , also known as the Prandtl–Tomlinson Model , is one of the most popular models in nanotribology widely used as the basis for many investigations of frictional mechanisms on the atomic scale . Essentially, a nanotip is dragged by a spring over a corrugated energy landscape. A "frictional parameter" η can be introduced to describe the ratio between the energy corrugation and the elastic energy stored in the spring. If the tip-surface interaction is described by a sinusoidal potential with amplitude V 0 and periodicity a then
where k is the spring constant.
If η <1, the tip slides continuously across the landscape ( superlubricity regime). If η >1, the tip motion consists in abrupt jumps between the minima of the energy landscape ( stick-slip regime). [ 1 ] [ 2 ]
The name "Tomlinson model" is, however, historically incorrect: the paper by Tomlinson that is often cited in this context [ 3 ] did not contain the model known as the "Tomlinson model" and suggests an adhesive contribution to friction. In reality it was Ludwig Prandtl who suggested in 1928 this model to describe the plastic deformations in crystals as well as the dry friction. [ 4 ] [ 5 ] In the meantime, many researchers still call this model the "Prandtl–Tomlinson Model".
In Russia this model was introduced by the Soviet physicists Yakov Frenkel and T. Kontorova. The Frenkel defect became firmly fixed in the physics of solids and liquids. In the 1930s, this research was supplemented with works on the theory of plastic deformation . Their theory, now known as the Frenkel–Kontorova model , is important in the study of dislocations . [ 6 ]
This nanotechnology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tomlinson_model |
Tommy Bonnesen (March 27, 1873, in Copenhagen – March 14, 1935) was a Danish mathematician and professor of geometry at the Polytechnical School . He is most known for Bonnesen's inequality .
He graduated from the Metropolitanskolen in 1892 and then began mathematical and natural science studies at University of Copenhagen . These studies were completed in 1896 with a master's conference in mathematics. In 1898 , he received the university's gold medal for solving a prize problem, and in 1902 , he earned his doctoral degree with the dissertation Analytical Studies on Non-Euclidean Geometry .
From 1906 to 1918 , Tommy Bonnesen was a highly respected and beloved rector at Østre Borgerdyd Gymnasium|Østre Borgerdyd School. In connection with his teaching, he wrote several mathematics textbooks for high schools. He also frequently wrote on geometric topics in the Mathematical Journal – a journal published for many years until 1952 by the Danish Mathematical Society . From 1919 to 1935 , Bonnesen, together with Harald Bohr , was an editor of this journal.
Tommy Bonnesen was a bright presence in the mathematical community of his time, known for his infectious humor and sharp wit. In 1917 , he was appointed professor of descriptive geometry at the Polytechnical School, where he worked until his death.
Tommy Bonnesen is internationally known for his work in convex geometry, and a generalization of the isoperimetric inequality , which establishes a relationship between the radii of inscribed and circumscribed circles for convex figures, bears his name. Together with Werner Fenchel , Bonnesen published the renowned mathematical classic Theory of Convex Bodies in 1934 through Springer-Verlag . The book was published in English translation, Theory of Convex Bodies , in 1987 .
Tommy Bonnesen was elected to the Royal Danish Academy of Sciences and Letters in 1930 . | https://en.wikipedia.org/wiki/Tommy_Bonnesen |
Tommy Gate is an American brand of hydraulic liftgate , or tail lift , manufactured by Woodbine Manufacturing Company. The company was formed in 1965 by Delbert "Bus" Brown and its production facility is located in Woodbine, Iowa . [ 1 ] [ 2 ]
Prior to founding Woodbine Manufacturing Company, Delbert Brown manufactured farming equipment under the name of Brown Manufacturing Company . After inventing what was then one of the first trenching machines, Brown Manufacturing Company was sold to Omaha Steel Works . Three years later, Brown founded Woodbine Manufacturing Company and launched the Tommy Gate brand. [ 3 ]
The Woodbine manufacturing facility was initially built in 1965 to occupy 70,000 square feet of production space. It expanded in 1980 to 90,000 square feet and once again in 2000 when it grew to 140,000 square feet. The most recent expansion, completed in 2011, grew the plant to an overall 200,000 square feet (including 40,000 square feet of warehouse space). [ 4 ]
Tommy Gate manufactures a variety of hydraulic liftgates for trucks and other vehicles. Their main product lines include:
Parallel-arm : Versatile and capable of handling heavy loads.
Rail-gate : Ideal for low-clearance items.
Tuckunder : Compact design for smaller vehicles.
They also offer specialized liftgates like dump-through, level-ride, and side-loader models. Tommy Gate focuses on durability, reliability, and ease of use, with options for customization like remote controls and platform extensions. | https://en.wikipedia.org/wiki/Tommy_Gate |
The reflected binary code ( RBC ), also known as reflected binary ( RB ) or Gray code after Frank Gray , is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representation of the decimal value "1" in binary would normally be " 001 ", and "2" would be " 010 ". In Gray code, these values are represented as " 001 " and " 011 ". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.
Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. [ 3 ]
Many devices indicate position by closing and opening switches. If that device uses natural binary codes , positions 3 and 4 are next to each other but all three bits of the binary representation differ:
The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce , the transition might look like 011 — 001 — 101 — 100 . When the switches appear to be in position 001 , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic , then the sequential system may store a false value.
This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers , or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance , [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] single-distance , single-step , monostrophic [ 9 ] [ 10 ] [ 7 ] [ 8 ] or syncopic codes , [ 9 ] in reference to the Hamming distance of 1 between adjacent codes.
In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code , or BRGC . Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. [ 11 ] [ 12 ] [ 13 ] Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". [ 14 ] He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
In the standard encoding of the Gray code the least significant bit follows a repetitive pattern of 2 on, 2 off (... 11001100 ...); the next digit a pattern of 4 on, 4 off; the i -th least significant bit a pattern of 2 i on 2 i off. The most significant digit is an exception to this: for an n -bit Gray code, the most significant digit follows the pattern 2 n −1 on, 2 n −1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 n −2 places. The four-bit version of this is shown below:
For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code. [ 15 ]
In modern digital communications , Gray codes play an important role in error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Despite the fact that Stibitz described this code [ 11 ] [ 12 ] [ 13 ] before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; [ 16 ] [ 17 ] one of those also lists "minimum error code" and "cyclic permutation code" among the names. [ 17 ] A 1954 patent application refers to "the Bell Telephone Gray code". [ 18 ] Other names include "cyclic binary code", [ 12 ] "cyclic progression code", [ 19 ] [ 12 ] "cyclic permuting binary" [ 20 ] or "cyclic permuted binary" (CPB). [ 21 ] [ 22 ]
The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray . [ 13 ] [ 23 ] [ 24 ] [ 25 ]
Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle , a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872. [ 26 ] [ 13 ]
It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. [ 27 ] [ 28 ] [ 29 ] [ 30 ] Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes. [ 31 ]
Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American . [ 32 ]
The code also forms a Hamiltonian cycle on a hypercube , where each bit is seen as one dimension.
When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 [ 33 ] or 1876, [ 34 ] [ 35 ] he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, [ 36 ] [ 37 ] [ 38 ] and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. [ 13 ] This code became known as Baudot code [ 39 ] and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932. [ 40 ] [ 41 ] [ 38 ]
About the same time, the German-Austrian Otto Schäffler [ de ] [ 42 ] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874. [ 43 ] [ 13 ]
Frank Gray , who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube -based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, [ 14 ] and the name of Gray stuck to the codes. The " PCM tube " apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code. [ 44 ]
Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.
Gray codes are used in linear and rotary position encoders ( absolute encoders and quadrature encoders ) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.
For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.
Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.
In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.
Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms . [ 15 ] They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.
Gray codes are also used in labelling the axes of Karnaugh maps since 1953 [ 45 ] [ 46 ] [ 47 ] as well as in Händler circle graphs since 1958, [ 48 ] [ 49 ] [ 50 ] [ 51 ] both graphical methods for logic circuit minimization .
In modern digital communications , 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.
If a system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.
A balanced Gray code can be constructed, [ 52 ] that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.
George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already. [ 11 ] [ 12 ] [ 13 ]
A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. [ 53 ] The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.
Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, [ nb 1 ] it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous. [ 54 ]
To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, [ 55 ] add one to it with a standard binary adder, and then convert the result back to Gray code. [ 56 ] Other methods of counting in Gray code are discussed in a report by Robert W. Doran , including taking the output from the first latches of the master-slave flip flops in a binary ripple counter. [ 57 ]
As the execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs. [ 58 ] [ 59 ]
The binary-reflected Gray code list for n bits can be generated recursively from the list for n − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0 , prefixing the entries in the reflected list with a binary 1 , and then concatenating the original list with the reversed list. [ 13 ] For example, generating the n = 3 list from the n = 2 list:
The one-bit Gray code is G 1 = ( 0,1 ). This can be thought of as built recursively as above from a zero-bit Gray code G 0 = ( Λ ) consisting of a single entry of zero length. This iterative process of generating G n +1 from G n makes the following properties of the standard reflecting code clear:
These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the n th Gray code is obtained by computing n ⊕ ⌊ n 2 ⌋ {\displaystyle n\oplus \left\lfloor {\tfrac {n}{2}}\right\rfloor } . Prepending a 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position i {\displaystyle i} of codewords are inverted, the order of neighbouring blocks of 2 i {\displaystyle 2^{i}} codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed
If bit 1 is inverted, blocks of 2 codewords change order:
If bit 2 is inverted, blocks of 4 codewords reverse order:
Thus, performing an exclusive or on a bit b i {\displaystyle b_{i}} at position i {\displaystyle i} with the bit b i + 1 {\displaystyle b_{i+1}} at position i + 1 {\displaystyle i+1} leaves the order of codewords intact if b i + 1 = 0 {\displaystyle b_{i+1}={\mathtt {0}}} , and reverses the order of blocks of 2 i + 1 {\displaystyle 2^{i+1}} codewords if b i + 1 = 1 {\displaystyle b_{i+1}={\mathtt {1}}} . Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.
A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming g i {\displaystyle g_{i}} is the i {\displaystyle i} th Gray-coded bit ( g 0 {\displaystyle g_{0}} being the most significant bit), and b i {\displaystyle b_{i}} is the i {\displaystyle i} th binary-coded bit ( b 0 {\displaystyle b_{0}} being the most-significant bit), the reverse translation can be given recursively: b 0 = g 0 {\displaystyle b_{0}=g_{0}} , and b i = g i ⊕ b i − 1 {\displaystyle b_{i}=g_{i}\oplus b_{i-1}} . Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.
To construct the binary-reflected Gray code iteratively, at step 0 start with the c o d e 0 = 0 {\displaystyle \mathrm {code} _{0}={\mathtt {0}}} , and at step i > 0 {\displaystyle i>0} find the bit position of the least significant 1 in the binary representation of i {\displaystyle i} and flip the bit at that position in the previous code c o d e i − 1 {\displaystyle \mathrm {code} _{i-1}} to get the next code c o d e i {\displaystyle \mathrm {code} _{i}} . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... [ nb 2 ] See find first set for efficient algorithms to compute these values.
The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist. [ 60 ] [ 55 ] [ nb 1 ]
On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set . If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.
In practice, "Gray code" almost always refers to a binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).
It is possible to construct binary Gray codes with n bits with a length of less than 2 n , if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. [ 61 ] OEIS sequence A290772 [ 62 ] gives the number of possible Gray sequences of length 2 n that include zero and use the minimum number of bits.
0 → 000 1 → 001 2 → 002 10 → 012 11 → 011 12 → 010 20 → 020 21 → 021 22 → 022 100 → 122 101 → 121 102 → 120 110 → 110 111 → 111 112 → 112 120 → 102 121 → 101 122 → 100 200 → 200 201 → 201 202 → 202 210 → 212 211 → 211 212 → 210 220 → 220 221 → 221
There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the n -ary Gray code , also known as a non-Boolean Gray code . As the name implies, this type of Gray code uses non- Boolean values in its encodings.
For example, a 3-ary ( ternary ) Gray code would use the values 0,1,2. [ 31 ] The ( n , k )- Gray code is the n -ary Gray code with k digits. [ 63 ] The sequence of elements in the (3, 2)-Gray code is: 00,01,02,12,11,10,20,21,22. The ( n , k )-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively . An algorithm to iteratively generate the ( N , k )-Gray code is presented (in C ):
There are other Gray code algorithms for ( n , k )-Gray codes. The ( n , k )-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, [ 63 ] lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from n − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.
Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.
See also Skew binary number system , a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.
Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity". [ 52 ] In balanced Gray codes , the number of changes in different coordinate positions are as close as possible. To make this more precise, let G be an R -ary complete Gray cycle having transition sequence ( δ k ) {\displaystyle (\delta _{k})} ; the transition counts ( spectrum ) of G are the collection of integers defined by
λ k = | { j ∈ Z R n : δ j = k } | , for k ∈ Z n {\displaystyle \lambda _{k}=|\{j\in \mathbb {Z} _{R^{n}}:\delta _{j}=k\}|\,,{\text{ for }}k\in \mathbb {Z} _{n}}
A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have λ k = R n n {\displaystyle \lambda _{k}={\tfrac {R^{n}}{n}}} for all k . Clearly, when R = 2 {\displaystyle R=2} , such codes exist only if n is a power of 2. [ 64 ] If n is not a power of 2, it is possible to construct well-balanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either 2 ⌊ 2 n 2 n ⌋ {\displaystyle 2\left\lfloor {\tfrac {2^{n}}{2n}}\right\rfloor } or 2 ⌈ 2 n 2 n ⌉ {\displaystyle 2\left\lceil {\tfrac {2^{n}}{2n}}\right\rceil } . [ 52 ] Gray codes can also be exponentially balanced if all of their transition counts are adjacent powers of two, and such codes exist for every power of two. [ 65 ]
For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced: [ 52 ]
whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight: [ 52 ]
We will now show a construction [ 66 ] and implementation [ 67 ] for well-balanced binary Gray codes which allows us to generate an n -digit balanced Gray code for every n . The main principle is to inductively construct an ( n + 2)-digit Gray code G ′ {\displaystyle G'} given an n -digit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of G = g 0 , … , g 2 n − 1 {\displaystyle G=g_{0},\ldots ,g_{2^{n}-1}} into an even number L of non-empty blocks of the form
{ g 0 } , { g 1 , … , g k 2 } , { g k 2 + 1 , … , g k 3 } , … , { g k L − 2 + 1 , … , g − 2 } , { g − 1 } {\displaystyle \left\{g_{0}\right\},\left\{g_{1},\ldots ,g_{k_{2}}\right\},\left\{g_{k_{2}+1},\ldots ,g_{k_{3}}\right\},\ldots ,\left\{g_{k_{L-2}+1},\ldots ,g_{-2}\right\},\left\{g_{-1}\right\}}
where k 1 = 0 {\displaystyle k_{1}=0} , k L − 1 = − 2 {\displaystyle k_{L-1}=-2} , and k L ≡ − 1 ( mod 2 n ) {\displaystyle k_{L}\equiv -1{\pmod {2^{n}}}} ). This partition induces an ( n + 2 ) {\displaystyle (n+2)} -digit Gray code given by
If we define the transition multiplicities
m i = | { j : δ k j = i , 1 ≤ j ≤ L } | {\displaystyle m_{i}=\left|\left\{j:\delta _{k_{j}}=i,1\leq j\leq L\right\}\right|}
to be the number of times the digit in position i changes between consecutive blocks in a partition, then for the ( n + 2)-digit Gray code induced by this partition the transition spectrum λ i ′ {\displaystyle \lambda '_{i}} is
λ i ′ = { 4 λ i − 2 m i , if 0 ≤ i < n L , otherwise {\displaystyle \lambda '_{i}={\begin{cases}4\lambda _{i}-2m_{i},&{\text{if }}0\leq i<n\\L,&{\text{ otherwise }}\end{cases}}}
The delicate part of this construction is to find an adequate partitioning of a balanced n -digit Gray code such that the code induced by it remains balanced, but for this only the transition multiplicities matter; joining two consecutive blocks over a digit i {\displaystyle i} transition and splitting another block at another digit i {\displaystyle i} transition produces a different Gray code with exactly the same transition spectrum λ i ′ {\displaystyle \lambda '_{i}} , so one may for example [ 65 ] designate the first m i {\displaystyle m_{i}} transitions at digit i {\displaystyle i} as those that fall between two blocks. Uniform codes can be found when R ≡ 0 ( mod 4 ) {\displaystyle R\equiv 0{\pmod {4}}} and R n ≡ 0 ( mod n ) {\displaystyle R^{n}\equiv 0{\pmod {n}}} , and this construction can be extended to the R -ary case as well. [ 66 ]
Long run (or maximum gap ) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum run-length of any bit remains unchanged for as long as possible. [ 68 ]
Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors. [ 69 ] If we define the weight of a binary string to be the number of 1s in the string, then although we clearly cannot have a Gray code with strictly increasing weight, we may want to approximate this by having the code run through two adjacent weights before reaching the next one.
We can formalize the concept of monotone Gray codes as follows: consider the partition of the hypercube Q n = ( V n , E n ) {\displaystyle Q_{n}=(V_{n},E_{n})} into levels of vertices that have equal weight, i.e.
V n ( i ) = { v ∈ V n : v has weight i } {\displaystyle V_{n}(i)=\{v\in V_{n}:v{\text{ has weight }}i\}}
for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} . These levels satisfy | V n ( i ) | = ( n i ) {\displaystyle |V_{n}(i)|=\textstyle {\binom {n}{i}}} . Let Q n ( i ) {\displaystyle Q_{n}(i)} be the subgraph of Q n {\displaystyle Q_{n}} induced by V n ( i ) ∪ V n ( i + 1 ) {\displaystyle V_{n}(i)\cup V_{n}(i+1)} , and let E n ( i ) {\displaystyle E_{n}(i)} be the edges in Q n ( i ) {\displaystyle Q_{n}(i)} . A monotonic Gray code is then a Hamiltonian path in Q n {\displaystyle Q_{n}} such that whenever δ 1 ∈ E n ( i ) {\displaystyle \delta _{1}\in E_{n}(i)} comes before δ 2 ∈ E n ( j ) {\displaystyle \delta _{2}\in E_{n}(j)} in the path, then i ≤ j {\displaystyle i\leq j} .
An elegant construction of monotonic n -digit Gray codes for any n is based on the idea of recursively building subpaths P n , j {\displaystyle P_{n,j}} of length 2 ( n j ) {\displaystyle 2\textstyle {\binom {n}{j}}} having edges in E n ( j ) {\displaystyle E_{n}(j)} . [ 69 ] We define P 1 , 0 = ( 0 , 1 ) {\displaystyle P_{1,0}=({\mathtt {0}},{\mathtt {1}})} , P n , j = ∅ {\displaystyle P_{n,j}=\emptyset } whenever j < 0 {\displaystyle j<0} or j ≥ n {\displaystyle j\geq n} , and
P n + 1 , j = 1 P n , j − 1 π n , 0 P n , j {\displaystyle P_{n+1,j}={\mathtt {1}}P_{n,j-1}^{\pi _{n}},{\mathtt {0}}P_{n,j}}
otherwise. Here, π n {\displaystyle \pi _{n}} is a suitably defined permutation and P π {\displaystyle P^{\pi }} refers to the path P with its coordinates permuted by π {\displaystyle \pi } . These paths give rise to two monotonic n -digit Gray codes G n ( 1 ) {\displaystyle G_{n}^{(1)}} and G n ( 2 ) {\displaystyle G_{n}^{(2)}} given by
G n ( 1 ) = P n , 0 P n , 1 R P n , 2 P n , 3 R ⋯ and G n ( 2 ) = P n , 0 R P n , 1 P n , 2 R P n , 3 ⋯ {\displaystyle G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdots {\text{ and }}G_{n}^{(2)}=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdots }
The choice of π n {\displaystyle \pi _{n}} which ensures that these codes are indeed Gray codes turns out to be π n = E − 1 ( π n − 1 2 ) {\displaystyle \pi _{n}=E^{-1}\left(\pi _{n-1}^{2}\right)} . The first few values of P n , j {\displaystyle P_{n,j}} are shown in the table below.
These monotonic Gray codes can be efficiently implemented in such a way that each subsequent element can be generated in O ( n ) time. The algorithm is most easily described using coroutines .
Monotonic codes have an interesting connection to the Lovász conjecture , which states that every connected vertex-transitive graph contains a Hamiltonian path. The "middle-level" subgraph Q 2 n + 1 ( n ) {\displaystyle Q_{2n+1}(n)} is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism ) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for n ≤ 15 {\displaystyle n\leq 15} , and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839 N , where N is the number of vertices in the middle-level subgraph. [ 70 ]
Another type of Gray code, the Beckett–Gray code , is named for Irish playwright Samuel Beckett , who was interested in symmetry . His play " Quad " features four actors and is divided into sixteen time periods. Each period ends with one of the four actors entering or leaving the stage. The play begins and ends with an empty stage, and Beckett wanted each subset of actors to appear on stage exactly once. [ 71 ] Clearly the set of actors currently on stage can be represented by a 4-bit binary Gray code. Beckett, however, placed an additional restriction on the script: he wished the actors to enter and exit so that the actor who had been on stage the longest would always be the one to exit. The actors could then be represented by a first in, first out queue , so that (of the actors onstage) the actor being dequeued is always the one who was enqueued first. [ 71 ] Beckett was unable to find a Beckett–Gray code for his play, and indeed, an exhaustive listing of all possible sequences reveals that no such code exists for n = 4. It is known today that such codes do exist for n = 2, 5, 6, 7, and 8, and do not exist for n = 3 or 4. An example of an 8-bit Beckett–Gray code can be found in Donald Knuth 's Art of Computer Programming . [ 13 ] According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than 9500 solutions for the case n = 7 have been found. [ 72 ]
Snake-in-the-box codes, or snakes , are the sequences of nodes of induced paths in an n -dimensional hypercube graph , and coil-in-the-box codes, [ 73 ] or coils , are the sequences of nodes of induced cycles in a hypercube. Viewed as Gray codes, these sequences have the property of being able to detect any single-bit coding error. Codes of this type were first described by William H. Kautz in the late 1950s; [ 5 ] since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.
Yet another kind of Gray code is the single-track Gray code (STGC) developed by Norman B. Spedding [ 74 ] [ 75 ] and refined by Hiltgen, Paterson and Brandestini in Single-track Gray Codes (1996). [ 76 ] [ 77 ] The STGC is a cyclical list of P unique binary encodings of length n such that two consecutive words differ in exactly one position, and when the list is examined as a P × n matrix , each column is a cyclic shift of the first column. [ 78 ]
The name comes from their use with rotary encoders , where a number of tracks are being sensed by contacts, resulting for each in an output of 0 or 1 . To reduce noise due to different contacts not switching at exactly the same moment in time, one preferably sets up the tracks so that the data output by the contacts are in Gray code. To get high angular accuracy, one needs lots of contacts; in order to achieve at least 1° accuracy, one needs at least 360 distinct positions per revolution, which requires a minimum of 9 bits of data, and thus the same number of contacts.
If all contacts are placed at the same angular position, then 9 tracks are needed to get a standard BRGC with at least 1° accuracy. However, if the manufacturer moves a contact to a different angular position (but at the same distance from the center shaft), then the corresponding "ring pattern" needs to be rotated the same angle to give the same output. If the most significant bit (the inner ring in Figure 1) is rotated enough, it exactly matches the next ring out. Since both rings are then identical, the inner ring can be cut out, and the sensor for that ring moved to the remaining, identical ring (but offset at that angle from the other sensor on that ring). Those two sensors on a single ring make a quadrature encoder. That reduces the number of tracks for a "1° resolution" angular encoder to 8 tracks. Reducing the number of tracks still further cannot be done with BRGC.
For many years, Torsten Sillke [ 79 ] and other mathematicians believed that it was impossible to encode position on a single track such that consecutive positions differed at only a single sensor, except for the 2-sensor, 1-track quadrature encoder. So for applications where 8 tracks were too bulky, people used single-track incremental encoders (quadrature encoders) or 2-track "quadrature encoder + reference notch" encoders.
Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible. [ 74 ] Although it is not possible to distinguish 2 n positions with n sensors on a single track, it is possible to distinguish close to that many. Etzion and Paterson conjecture that when n is itself a power of 2, n sensors can distinguish at most 2 n − 2 n positions and that for prime n the limit is 2 n − 2 positions. [ 80 ] The authors went on to generate a 504-position single track code of length 9 which they believe is optimal. Since this number is larger than 2 8 = 256, more than 8 sensors are required by any code, although a BRGC could distinguish 512 positions with 9 sensors.
An STGC for P = 30 and n = 5 is reproduced here:
Each column is a cyclic shift of the first column, and from any row to the next row only one bit changes. [ 81 ] The single-track nature (like a code chain) is useful in the fabrication of these wheels (compared to BRGC), as only one track is needed, thus reducing their cost and size.
The Gray code nature is useful (compared to chain codes , also called De Bruijn sequences ), as only one sensor will change at any one time, so the uncertainty during a transition between two discrete states will only be plus or minus one unit of angular measurement the device is capable of resolving. [ 82 ]
Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams, [ 83 ] [ user-generated source? ] based on previous work, [ 80 ] discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site Thingiverse . This device [ 84 ] was designed by etzenseep (Florian Bauer) in September 2022.
An STGC for P = 360 and n = 9 is reproduced here:
Two-dimensional Gray codes are used in communication to minimize the number of bit errors in quadrature amplitude modulation (QAM) adjacent points in the constellation . In a typical encoding the horizontal and vertical adjacent constellation points differ by a single bit, and diagonal adjacent points differ by 2 bits. [ 85 ]
Two-dimensional Gray codes also have uses in location identifications schemes, where the code would be applied to area maps such as a Mercator projection of the earth's surface and an appropriate cyclic two-dimensional distance function such as the Mannheim metric be used to calculate the distance between two encoded locations, thereby combining the characteristics of the Hamming distance with the cyclic continuation of a Mercator projection. [ 86 ]
If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit Gray code, the resulting code will be an "excess Gray code". This code shows the property of counting backwards in those extracted bits if the original value is further increased. Reason for this is that Gray-encoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.
Example: The highest 3-bit Gray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in Gray as 1100. The last 3 bits do not overflow and count backwards if you further increase the original 4 bit code.
When working with sensors that output multiple, Gray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.
The bijective mapping { 0 ↔ 00 , 1 ↔ 01 , 2 ↔ 11 , 3 ↔ 10 } establishes an isometry between the metric space over the finite field Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} with the metric given by the Hamming distance and the metric space over the finite ring Z 4 {\displaystyle \mathbb {Z} _{4}} (the usual modular arithmetic ) with the metric given by the Lee distance . The mapping is suitably extended to an isometry of the Hamming spaces Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} . Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Gray-map images in Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} of ring-linear codes from Z 4 {\displaystyle \mathbb {Z} _{4}} . [ 87 ] [ 88 ]
There are a number of binary codes similar to Gray codes, including:
The following binary-coded decimal (BCD) codes are Gray code variants as well: | https://en.wikipedia.org/wiki/Tompkins_code_I |
The reflected binary code ( RBC ), also known as reflected binary ( RB ) or Gray code after Frank Gray , is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representation of the decimal value "1" in binary would normally be " 001 ", and "2" would be " 010 ". In Gray code, these values are represented as " 001 " and " 011 ". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.
Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. [ 3 ]
Many devices indicate position by closing and opening switches. If that device uses natural binary codes , positions 3 and 4 are next to each other but all three bits of the binary representation differ:
The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce , the transition might look like 011 — 001 — 101 — 100 . When the switches appear to be in position 001 , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic , then the sequential system may store a false value.
This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers , or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance , [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] single-distance , single-step , monostrophic [ 9 ] [ 10 ] [ 7 ] [ 8 ] or syncopic codes , [ 9 ] in reference to the Hamming distance of 1 between adjacent codes.
In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code , or BRGC . Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. [ 11 ] [ 12 ] [ 13 ] Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". [ 14 ] He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
In the standard encoding of the Gray code the least significant bit follows a repetitive pattern of 2 on, 2 off (... 11001100 ...); the next digit a pattern of 4 on, 4 off; the i -th least significant bit a pattern of 2 i on 2 i off. The most significant digit is an exception to this: for an n -bit Gray code, the most significant digit follows the pattern 2 n −1 on, 2 n −1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 n −2 places. The four-bit version of this is shown below:
For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code. [ 15 ]
In modern digital communications , Gray codes play an important role in error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Despite the fact that Stibitz described this code [ 11 ] [ 12 ] [ 13 ] before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; [ 16 ] [ 17 ] one of those also lists "minimum error code" and "cyclic permutation code" among the names. [ 17 ] A 1954 patent application refers to "the Bell Telephone Gray code". [ 18 ] Other names include "cyclic binary code", [ 12 ] "cyclic progression code", [ 19 ] [ 12 ] "cyclic permuting binary" [ 20 ] or "cyclic permuted binary" (CPB). [ 21 ] [ 22 ]
The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray . [ 13 ] [ 23 ] [ 24 ] [ 25 ]
Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle , a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872. [ 26 ] [ 13 ]
It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. [ 27 ] [ 28 ] [ 29 ] [ 30 ] Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes. [ 31 ]
Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American . [ 32 ]
The code also forms a Hamiltonian cycle on a hypercube , where each bit is seen as one dimension.
When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 [ 33 ] or 1876, [ 34 ] [ 35 ] he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, [ 36 ] [ 37 ] [ 38 ] and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. [ 13 ] This code became known as Baudot code [ 39 ] and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932. [ 40 ] [ 41 ] [ 38 ]
About the same time, the German-Austrian Otto Schäffler [ de ] [ 42 ] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874. [ 43 ] [ 13 ]
Frank Gray , who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube -based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, [ 14 ] and the name of Gray stuck to the codes. The " PCM tube " apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code. [ 44 ]
Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.
Gray codes are used in linear and rotary position encoders ( absolute encoders and quadrature encoders ) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.
For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.
Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.
In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.
Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms . [ 15 ] They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.
Gray codes are also used in labelling the axes of Karnaugh maps since 1953 [ 45 ] [ 46 ] [ 47 ] as well as in Händler circle graphs since 1958, [ 48 ] [ 49 ] [ 50 ] [ 51 ] both graphical methods for logic circuit minimization .
In modern digital communications , 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.
If a system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.
A balanced Gray code can be constructed, [ 52 ] that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.
George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already. [ 11 ] [ 12 ] [ 13 ]
A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. [ 53 ] The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.
Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, [ nb 1 ] it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous. [ 54 ]
To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, [ 55 ] add one to it with a standard binary adder, and then convert the result back to Gray code. [ 56 ] Other methods of counting in Gray code are discussed in a report by Robert W. Doran , including taking the output from the first latches of the master-slave flip flops in a binary ripple counter. [ 57 ]
As the execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs. [ 58 ] [ 59 ]
The binary-reflected Gray code list for n bits can be generated recursively from the list for n − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0 , prefixing the entries in the reflected list with a binary 1 , and then concatenating the original list with the reversed list. [ 13 ] For example, generating the n = 3 list from the n = 2 list:
The one-bit Gray code is G 1 = ( 0,1 ). This can be thought of as built recursively as above from a zero-bit Gray code G 0 = ( Λ ) consisting of a single entry of zero length. This iterative process of generating G n +1 from G n makes the following properties of the standard reflecting code clear:
These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the n th Gray code is obtained by computing n ⊕ ⌊ n 2 ⌋ {\displaystyle n\oplus \left\lfloor {\tfrac {n}{2}}\right\rfloor } . Prepending a 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position i {\displaystyle i} of codewords are inverted, the order of neighbouring blocks of 2 i {\displaystyle 2^{i}} codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed
If bit 1 is inverted, blocks of 2 codewords change order:
If bit 2 is inverted, blocks of 4 codewords reverse order:
Thus, performing an exclusive or on a bit b i {\displaystyle b_{i}} at position i {\displaystyle i} with the bit b i + 1 {\displaystyle b_{i+1}} at position i + 1 {\displaystyle i+1} leaves the order of codewords intact if b i + 1 = 0 {\displaystyle b_{i+1}={\mathtt {0}}} , and reverses the order of blocks of 2 i + 1 {\displaystyle 2^{i+1}} codewords if b i + 1 = 1 {\displaystyle b_{i+1}={\mathtt {1}}} . Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.
A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming g i {\displaystyle g_{i}} is the i {\displaystyle i} th Gray-coded bit ( g 0 {\displaystyle g_{0}} being the most significant bit), and b i {\displaystyle b_{i}} is the i {\displaystyle i} th binary-coded bit ( b 0 {\displaystyle b_{0}} being the most-significant bit), the reverse translation can be given recursively: b 0 = g 0 {\displaystyle b_{0}=g_{0}} , and b i = g i ⊕ b i − 1 {\displaystyle b_{i}=g_{i}\oplus b_{i-1}} . Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.
To construct the binary-reflected Gray code iteratively, at step 0 start with the c o d e 0 = 0 {\displaystyle \mathrm {code} _{0}={\mathtt {0}}} , and at step i > 0 {\displaystyle i>0} find the bit position of the least significant 1 in the binary representation of i {\displaystyle i} and flip the bit at that position in the previous code c o d e i − 1 {\displaystyle \mathrm {code} _{i-1}} to get the next code c o d e i {\displaystyle \mathrm {code} _{i}} . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... [ nb 2 ] See find first set for efficient algorithms to compute these values.
The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist. [ 60 ] [ 55 ] [ nb 1 ]
On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set . If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.
In practice, "Gray code" almost always refers to a binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).
It is possible to construct binary Gray codes with n bits with a length of less than 2 n , if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. [ 61 ] OEIS sequence A290772 [ 62 ] gives the number of possible Gray sequences of length 2 n that include zero and use the minimum number of bits.
0 → 000 1 → 001 2 → 002 10 → 012 11 → 011 12 → 010 20 → 020 21 → 021 22 → 022 100 → 122 101 → 121 102 → 120 110 → 110 111 → 111 112 → 112 120 → 102 121 → 101 122 → 100 200 → 200 201 → 201 202 → 202 210 → 212 211 → 211 212 → 210 220 → 220 221 → 221
There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the n -ary Gray code , also known as a non-Boolean Gray code . As the name implies, this type of Gray code uses non- Boolean values in its encodings.
For example, a 3-ary ( ternary ) Gray code would use the values 0,1,2. [ 31 ] The ( n , k )- Gray code is the n -ary Gray code with k digits. [ 63 ] The sequence of elements in the (3, 2)-Gray code is: 00,01,02,12,11,10,20,21,22. The ( n , k )-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively . An algorithm to iteratively generate the ( N , k )-Gray code is presented (in C ):
There are other Gray code algorithms for ( n , k )-Gray codes. The ( n , k )-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, [ 63 ] lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from n − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.
Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.
See also Skew binary number system , a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.
Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity". [ 52 ] In balanced Gray codes , the number of changes in different coordinate positions are as close as possible. To make this more precise, let G be an R -ary complete Gray cycle having transition sequence ( δ k ) {\displaystyle (\delta _{k})} ; the transition counts ( spectrum ) of G are the collection of integers defined by
λ k = | { j ∈ Z R n : δ j = k } | , for k ∈ Z n {\displaystyle \lambda _{k}=|\{j\in \mathbb {Z} _{R^{n}}:\delta _{j}=k\}|\,,{\text{ for }}k\in \mathbb {Z} _{n}}
A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have λ k = R n n {\displaystyle \lambda _{k}={\tfrac {R^{n}}{n}}} for all k . Clearly, when R = 2 {\displaystyle R=2} , such codes exist only if n is a power of 2. [ 64 ] If n is not a power of 2, it is possible to construct well-balanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either 2 ⌊ 2 n 2 n ⌋ {\displaystyle 2\left\lfloor {\tfrac {2^{n}}{2n}}\right\rfloor } or 2 ⌈ 2 n 2 n ⌉ {\displaystyle 2\left\lceil {\tfrac {2^{n}}{2n}}\right\rceil } . [ 52 ] Gray codes can also be exponentially balanced if all of their transition counts are adjacent powers of two, and such codes exist for every power of two. [ 65 ]
For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced: [ 52 ]
whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight: [ 52 ]
We will now show a construction [ 66 ] and implementation [ 67 ] for well-balanced binary Gray codes which allows us to generate an n -digit balanced Gray code for every n . The main principle is to inductively construct an ( n + 2)-digit Gray code G ′ {\displaystyle G'} given an n -digit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of G = g 0 , … , g 2 n − 1 {\displaystyle G=g_{0},\ldots ,g_{2^{n}-1}} into an even number L of non-empty blocks of the form
{ g 0 } , { g 1 , … , g k 2 } , { g k 2 + 1 , … , g k 3 } , … , { g k L − 2 + 1 , … , g − 2 } , { g − 1 } {\displaystyle \left\{g_{0}\right\},\left\{g_{1},\ldots ,g_{k_{2}}\right\},\left\{g_{k_{2}+1},\ldots ,g_{k_{3}}\right\},\ldots ,\left\{g_{k_{L-2}+1},\ldots ,g_{-2}\right\},\left\{g_{-1}\right\}}
where k 1 = 0 {\displaystyle k_{1}=0} , k L − 1 = − 2 {\displaystyle k_{L-1}=-2} , and k L ≡ − 1 ( mod 2 n ) {\displaystyle k_{L}\equiv -1{\pmod {2^{n}}}} ). This partition induces an ( n + 2 ) {\displaystyle (n+2)} -digit Gray code given by
If we define the transition multiplicities
m i = | { j : δ k j = i , 1 ≤ j ≤ L } | {\displaystyle m_{i}=\left|\left\{j:\delta _{k_{j}}=i,1\leq j\leq L\right\}\right|}
to be the number of times the digit in position i changes between consecutive blocks in a partition, then for the ( n + 2)-digit Gray code induced by this partition the transition spectrum λ i ′ {\displaystyle \lambda '_{i}} is
λ i ′ = { 4 λ i − 2 m i , if 0 ≤ i < n L , otherwise {\displaystyle \lambda '_{i}={\begin{cases}4\lambda _{i}-2m_{i},&{\text{if }}0\leq i<n\\L,&{\text{ otherwise }}\end{cases}}}
The delicate part of this construction is to find an adequate partitioning of a balanced n -digit Gray code such that the code induced by it remains balanced, but for this only the transition multiplicities matter; joining two consecutive blocks over a digit i {\displaystyle i} transition and splitting another block at another digit i {\displaystyle i} transition produces a different Gray code with exactly the same transition spectrum λ i ′ {\displaystyle \lambda '_{i}} , so one may for example [ 65 ] designate the first m i {\displaystyle m_{i}} transitions at digit i {\displaystyle i} as those that fall between two blocks. Uniform codes can be found when R ≡ 0 ( mod 4 ) {\displaystyle R\equiv 0{\pmod {4}}} and R n ≡ 0 ( mod n ) {\displaystyle R^{n}\equiv 0{\pmod {n}}} , and this construction can be extended to the R -ary case as well. [ 66 ]
Long run (or maximum gap ) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum run-length of any bit remains unchanged for as long as possible. [ 68 ]
Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors. [ 69 ] If we define the weight of a binary string to be the number of 1s in the string, then although we clearly cannot have a Gray code with strictly increasing weight, we may want to approximate this by having the code run through two adjacent weights before reaching the next one.
We can formalize the concept of monotone Gray codes as follows: consider the partition of the hypercube Q n = ( V n , E n ) {\displaystyle Q_{n}=(V_{n},E_{n})} into levels of vertices that have equal weight, i.e.
V n ( i ) = { v ∈ V n : v has weight i } {\displaystyle V_{n}(i)=\{v\in V_{n}:v{\text{ has weight }}i\}}
for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} . These levels satisfy | V n ( i ) | = ( n i ) {\displaystyle |V_{n}(i)|=\textstyle {\binom {n}{i}}} . Let Q n ( i ) {\displaystyle Q_{n}(i)} be the subgraph of Q n {\displaystyle Q_{n}} induced by V n ( i ) ∪ V n ( i + 1 ) {\displaystyle V_{n}(i)\cup V_{n}(i+1)} , and let E n ( i ) {\displaystyle E_{n}(i)} be the edges in Q n ( i ) {\displaystyle Q_{n}(i)} . A monotonic Gray code is then a Hamiltonian path in Q n {\displaystyle Q_{n}} such that whenever δ 1 ∈ E n ( i ) {\displaystyle \delta _{1}\in E_{n}(i)} comes before δ 2 ∈ E n ( j ) {\displaystyle \delta _{2}\in E_{n}(j)} in the path, then i ≤ j {\displaystyle i\leq j} .
An elegant construction of monotonic n -digit Gray codes for any n is based on the idea of recursively building subpaths P n , j {\displaystyle P_{n,j}} of length 2 ( n j ) {\displaystyle 2\textstyle {\binom {n}{j}}} having edges in E n ( j ) {\displaystyle E_{n}(j)} . [ 69 ] We define P 1 , 0 = ( 0 , 1 ) {\displaystyle P_{1,0}=({\mathtt {0}},{\mathtt {1}})} , P n , j = ∅ {\displaystyle P_{n,j}=\emptyset } whenever j < 0 {\displaystyle j<0} or j ≥ n {\displaystyle j\geq n} , and
P n + 1 , j = 1 P n , j − 1 π n , 0 P n , j {\displaystyle P_{n+1,j}={\mathtt {1}}P_{n,j-1}^{\pi _{n}},{\mathtt {0}}P_{n,j}}
otherwise. Here, π n {\displaystyle \pi _{n}} is a suitably defined permutation and P π {\displaystyle P^{\pi }} refers to the path P with its coordinates permuted by π {\displaystyle \pi } . These paths give rise to two monotonic n -digit Gray codes G n ( 1 ) {\displaystyle G_{n}^{(1)}} and G n ( 2 ) {\displaystyle G_{n}^{(2)}} given by
G n ( 1 ) = P n , 0 P n , 1 R P n , 2 P n , 3 R ⋯ and G n ( 2 ) = P n , 0 R P n , 1 P n , 2 R P n , 3 ⋯ {\displaystyle G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdots {\text{ and }}G_{n}^{(2)}=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdots }
The choice of π n {\displaystyle \pi _{n}} which ensures that these codes are indeed Gray codes turns out to be π n = E − 1 ( π n − 1 2 ) {\displaystyle \pi _{n}=E^{-1}\left(\pi _{n-1}^{2}\right)} . The first few values of P n , j {\displaystyle P_{n,j}} are shown in the table below.
These monotonic Gray codes can be efficiently implemented in such a way that each subsequent element can be generated in O ( n ) time. The algorithm is most easily described using coroutines .
Monotonic codes have an interesting connection to the Lovász conjecture , which states that every connected vertex-transitive graph contains a Hamiltonian path. The "middle-level" subgraph Q 2 n + 1 ( n ) {\displaystyle Q_{2n+1}(n)} is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism ) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for n ≤ 15 {\displaystyle n\leq 15} , and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839 N , where N is the number of vertices in the middle-level subgraph. [ 70 ]
Another type of Gray code, the Beckett–Gray code , is named for Irish playwright Samuel Beckett , who was interested in symmetry . His play " Quad " features four actors and is divided into sixteen time periods. Each period ends with one of the four actors entering or leaving the stage. The play begins and ends with an empty stage, and Beckett wanted each subset of actors to appear on stage exactly once. [ 71 ] Clearly the set of actors currently on stage can be represented by a 4-bit binary Gray code. Beckett, however, placed an additional restriction on the script: he wished the actors to enter and exit so that the actor who had been on stage the longest would always be the one to exit. The actors could then be represented by a first in, first out queue , so that (of the actors onstage) the actor being dequeued is always the one who was enqueued first. [ 71 ] Beckett was unable to find a Beckett–Gray code for his play, and indeed, an exhaustive listing of all possible sequences reveals that no such code exists for n = 4. It is known today that such codes do exist for n = 2, 5, 6, 7, and 8, and do not exist for n = 3 or 4. An example of an 8-bit Beckett–Gray code can be found in Donald Knuth 's Art of Computer Programming . [ 13 ] According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than 9500 solutions for the case n = 7 have been found. [ 72 ]
Snake-in-the-box codes, or snakes , are the sequences of nodes of induced paths in an n -dimensional hypercube graph , and coil-in-the-box codes, [ 73 ] or coils , are the sequences of nodes of induced cycles in a hypercube. Viewed as Gray codes, these sequences have the property of being able to detect any single-bit coding error. Codes of this type were first described by William H. Kautz in the late 1950s; [ 5 ] since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.
Yet another kind of Gray code is the single-track Gray code (STGC) developed by Norman B. Spedding [ 74 ] [ 75 ] and refined by Hiltgen, Paterson and Brandestini in Single-track Gray Codes (1996). [ 76 ] [ 77 ] The STGC is a cyclical list of P unique binary encodings of length n such that two consecutive words differ in exactly one position, and when the list is examined as a P × n matrix , each column is a cyclic shift of the first column. [ 78 ]
The name comes from their use with rotary encoders , where a number of tracks are being sensed by contacts, resulting for each in an output of 0 or 1 . To reduce noise due to different contacts not switching at exactly the same moment in time, one preferably sets up the tracks so that the data output by the contacts are in Gray code. To get high angular accuracy, one needs lots of contacts; in order to achieve at least 1° accuracy, one needs at least 360 distinct positions per revolution, which requires a minimum of 9 bits of data, and thus the same number of contacts.
If all contacts are placed at the same angular position, then 9 tracks are needed to get a standard BRGC with at least 1° accuracy. However, if the manufacturer moves a contact to a different angular position (but at the same distance from the center shaft), then the corresponding "ring pattern" needs to be rotated the same angle to give the same output. If the most significant bit (the inner ring in Figure 1) is rotated enough, it exactly matches the next ring out. Since both rings are then identical, the inner ring can be cut out, and the sensor for that ring moved to the remaining, identical ring (but offset at that angle from the other sensor on that ring). Those two sensors on a single ring make a quadrature encoder. That reduces the number of tracks for a "1° resolution" angular encoder to 8 tracks. Reducing the number of tracks still further cannot be done with BRGC.
For many years, Torsten Sillke [ 79 ] and other mathematicians believed that it was impossible to encode position on a single track such that consecutive positions differed at only a single sensor, except for the 2-sensor, 1-track quadrature encoder. So for applications where 8 tracks were too bulky, people used single-track incremental encoders (quadrature encoders) or 2-track "quadrature encoder + reference notch" encoders.
Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible. [ 74 ] Although it is not possible to distinguish 2 n positions with n sensors on a single track, it is possible to distinguish close to that many. Etzion and Paterson conjecture that when n is itself a power of 2, n sensors can distinguish at most 2 n − 2 n positions and that for prime n the limit is 2 n − 2 positions. [ 80 ] The authors went on to generate a 504-position single track code of length 9 which they believe is optimal. Since this number is larger than 2 8 = 256, more than 8 sensors are required by any code, although a BRGC could distinguish 512 positions with 9 sensors.
An STGC for P = 30 and n = 5 is reproduced here:
Each column is a cyclic shift of the first column, and from any row to the next row only one bit changes. [ 81 ] The single-track nature (like a code chain) is useful in the fabrication of these wheels (compared to BRGC), as only one track is needed, thus reducing their cost and size.
The Gray code nature is useful (compared to chain codes , also called De Bruijn sequences ), as only one sensor will change at any one time, so the uncertainty during a transition between two discrete states will only be plus or minus one unit of angular measurement the device is capable of resolving. [ 82 ]
Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams, [ 83 ] [ user-generated source? ] based on previous work, [ 80 ] discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site Thingiverse . This device [ 84 ] was designed by etzenseep (Florian Bauer) in September 2022.
An STGC for P = 360 and n = 9 is reproduced here:
Two-dimensional Gray codes are used in communication to minimize the number of bit errors in quadrature amplitude modulation (QAM) adjacent points in the constellation . In a typical encoding the horizontal and vertical adjacent constellation points differ by a single bit, and diagonal adjacent points differ by 2 bits. [ 85 ]
Two-dimensional Gray codes also have uses in location identifications schemes, where the code would be applied to area maps such as a Mercator projection of the earth's surface and an appropriate cyclic two-dimensional distance function such as the Mannheim metric be used to calculate the distance between two encoded locations, thereby combining the characteristics of the Hamming distance with the cyclic continuation of a Mercator projection. [ 86 ]
If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit Gray code, the resulting code will be an "excess Gray code". This code shows the property of counting backwards in those extracted bits if the original value is further increased. Reason for this is that Gray-encoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.
Example: The highest 3-bit Gray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in Gray as 1100. The last 3 bits do not overflow and count backwards if you further increase the original 4 bit code.
When working with sensors that output multiple, Gray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.
The bijective mapping { 0 ↔ 00 , 1 ↔ 01 , 2 ↔ 11 , 3 ↔ 10 } establishes an isometry between the metric space over the finite field Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} with the metric given by the Hamming distance and the metric space over the finite ring Z 4 {\displaystyle \mathbb {Z} _{4}} (the usual modular arithmetic ) with the metric given by the Lee distance . The mapping is suitably extended to an isometry of the Hamming spaces Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} . Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Gray-map images in Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} of ring-linear codes from Z 4 {\displaystyle \mathbb {Z} _{4}} . [ 87 ] [ 88 ]
There are a number of binary codes similar to Gray codes, including:
The following binary-coded decimal (BCD) codes are Gray code variants as well: | https://en.wikipedia.org/wiki/Tompkins_code_II |
In fluid dynamics , the Toms effect is a reduction of the drag of a turbulent flow thought a pipeline when polymer solutions are added. [ 1 ]
In 1948, B. A. Toms discovered [ 2 ] by experiments that the addition of a small amount of polymer into a Newtonian solvent ( parts per million by weight), which results in a non-Newtonian fluid solution, can reduce the skin frictional drag on a stationary surface by up to 80% when turbulence is present. [ 1 ]
This technology has been successfully implemented to reduce pumping cost for oil pipelines, to increase the flow rate in fire fighting equipment and to help irrigation and drainage. [ 3 ] [ 4 ] It also has potential applications in the design of ship and submarine hulls to achieve an increased speed and reduced energy cost. | https://en.wikipedia.org/wiki/Toms_effect |
Tonalli (see also: Tonal ) plays a multiplicity of roles; acting as a day sign, body part, and a symbol of the sun's warmth. Ancient Nahua people believed that it was located in the hair and the fontanel area of one's skull, and that the tonalli provided the “vigor and energy for growth and development”. [ 1 ] It often overlaps with the force of teyolía which was often considered both an animating force ( soul ) and the physical heart in various Mesoamerican cultures .
The root “tona” acts as a verb to mean "to irradiate or make warm with sun”. [ 2 ]
In the Ancient Nahua belief, the tonalli is bestowed upon a child in utero by the aged deities known as Ometecuhtli and Omecihuatl , or the “Lord and Lady of Duality”. [ 3 ] The implementation of tonalli is conducted through a process known as Fire Drilling.
It is believed that the old deities, Ometecuhtli and Omecihuatl transferred tonalli to human fetuses by “simultaneously breath[ing] the tonalli into the child and ignit[ing] a fire in its chest”. [ 4 ] This Fire Drilling process involves an upright wooden piece being twirled rapidly on a flat base. It produces heat through friction, although this seemingly simple instrument requires considerable skill to make anything but smoke. The fire maker blows on an ignited spark to fan it into a vigorous flame, and the breathing (or blowing air) and friction in the chest animate an infant. [ 4 ] The Franciscan friars connected this idea of Fire Drilling, namely, the conception of tonalli as breath, to Christianity as the infusion of breath into the body recalls the beginning of Genesis , where God the Father breathes life into Adam.
The Nahua people of Mesoamerica believed that the soul comprised three entities : Tonalli, Teyolía, and Ihíyotl, three souls in the body. Tonalli is located in fontanel area of the skull. Teyolía is located in the heart and Ihíyotl is in the liver. Each of these souls has its own functions and protective deities. But there are important differences. The Tonalli is the soul that enters and leaves the body. In Atla in the northern Sierra de Puebla, the inhabitants believe this is the soul that travels while you sleep at night, and then comes back. [ 5 ] This is the soul that leaves and comes back every time you sneeze, or whenever you yawn, or even when you are startled. The Nahua believed that it was not good to sneeze and keep talking, because it causes your tonalli to leave and once your Tonalli leaves, you have to wait for a period of time before it returns. At that moment, anything can enter your body. However, the soul of the heart (Teyolía) and the soul of the liver (Ihiíyotl) only leave your body when you die; those two souls will exit only at the exact moment of your death. [ 6 ]
Along with the heart, the ancient Mexica took blood from sacrificed prisoners and offered its fructifying force to the gods. On public and private ritual occasions, people drew blood from their ears, tongues, or calves, splattered it on pieces of paper, and gave it to the spirits as tokens of thanks for benefits received or as requests for future favors. The gods could be coaxed or rewarded, less by the physical fluid itself than by the tonalli (life force) it carried and transferred to them. [ 7 ] Among most modern Nahuatl speakers, the state of the life force may even be determined from the movement of blood in the body, whether this movement is experienced as a tic, a pulse, or a muscular movement. [ 8 ] When blood drains from its proper course, the person loses his/her life force, and, essentially, soul.
Tonalli was highly valued in society and sought after in warfare and ritual sacrifice. The hair that covered the head, especially the fontanel area, was a major receptacle of tonalli, and it was believed that hair prevented the tonalli from leaving the body. During times of war, when a warrior would take a prisoner captive, the warrior would often seize the captive by the hair (the fontanel area). It was believed that the fortitude and valor of a warrior resided, in part, in the hair, and there are many pictorial scenes showing Aztec warriors grabbing the hair of enemies. The hair of warriors captured in battle was kept by the captors in order to increase their tonalli. The severed heads of enemy warriors were a supreme prize for the city, which gained more tonalli through the ceremonial use of heads. [ 9 ] The tonalli embedded within skulls of enemy warriors and captives were also offered as gifts to the gods in the temple complexes. This was believed to be an offering to the gods in the form of a type of debt repayment. [ 1 ]
The essence of tonalli was a force that could transcend the limits of the human body. Parts of the tonalli could reside outside the body in objects and animals. For example, tlacopatli beads were often left in the temple and represented a substitute for a child unable to go to the temple school, due to age restrictions. These beads served to contain the tonalli and do penitence for the underaged child. [ 10 ] This points to the belief of the physicality of the soul and the embodiment of the tonalli. Stones were also used as repositories of the soul in a different sense. As tonalli was considered to be an indication of destiny or fate, the possession of gemstones served as direct evidence of those with positive fates despite low births and the auspicious destinies of members of the elite class. [ 10 ]
The tonalli operated within a complex that involved the god of the birth date and the human. The soul of the individuality of a person, resided inside; but the god of the tonalli resided outside [ 11 ] The co-essence of tonalli in a human body is subject to the power of an external god/time that could lodge in a person's body. The dynamics of this joint essence relationship made it necessary for humans to implore each god of birth dates for internal strength, health and good fortune. [ 12 ]
Tonalli, along with teyolia and ihiyotl , was believed to direct the physiological process of the human body. It gave a person character, and was highly valued by the family and sought after in warfare and ritual sacrifice . It was believed that tonalli could be taken from a human body and either offered to the gods as a form of debt payment or acquired by the ritual person who touched the physical entity in which they resided. [ 1 ] The concept of tonalli was not only limited to human beings. It infiltrated animals, gods, plants, and objects used in rituals. [ 2 ]
The tonalli also determines the sign under which a person is born and informs fortune , character, and name. Tonalli conveyed astrological signs and names through birthdays, and in the Mexica divinatory system, a person's birthday fell on one of the 260 name days in a special calendar . Individuals followed the path or code of conduct demanded by the tonalli and the day sign. This calendar was notable because it was used solely for divination and celebrating rituals in the deities’ honor. The calendrical name of a given person transmitted a character and fate to both men and women. A person born on the first day of the 260-day cycle, would be named One Crocodile and was given a positive character that would bring about authority, wealth, and fame. [ 13 ] It is important to note that inauspicious day signs could be ameliorated through rituals such as first baths, and life events.
It was believed that individuals possess free will within the constraints imposed by their tonalli. One is born with either favorable or unfavorable tonalli and with a corresponding predetermined character. While this places certain constraints upon what one may accomplish, one freely chooses what to make of one's tonalli within these limits. Someone born with favorable tonalli may squander it through improper action; someone with unfavorable tonalli may neutralize its adverse effects through knowledge of the sacred calendar and careful selection of actions. [ 14 ]
As an animating force within both the human and spiritual world, retaining tonalli provides consciousness and personality. The concept of soul loss was inherent to the ancient Central Mexican understanding of aging and death . The loss of tonalli, in its various stages, is known by states in which the person suffers slowed, impaired, or complete loss of consciousness. As the understanding of tonalli is reliant upon conceptions of warmth, heat, and the sun, the absence of tonalli is felt as fluctuations in internal temperature. The inability to re-implant the life force leads to a decline in health and ultimately death. The modern Nahua and the Mexica have identified the tonalli's departure as the cause of illnesses with the same general and observable symptoms as death and dying. [ 3 ] | https://en.wikipedia.org/wiki/Tonalli |
In mathematics , the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R 2 to be an integrable function . It is often used to establish that Fubini's theorem may be applied to ƒ . It is named for Leonida Tonelli and E. W. Hobson .
More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R 2 , and either of the two iterated integrals
or
is finite, then ƒ is Lebesgue-integrable on R 2 . [ 1 ]
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tonelli–Hobson_test |
The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r 2 ≡ n (mod p ), where p is a prime : that is, to find a square root of n modulo p .
Tonelli–Shanks cannot be used for composite moduli: finding square roots modulo composite numbers is a computational problem equivalent to integer factorization . [ 1 ]
An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli [ 2 ] [ 3 ] in 1891. The version discussed here was developed independently by Daniel Shanks in 1973, who explained:
My tardiness in learning of these historical references was because I had lent Volume 1 of Dickson's History to a friend and it was never returned. [ 4 ]
According to Dickson, [ 3 ] Tonelli's algorithm can take square roots of x modulo prime powers p λ apart from primes.
Given a non-zero n {\displaystyle n} and a prime p > 2 {\displaystyle p>2} (which will always be odd), Euler's criterion tells us that n {\displaystyle n} has a square root (i.e., n {\displaystyle n} is a quadratic residue ) if and only if:
In contrast, if a number z {\displaystyle z} has no square root (is a non-residue), Euler's criterion tells us that:
It is not hard to find such z {\displaystyle z} , because half of the integers between 1 and p − 1 {\displaystyle p-1} have this property. So we assume that we have access to such a non-residue.
By (normally) dividing by 2 repeatedly, we can write p − 1 {\displaystyle p-1} as Q 2 S {\displaystyle Q2^{S}} , where Q {\displaystyle Q} is odd. Note that if we try
then R 2 ≡ n Q + 1 = ( n ) ( n Q ) ( mod p ) {\displaystyle R^{2}\equiv n^{Q+1}=(n)(n^{Q}){\pmod {p}}} . If t ≡ n Q ≡ 1 ( mod p ) {\displaystyle t\equiv n^{Q}\equiv 1{\pmod {p}}} , then R {\displaystyle R} is a square root of n {\displaystyle n} . Otherwise, for M = S {\displaystyle M=S} , we have R {\displaystyle R} and t {\displaystyle t} satisfying:
If, given a choice of R {\displaystyle R} and t {\displaystyle t} for a particular M {\displaystyle M} satisfying the above (where R {\displaystyle R} is not a square root of n {\displaystyle n} ), we can easily calculate another R {\displaystyle R} and t {\displaystyle t} for M − 1 {\displaystyle M-1} such that the above relations hold, then we can repeat this until t {\displaystyle t} becomes a 2 0 {\displaystyle 2^{0}} -th root of 1, i.e., t = 1 {\displaystyle t=1} . At that point R {\displaystyle R} is a square root of n {\displaystyle n} .
We can check whether t {\displaystyle t} is a 2 M − 2 {\displaystyle 2^{M-2}} -th root of 1 by squaring it M − 2 {\displaystyle M-2} times and check whether it is 1. If it is, then we do not need to do anything, as the same choice of R {\displaystyle R} and t {\displaystyle t} works. But if it is not, t 2 M − 2 {\displaystyle t^{2^{M-2}}} must be -1 (because squaring it gives 1, and there can only be two square roots 1 and -1 of 1 modulo p {\displaystyle p} ).
To find a new pair of R {\displaystyle R} and t {\displaystyle t} , we can multiply R {\displaystyle R} by a factor b {\displaystyle b} , to be determined. Then t {\displaystyle t} must be multiplied by a factor b 2 {\displaystyle b^{2}} to keep R 2 ≡ n t ( mod p ) {\displaystyle R^{2}\equiv nt{\pmod {p}}} . So, when t 2 M − 2 {\displaystyle t^{2^{M-2}}} is -1, we need to find a factor b 2 {\displaystyle b^{2}} so that t b 2 {\displaystyle tb^{2}} is a 2 M − 2 {\displaystyle 2^{M-2}} -th root of 1, or equivalently b 2 {\displaystyle b^{2}} is a 2 M − 2 {\displaystyle 2^{M-2}} -th root of -1.
The trick here is to make use of z {\displaystyle z} , the known non-residue. The Euler's criterion applied to z {\displaystyle z} shown above says that z Q {\displaystyle z^{Q}} is a 2 S − 1 {\displaystyle 2^{S-1}} -th root of -1. So by squaring z Q {\displaystyle z^{Q}} repeatedly, we have access to a sequence of 2 i {\displaystyle 2^{i}} -th root of -1. We can select the right one to serve as b {\displaystyle b} . With a little bit of variable maintenance and trivial case compression, the algorithm below emerges naturally.
Operations and comparisons on elements of the multiplicative group of integers modulo p Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } are implicitly mod p .
Inputs :
Outputs :
Algorithm :
Once you have solved the congruence with r the second solution is − r ( mod p ) {\displaystyle -r{\pmod {p}}} . If the least i such that t 2 i = 1 {\displaystyle t^{2^{i}}=1} is M , then no solution to the congruence exists, i.e. n is not a quadratic residue.
This is most useful when p ≡ 1 (mod 4).
For primes such that p ≡ 3 (mod 4), this problem has possible solutions r = ± n p + 1 4 ( mod p ) {\displaystyle r=\pm n^{\frac {p+1}{4}}{\pmod {p}}} . If these satisfy r 2 ≡ n ( mod p ) {\displaystyle r^{2}\equiv n{\pmod {p}}} , they are the only solutions. If not, r 2 ≡ − n ( mod p ) {\displaystyle r^{2}\equiv -n{\pmod {p}}} , n is a quadratic non-residue, and there are no solutions.
We can show that at the start of each iteration of the loop the following loop invariants hold:
Initially:
At each iteration, with M' , c' , t' , R' the new values replacing M , c , t , R :
From t 2 M − 1 = 1 {\displaystyle t^{2^{M-1}}=1} and the test against t = 1 at the start of the loop, we see that we will always find an i in 0 < i < M such that t 2 i = 1 {\displaystyle t^{2^{i}}=1} . M is strictly smaller on each iteration, and thus the algorithm is guaranteed to halt. When we hit the condition t = 1 and halt, the last loop invariant implies that R 2 = n .
We can alternately express the loop invariants using the order of the elements:
Each step of the algorithm moves t into a smaller subgroup by measuring the exact order of t and multiplying it by an element of the same order.
Solving the congruence r 2 ≡ 5 (mod 41). 41 is prime as required and 41 ≡ 1 (mod 4). 5 is a quadratic residue by Euler's criterion: 5 41 − 1 2 = 5 20 = 1 {\displaystyle 5^{\frac {41-1}{2}}=5^{20}=1} (as before, operations in ( Z / 41 Z ) × {\displaystyle (\mathbb {Z} /41\mathbb {Z} )^{\times }} are implicitly mod 41).
Indeed, 28 2 ≡ 5 (mod 41) and (−28) 2 ≡ 13 2 ≡ 5 (mod 41). So the algorithm yields the two solutions to our congruence.
The Tonelli–Shanks algorithm requires (on average over all possible input (quadratic residues and quadratic nonresidues))
modular multiplications, where m {\displaystyle m} is the number of digits in the binary representation of p {\displaystyle p} and k {\displaystyle k} is the number of ones in the binary representation of p {\displaystyle p} . If the required quadratic nonresidue z {\displaystyle z} is to be found by checking if a randomly taken number y {\displaystyle y} is a quadratic nonresidue, it requires (on average) 2 {\displaystyle 2} computations of the Legendre symbol . [ 5 ] The average of two computations of the Legendre symbol are explained as follows: y {\displaystyle y} is a quadratic residue with chance p + 1 2 p = 1 + 1 p 2 {\displaystyle {\tfrac {\tfrac {p+1}{2}}{p}}={\tfrac {1+{\tfrac {1}{p}}}{2}}} , which is smaller than 1 {\displaystyle 1} but ≥ 1 2 {\displaystyle \geq {\tfrac {1}{2}}} , so we will on average need to check if a y {\displaystyle y} is a quadratic residue two times.
This shows essentially that the Tonelli–Shanks algorithm works very well if the modulus p {\displaystyle p} is random, that is, if S {\displaystyle S} is not particularly large with respect to the number of digits in the binary representation of p {\displaystyle p} . As written above, Cipolla's algorithm works better than Tonelli–Shanks if (and only if) S ( S − 1 ) > 8 m + 20 {\displaystyle S(S-1)>8m+20} .
However, if one instead uses Sutherland's algorithm to perform the discrete logarithm computation in the 2-Sylow subgroup of F p ∗ {\displaystyle \mathbb {F} _{p}^{\ast }} , one may replace S ( S − 1 ) {\displaystyle S(S-1)} with an expression that is asymptotically bounded by O ( S log S / log log S ) {\displaystyle O(S\log S/\log \log S)} . [ 6 ] Explicitly, one computes e {\displaystyle e} such that c e ≡ n Q {\displaystyle c^{e}\equiv n^{Q}} and then R ≡ c − e / 2 n ( Q + 1 ) / 2 {\displaystyle R\equiv c^{-e/2}n^{(Q+1)/2}} satisfies R 2 ≡ n {\displaystyle R^{2}\equiv n} (note that e {\displaystyle e} is a multiple of 2 because n {\displaystyle n} is a quadratic residue).
The algorithm requires us to find a quadratic nonresidue z {\displaystyle z} . There is no known deterministic algorithm that runs in polynomial time for finding such a z {\displaystyle z} . However, if the generalized Riemann hypothesis is true, there exists a quadratic nonresidue z < 2 ln 2 p {\displaystyle z<2\ln ^{2}{p}} , [ 7 ] making it possible to check every z {\displaystyle z} up to that limit and find a suitable z {\displaystyle z} within polynomial time . Keep in mind, however, that this is a worst-case scenario; in general, z {\displaystyle z} is found in on average 2 trials as stated above.
The Tonelli–Shanks algorithm can (naturally) be used for any process in which square roots modulo a prime are necessary. For example, it can be used for finding points on elliptic curves . It is also useful for the computations in the Rabin cryptosystem and in the sieving step of the quadratic sieve .
Tonelli–Shanks can be generalized to any cyclic group (instead of ( Z / p Z ) × {\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{\times }} ) and to k th roots for arbitrary integer k , in particular to taking the k th root of an element of a finite field . [ 8 ]
If many square-roots must be done in the same cyclic group and S is not too large, a table of square-roots of the elements of 2-power order can be prepared in advance and the algorithm simplified and sped up as follows.
According to Dickson's "Theory of Numbers" [ 3 ]
A. Tonelli [ 9 ] gave an explicit formula for the roots of x 2 = c ( mod p λ ) {\displaystyle x^{2}=c{\pmod {p^{\lambda }}}} [ 3 ]
The Dickson reference shows the following formula for the square root of x 2 mod p λ {\displaystyle x^{2}{\bmod {p^{\lambda }}}} .
Noting that 23 2 mod 29 3 ≡ 529 {\displaystyle 23^{2}{\bmod {29^{3}}}\equiv 529} and noting that β = 7 ⋅ 29 2 {\displaystyle \beta =7\cdot 29^{2}} then
To take another example: 2333 2 mod 29 3 ≡ 4142 {\displaystyle 2333^{2}{\bmod {29^{3}}}\equiv 4142} and
Dickson also attributes the following equation to Tonelli:
Using p = 23 {\displaystyle p=23} and using the modulus of p 3 {\displaystyle p^{3}} the math follows:
First, find the modular square root mod p {\displaystyle p} which can be done by the regular Tonelli algorithm for one or the other roots:
And applying Tonelli's equation (see above):
Dickson's reference [ 3 ] clearly shows that Tonelli's algorithm works on moduli of p λ {\displaystyle p^{\lambda }} . | https://en.wikipedia.org/wiki/Tonelli–Shanks_algorithm |
Tong Dizhou ( Chinese : 童第周 ; May 28, 1902 – March 30, 1979) was a Chinese embryologist known for his contributions to the field of cloning . He was a vice president of Chinese Academy of Sciences .
Born in Yinxian , Zhejiang province, Tong graduated from Fudan University in 1924 with a degree in biology, and received a PhD in zoology in 1930 from Free University Brussels .
In 1963, Tong inserted DNA of a male carp into the egg of a female carp and became the first to successfully clone a fish. [ 1 ] He is regarded as "the father of China's clone".
Tong was also an academician at the Chinese Academy of Sciences and the first director of its Institute of Oceanology from its founding in 1950 until 1978.
Tong died on 30 March 1979 at Beijing Hospital in Beijing.
This article about a biologist is a stub . You can help Wikipedia by expanding it .
This biographical article about a Chinese scientist is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tong_Dizhou |
Tonic in physiology refers to a physiological response which is slow and may be graded. This term is typically used in opposition to a fast response. For instance, tonic muscles are contrasted by the more typical and much faster twitch muscles, while tonic sensory nerve endings are contrasted to the much faster phasic sensory nerve endings.
Tonic muscles are much slower than twitch fibers in terms of time from stimulus to full activation, time to full relaxation upon cessation of stimuli, and maximal shortening velocity. [ 1 ] These muscles are rarely found in mammals (only in the muscles moving the eye and in the middle ear), but are common in reptiles and amphibians. [ 1 ]
Tonic receptors adapt slowly to a stimulus [ 2 ] and continues to produce action potentials over the duration of the stimulus. [ 3 ] In this way it conveys information about the duration of the stimulus. In contrast, phasic receptors adapt rapidly to a stimulus. The response of the cell diminishes very quickly and then stops. [ 2 ] It does not provide information on the duration of the stimulus; [ 3 ] instead some of them convey information on rapid changes in stimulus intensity and rate. [ 4 ] Examples of tonic receptors are pain receptors , the joint capsule , muscle spindle , [ 3 ] and the Ruffini corpuscle .
This medical article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tonic_(physiology) |
In chemical biology , tonicity is a measure of the effective osmotic pressure gradient; the water potential of two solutions separated by a partially-permeable cell membrane . Tonicity depends on the relative concentration of selective membrane-impermeable solutes across a cell membrane which determine the direction and extent of osmotic flux . It is commonly used when describing the swelling-versus-shrinking response of cells immersed in an external solution.
Unlike osmotic pressure, tonicity is influenced only by solutes that cannot cross the membrane, as only these exert an effective osmotic pressure. Solutes able to freely cross the membrane do not affect tonicity because they will always equilibrate with equal concentrations on both sides of the membrane without net solvent movement. It is also a factor affecting imbibition .
There are three classifications of tonicity that one solution can have relative to another: hypertonic , hypotonic , and isotonic . [ 1 ] A hypotonic solution example is distilled water.
A hypertonic solution has a greater concentration of non-permeating solutes than another solution. [ 2 ] In biology, the tonicity of a solution usually refers to its solute concentration relative to that of another solution on the opposite side of a cell membrane ; a solution outside of a cell is called hypertonic if it has a greater concentration of solutes than the cytosol inside the cell. When a cell is immersed in a hypertonic solution, osmotic pressure tends to force water to flow out of the cell in order to balance the concentrations of the solutes on either side of the cell membrane. The cytosol is conversely categorized as hypotonic, opposite of the outer solution. [ 3 ] [ 4 ]
When plant cells are in a hypertonic solution, the flexible cell membrane pulls away from the rigid cell wall , but remains joined to the cell wall at points called plasmodesmata . The cells often take on the appearance of a pincushion , and the plasmodesmata almost cease to function because they become constricted, a condition known as plasmolysis . In plant cells the terms isotonic, hypotonic and hypertonic cannot strictly be used accurately because the pressure exerted by the cell wall significantly affects the osmotic equilibrium point. [ 5 ]
Some organisms have evolved intricate methods of circumventing hypertonicity. For example, saltwater is hypertonic to the fish that live in it. Because the fish need a large surface area in their gills in contact with seawater for gas exchange , they lose water osmotically to the sea from gill cells. They respond to the loss by drinking large amounts of saltwater, and actively excreting the excess salt. [ 6 ] This process is called osmoregulation . [ 7 ]
A hypotonic solution has a lower concentration of solutes than another solution. In biology, a solution outside of a cell is called hypotonic if it has a lower concentration of solutes relative to the cytosol . Due to osmotic pressure , water diffuses into the cell, and the cell often appears turgid , or bloated.
For cells without a cell wall such as animal cells, if the gradient is large enough, the uptake of excess water can produce enough pressure to induce cytolysis , or rupturing of the cell.
When plant cells are in a hypotonic solution, the central vacuole takes on extra water and pushes the cell membrane against the cell wall. Due to the rigidity of the cell wall, it pushes back, preventing the cell from bursting. This is called turgor pressure . [ 8 ]
A solution is isotonic when its effective osmole concentration is the same as that of another solution. In biology, the solutions on either side of a cell membrane are isotonic if the concentration of solutes outside the cell is equal to the concentration of solutes inside the cell. In this case the cell neither swells nor shrinks because there is no concentration gradient to induce the diffusion of large amounts of water across the cell membrane. Water molecules freely diffuse through the plasma membrane in both directions, and as the rate of water diffusion is the same in each direction, the cell will neither gain nor lose water.
An iso-osmolar solution can be hypotonic if the solute is able to penetrate the cell membrane. For example, an iso-osmolar urea solution is hypotonic to red blood cells, causing their lysis . This is due to urea entering the cell down its concentration gradient, followed by water. The osmolarity of normal saline , 9 grams NaCl dissolved in water to a total volume of one liter, is a close approximation to the osmolarity of NaCl in blood (about 290 mOsm / L ). Thus, normal saline is almost isotonic to blood plasma. Neither sodium nor chloride ions can freely pass through the plasma membrane, unlike urea . | https://en.wikipedia.org/wiki/Tonicity |
In physics , a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the system's physics. It is named after physicists Lewi Tonks , who developed a classical model in 1936, and Marvin D. Girardeau who generalized it to the quantum regime. [ 1 ] It is not a Bose–Einstein condensate as it does not demonstrate any of the necessary characteristics, such as off-diagonal long-range order or a unitary two-body correlation function , even in a thermodynamic limit and as such cannot be described by a macroscopically occupied orbital (order parameter) in the Gross–Pitaevskii formulation.
The Tonks–Girardeau gas is a particular case of the Lieb–Liniger model . [ 1 ]
A row of bosons all confined to a one-dimensional line cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a traffic jam : the motion of each boson is strongly correlated with that of its two neighbors. This can be thought of as the large- c limit of the delta Bose gas .
Because the particles cannot exchange places, their behavior might be expected to be fermionic , but their behavior differs from that of fermions in several important ways: the particles can all occupy the same momentum state, which corresponds to neither Bose-Einstein nor Fermi–Dirac statistics . This is the phenomenon of bosonization which happens in 1+1 dimensions.
In the case of a Tonks–Girardeau gas (TG), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the ' fermionization ' of bosons. Tonks–Girardeau gas matches quantum Nonlinear Schrödinger equation for infinite repulsion, which can be efficiently analyzed by quantum inverse scattering method . This relation helps to study correlation functions. The correlation functions can be described by an Integrable system . In a simple case, it is a Painlevé transcendent . The quantum correlation functions of a Tonks–Girardeau gas can be described by means of classical, completely integrable, differential equations. [ 2 ] Thermodynamics of Tonks–Girardeau gas was described by Chen Ning Yang .
The first example of TGs came in 2004 when Paredes and coworkers created an array of such gases using an optical lattice . [ 3 ] In a different experiment, Kinoshita and coworkers observed a strongly correlated 1D Tonks–Girardeau gas. [ 4 ]
The optical lattice is formed by six intersecting laser beams, which generate an interference pattern. The beams are arranged as standing waves along three orthogonal directions. This results in an array of optical dipole traps where atoms are stored in the intensity maxima of the interference pattern.
The researchers loaded ultracold rubidium atoms into one-dimensional tubes formed by a two-dimensional lattice (the third standing wave is initially off). This lattice is strong so that the atoms have insufficient energy to tunnel between neighboring tubes. The interaction is too low for the transition to the TG regime. For that, the third axis of the lattice is used. It is set to a lower intensity and shorter time than the other two, so that tunneling in this direction is possible. For increasing intensity of the third lattice, atoms in the same lattice well are more and more tightly trapped, which increases the collisional energy. When the collisional energy becomes much bigger than the tunneling energy, the atoms can still tunnel into empty lattice wells, but not into or across occupied ones.
This technique has been used by other researchers to obtain an array of one-dimensional Bose gases in the Tonks-Girardeau regime. However, the fact that an array of gases is observed only allows the measurement of averaged quantities. Moreover, the temperatures and chemical potential between the different tubes are dispersed, which wash out many effects. For instance, this configuration does not allow probing of system fluctuations. Thus it proved interesting to produce a single Tonks–Girardeau gas. In 2011 one team created a single one-dimensional TG gas by trapping rubidium atoms magnetically in the vicinity of a microstructure. Thibaut Jacqmin et al. measured density fluctuations in that single strongly interacting gas. Those fluctuations proved to be sub-Poissonian , as expected for a Fermi gas . [ 5 ] | https://en.wikipedia.org/wiki/Tonks–Girardeau_gas |
Tonmeister is a job description in the music and recording industries that describes a so-called "sound master" (a literal translation of the German Tonmeister ): a person who creates recordings or broadcasts of music who is also both musically trained (in classical and non-classical genres) and has theoretical and practical knowledge.
The word tonmeister was trademarked in 1996 by the University of Surrey , United Kingdom. Also within the UK, the SAE Institute registered the term SAE Tonmeister . The title has been abbreviated to tonmeister in their registrations in several other countries, not including Germany, Switzerland or Austria. Members of the VDT may call themselves Tonmeister VDT .
The concept of a tonmeister dates back to 1946, [ 1 ] when Arnold Schoenberg wrote a letter to the Chancellor of the University of Chicago suggesting a course to train "soundmen". Schoenberg wrote, "soundmen will be trained in music, acoustics, physics, mechanics and related fields to a degree enabling them to control and improve the sonority of recordings, radio broadcasts and sound films". [ 2 ] It was also in this year that the University of Music Detmold in Germany started the first Tonmeister course. [ 3 ] | https://en.wikipedia.org/wiki/Tonmeister |
In musical tuning and harmony , the Tonnetz (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. [ 1 ] Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.
The Tonnetz originally appeared in Leonhard Euler 's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae . Euler's Tonnetz , pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a major third above F). Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst , discusses the relationships between keys, presenting them in a network analogous to Euler's Tonnetz , but showing keys rather than notes. [ 2 ] The Tonnetz itself was rediscovered in 1858 by Ernst Naumann in his Harmoniesystem in dualer Entwickelung ., [ 3 ] and was disseminated in an 1866 treatise of Arthur von Oettingen . Oettingen and the influential musicologist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann ) explored the capacity of the space to chart harmonic modulation between chords and motion between keys. Similar understandings of the Tonnetz appeared in the work of many late-19th century German music theorists. [ 4 ]
Oettingen and Riemann both conceived of the relationships in the chart being defined through just intonation , which uses pure intervals. One can extend out one of the horizontal rows of the Tonnetz indefinitely, to form a never-ending sequence of perfect fifths: F-C-G-D-A-E-B-F♯-C♯-G♯-D♯-A♯-E♯-B♯-F𝄪-C𝄪-G𝄪- (etc.) Starting with F, after 12 perfect fifths, one reaches E♯. Perfect fifths in just intonation are slightly larger than the compromised fifths used in equal temperament tuning systems more common in the present. This means that when one stacks 12 fifths starting from F, the E♯ we arrive at will not be seven octaves above the F we started with. Oettingen and Riemann's Tonnetz thus extended on infinitely in every direction without actually repeating any pitches. In the twentieth century, composer-theorists such as Ben Johnston and James Tenney continued to developed theories and applications involving just-intoned Tonnetze .
The appeal of the Tonnetz to 19th-century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C♯-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E major (E-G♯-B) is diagonally across the E vertex, and shares no other vertices. One important point is that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared vertices, the more shared pitches the chord will have. This provides a visualization of the principle of parsimonious voice-leading, in which motions between chords are considered smoother when fewer pitches change. This principle is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships. [ 4 ]
Neo-Riemannian music theorists David Lewin and Brian Hyer revived the Tonnetz to further explore properties of pitch structures. [ 4 ] Modern music theorists generally construct the Tonnetz in equal temperament [ 4 ] and without distinction between octave transpositions of a pitch (i.e., using pitch classes ). Under equal temperament, the never-ending series of ascending fifths mentioned earlier becomes a cycle. Neo-Riemannian theorists typically assume enharmonic equivalence (in other words, A♭ = G♯), and so the two-dimensional plane of the 19th-century Tonnetz cycles in on itself in two different directions, and is mathematically isomorphic to a torus .
Neo-Riemannian theorists have also used the Tonnetz to visualize non-tonal triadic relationships. For example, the diagonal going up and to the left from C in the diagram at the beginning of the article forms a division of the octave in three major thirds : C-A♭-E-C (the E is actually an F♭, and the final C a D♭♭). Richard Cohn argues that while a sequence of triads built on these three pitches (C major, A♭ major, and E major) cannot be adequately described using traditional concepts of functional harmony, this cycle has smooth voice leading and other important group properties which can be easily observed on the Tonnetz . [ 5 ]
The harmonic table note layout is a note layout that is topologically equivalent to the Tonnetz , and is used on several music instruments that allow playing major and minor chords with a single finger.
The Tonnetz can be overlayed on the Wicki–Hayden note layout , where the major second can be found half way the major third.
The Tonnetz is the dual graph of Schoenberg 's chart of the regions , [ 6 ] and of course vice versa . Research into music cognition has demonstrated that the human brain uses a "chart of the regions" to process tonal relationships. [ 7 ] | https://en.wikipedia.org/wiki/Tonnetz |
Anthony Frank Marchington (2 December 1955 [ 1 ] – 16 October 2011 [ 2 ] ) was an English biotechnology entrepreneur and businessman, famous as the co-founder of Oxford Molecular, and the former owner of the famous Class A3 4472 Flying Scotsman locomotive. [ 3 ]
Born in Buxton , Derbyshire , [ 1 ] he was brought up on the family farm in Buxworth . He passed his motorcycle test at the age of 16, having learned to ride his father's 1914 Bradbury motorcycle and sidecar combination. [ 4 ] He attended New Mills Grammar School .
He gained his bachelor's degree, master's and D.Phil. at Brasenose College, Oxford . [ 4 ]
While at Oxford, Marchington befriended and later lodged with American Walter Hooper , the last personal secretary of the writer C. S. Lewis . Through this relationship Marchington shared a lectern with Hooper in 1975 in North Carolina , [ 5 ] co-wrote the script of Through Joy and Beyond (the 1977 documentary life of Lewis [ 6 ] ), and created the Lewis bonfire hoax letter, sent to Christianity and Literature in 1978. [ 7 ]
Marchington began his career as a product manager with ICI Agrochemicals in 1983, becoming marketing manager for South America in 1986. [ 1 ]
In 1988, he started several companies in the areas of intellectual property, drug discovery and biotechnology . [ 6 ] As these expanded, in the same year, under his tutor Professor Graham Richards , Marchington co-founded Oxford Molecular Ltd. (later to become Oxford Molecular Plc.). Worth £450 million at its peak, it was eventually sold for £70million. [ 8 ]
A former member of the Department of Trade and Industry 's Competitiveness Advisory Group, from 2000 Marchington's entrepreneurial activities included: running Marchington Consulting, based at the Sheffield Bioincubator ; [ 9 ] CEO at Savyon Diagnostics; and co-founded, as chairman and director, Venture Hothouse Ltd. [ 10 ] From 2010, Marchington was CEO at Oxford Medical Diagnostics. [ 11 ]
Marchington was made a Freeman of the City of London in 1997 [ 1 ] and was an honorary fellow of St Edmund Hall, Oxford .
A steam fan from a young age, aged 22 Marchington bought his first steamroller from haulage contractor and scrap dealer Ted Eansworth in Chesterfield . The collection that he started with his father eventually became the Buxworth Steam Group, which comprised a full working Victorian fair , and raised revenue through offering them for hire: [ 4 ] [ 12 ]
Marchington's Buxworth Steam Group was the star of the 1985 BBC documentary 'A Gambol on Steam', which featured his first steam rally in the group, hosted at Lyme Park , and featured exhibits from names such as Fred Dibnah in addition to his current collection of a 1904 Fowler D2 steamroller and his pair of Fowler BB1 ploughing engines ('Fame' and 'Fortune') and was one of the largest rallies of its time. [ 14 ]
In 1996, Marchington bought the famous LNER steam locomotive Class A3 4472 Flying Scotsman at a cost of £1.5M. After a three-year restoration which cost an additional £1M, she returned to steam in 1999. [ 15 ] She made an appearance on Peak Rail in summer 2000, together with most of the Buxworth Steam Group collection. [ 12 ]
In 1997, Marchington purchased LNER Class A4 4464 Bittern from the family of Geoff Drury, which he also based at the Southall Railway Centre . However, after the completion of the £1 million over-budget restoration of Flying Scotsman was complete, he sold Bittern in 2000 to Jeremy Hosking , who moved her to the Mid-Hants Railway in Hampshire in January 2001, for a major restoration. [ 16 ] Despite this, the ownership of both Bittern and Flying Scotsman meant that he is still the only ever private owner to own two Gresley Pacific class locomotives.
With Flying Scotsman's regular use on the VSOE Pullman , in 2002, Marchington proposed a business plan, which included the construction of a 'Flying Scotsman Village' in Edinburgh , to create revenue from associated branding. After floating on OFEX as 'Flying Scotsman Plc.' in the same year, [ 6 ] in 2003 Edinburgh City Council turned down the village plans, and in September 2003 Marchington was declared bankrupt. [ 17 ] This resulted in the sale of most of the assets of the Buxworth Steam Group, including The Iron Maiden to Graeme Atkinson, who displays the engine alongside a collection of other engines and fair organs as part of the Scarborough Fair Collection , at his holiday park in Lebberston , near Scarborough, North Yorkshire . [ 18 ]
At the company's AGM in October 2003, CEO Peter Butler announced losses of £474,619, and with a £1.5M overdraft at Barclays Bank , stated that the company only had enough cash to trade until April 2004. The company's shares were suspended from OFEX on 3 November 2003, after it failed to declare interim results. [ 17 ]
With the locomotive effectively placed up for sale, after a high-profile national campaign it was bought in April 2004 by the National Railway Museum in York , [ 19 ] and it is now part of the National Collection.
Marchington's time with the Flying Scotsman was documented in the Channel 4 documentaries A Steamy Affair: The Story of Flying Scotsman , directed by former Blue Peter presenter Simon Groom . [ 6 ]
Marchington met his second wife Caroline after he and his father offered her a lift on their steam engine to the Devonshire Arms, Peak Forest, the local public house on the A6 road , where they were staying that night. [ 4 ] The couple had two children, and family homes in Buxton, Derbyshire and Oxfordshire. [ 12 ] He also had two children from a previous marriage. His passion for vintage restoration continued with his family for the rest of his life, even after the sagas of Buxworth Steam Group and Flying Scotsman .
After meeting Jim Daniel, the Grand Secretary of the United Grand Lodge of England , at a dinner of Brasenose College members, Marchington was initiated as a Freemason at Oxford-based Apollo University Lodge number 357 in January 1991. He was passed and raised the following year, and went into the chair in November 1996. Marchington celebrated his installation as Worshipful Master of the Lodge with the commissioning of a set of limited edition glass tankards, engraved with the square and compasses on one side and the Flying Scotsman on the other.
Marchington joined a number of other Masonic Orders, including the Holy Royal Arch , the Order of the Red Cross of Constantine , the Order of Mark Master Masons , and the Royal Ark Mariners. He was appointed a Provincial Grand Steward for Oxfordshire in 1997, and became Oxfordshire's Assistant Provincial Grand Master in 1998. [ 4 ]
After a long period of treatment, Marchington died of cancer at Buxton's Cottage Hospital on 16 October 2011. [ 2 ] | https://en.wikipedia.org/wiki/Tony_Marchington |
Anthony Frederick Orchard (13 March 1941 – 19 August 2005) was a pioneer of inorganic chemistry . [ 1 ] [ 2 ] His research contributed to laying the foundations of much modern consumer electronic technology . [ 3 ]
Tony Orchard was born in Carmarthen , Wales , and moved to Swansea . He studied Chemistry first at Wadham College, Oxford as an undergraduate and then towards a DPhil doctoral degree in theoretical inorganic chemistry at Merton College, Oxford . He left Merton College before he had completed his doctorate at the age of 26 to become a Fellow in Inorganic Chemistry at University College in Oxford . [ 4 ] He stayed at University College until his death.
During the 1970s, Orchard led a group of researchers working in the area of photoelectron spectroscopy . This enabled scientists to examine the electronic structure of materials. The research was important for technological innovations in modern electronics , helping with the development of advances such as the personal computer and mobile phone . He published the book Magnetochemistry in 2003. [ 5 ]
As well as his research contributions, Orchard also helped to improve the system of undergraduate applications for chemistry at Oxford University . [ 2 ]
Tony Orchard was an amateur sportsman, playing tennis and snooker . [ 2 ] At an early age, he won snooker games with the later world champions Terry Griffiths and Ray Reardon .
Orchard's friends included former United States president Bill Clinton , who he met during the 1960s when Clinton was studying at University College as a Rhodes Scholar .
Orchard was married to his wife Jeanne and later divorced, with two sons and two daughters. He died aged 64 of colon cancer . | https://en.wikipedia.org/wiki/Tony_Orchard |
Professor Anthony Peter Francis Turner , FRSC , usually known as Tony Turner , is a British academic specialising in the fields of biosensors and bioelectronics .
Professor Anthony (Tony) Turner is an Emeritus Professor of Cranfield University in England , where he was previously the Distinguished Professor of Biotechnology (until 2010) and Principal of Cranfield University at Silsoe . He remained Innovations Director for Cranfield Ventures Ltd (until 2014), with responsibility for licensing and spin offs from Cranfield University. He joined Linköping University (Sweden) in 2010, to help re-establish the university in the field of Biosensors and Bioelectronics, but retired the end of 2018 to focus on translational research and technology transfer in the Skåne region of Southern Sweden. He finally retired completely from working life in July 2021.
In 1996, he was elected Fellow of the Royal Society of Chemistry , which awarded him the Theophilus Redwood Medal in 2011 for his outstanding contribution to analytical science and especially for "his pioneering work which has led to the development of home blood glucose monitoring technology". [ 1 ] He received a higher doctorate from the University of Kent at Canterbury in 2001 for his "outstanding contribution to biosensors" and was presented with an honorary doctorate for his "contribution to higher education" by Bedfordshire University in 2008. He was elected as a Foreign Associate of the National Academy of Engineering in the USA, in 2006, for his exceptional contribution to "glucose sensors, environmental monitors and synthetic recognition molecules" and elected to the Royal Swedish Academy of Engineering Sciences (IVA) in 2013. In 2016, he was awarded the Ukraine’s highest academic distinction, the Vernadsky Gold Medal from the National Academy of Sciences of Ukraine , for his "outstanding achievements in the field of bioelectronics", and in the same year, the Datta Medal by the Federation of European Biochemical Societies .
He helped found the UK’s first Biotechnology Centre in 1981 at Cranfield University . There he led the Biosensors Group that went on to develop, in collaboration with Oxford University , a biosensor that transformed the lives of people with diabetes . The principle they established, of using a mediator in a disposable electrochemical glucose biosensor, became the technology of choice for this US$15 billion a year industry. He played a major role in consolidating the field of biosensors as an academic discipline by co-founding the Journal of Biosensors (Elsevier) in 1985 (renamed the Journal of Biosensors and Bioelectronics in 1991), publishing the first text book on Biosensors in 1987 [ 2 ] and establishing the World Congress on Biosensors in 1990. He edited Biosensors and Bioelectronics until 2019 (and remains Founding Editor-In-Chief) and was Executive Chair of the World Congress until 2021. His team has been ranked in the top ten in the world for biosensor development by a variety of international commentators. In academic circles, his name is synonymous with the field of Biosensors, but he has also specialised in leveraging IP and driven numerous biosensor start-ups over the past four decades.
Professor Turner has over 750 publications and patents in the field of biosensors and biomimetic sensors and a G.S. (2024) h-index of 98, with his most popular paper receiving over 2,700 citations. [ 3 ]
Personal page @Cranfield University == External links== | https://en.wikipedia.org/wiki/Tony_Turner_(scientist) |
A tool is an object that can extend an individual's ability to modify features of the surrounding environment or help them accomplish a particular task. Although many animals use simple tools , only human beings, whose use of stone tools dates back hundreds of millennia, have been observed using tools to make other tools.
Early human tools, made of such materials as stone , bone , and wood , were used for the preparation of food , hunting , the manufacture of weapons , and the working of materials to produce clothing and useful artifacts and crafts such as pottery , along with the construction of housing , businesses , infrastructure , and transportation . The development of metalworking made additional types of tools possible. Harnessing energy sources , such as animal power , wind , or steam , allowed increasingly complex tools to produce an even larger range of items, with the Industrial Revolution marking an inflection point in the use of tools. The introduction of widespread automation in the 19th and 20th centuries allowed tools to operate with minimal human supervision, further increasing the productivity of human labor .
By extension, concepts that support systematic or investigative thought are often referred to as "tools" or "toolkits".
While a common-sense understanding of the meaning of tool is widespread, several formal definitions have been proposed.
In 1981, Benjamin Beck published a widely used definition of tool use. [ 1 ] This has been modified to:
The external employment of an unattached or manipulable attached environmental object to alter more efficiently the form, position, or condition of another object, another organism, or the user itself, when the user holds and directly manipulates the tool during or prior to use and is responsible for the proper and effective orientation of the tool. [ 2 ]
Other, briefer definitions have been proposed:
An object carried or maintained for future use.
The use of physical objects other than the animal's own body or appendages as a means to extend the physical influence realized by the animal.
An object that has been modified to fit a purpose ... [or] An inanimate object that one uses or modifies in some way to cause a change in the environment, thereby facilitating one's achievement of a target goal.
Anthropologists believe that the use of tools was an important step in the evolution of mankind . [ 6 ] Because tools are used extensively by both humans (Homo sapiens) and wild chimpanzees , it is widely assumed that the first routine use of tools took place prior to the divergence between the two ape species. [ 7 ] These early tools, however, were likely made of perishable materials such as sticks, or consisted of unmodified stones that cannot be distinguished from other stones as tools.
Stone artifacts date back to about 2.5 million years ago. [ 8 ] However, a 2010 study suggests the hominin species Australopithecus afarensis ate meat by carving animal carcasses with stone implements. This finding pushes back the earliest known use of stone tools among hominins to about 3.4 million years ago. [ 9 ] Finds of actual tools date back at least 2.6 million years in Ethiopia . [ 10 ] One of the earliest distinguishable stone tool forms is the hand axe .
Up until recently, weapons found in digs were the only tools of "early man" that were studied and given importance. Now, more tools are recognized as culturally and historically relevant. As well as hunting, other activities required tools such as preparing food, "...nutting, leatherworking , grain harvesting and woodworking..." [ 11 ] Included in this group are "flake stone tools".
Tools are the most important items that the ancient humans used to climb to the top of the food chain ; by inventing tools, they were able to accomplish tasks that human bodies could not, such as using a spear or bow to kill prey , since their teeth were not sharp enough to pierce many animals' skins. "Man the hunter" as the catalyst for Hominin change has been questioned. Based on marks on the bones at archaeological sites, it is now more evident that pre-humans were scavenging off of other predators' carcasses rather than killing their own food. [ 12 ]
Many tools were made in prehistory or in the early centuries of recorded history, but archaeological evidence can provide dates of development and use. [ 13 ] [ 14 ] [ 15 ]
Several of the six classic simple machines ( wheel and axle , lever , pulley , inclined plane , wedge , and screw ) were invented in Mesopotamia . [ 16 ] The wheel and axle mechanism first appeared with the potter's wheel , invented in what is now Iraq during the 5th millennium BC. [ 17 ] This led to the invention of the wheeled vehicle in Mesopotamia during the early 4th millennium BC. [ 18 ] The lever was used in the shadoof water-lifting device, the first crane machine, which appeared in Mesopotamia c. 3000 BC , [ 19 ] and then in ancient Egyptian technology c. 2000 BC . [ 20 ] The earliest evidence of pulleys date back to Mesopotamia in the early 2nd millennium BC. [ 21 ]
The screw , the last of the simple machines to be invented, [ 22 ] first appeared in Mesopotamia during the Neo-Assyrian period (911–609 BC). [ 21 ] The Assyrian King Sennacherib (704–681 BC) claims to have invented automatic sluices and to have been the first to use water screw pumps , of up to 30 tons weight, which were cast using two-part clay molds rather than by the ' lost wax ' process. [ 23 ] The Jerwan Aqueduct ( c. 688 BC) is made with stone arches and lined with waterproof concrete. [ 24 ] The earliest evidence of water wheels and watermills date back to the ancient Near East in the 4th century BC, [ 25 ] specifically in the Persian Empire before 350 BC, in the regions of Mesopotamia (Iraq) and Persia (Iran). [ 26 ] This pioneering use of water power constituted perhaps the first use of mechanical energy . [ 27 ]
Mechanical devices experienced a major expansion in their use in Ancient Greece and Ancient Rome with the systematic employment of new energy sources, especially waterwheels . Their use expanded through the Dark Ages with the addition of windmills .
Machine tools occasioned a surge in producing new tools in the Industrial Revolution . Pre-industrial machinery was built by various craftsmen— millwrights built water and windmills, carpenters made wooden framing, and smiths and turners made metal parts. Wooden components had the disadvantage of changing dimensions with temperature and humidity, and the various joints tended to rack (work loose) over time. As the Industrial Revolution progressed, machines with metal parts and frames became more common. [ 28 ] [ 29 ]
Other important uses of metal parts were in firearms and threaded fasteners, such as machine screws, bolts, and nuts. There was also the need for precision in making parts. Precision would allow better working machinery, interchangeability of parts, and standardization of threaded fasteners. The demand for metal parts led to the development of several machine tools . They have their origins in the tools developed in the 18th century by makers of clocks and watches and scientific instrument makers to enable them to batch-produce small mechanisms. Before the advent of machine tools, metal was worked manually using the basic hand tools of hammers, files, scrapers, saws, and chisels. Consequently, the use of metal machine parts was kept to a minimum. Hand methods of production were very laborious and costly and precision was difficult to achieve. [ 30 ] [ 31 ] With their inherent precision, machine tools enabled the economical production of interchangeable parts . [ 28 ] [ 29 ] [ 32 ]
Examples of machine tools include: [ 28 ] [ 29 ]
Advocates of nanotechnology expect a similar surge as tools become microscopic in size. [ 33 ] [ 34 ]
One can classify tools according to their basic functions:
Some tools may be combinations of other tools. An alarm-clock is for example a combination of a measuring tool (the clock) and a perception tool (the alarm). This enables the alarm-clock to be a tool that falls outside of all the categories mentioned above.
There is some debate on whether to consider protective gear items as tools, because they do not directly help perform work, just protect the worker like ordinary clothing. They do meet the general definition of tools and in many cases are necessary for the completion of the work. Personal protective equipment includes such items as gloves , safety glasses , ear defenders and biohazard suits. [ 38 ]
Often, by design or coincidence, a tool may share key functional attributes with one or more other tools. In this case, some tools can substitute for other tools, either as a makeshift solution or as a matter of practical efficiency. "One tool does it all" is a motto of some importance for workers who cannot practically carry every specialized tool to the location of every work task, such as a carpenter who does not necessarily work in a shop all day and needs to do jobs in a customer's house. Tool substitution may be divided broadly into two classes: substitution "by-design", or "multi-purpose", and substitution as makeshift. Substitution "by-design" would be tools that are designed specifically to accomplish multiple tasks using only that one tool.
Substitution is "makeshift" when human ingenuity comes into play and a tool is used for an unintended purpose, such as using a long screwdriver to separate a cars control arm from a ball joint, instead of using a tuning fork. In many cases, the designed secondary functions of tools are not widely known. For example, many wood-cutting hand saws integrate a square by incorporating a specially-shaped handle, that allows 90° and 45° angles to be marked by aligning the appropriate part of the handle with an edge, and scribing along the back edge of the saw. The latter is illustrated by the saying "All tools can be used as hammers". Nearly all tools can be used to function as a hammer, [ 39 ] even though few tools are intentionally designed for it and even fewer work as well as the original.
Tools are often used to substitute for many mechanical apparatuses, especially in older mechanical devices. In many cases a cheap tool could be used to occupy the place of a missing mechanical part. A window roller in a car could be replaced with pliers . A transmission shifter or ignition switch would be able to be replaced with a screwdriver. Again, these would be considered tools that are being used for their unintended purposes, substitution as makeshift. Tools such as a rotary tool would be considered the substitution "by-design", or "multi-purpose". This class of tools allows the use of one tool that has at least two different capabilities. "Multi-purpose" tools are basically multiple tools in one device/tool. Tools such as this are often power tools that come with many different attachments like a rotary tool does, so one could say that a power drill is a "multi-purpose" tool. [ 40 ]
A multi-tool is a hand tool that incorporates several tools into a single, portable device; the Swiss Army knife represents one of the earliest examples. [ 41 ] Other tools have a primary purpose but also incorporate other functionality – for example, lineman's pliers incorporate a gripper and cutter and are often used as a hammer; [ 39 ] and some hand saws incorporate a square in the right-angle between the blade's dull edge and the saw's handle. This would also be the category of "multi-purpose" tools, since they are also multiple tools in one (multi-use and multi-purpose can be used interchangeably – compare hand axe ). These types of tools were specifically made to catch the eye of many different craftsman who traveled to do their work. To these workers these types of tools were revolutionary because they were one tool or one device that could do several different things. With this new revolution of tools, the traveling craftsman would not have to carry so many tools with them to job sites, in that their space would be limited to the vehicle or to the beast of burden they were driving. Multi-use tools solve the problem of having to deal with many different tools.
Tool use by animals is a phenomenon in which an animal uses any kind of tool in order to achieve a goal such as acquiring food and water, grooming , defense, communication , recreation or construction . [ 42 ] Originally thought to be a skill possessed only by humans , some tool use requires a sophisticated level of cognition. [ 43 ] There is considerable discussion about the definition of what constitutes a tool and therefore which behaviours can be considered true examples of tool use. [ 42 ] [ 44 ] Observation has confirmed that a number of species can use tools including monkeys , apes , elephants , several birds, and sea otters . Now the unique relationship of humans with tools is considered to be that we are the only species that uses tools to make other tools. [ 42 ] [ 45 ]
Primates are well known for using tools for hunting or gathering food and water, cover for rain, and self-defense. Chimpanzees have often been the object of study in regard to their usage of tools, most famously by Jane Goodall ; these animals are closely related to humans. Wild tool-use in other primates, especially among apes and monkeys , is considered relatively common, though its full extent remains poorly documented, as many primates in the wild are mainly only observed distantly or briefly when in their natural environments and living without human influence. [ 42 ] [ 44 ] Some novel tool-use by primates may arise in a localized or isolated manner within certain unique primate cultures , being transmitted and practiced among socially connected primates through cultural learning . [ 43 ] Many famous researchers, such as Charles Darwin in his book The Descent of Man , mentioned tool-use in monkeys (such as baboons ). [ 42 ] [ 44 ] [ 46 ]
Among other mammals , both wild and captive elephants are known to create tools using their trunks and feet, mainly for swatting flies, scratching, plugging up waterholes that they have dug (to close them up again so the water does not evaporate), and reaching food that is out of reach. Many other social mammals particularly have been observed engaging in tool-use. A group of dolphins in Shark Bay uses sea sponges to protect their beaks while foraging. Sea otters will use rocks or other hard objects to dislodge food (such as abalone ) and break open shellfish . Many or most mammals of the order Carnivora have been observed using tools, often to trap or break open the shells of prey, as well as for scratching. [ 42 ] [ 44 ] [ 46 ]
Corvids (such as crows , ravens and rooks ) are well known for their large brains (among birds ) and tool use. New Caledonian crows are among the only animals that create their own tools. They mainly manufacture probes out of twigs and wood (and sometimes metal wire) to catch or impale larvae . Tool use in some birds may be best exemplified in nest intricacy. Tailorbirds manufacture 'pouches' to make their nests in. Some birds, such as weaver birds , build complex nests utilizing a diverse array of objects and materials, many of which are specifically chosen by certain birds for their unique qualities. Woodpecker finches insert twigs into trees in order to catch or impale larvae. Parrots may use tools to wedge nuts so that they can crack open the outer shell of nuts without launching away the inner contents. Some birds take advantage of human activity, such as carrion crows in Japan, which drop nuts in front of cars to crack them open. [ 42 ] [ 44 ] [ 46 ]
Several species of fish use tools to hunt and crack open shellfish, extract food that is out of reach, or clear an area for nesting. Among cephalopods (and perhaps uniquely or to an extent unobserved among invertebrates ), octopuses are known to use tools relatively frequently, such as gathering coconut shells to create a shelter or using rocks to create barriers. [ 42 ] [ 44 ] [ 46 ]
By extension, concepts which support systematic or investigative thought are often referred to as "tools", for example Vanessa Dye refers to "tools of reflection" and "tools to help sharpen your professional practice" for trainee teachers, [ 47 ] illustrating the connection between physical and conceptual tools by quoting the French scientist Claude Bernaud :
we must change [our ideas] when they have served their purpose, as we change a blunt lancet that we have used long enough. [ 47 ]
Similarly, a decision-making process "developed to help women and their partners make confident and informed decisions when planning where to give birth" is described as a "Birth Choice tool":
The tool encourages women to consider out-of-hospital settings where appropriate, [ 48 ]
and the idea of a "toolkit" is used by the International Labour Organization to describe a set of processes applicable to improving global labour relations . [ 49 ]
A telephone is a communication tool that interfaces between two people engaged in conversation at one level. It also interfaces between each user and the communication network at another level. It is in the domain of media and communications technology that a counter-intuitive aspect of our relationships with our tools first began to gain popular recognition. John M. Culkin famously said, "We shape our tools and thereafter our tools shape us". [ 50 ] One set of scholars expanded on this to say: "Humans create inspiring and empowering technologies but also are influenced, augmented, manipulated, and even imprisoned by technology". [ 51 ] | https://en.wikipedia.org/wiki/Tool |
Toolkits for user innovation and custom design are coordinated sets of “user-friendly” design tools . They are designed to support users who may wish to develop products or services for their own use. [ 1 ] [ 2 ] [ 3 ] The problem toolkits are developed to solve is that, while user designers may know their own needs better than do producers, their technical design skills may be less than those of producer-employed developers. [ 4 ] For example, expert users of tennis rackets – or expert users of custom integrated circuits – generally know more than producers do about the function they want a product (or service) to serve. However, they are often not as good as producer engineers at actually designing the product they need.
Toolkits for user innovation (or design customization) solve this problem in two steps. First, they divide the total set of design problems facing product designers into two categories:
Toolkits then offer easy-to-use tools to enable user designers to solve type (1) problems without needing to have technical skills equal to those of producer engineers. Type (2) problems are then assigned either toolkit software for automatic solution or to producers’ technical design specialists.
To illustrate the basic concepts of a toolkit for product innovation and product customization by users, consider a house owner who wants to self-design a custom deck that is “just right” for his or her specific backyard physical setting and planned deck usages. The house owner will know the functions they want their custom deck to serve – outdoor barbecues for up to 10 people, play space for their kids, etc. But suppose that these users – like the vast majority of deck users – do not actually have the architectural and engineering skills required to create a complete, buildable design for the deck they want. A “deck design” toolkit solves this problem by inviting the DIY user to design only the top surface of deck they want by simply sketching the deck size and shape they have in mind onto a computer screen. Note that this top surface is the only portion of the deck that will directly interact with planned user activities on the deck – and so accordingly is the only aspect of deck design for which specialized user knowledge of intended deck uses is relevant.
The toolkit then further helps the user to check deck surface functionality via a simulation of just that part of the deck design. For example, users might be asked to select and place images of items like chairs and tables on the deck surface they have sketched out. This will assist them to determine whether the size and shape of the deck surface they have designed will actually be adequate for their intended uses. If not, they can easily make adjustments directly on the computer screen to solve the usage problems they see, and then assess again to see if they are now satisfied. Again, in this simple example, deck size and shape are the only function-related design problems for which user knowledge is essential.
Next, software incorporated into the deck design toolkit takes over and automatically does “everything else” required to design a complete deck. That is, it designs the structure of a deck suitable to support the size and shape of the deck surface the user has designed. Solving this second set of design problems requires specialized structural engineering knowledge that the user is unlikely to have – for example, how many concrete foundation piers a deck will require for long-term safety and stability. However, since solving these problems does not require users’ special knowledge regarding intended use, it can be performed by the producer without further user inputs – in this case automatically by means of producer-designed software.
Traditionally, in the product development process, producer firms assign market research specialists to determine users’ needs. Marketing thus developed numerous “voice of the customer” market research techniques [ 5 ] aiming to identify user’s needs and preferences. These then become goals for new product developers to meet. Advances in technology such as computerization of production have more recently enabled many firms to abandon the “one size fits all” model, and to offer customers “configurators” via which they can to some extent adapt products to their individual preferences. [ 6 ] [ 7 ] In 2002, von Hippel and Katz conceptualized the role of toolkits in the innovation process in a more general way that extends far beyond customization. A toolkit lets the producer actually abandon the attempt to understand user needs in detail in favor of transferring need-related aspects of product and service development to users.
Today, toolkits for user innovation are routinely used in fields ranging from neural network design to the design of new biological systems in synthetic biology. Research is also going on with respect to improving their function. For example, it has been found that the provision of starting solutions, [ 8 ] a transparent process, [ 9 ] an appropriate solution space, [ 10 ] and peer feedback [ 11 ] [ 12 ] can increase toolkit utility as well as toolkit users’ satisfaction.
Toolkits can be designed to support end user innovation and/or end user customization. In the case of the deck design illustration just provided, the toolkit is clearly capable of deck customization only. However, the same concepts are widely applied today in many fields. For example, innovation toolkits have been designed that enable users with no knowledge of solid-state physics or semiconductor design to create custom semiconductors that will carry out exactly the functions they specify, ranging from the simulation of artificial life to the design of very novel controllers for autonomous electric vehicles. To accomplish this, semiconductor architectures like “sea of (digital logic) gates” are designed to separate out the physics of gate design from function design. Similarly, toolkits have been designed that enable designers with very little knowledge of manufacturing to design producible parts for their projects. Again, as in the deck design case, design of the function of the parts is left to user expertise assisted by CAD . Then, producer expertise, encoded into toolkit software often called CAM (Computer Assisted Manufacturing), is used to convert the user functional part designs into designs that are well-suited to high-quality and economical manufacturing in volume.
The value of toolkits to professional designers is very clear. Because of the very large functional and economic advantages they offer to product designers, CAD-CAM toolkits have largely swept away earlier manual design processes in all fields where design decisions can be digitized. Toolkits are also increasingly being offered to consumers to enable them to customize the products they purchase. The deck design toolkit discussed earlier is an example of this.
For consumers, the value of toolkits has two major elements: the self-perceived quality of the design that users create for themselves, and the process fun and learning that consumers can gain by engaging in custom design using a toolkit.
With respect to self-perceived quality, there is strong research evidence that consumers prefer their self-designed products over products of similar function designed by professional designers. For example, Franke and Piller [ 13 ] conducted an experiment in which they offered one group of individuals the opportunity to customize and buy a watch they designed themselves with the aid of a toolkit. A second group was offered the opportunity to buy any one of the three most successful watches in the market, designed by professional designers. The objective quality of all watches (quality of materials, functions, etc.) was identical in both groups. However, it turned out that toolkit users were willing to pay 100% more to buy the watch they had designed themselves than members of the second group were willing to pay for any of the professionally-designed watches offered to them for purchase. This high additional value has been confirmed in several studies using different toolkits and in different product areas such as breakfast cereals, carving skis, mobile phone covers, fountain pens, kitchens, newspapers, scarves, and t-shirts. [ 14 ]
Several factors contribute to the value increment toolkit-assisted self-design provides to consumer-designers. The most obvious reason for increased subjective value is the increased preference fit. Toolkits allow to tailoring the product to the users’ individual preferences. Given that these preferences are different and significantly matter to users, the individualization possibility will lead to a higher subjective value. [ 15 ] The process of using toolkits to quickly try many design variations also enables consumers to more deeply understand their own preferences. Even short self‐design processes with a relatively simple toolkit bring about significant and time‐stable enhancements of preference insight and allows users to obtain high value from new individual products. [ 16 ] [ 17 ]
Another quality of a custom product is uniqueness. Toolkits enable customers to more easily differentiate themselves from others through their own custom product design, a second driver of product value). [ 18 ] Finally, the creative process itself is also valued by many people. With a toolkit, the customer is the designer, not just a consumer. Experiments confirm the importance of the “I designed it myself” effect in the toolkit-assisted custom product design. [ 19 ] | https://en.wikipedia.org/wiki/Toolkits_for_user_innovation |
Tools for Thought: The History and Future of Mind-Expanding Technology is a work of "retrospective futurism " in which Smart Mobs author Howard Rheingold looked at the history of computing and then attempted to predict what the networked world might look like in the mid-1990s. The book covers the groundbreaking work of thinkers like Alan Turing , John von Neumann , and J.C.R. Licklider , as well as Xerox PARC , Apple Computer , and Microsoft (when Microsoft was "aiming for the hundred-million-dollar category"). Rheingold wrote that the impetus behind Tools for Thought was to understand where " mind-amplifying technology" was going by understanding where it came from.
This article about a computer book or series of books is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tools_for_Thought |
In astrophysics , Toomre's stability criterion (also known as the Safronov–Toomre criterion ) is a relationship between parameters of a differentially rotating , gaseous accretion disc which can be used to determine approximately whether the system is stable. In the case of a stationary gas, the Jeans stability criterion can be used to compare the strength of gravity with that of thermal pressure . In the case of a differentially rotating disk, the shear force can provide an additional stabilizing force.
The Toomre criterion for a disk to be stable can be expressed as,
where c s {\displaystyle c_{s}} is the speed of sound (and measure of the thermal pressure), κ {\displaystyle \kappa } is the epicyclic frequency , G is Newton's gravitational constant , and Σ {\displaystyle \Sigma } is the surface density of the disk. [ 1 ]
The Toomre Q parameter is often defined as the left-hand side of Eq. 1 ,
The stability criterion can then simply be stated as, Q > 1 {\displaystyle Q>1} for a disk to be stable against collapse.
The previous discussion was for a gaseous disk, but a similar analysis can be applied to a disk of stars (for example, the disk of a galaxy), yielding a kinematic Q parameter, [ 1 ]
where σ R {\displaystyle \sigma _{R}} is the radial velocity dispersion, and κ {\displaystyle \kappa } is the local epicyclic frequency .
Many astrophysical objects result from the gravitational collapse of gaseous objects (for example, star formation occurs when molecular clouds collapse under gravity), and thus the stability of gaseous systems is of great interest. In general, a physical system is 'stable' if: 1) It is in equilibrium (there is a balance of forces such that the system is static), and 2) small deviations from equilibrium will tend to damp out, so that the system tends to return to equilibrium.
The most basic gravitational stability analysis is the Jeans criteria , which addresses the balance between self-gravity and thermal pressure in a gas. In terms of the two above stability conditions, the system is stable if: i) thermal pressure balances the force of gravity, and ii) if the system is compressed slightly, the outward pressure force must become stronger than the inward gravitational force - to return the system to equilibrium. In the Jeans case, the stability criterion is size dependent, resulting in the concept of a Jeans length and Jeans mass.
The Toomre analysis , first studied by Viktor Safronov in the 1960s, [ 2 ] considers not only gravity and pressure, but also shear forces from differential rotation. Conceptually, if a fluid is differentially rotating (such as in the keplerian motion of an astrophysical disk), gravity not only has to overcome the internal pressure of the gas, but also needs to halt the relative motion between two parcels of fluid, allowing them to collapse together.
The analysis was expanded upon by Alar Toomre in 1964, [ 1 ] and presented in a more general and comprehensive framework.
Recent high-resolution observations have provided direct evidence supporting the role of the Toomre's stability criterion in regulating star formation and clump formation in galaxies. In particular, a study by Puschnig et al. (2023) [ 3 ] analyzed the clumpy star-forming galaxy SDSS J125013.84+073444.5 (LARS 8) using high-resolution interferometric observations from NOEMA and optical integral-field spectroscopy from MUSE , enabling a spatially resolved Toomre-Q analysis on sub-kiloparsec scales.
They found that, aside from its central region (within approximately 500 pc), the rotating molecular disk of LARS 8 is globally Toomre-unstable ( Q < 1 {\displaystyle Q<1} ) across a wide radial range. This gravitational instability is correlated with the presence of massive molecular clumps, which appear to be virialized and follow a mass–size scaling relation consistent with that observed in giant molecular clouds of the Milky Way , despite the significantly different global environment.
Furthermore, the spatial variation in gas depletion time across the galaxy—dropping from over 1 Gyr in the central regions to less than 100 Myr in the outer disk—aligns closely with the measured instability pattern. The regions with the shortest depletion times (and highest star formation rates) coincide with those where the Toomre Q parameter falls well below unity, emphasizing the predictive power of the criterion.
To refine the analysis of disk stability, Puschnig et al. also highlight the importance of accounting for both stellar and gaseous components in the gravitational stability budget. This approach builds on the theoretical framework developed by A. Romeo and collaborators (Romeo, Burkert & Agertz 2010; Romeo & Wiegert 2011; Romeo & Agertz 2014), who introduced a multi-component stability parameter that effectively incorporates the combined effect of stars and gas, weighted by their respective velocity dispersions and disk thicknesses. [ 4 ] [ 5 ] [ 6 ]
Romeo's criterion provides a more realistic diagnostic, particularly for galaxies with significant stellar mass components or non-negligible turbulent pressure. While the analysis of LARS 8 focused primarily on the gas-dominated component, the inclusion of stellar dynamics is essential for interpreting the stability of more evolved or massive disk galaxies. Puschnig et al. note that applying such a combined Q formalism to future observations will be crucial for understanding the full range of dynamical processes governing galactic evolution.
Overall, these findings reinforce the view that large-scale gravitational instability, as quantified by Toomre's Q parameter (and its multi-component extensions), is a key driver of star formation and morphological evolution in galaxies. The results lend strong observational support to theoretical models of clump-formation triggered by gravitational instabilities in gas-rich galaxies. | https://en.wikipedia.org/wiki/Toomre's_stability_criterion |
In geometry , the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence.
The first stage of the design is a single "toothpick", or line segment. Each stage after the first is formed by taking the previous design and, for every exposed toothpick end, placing another toothpick centered at a right angle on that end. [ 1 ]
This process results in a pattern of growth in which the number of segments at stage n oscillates with a fractal pattern between 0.45 n 2 and 0.67 n 2 . If T ( n ) denotes the number of segments at stage n , then values of n for which T ( n )/ n 2 is near its maximum occur when n is near a power of two, while the values for which it is near its minimum occur near numbers that are approximately 1.43 times a power of two. [ 2 ] The structure of stages in the toothpick sequence often resemble the T-square fractal, or the arrangement of cells in the Ulam–Warburton cellular automaton . [ 1 ]
All of the bounded regions surrounded by toothpicks in the pattern, but not themselves crossed by toothpicks, must be squares or rectangles. [ 1 ] It has been conjectured that every open rectangle in the toothpick pattern (that is, a rectangle that is completely surrounded by toothpicks, but has no toothpick crossing its interior) has side lengths and areas that are powers of two , with one of the side lengths being at most two. [ 3 ] | https://en.wikipedia.org/wiki/Toothpick_sequence |
A top-lit updraft gasifier (also known as a TLUD ) is a micro-kiln used to produce charcoal, especially biochar , and heat for cooking. [ 1 ] A TLUD pyrolyzes organic material, including wood or manure, and uses a reburner to eliminate volatile byproducts of pyrolization. The process leaves mostly carbon as a residue, which can be incorporated into soil to create terra preta .
Dr Thomas B Reed and the Norwegian architect Paal Wendelbo independently developed the working idea of a TLUD gasifier in the 1990s. [ 2 ]
A TLUD gasifier is a considerable improvement on the rocket stove , being a more efficient way to achieve smoke-free combustion of the fuel.
A TLUD gasifier stove [ 3 ] is commonly constructed with two concentric cylindrical containers.
The inner cylinder is the fuel pot. The fuel pot has holes in the base. These holes are the primary air inlet. The fuel pot also has holes on the neck, like the skirt, serving as a secondary air inlet.
The outer cylinder has holes near the bottom on the sides. During combustion, air enters these holes, either by natural air draft or forced with a DC fan depending on requirement and construction model.
Any biomass with less than 20% water content can be used as fuel. The user fills the fuel pot up to the neck, just below the secondary air inlet holes. The user ignites the top layer of fuel for the pyrolysis to start. Air then flows in through the primary and secondary air inlets. The primary inlet helps the draft of pyrolysed wood gas flow up.
The secondary air inlet blows hot air by the time it travels around the fuel pot. The secondary inlet above the fuel layer helps burn the wood gas.
YouTube has many instructional videos, with further explanation on other websites. A range of construction plans to make TLUD gasifier stoves
This agriculture article is a stub . You can help Wikipedia by expanding it .
This sustainability -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Top-lit_updraft_gasifier |
TopFIND is the T ermini o riented p rotein F unction I nferred D atabase ( TopFIND ) is an integrated knowledgebase focused on protein termini, their formation by proteases and functional implications. It contains information about the processing and the processing state of proteins and functional implications thereof derived from research literature, contributions by the scientific community and biological databases . [ 2 ]
Among the most fundamental characteristics of a protein are the N- and C-termini defining the start and end of the polypeptide chain . While genetically encoded, protein termini isoforms are also often generated during translation , following which, termini are highly dynamic, being frequently trimmed at their ends by a large array of exopeptidases. Neo-termini can also be generated by endopeptidases after precise and limited proteolysis, termed processing. Necessary for the maturation of many proteins, processing can also occur afterwards, often resulting in dramatic functional consequences. Aberrant proteolysis can cause wide range of diseases like arthritis [ 3 ] or cancer. [ 4 ] Hence, proteolytic generation of pleiotrophic stable forms of proteins, the universal susceptibility of proteins to proteolysis, and its irreversibility, distinguishes proteolysis from many highly studied posttranslational modifications. Proteases are tightly interconnected in the protease web [ 5 ] [ 6 ] and their aberrant activity in disease can lead to diagnostic fragment profiles with characteristic protein termini. [ 7 ] Following proteolysis, the newly formed protein termini can be further modified, [ 8 ] a process that affects protein function and stability. [ 9 ]
TopFIND is a resource for comprehensive coverage of protein N- and C-termini discovered by all available in silico, in vitro as well as in vivo methodologies. It makes use of existing knowledge by seamless integration of data from UniProt and MEROPS and provides access to new data from community submission and manual literature curating. It renders modifications of protein termini, such as acetylation and citrullination, easily accessible and searchable and provides the means to identify and analyse extend and distribution of terminal modifications across a protein. Since its inception TopFIND has been expanded to further species. [ 1 ]
The data is presented to the user with a strong emphasis on the relation to curated background information and underlying evidence that led to the observation of a terminus, its modification or proteolytic cleavage. In brief the protein information, its domain structure, protein termini, terminus modifications and proteolytic processing of and by other proteins is listed. All information is accompanied by metadata like its original source, method of identification, confidence measurement or related publication. A positional cross correlation evaluation matches termini and cleavage sites with protein features (such as amino acid variants) and domains to highlight potential effects and dependencies in a unique way. Also, a network view of all proteins showing their functional dependency as protease , substrate or protease inhibitor tied in with protein interactions is provided for the easy evaluation of network wide effects. A powerful yet user friendly filtering mechanism allows the presented data to be filtered based on parameters like methodology used, in vivo relevance, confidence or data source (e.g. limited to a single laboratory or publication). This provides means to assess physiological relevant data and to deduce functional information and hypotheses relevant to the bench scientist. In a later release analysis tools for the evaluation of proteolytic pathways in experimental data have been added. [ 10 ] | https://en.wikipedia.org/wiki/TopFIND |
Top International Managers in Engineering (T.I.M.E.), formerly Top Industrial Managers for Europe, is a network of fifty-seven engineering schools , faculties and technical universities . The oldest European network of engineering schools in its field, the T.I.M.E. Association promotes graduate student exchanges and double degrees throughout Europe and the world to enable students to achieve a broader, high-level scientific engineering education with in-depth intercultural experience. [ 1 ]
Several hundreds of graduate students per year participate in T.I.M.E. mobility activities and pursue double degrees (at Master and Doctorate levels). Double degrees require the participating student to spend more than three semesters in another member university and at least the same in her/his home university, in order to be awarded two full degrees.
The T.I.M.E. network includes primarily graduate engineering schools and technical universities from Europe, but increasing numbers of members are now from other continents.
In 1989, the T.I.M.E. network was created at the École Centrale Paris . Its main purpose was to coordinate European double degree and exchange programmes in engineering at the Master's level. The T.I.M.E. network had 16 founding members, each a leading engineering institution in its respective country. The T.I.M.E. Association was formally incorporated as a not-for-profit body under French law in 1997, with 29 members. Current membership is 57 institutions from 23 countries. [ 2 ] [ 3 ]
TIME members include the following engineering schools and faculties and technical universities: [ 4 ]
Australia :
Austria :
Belgium :
Brazil :
China :
Czech Republic :
Denmark :
Spain :
France :
Germany :
Greece :
Hungary :
Italy :
Japan :
Norway :
Poland :
Portugal :
Russia :
Sweden :
Turkey : | https://en.wikipedia.org/wiki/Top_International_Managers_in_Engineering |
Top Tier Detergent Gasoline and Top Tier Diesel Fuel are performance specifications and trademarks designed and supported by several automakers. [ 1 ] BMW , General Motors , Fiat Chrysler Automobiles , Ford , Acura / Honda , Toyota , Volkswagen , Mercedes-Benz , Navistar , Audi , and Volvo support the gasoline standard, while General Motors , Volkswagen , Detroit Diesel , and Navistar support the diesel standard (as of 2018 [update] ). Top Tier fuels must maintain levels of detergent additives that are believed to result in a higher standard of engine cleanliness and performance as compared to the United States Environmental Protection Agency (EPA) requirement. In addition, Top Tier fuels may not contain metallic additives, which can harm the vehicle emission system and create pollutants. As of 2018, Top Tier Detergent Gasoline is available from 61 licensed retail brands, and Top Tier Diesel Fuel is available from 5 licensed retail brands.
Licensed Top Tier fuel retailers use a higher level of detergent additive which can increase fuel economy and optimal engine performance. According to an automotive industry spokesman, the regular use of this type of gasoline results in improved engine life. [ 2 ]
The Top Tier standards must apply to all grades of gasoline or diesel that a company sells, whether it is economy (low-octane) or premium (high-octane). [ 3 ]
Detergent additives serve to prevent the buildup of engine "gunk," which can cause a host of mechanical problems. Automotive journalist Craig Cole writes, "Gasoline is an impure substance refined from a very impure base stock –crude oil. It’s an explosive hydrocarbon cocktail containing all kinds of different chemicals. In addition to its own molecular variability, refiners and retailers incorporate additional substances into the mix, from ethanol alcohol to octane enhancers." [ 4 ]
While General Motors' fuels engineer Andrew Buczynsky claims that no one has identified the exact molecule in gasoline that causes engine buildup, he suggests using Top Tier Detergent Gasoline to keep one's engine cleaner. [ 5 ] Engine gunk typically builds up in fuel injectors and on intake valves, and if severe can result in reduced fuel efficiency , acceleration, and power. Left unchecked, engine gunk can also contribute to increased emissions, rough idling, and tendency to stall, and can therefore increase required motor repairs. [ 6 ] [ 7 ] When fuel injectors accumulate deposits, they do not distribute fuel evenly, creating pockets of too much fuel and too little fuel. Too-little fuel around the spark plug dampens the combustion that drives the piston downward and may cause a misfire. When the frequency of misfires reaches a certain point, the on-board computer turns on the "service engine" light on the dash. The repair for this type of problem depends on the severity of the deposits. In milder cases, a mechanic may solve the problem by adding a can of fuel-injector cleaner into the gas tank. However, in some cases, the fuel injectors must be replaced. Deposits formed on the intake valves may be removed via walnut shell blasting. In severe cases, a more costly cylinder-head rebuild may be necessary. [ 5 ]
Certain forms of sulfur that refiners or pipelines may leave in finished gasoline, such as mercaptans and hydrogen sulfide , can contaminate fuel sending units and lead to erratic dashboard fuel gauge readings, which may be expensive to repair. However, this problem has become less common since 2006, since manufacturers have been making these units with improved alloys that are less affected by these forms of sulfur. [ 8 ]
Chris Martin at Honda states, "We've supported it [Top Tier gasoline] because we've seen a benefit from it for our consumers in the long run. . . We don't require that our vehicle owners use Top Tier gas [but it helps] make sure the engines are going to last as long as they could." [ 9 ]
To be certified as Top Tier, a gasoline must pass a series of performance tests that demonstrate specified levels of: 1) deposit control on intake valves; 2) deposit control on fuel injectors; 3) deposit control on combustion chambers; 4) prevention of intake-valve sticking. [ 4 ] Gasoline marketers agree when they sign on to Top Tier program that all their grades of gasoline meet these standards. [ 10 ] However, premium grade gasoline may have yet higher levels of detergent additives. [ 11 ] Typically, Top Tier gasoline will contain two to three times the amount of detergent additives currently required by the EPA . [ 12 ] The extra additives are estimated to cost less than a cent per gallon. [ 13 ]
In addition to the detergent additive requirement, Top Tier gasoline cannot contain metallic additives, because they can be harmful to a car's emissions-control systems. [ 4 ]
According to its auto industry research and to automotive journalists, all vehicles will benefit from using Top Tier Detergent Gasoline over gasoline meeting the basic EPA standard. [ 10 ] [ 3 ] [ 14 ] New vehicles will supposedly benefit by keeping their engine clean and running optimally, while older vehicles may benefit with increased engine performance and prolonged vehicle life.
In the late 1980s, automakers became concerned with fuel additives as more advanced fuel injection technology became widely used in new cars. The injectors often became clogged, and the problem was found to be inadequate levels of detergent additives in some gasoline. The automakers began to recommend specific brands of gas with adequate content to their customers. But some fuel marketers were still not using detergents, and in a move supported by the auto industry, the federal government mandated specific levels of additives. The U.S. Environmental Protection Agency (EPA) introduced the minimum gasoline detergent standard [ 15 ] in 1995.
However, the new regulations had unintended consequences. The new EPA standards required lower levels of detergent additives than were then being used by a few major fuel marketers. When the new regulations came in, most gasoline marketers who had previously provided higher levels of detergents reduced the level of detergents in their gasolines to meet the new standard. [ 16 ] The EPA detergent additive levels were designed to meet emissions standards but not engine longevity standards. Automakers said they were seeing persistent problems such as clogged fuel injectors, and contaminated combustion chambers, resulting in higher emissions and lower fuel economy. [ 17 ]
By 2002, the automakers said their repair records suggested that the EPA standard for detergents was not high enough, but the EPA was not responsive when they asked them to increase the standards. These concerns were heightened by plans to introduce a new generation of vehicles that would meet the EPA's “Tier Two” environmental standards for reduced emissions. These vehicles require higher levels of detergents to avoid reduced performance. [ 16 ] Cars with gasoline direct injection (GDI) have been especially prone to carbon buildup, and car makers recommend fuels with higher detergent levels to combat the problem. [ 18 ] At first GDI was mainly available in high-end autos, but it is now being used in mid-range cars and economy cars, such as the Hyundai Sonata, Ford Focus and Hyundai Accent. [ 19 ] [ 20 ] [ 21 ]
In 2004 representatives of BMW , General Motors , Honda , and Toyota got together to specify what makes a good fuel. Using recommendations from the Worldwide Fuel Charter, a global committee of automakers and engine manufacturers, they established a proprietary standard for a class of gasoline called "TOP TIER" Detergent Gasoline The new standard required increased levels of detergents, and restricted metallic content. Volkswagen / Audi joined the group of automakers in 2007. Gas brands can participate and get a TOP TIER license if they meet certain standards, which includes performance tests for intake valve and combustion chamber deposits, fuel injector fouling, and intake valve sticking. Additive manufacturers pay for the testing, the cost of which varies from year to year, while gasoline companies pay an annual fee based on the number of stations it operates to participate in the program. [ 22 ]
In addition to higher detergent levels, Top Tier standards also require that gasoline be free of metallic additives, which can be harmful to the emissions control systems in cars. [ 3 ] [ 23 ]
In October 2017 a Top Tier Diesel Fuel program was launched. [ 24 ]
Most of the fuel experts and auto mechanics who have publicly commented on Top Tier gasoline recommend it [ citation needed ] . A 2007 USA Today article quoted three critics who say it has little or no benefit, but the same article quoted three endorsers of the new standard. [ 11 ] Tom Magliozzi , co-host of NPR's weekly radio show, Car Talk , said that using top tier detergent gasoline is only critical on high-end vehicles. For other vehicles, he and another source said that periodic use of a concentrated engine cleaner every 100,000 miles will "often" clean out carbon buildup. [ 25 ] However, journalist and automotive mechanics instructor Jim Kerr says that with some brands of gasoline, deposits can build up on intake valves in less than 10,000 kilometers (6200 miles). [ 7 ] And General Motors fuels engineer Andrew Buczynsky says the various engine-cleaning additives available at auto-parts stores should be used with caution. He said some work but most do not, and that care must be taken when using these additives because some may contaminate the catalytic converter. Also, if too much is used, the additive may cling to valve stems and cause them to hang open. [ 5 ]
Most mechanics agree that consistent use of a fuel with adequate cleaning ability is best [ citation needed ] . Magliozzi's co-host, Ray Magliozzi, said that to be sure of preventing buildup of fuel injectors and valves, motorists should use Top Tier gasoline "at least most of the time." [ 25 ] [ 26 ] Several others agree: Mechanic Pam Oakes says Top Tier gas is effective in cleaning carbon from engines and is worth buying. She says she's seen the difference it can make and recommends it to all of her customers. [ 27 ] Westside Autos in Clive, Iowa, and Motor Age columnist Larry Hammer also recommend Top Tier for removing carbon build-up, adding that a cleaner engine will also burn fuel more cleanly and therefore produce less emissions. [ 28 ] Automotive mechanics instructor Jim Kerr concurs: "All gasoline is not created equal . . . Top Tier does have benefits." [ 10 ]
In 2004 the standard was adopted by ten gasoline distributors. Chevron and QuikTrip were first, [ 29 ] followed that same year by 76 Stations , Conoco , Phillips 66 , Road Ranger , Kwik Trip / Kwik Star , Shell , and MFA Oil Company. Since then, many more gasoline distributors have met the proprietary standard and TOP TIER gasoline can now be found in gas stations all over the U.S. and Canada. [ 30 ] [ 31 ]
Top Tier is also available from select brands in Canada, El Salvador, Guatemala, Honduras, Mexico, Panama, and Puerto Rico. Meeting this standard allows gasoline marketers to differentiate themselves from their competition. All stations selling the brand must meet Top Tier standards before the brand is qualified. They must pass separate tests measuring the ability of their gasoline to keep intake valves, combustion chambers, fuel injectors clean, and to prevent intake valves from sticking. [ 32 ]
As of April 2022, there are two Top Tier Diesel brands [ 33 ] and more than 60 Top Tier gasoline brands [ 32 ] in the US and Canada. | https://en.wikipedia.org/wiki/Top_Tier_Detergent_Gasoline |
A top drive is a mechanical device on a drilling rig that provides clockwise torque to the drill string to drill a borehole . It is an alternative to the rotary table and kelly drive . It is located at the swivel's place below the traveling block and moves vertically up and down the derrick. [ 1 ]
The top drive allows the drilling rig to drill the longer section of a stand of drill pipe in one operation. [ 1 ] A rotary table type rig can only drill 30-foot (9.1 m) (single drill pipe) sections of drill pipe whereas a top drive can drill 60–90-foot (18–27 m) stands (double and triple drill pipe respectively, a triple being three joints of drillpipe screwed together), depending on the drilling rig size. Handling longer sections of drill pipe enables a drilling rig to make greater daily progress because up to 90-foot (27 m) can be drilled at a time, thus requiring fewer "connections" to add another 30-foot (9.1 m) of drill pipe. [ 2 ] Another advantage of top drive systems is time efficiency. [ 1 ] When the bit progresses under a kelly drive, the entire string must be withdrawn from the well bore for the length of the kelly in order to add one more length of drill pipe. With a top drive, the draw works only has to pick up a new stand from the rack and make up two joints. Making fewer and quicker connections reduces the risk of a stuck string [ 3 ] from annulus clogging while drilling fluid is not being pumped. [ 4 ]
Several different kinds of top drives exist, and are usually classified based on the "Safe Working Load" (SWL) of the equipment and the size and type of motor used to rotate the drillpipe. For offshore and heavy duty use, a 1000 short ton unit would be used, whereas a smaller land rig may only require a 500 short ton device. Various sizes of hydraulic motors, or AC or DC electric motors, are available. [ citation needed ]
The American Petroleum Institute has set standards for top drives in a number of its publications including:
The International Organization for Standardization publishes a standard relating to top drives in:
Notable manufacturers of top drives:
This article related to natural gas, petroleum or the petroleum industry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Top_drive |
Top trading cycle (TTC) is an algorithm for trading indivisible items without using money. It was developed by David Gale and published by Herbert Scarf and Lloyd Shapley . [ 1 ] : 30–31
The basic TTC algorithm is illustrated by the following house allocation problem . There are n {\displaystyle n} students living in the student dormitories. Each student lives in a single house. Each student has a preference relation on the houses, and some students prefer the houses assigned to other students. This may lead to mutually-beneficial exchanges. For example, if student 1 prefers the house allocated to student 2 and vice versa, both of them will benefit by exchanging their houses. The goal is to find a core-stable allocation – a re-allocation of houses to students, such that all mutually-beneficial exchanges have been realized (i.e., no group of students can together improve their situation by exchanging their houses).
The algorithm works as follows.
The algorithm must terminate, since in each iteration we remove at least one agent. It can be proved that this algorithm leads to a core-stable allocation.
For example, [ 2 ] : 223–224 suppose the agents' preference ordering is as follows (where only the at most 4 top choices are relevant):
In the first iteration, the only top-trading-cycle is {3} (it is a cycle of length 1), so agent 3 keeps his current house and leaves the market.
In the second iteration, agent 1's top house is 2 (since house 3 is unavailable). Similarly, agent 2's top house is 5 and agent 5's top house is 1. Hence, {1,2,5} is a top-trading-cycle. It is implemented: agent 1 gets house 2, agent 2 gets house 5 and agent 5 gets house 1. These three agents leave the market.
In the third iteration, the top-trading-cycle {4,6} is, so agents 4 and 6 exchange their houses. There are no more agents left, so the game is over. The final allocation is:
This allocation is core-stable, since no coalition can improve its situation by mutual exchange.
The same algorithm can be used in other situations, for example: [ 2 ] suppose there are 7 doctors that are assigned to night-shifts; each doctor is assigned to a night-shift in one day of the week. Some doctors prefer the shifts given to other doctors. The TTC algorithm can be used here to attain a maximal mutually-beneficial exchange.
TTC is a truthful mechanism . This was proved by Alvin Roth . [ 3 ]
When the preferences are strict (there are no indifferences), TTC always finds a strictly Pareto-efficient allocation. Moreover, it always finds a core-stable allocation. Moreover, with strict preferences, there is a unique core-stable allocation, and it is the one found by TTC.
In the strict preferences domain, TTC is the only mechanism that satisfies Individual rationality, Pareto efficiency and Strategy-proofness. [ 4 ] [ 5 ]
The original TTC algorithm assumed that the preferences are strict, so that each agent always has a single top house. In realistic settings, agents may be indifferent between houses, and an agent may have two or more top houses. Several different algorithms have been suggested for this setting. [ 6 ] [ 7 ] They were later generalized in several ways. [ 8 ] [ 9 ] [ 10 ] The general scheme is as follows.
The mechanisms differ in the selection rule used in Step 4. The selection rule should satisfy several conditions: [ 9 ]
If the selection rule satisfies Uniqueness and Termination, the resulting mechanism yields an allocation that is Pareto-efficient and in the weak core (no subset of agents can get a strictly better house for all of them by trading among themselves). Weak core also implies that it is individually-rational. If, in addition, the selection rule satisfies Persistence, Independence of unsatisfied agents, and some other technical conditions, the resulting mechanism is strategyproof .
A particular selection rule that satisfies these conditions is the Highest Priority Object (HPO) rule. It assumes a pre-determined priority-ordering on the houses. It works as follows. [ 9 ]
When the rule terminates, each all agents are labeled, and every labeled agent has a unique outgoing edge. The rule guarantees that, at each iteration, all cycles contain at least one unsatisfied agent. Therefore, in each iteration, at least one new agent becomes satisfied. Therefore, the algorithm ends after at most n iterations. The run-time of each iteration is O ( n log n + n γ ) {\displaystyle O(n\log {n}+n\gamma )} , where γ {\displaystyle \gamma } is the maximum size of an indifference class. Therefore, the total run-time is O ( n 2 log n + n 2 γ ) {\displaystyle O(n^{2}\log {n}+n^{2}\gamma )} .
The TTC algorithm has been extended in various ways.
1. A setting in which, in addition to students already living in houses, there are also new students without a house, and vacant houses without a student. [ 11 ]
2. The school choice setting. [ 12 ] The New Orleans Recovery School District adopted school choice version of TTC in 2012. [ 13 ]
3. The kidney exchange setting: Top Trading Cycles and Chains (TTCC). [ 14 ] | https://en.wikipedia.org/wiki/Top_trading_cycle |
Topera, Inc. is a cardiac arrhythmia mapping company for targeting catheter ablation company launched in San Diego, California and specializes in mapping electrical signals of the heart. [ 1 ] [ 2 ] [ 3 ] Topera's headquarters are located in Palo Alto, California. The company uses 3D analysis and mapping to detect the sources of atrial fibrillation , atrial flutter , and atrial tachycardia and ventricular tachycardia to identify targets for catheter ablation. [ 2 ]
In 2010, Dr. Sanjiv Narayan and Dr. Ruchir Sehra founded Topera in San Diego, California . [ 3 ] [ 4 ] [ 5 ] Narayan founded the company to commercialize the technology he developed which maps irregular heartbeats. [ 6 ] [ 7 ] Prior to founding the company, he had collected cardiac electrophysiological data from patients and wrote software to code and analyze collected data. [ 8 ] He conducted clinical studies to prove that conventional pulmonary vein isolation (PVI) plus targeting and ablating rotors and focal impulses would increase the success rate of single procedure atrial fibrillation ablations. [ 8 ] His training includes a master's degree in software engineering, a clinical fellowship at Harvard Medical School , and a position as a faculty tutor for the Harvard-MIT Division of Health Sciences and Technology program . [ 6 ]
In 2011, Topera opened an office in Lexington, Massachusetts. [ 6 ] [ 9 ] Edward Kerslake, former corporate vice president of Boston Scientific became the CEO of Topera in 2010. [ 6 ] In 2011, the firm created a management advisory board. [ 10 ]
In 2012, the FDA cleared Topera's 3D mapping and analysis system, RhythmView. The system was presented at the Heart Rhythm Society's 2012 scientific sessions in Boston, Massachusetts . [ 11 ] [ 12 ]
The company raised $2.75 million in its seed funding round in 2012; [ 13 ] A few months later, it received an additional $3.77 million in partial close funding. [ 14 ] In May 2013, the company closed on $25 million in a C series of funding led by New Enterprise Associates . [ 15 ] [ 16 ]
Topera developed a 3D mapping tool to assists physicians in identifying the electrical source of complex cardiac arrhythmias. [ 4 ] The FIRMap catheter, used with the RhythmView workstation, received CE clearance and FDA clearance in 2013. [ 17 ] [ 18 ] [ 19 ] [ 20 ] The tip of the catheter has a spherical wire basket that has 64 evenly placed electrodes over the 8 splines that make up the basket. The basket expands, capturing the contours of the heart chambers and creating a panoramic map of the electrical heart activity. [ 19 ] [ 20 ] This information is sent to the workstation and creates a near real-time 3D reconstruction of the heart and its electrophysiological activity. [ 19 ] The data from the workstation is used to help diagnose the source of atrial fibrillation , atrial flutter , and atrial tachycardia and ventricular tachycardia . [ 2 ] [ 19 ] [ 21 ]
Prior to the company's FIRMap catheter being commercially available, the workstation was compatible with other multi-polar mapping catheters. [ 17 ] [ 19 ]
The Focal Impulse and Rotor Modulation procedure decreases procedure times and reduces re-ablation rates by targeting the source of arrhythmia. [ 20 ] [ 22 ]
Narayan and six other independent clinical investigators performed clinical trials that followed post-ablation procedure patients for a year. Published as a comprehensive study in August 2012, the trials showed that 88 percent of the patients in the trial who received the FIRM (Focal Impulse and Rotor Modulation) procedure had a successful termination or slowing of their atrial fibrillation. Long-term success was achieved in 82 percent of patients with a single FIRM procedure. [ 23 ]
Ten centers reported that Topera Medical's FIRM mapping system identifies patient specific sources of atrial fibrillation (rotors), and that ablation of the rotors improves treatment success with a single procedure. [ 24 ]
A second, broader clinical study at ten centers independently confirmed the findings of the first study with a success rate of 80.5% where FIRM was used. [ 25 ]
The first and second set of trials reported success after the patients were followed for one year. At the beginning of 2014, three-year results showed a success rate of 78% compared to the traditional PVI success rate of 39%. [ 26 ]
Other published studies have shown that rotors are located in both atria of the heart and in locations that are not targeted and ablated in traditional procedures. [ 24 ] [ 27 ] Eliminating rotors increases success rates, even in patients with whom traditional techniques are less successful because of pre-existing conditions such as persistent atrial fibrillation, obstructive sleep apnea, metabolic syndrome, or body mass index. [ 28 ] [ 29 ] [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ] [ 35 ] [ 36 ] [ 37 ] [ 38 ]
Published studies reporting success rates with Topera's FIRM-guided ablations are based on single procedure results, while other published reports reflect the outcome of multiple ablation procedures. [ 24 ] [ 25 ] [ 39 ] [ 40 ] [ 41 ] [ 42 ] [ 43 ] For this reason it is difficult to compare relative success rates. One study reported success rates after a single, as well as those after multiple, procedures. The cumulative long-term success after a mean of 2 procedures was 63% while long-term success after single-procedure was only 29%. [ 44 ] | https://en.wikipedia.org/wiki/Topera_Medical |
Topical steroid withdrawal , also known as red burning skin and steroid dermatitis , has been reported in people who apply topical steroids for 2 weeks or longer and then discontinue use. [ 4 ] [ 5 ] [ 2 ] [ 1 ] Symptoms affect the skin and include redness, a burning sensation, and itchiness, [ 2 ] which may then be followed by peeling. [ 2 ]
This condition generally requires the daily application of a topical steroid for more than 2 weeks but sometimes can occur with even less steroid use. It appears to be a specific adverse effect of topical corticosteroid use. [ 6 ] People with atopic dermatitis are most at risk. [ 3 ]
Treatment involves discontinuing the use of topical steroids, [ 2 ] either gradually or suddenly. [ 2 ] Counselling and cold compresses may also help. [ 2 ] Thousands of people congregate in online communities to support one another throughout the healing process, and cases have been reported in both adults and children. [ 2 ] [ 1 ] It was first described in 1979. [ 3 ]
Before discontinuation , steroid dermatitis is characterised by spreading dermatitis and worsening skin inflammation , which requires a stronger topical steroid to get the same result as the first prescription. This cycle is known as steroid addiction syndrome. [ 7 ] When topical steroid medication is stopped, the skin experiences redness, burning, itchiness, scabs, hot skin, swelling, stinging, hives, or oozing. This is known as topical steroid withdrawal. After the withdrawal period is over, the atopic dermatitis can cease or is less severe than it was before. [ 8 ] Topical steroid withdrawal has also been reported in the male scrotum area. [ 9 ] Other symptoms include nerve pain, insomnia, excessive sweating, anxiety, depression, fatigue, eye problems, and frequent infections. [ citation needed ]
The duration of acute topical corticosteroid withdrawal is variable; the skin can take months to years to return to its original condition. [ 2 ] [ 10 ] The duration of steroid use may influence the recovery factor time, with the patients who used steroids for the longest reporting the slowest recovery.
To experience this withdrawal, it generally requires the misuse or application of a topical steroid daily for 2 to 4 months, depending on the potency of the topical corticosteroid. In some cases, this has been reported after as little as 2 weeks of use. [ 11 ] [ 1 ]
Historically, it was believed that cortisol was only produced by the adrenal glands, but research has shown that keratinocytes in human skin also produce cortisol. [ 12 ] Prolonged topical steroid (TS) application changes the glucocorticoid receptor (GR) expression pattern on the surface of lymphocytes; a patient experiencing resistance to a TS has a low ratio of GR-α to GR-β. In addition, the erythema characteristic of ‘red skin syndrome’ is due to a release of stored endothelial nitric oxide (NO) and subsequent vasodilation of dermal vessels. [ 1 ]
Diagnosis is based on a rash occurring within weeks of stopping long-term topical steroids. [ 2 ] Specific signs include "headlight sign" (redness of the lower part of the face but not the nose or the area around the mouth), "red sleeve" (a rebound eruption stopping abruptly at the lower arms and hands), and "elephant wrinkles" (reduced skin elasticity). [ 3 ]
Differentiating this condition from the skin condition that the steroids were originally used to treat can be difficult. [ 2 ] Red, burning skin may be misdiagnosed. [ 8 ]
This condition can be avoided by not using steroid creams for periods of time longer than 2 weeks. [ 2 ] [ 3 ]
Treatment involves ceasing all use of topical steroids, either gradually or suddenly. [ 2 ] Keeping affected areas dry and disinfected will speed healing. [ citation needed ] [ dubious – discuss ] Avoid moisturizer, as any dampness elongates the healing process and encourages bacterial growth. [ citation needed ] [ dubious – discuss ] Antihistamines may help for itchiness . [ 3 ] Immunosuppressants and light therapy may also help some people. [ 3 ] Psychological support is often recommended. [ 2 ] [ 3 ] [ 11 ] At this time, treatment options that have been documented in literature include tacrolimus , pimecrolimus , and dupilumab (Dupixent). Some physicians have also seen positive outcomes with oral doxycycline or topical clindamycin . [ 13 ]
The prevalence of the condition is unknown. [ 14 ] Many cases ranging from mild to severe have been reported in both adults and children. One survey of atopic dermatitis patients treated with topical steroids in Japan estimated that approximately 12% of adult patients may appear to be uncontrolled cases, although they are in fact addicted to a topical steroid. [ 8 ] [ clarification needed ]
A systematic review ( meta-analysis ) in accordance with evidence-based medicine frameworks and current research standards for clinical decision-making was performed in 2016 and was republished with updates in 2020. [ 15 ] | https://en.wikipedia.org/wiki/Topical_steroid_withdrawal |
In stereochemistry , topicity is the stereochemical relationship between substituents and the structure to which they are attached. Depending on the relationship, such groups can be heterotopic , homotopic , enantiotopic , or diastereotopic .
Homotopic groups in a chemical compound are equivalent groups. Two groups A and B are homotopic if the molecule remains achiral when the groups are interchanged with some other atom (such as bromine) while the remaining parts of the molecule stay fixed. Homotopic atoms are always identical, in any environment. Homotopic NMR-active nuclei have the same chemical shift in an NMR spectrum. For example, the four hydrogen atoms of methane (CH 4 ) are homotopic with one another, as are the two hydrogens or the two chlorines in dichloromethane (CH 2 Cl 2 ).
The stereochemical term enantiotopic refers to the relationship between two groups in a molecule which, if one or the other were replaced, would generate a chiral compound. The two possible compounds resulting from that replacement would be enantiomers .
For example, the two hydrogen atoms attached to the second carbon in butane are enantiotopic. Replacement of one hydrogen atom (colored blue) with a bromine atom will produce ( R )-2-bromobutane. Replacement of the other hydrogen atom (colored red) with a bromine atom will produce the enantiomer ( S )-2-bromobutane.
Enantiotopic groups are identical and indistinguishable except in chiral environments. For instance, the CH 2 hydrogens in ethanol (CH 3 CH 2 OH) are normally enantiotopic, but can be made different ( diastereotopic ) if combined with a chiral center, for instance by conversion to an ester of a chiral carboxylic acid such as lactic acid , or if coordinated to a chiral metal center, or if associated with an enzyme active site , since enzymes are constituted of chiral amino acids . Indeed, in the presence of the enzyme LADH , one specific hydrogen is removed from the CH 2 group during the oxidation of ethanol to acetaldehyde , and it gets replaced in the same place during the reverse reaction. The chiral environment needs not be optically pure for this effect.
Enantiotopic groups are mirror images of each other about an internal plane of symmetry. A chiral environment removes that symmetry. Enantiotopic pairs of NMR-active nuclei are also indistinguishable by NMR and produce a single signal.
Enantiotopic groups need not be attached to the same atom. For example, two hydrogen atoms adjacent to the carbonyl group in cis -2,6-dimethylcyclohexanone are enantiotopic; they are related by an internal plane of symmetry passing through the carbonyl group, but deprotonation on one side of the carbonyl group or on the other will generate compounds that are enantiomers. Similarly, the replacement of one or the other with deuterium will generate enantiomers.
The stereochemical term diastereotopic refers to the relationship between two groups in a molecule which, if replaced, would generate compounds that are diastereomers . Diastereotopic groups are often, but not always, identical groups attached to the same atom in a molecule containing at least one chiral center.
For example, the two hydrogen atoms of the CH 2 moiety in ( S )-2-bromobutane are diastereotopic. Replacement of one hydrogen atom (colored blue) with a bromine atom will produce ( 2S,3R )-2,3-dibromobutane. Replacement of the other hydrogen atom (colored red) with a bromine atom will produce the diastereomer ( 2S,3S )-2,3-dibromobutane.
In chiral molecules containing diastereotopic groups, such as in 2-bromobutane, there is no requirement for enantiomeric or optical purity ; no matter its proportion, each enantiomer will generate enantiomeric sets of diastereomers upon substitution of diastereotopic groups (though, as in the case of substitution by bromine in 2-bromobutane, meso isomers have, strictly speaking, no enantiomer).
Diastereotopic groups are not mirror images of one another about any plane. They are always different, in any environment, but may not be distinguishable. For instance, both pairs of CH 2 hydrogens in ethyl phenylalaninate hydrochloride (PhCH 2 CH(NH 3 + )COOCH 2 CH 3 Cl − ) are diastereotopic and both give pairs of distinct 1 H-NMR signals in DMSO-d 6 at 300 MHz, [ 1 ] but in the similar ethyl 2-nitrobutanoate (CH 3 CH 2 CH(NO 2 )COOCH 2 CH 3 ), only the CH 2 group next to the chiral center gives distinct signals from its two hydrogens with the same instrument in CDCl 3 . [ 2 ] Such signals are often complex because of small differences in chemical shift, overlap and an additional strong coupling between geminal hydrogens. On the other hand, the two CH 3 groups of ipsenol, which are three bonds away from the chiral center, give separate 1 H doublets at 300 MHz and separate 13 C-NMR signals in CDCl 3 , [ 3 ] but the diastereotopic hydrogens in ethyl alaninate hydrochloride (CH 3 CH(NH 3 + )COOCH 2 CH 3 Cl − ), also three bonds away from the chiral center, show barely distinguishable 1 H-NMR signals in DMSO-d 6 . [ 4 ]
Diastereotopic groups also arise in achiral molecules. For instance, any one pair of CH 2 hydrogens in 3-pentanol (Figure 1) are diastereotopic, as the two CH 2 carbons are enantiotopic . Substitution of any one of the four CH 2 hydrogens creates two chiral centers at once, and the two possible hydrogen substitution products at any one CH 2 carbon will be diastereomers. This kind of relationship is often easier to detect in cyclic molecules. For instance, any pair of CH 2 hydrogens in cyclopentanol (Figure 2) are similarly diastereotopic, and this is easily discerned as one of the hydrogens in the pair will be cis to the OH group (on the same side of the ring face) while the other will be trans to it (on the opposite side).
The term diastereotopic is also applied to identical groups attached to the same end of an alkene moiety which, if replaced, would generate geometric isomers (also falling in the category of diastereomers). Thus, the CH 2 hydrogens of propene are diastereotopic, one being cis to the CH 3 group, and the other being trans to it, and replacement of one or the other with CH 3 would generate cis- or trans- -2- butene .
Diastereotopicity is not limited to organic molecules, nor to groups attached to carbon, nor to molecules with chiral tetrahedral ( sp 3 -hybridized) centers: for instance, the pair of hydrogens in any CH 2 or NH 2 group in tris (ethylenediamine)chromium(III) ion (Cr(en) 3 3+ ), where the metal center is chiral, are diastereotopic (Figure 2).
The terms enantiotopic and diastereotopic can also be applied to the faces of planar groups (especially carbonyl groups and alkene moieties). See Cahn-Ingold-Prelog priority rule .
Heterotopic groups are those that when substituted are structurally different. They are neither diastereotopic or enantiotopic nor homotopic. [ 5 ] | https://en.wikipedia.org/wiki/Topicity |
In mathematical economics , Topkis's theorem is a result that is useful for establishing comparative statics . The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes. The result states that if f is supermodular in ( x , θ ), and D is a lattice , then x ∗ ( θ ) = arg max x ∈ D f ( x , θ ) {\displaystyle x^{*}(\theta )=\arg \max _{x\in D}f(x,\theta )} is nondecreasing in θ . The result is especially helpful for establishing comparative static results when the objective function is not differentiable. The result is named after Donald M. Topkis .
This example will show how using Topkis's theorem gives the same result as using more standard tools. The advantage of using Topkis's theorem is that it can be applied to a wider class of problems than can be studied with standard economics tools.
A driver is driving down a highway and must choose a speed, s . Going faster is desirable, but is more likely to result in a crash. There is some prevalence of potholes, p . The presence of potholes increases the probability of crashing. Note that s is a choice variable and p is a parameter of the environment that is fixed from the perspective of the driver. The driver seeks to max s U ( s , p ) {\displaystyle \max _{s}U(s,p)} .
We would like to understand how the driver's speed (a choice variable) changes with the amount of potholes:
If one wanted to solve the problem with standard tools such as the implicit function theorem , one would have to assume that the problem is well behaved: U (.) is twice continuously differentiable, concave in s , that the domain over which s is defined is convex, and that it there is a unique maximizer s ∗ ( p ) {\displaystyle s^{\ast }(p)} for every value of p and that s ∗ ( p ) {\displaystyle s^{\ast }(p)} is in the interior of the set over which s is defined. Note that the optimal speed is a function of the amount of potholes. Taking the first order condition, we know that at the optimum, U s ( s ∗ ( p ) , p ) = 0 {\displaystyle U_{s}(s^{\ast }(p),p)=0} . Differentiating the first order condition, with respect to p and using the implicit function theorem, we find that
or that
So,
If s and p are substitutes,
and hence
and more potholes causes less speeding. Clearly it is more reasonable to assume that they are substitutes.
The problem with the above approach is that it relies on the differentiability of the objective function and on concavity. We could get at the same answer using Topkis's theorem in the following way. We want to show that U ( s , p ) {\displaystyle U(s,p)} is submodular (the opposite of supermodular) in ( s , p ) {\displaystyle \left(s,p\right)} . Note that the choice set is clearly a lattice. The cross partial of U being negative, ∂ 2 U ∂ s ∂ p < 0 {\displaystyle {\frac {\partial ^{2}U}{\partial s\,\partial p}}<0} , is a sufficient condition. Hence if ∂ 2 U ∂ s ∂ p < 0 , {\displaystyle {\frac {\partial ^{2}U}{\partial s\,\partial p}}<0,} we know that ∂ s ∗ ( p ) ∂ p < 0 {\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}<0} .
Hence using the implicit function theorem and Topkis's theorem gives the same result, but the latter does so with fewer assumptions. | https://en.wikipedia.org/wiki/Topkis's_theorem |
TopoFlight is a three-dimensional flight planning software for photogrammetric flights.
Originally conceived by a team of experts in the mapping industry, it has been in use since 2003. The program is used to facilitate the planning of flight lines with the help of a Digital Terrain Model (DTM), to document the flight plan and transfer it into the flight management system of the camera (for instance SoftNav, TrackAir, ASCOT or CCNS4), to calculate the costs of photogrammetric flight and subsequent photogrammetric products with the aid of Microsoft Excel as well as flight parameters, and to complete the post checking of a flight (flying height, length overlap, and side lap). Coordinates that have been calculated can be exported to be used during flight. TopoFlight is able to work with frame, line, and LIDAR sensors.
As of 2017 [update] the software is at version 10.5.3. [ 1 ]
The TopoFlight Mission Planner has been upgraded to version 9.5. TopoFlight does use the edge of the image to calculate side overlap and length overlap. It actually calculates over the whole covered area by photos.
It is easier now to plan the flights with cameras, sensors and LiDAR systems.
Maps can be directly downloaded from Google into TopoFlight with user selectable resolution.
Due to a change in Google’s API, the Google Maps Tool in the TopoFlight Program had to be adapted. Improvements, like enhanced calculation of side lap, importing from TIFF DTM, importing of XYZOPK files for quality control and many other features have been completed. Large flight plans in fairly flat areas can be computed much faster now by switching OFF the ‘Precise Calculation' option.
Version 7
Version 6
Version 5
Version 4
Version 1
TopoFlight works with multiple layers. The generated flight plans are stored in the widely used "shape format" (ESRI/ARC VIEW).
Additional maps can be attached as reference files. These maps include:
The reference files can be in the following formats:
Best fit flying height – Calculation of the best flying height to achieve the desired image scale as well as minimum and maximum image scales for given flying height.
Coordinate transformations – Transformation of the coordinates from the local grid to another system (such as WGS84 ).
Calculation of image centers – calculation of coordinates of each image with image scales and overlap.
Calculation of the effective covered area by the images of each strip
Area of side lap – Calculation of the side lap between two neighboring flight lines.
Calculation of costs – Calculation of costs of flight and photogrammetric products which can be transferred into Excel. Custom forms can be later defined.
List of coordinates – Transfer of the flight parameters to Excel. Custom forms can later be defined.
Ground control points – The coordinates of existing ground control points can be imported and annotated. They can also be placed with a mouse click to show the surveyor where to paint and measure a new ground control point.
Exporting the flight plan – The plot can be exported either through SHAPE files, DXF format, or in TIFF format with a TFW header file.
Transfer to flight management system – coordinates can be exported to ASCOT, CCNS, or TrackAir.
Check overlap for aerial triangulation – Check if the minimal overlap is achieved over the whole strip area.
Create image indexes – The coordinates of image centers, stored in a text file can be read by TopoFlight.
TopoFlight is currently in use in 19 countries including the United States, Germany, Brazil, Mexico, Canada, Austria, Italy and others. | https://en.wikipedia.org/wiki/TopoFlight |
TopoFusion GPS Mapping software designed to plan and analyze trails using topographic maps and GPS tracks .
The software was created in 2002 by two brothers who were outdoor bikepacking enthusiasts and felt software could help them plan better trails . [ 3 ] They developed the first version of the software in 2002 and one included it as part of his doctorate dissertation on GPS Driven Trail Simulation and Network Production . [ 4 ] In 2004 the developers and one other jointly presented the paper Digital Trail Libraries which illustrated some of the graph theory algorithms used by the software. [ 5 ] [ 1 ] As of 2018 [update] the software remains supported with refined functionality and improved support for additional maps and GPS Devices. [ 6 ]
The software was designed to plan and analyze trails. [ 7 ] When used for planning proposed routes may be planned and checked against different maps, and the result(s) downloaded to a GPS tracking device . [ 7 ] Topofusion is particularly noted for eased of switch and combining maps and for capability of simultaneously managing multiple trails. [ 8 ] After a trail has been executed the resultant GPS log can be uploaded to TopoFusion and the actual route analyzed with the addition of any photographic images recorded on route. [ 7 ]
The product is marketed as a fully featured 'professional version and a more basic version with reduced functionality at lower cost. A fully featured trial version which is not time limited is available which restricts usability by watermarking map display tiles by overlaying the word 'DEMO'. [ 2 ] The software is available directly Microsoft Windows only, [ 9 ] however TopoFusion has claimed users have reported success using VMWare Fusion and Parallels emulation on Mac OS. [ 3 ] [ 10 ]
TopoFusion has been found useful by those engaged in the sport of geocaching . [ 11 ]
The software has been used in assisting analysis of GPS routes. A survey reported in 2004 of GPS tracking of motorists visiting the Acadia National Park in Maine , United States was assisted by use of Topofusion to review the scenes visited. [ 12 ] It has also been used in studies of agriculture transportation logistics. [ 1 ]
TopoFusion can also assist in determining where photographs have been taken on a trail and can geocoded photo the image or tag it onto a map. For this to be successful the digital camera's time must be synchronized with the GPS unit time, and both the GPS track and digital images made available to Topofusion. The time when the image was taken can then be matched to the time on the GPS log and this enables the image to be enhanced with geocode fields when Real-time geotagging was not available when the image was taken. TopoFusion can also optionally annotate maps with images. [ 13 ] | https://en.wikipedia.org/wiki/TopoFusion |
Topo-Chico is a brand of sparkling mineral water from Mexico . Topo-Chico is both naturally carbonated at the source and artificially carbonated. [ 1 ]
Topo-Chico has been sourced from and bottled in Monterrey, Mexico since 1895. The drink takes its name from the mountain Cerro del Topo Chico in Monterrey. [ 2 ] [ 3 ]
In 2017, The Coca-Cola Company purchased Topo-Chico for $220 million. [ 3 ] [ 4 ] The brand was originally popular in northern Mexico and Texas , with the Coca-Cola Company later helping popularize it across the United States. [ 3 ] The drink has a cult following . [ 3 ] [ 4 ] [ 5 ] [ 6 ]
According to Consumer Reports , in 2020 Topo Chico sparkling water had PFAS measured at 9.76 ppt, the highest PFAS content of any brand. [ 7 ] In 2021 Coca-Cola reduced the level of PFAS chemicals in Topo Chico mineral water, but levels are still above the maximum for bottled water recommended by experts. [ 8 ] [ 9 ]
Ranch water is a cocktail made with tequila, lime juice and Topo-Chico, over ice, a popular drink in Texas. [ 10 ] A similar drink, the Chilton, substitutes the lime for a lemon, the tequila for vodka, and adds salt on the rim. The drink allegedly derives its name from a doctor in Lubbock. [ 11 ]
In 2021, the Coca-Cola Co used its [ 12 ] sparkling mineral water brand Topo-Chico to launch a range of vegan friendly [ 13 ] alcoholic hard seltzers in the United Kingdom and in the United States with Molson Coors . [ 14 ] [ 15 ] [ 16 ] The range includes Tangy Lemon Lime, Tropical Mango and Cherry Acai flavors in the United Kingdom and flavors such as Tangy Lemon Lime, Tropical Mango, Strawberry Guava and Exotic Pineapple in the US.
In early 2022, Topo Chico ranch water launched their new Hard Seltzer Topo Chico Ranch Water in select markets, along with the national rollout of its variety pack. The product is now available in stores across Alabama, Arizona, California, Colorado, Georgia, New Mexico, Oklahoma, Tennessee, and Texas. [ 17 ]
Neither Topo Chico Hard Seltzer nor Topo Chico Ranch Water are made with mineral water from the original Topo Chico spring. Rather, they are “inspired by the taste” of the original drink.
In 2023, a New York resident sued Coca-Cola because its Topo Chico Margarita Hard Seltzers do not contain tequila and cited that the product's packaging was misleading about the contents of the beverage. The lawsuit was dismissed later that year. [ 18 ] In 2024, a Florida resident brought a similar suit against Coca-Cola, also citing that the product's packaging includes "false and misleading representations and omissions" suggesting that the product contains tequila. [ 19 ]
"Topo Chico" is the subject and title of the last song on Robert Ellis's 2019 album Texas Piano Man . [ 20 ]
Topo Chico is featured on the album cover of Seattle WA band iji's 2013 album UNLTD. COOL DRINKS . [ 21 ] | https://en.wikipedia.org/wiki/Topo_Chico |
Topochemical polymerization is a polymerization method performed by monomers aligned in the crystal state. In this process, the monomers are crystallised and polymerised under external stimuli such as heat, light, or pressure. Compared to traditional polymerisation, the movement of monomers was confined by the crystal lattice in topochemical polymerisation, giving rise to polymers with high crystallinity, tacticity , and purity. [ 1 ] Topochemical polymerisation can also be used to synthesise unique polymers such as polydiacetylene [ 2 ] that are otherwise hard to prepare.
Various reactions have been adopted in the field of topochemical polymerisation, such as [2+2], [ 3 ] [4+2], [ 4 ] [4+4], [ 5 ] and [3+2] [ 6 ] cycloaddition, linear addition between dienes, trienes, diacetylenes. Other than linear polymers, they can also be applied to the synthesis of two dimensional covalent networks. [ 7 ]
The term "topochemistry" was first introduced by Kohlschütter in 1919, referring to the chemical reactions driven by the molecular alignments within the crystal. [ 8 ] The prefix "topo" came from the Greek word "topos", which means "site". [ 9 ] These reactions quickly draw people's attention because of their high conversion as well as solvent/catalyst-free nature. However, the early studies were usually serendipitous.
In the 1960s, Schmidt's work on [2+2] photodimerization of cinnamic acids established the systematic approach to study the topochemical reactions. They proposed that only double bonds adopting coplanar and parallel orientation within a distance of 3.5-4.2 Å could react with each other in the crystal lattice. [ 10 ] This empirical rule was later referred to as Schmidt's criteria.
[2+2] cycle addition [ 3 ] and diacetylene polymerization [ 11 ] are among the early examples of topochemical polymerization. As shown in the figure, the formation of 1,3-diphenyl substituted cyclobutane derivatives was first studied in detail by Hasegawa and his coworkers in 1967. [ 12 ] A series of similar monomers had also been studied by them. In 1969, the 1,4-addition polymerization of diacetylene was confirmed by Wegner and his coworkers. [ 11 ] Restricted by the experimental condition, early researchers of topochemical polymerization usually characterized the reaction process and product with traditional chemical methods. The development of modern analysis technology such as single-crystal X-ray diffraction greatly facilitated the systematic study of topochemical polymerization and kept the popularity till these days.
In topochemical polymerization, little room is provided for the monomer to adjust their position. Thus, the reacting sites of the monomer should be pre-packed in a suitable manner. If [2+2] cycloaddition is involved in the polymerization, then the alignment of double bonds within the crystal should fulfill the aforementioned Schmidt's criteria. Sometimes multiple parameters should be considered. As shown in the figure, for example, the 1,4-polymerization of diacetylene requires the fine adjustment of angle as well as the monomer packing distance to achieve a satisfying reaction site distance d CC (distance between C1 and C4). [ 13 ]
The method invented by Schmidt is still the most promising way to investigate the structural criteria of polymerization. [ 10 ] In this approach, a series of monomers with different substituents are crystallized and characterized by single-crystal X-ray diffractometer. By comparing their polymerization reactivity and slightly different structure, the suitable range of lattice parameters can be derived. [ 14 ]
Though Schmidt's criteria are generally useful for predicting the topochemical reactivity, there are many instances of violation of these criteria. Many examples of smooth reaction of crystals that are not expected to be reactive based on Schmidt's criteria are reported. [ 15 ]
Various methods have been proposed to achieve the suitable alignment of monomers in the crystal. [ 1 ] These methods can be divided into two categories:
An obvious method is to introduce supramolecular interactions to the monomer. Popular choices include π - π stacking interactions, hydrogen/halogen bonding interactions, and Coulomb interactions. These interactions are sometimes inherent properties of reaction groups, such as π-π interaction between azide and acetylene group, or stacking force between biphenylethylene unit. Sometimes the side groups are introduced to form a network within the crystal.
The other strategy is to take advantage of the so-called "host-guest" assembly. In this case, the monomer is designed to link to a "host" molecule, while the host molecule is in charge of forming the ordered network. The host molecule stays intact during the polymerization. Such strategies simplify the synthesis of monomer. [ 16 ]
Although the movement of the mass center of the monomer is restricted by the crystal during the polymerization, the slight change of the bond length before and after the reaction give rise to the shifting of lattice parameters. Consider a real-life topochemical polymerization initiated by irradiation: if monomer beneath the surface polymerizes later due to the light absorption near the surface, the already polymerized layer will shrink or expand, causing unbalanced stress within the crystal. The crystal might break or even lose crystallinity if the stress isn't handled properly. [ 6 ]
Using elastic interaction such as weak hydrogen bonds is a common strategy to release the stress. It has been found that the bond length of the hydrogen bond in the crystal would change after polymerization, acting as cushion. [ 17 ] Another possible routine is to introduce "soft" parts (C-C or C-O bond free to rotate instead of rigid conjugated system) in the monomer molecule. But it will in turn increase the difficulty of crystallization. [ 18 ]
Light irradiation can initiate the reaction while avoiding exerting additional physical effects on the monomer crystal. It can be used in topochemical polymerization based on free radical mechanism such as 1,4-polymerization of diacetylene or diene polymerization. [ 1 ] UV light is widely used as initiation method as it does in conventional polymerization. In some circumstances, however, the polymerization initiated by UV light is so slow that unbalanced pressure will accumulate more easily as previously stated. γ-irradiation can trigger the reaction faster due to the shorter wavelength. Thus, it was proved to be a better choice than UV in various reactions such as topochemical polymerization of 1 , 3-diene carboxylic acid derivatives. [ 19 ]
Heat can be used to trigger the electrocyclization topochemical polymerization. For example, Kana M. Sureshan et al. have developed a series of bio-compatible polymer crystals based on [3+2] Topochemical Azide-Alkyne Cycloaddition (TAAC) reaction [ 20 ] [ 21 ] and [3+2] topochemical ene-azide cycloaddition (TEAC) reaction. [ 22 ] The monomers are polymerized by heating for a few days. Contrary to the light-initiated topochemical polymerization, the lower temperature and slower reaction rate would produce high quality polymer crystals. This is due to the fact that heat expansion is not obvious in lower temperature. [ 23 ]
Topochemical polymerization can also be triggered by pressure. It has been reported that the cocrystal of diododiacetylene (guest) and bispyridyl oxalamide (host) could be polymerized under pressure. [ 24 ] Interestingly, no polymerization was observed under light or heat due to the unfavorable distance between diacetylene units. The researcher postulated that the high pressure might "squeeze" the reactive site together and initiate the polymerization.
Tactic and stereoselective polymerizations are traditionally catalyzed by metal-organic complexes. Topochemical polymerization provides an additional choice. In addition, by changing the alignment of the monomer within the crystal, the tacticity/stereochemistry of the polymer product could be easily controlled. An intuitive example is shown in the figure. In topochemical polymerization of 1 , 3-diene carboxylic acid derivatives, polymers with four different configurations can be prepared. Their structural relationships with the monomer packing are also shown in the figure. [ 25 ]
Single crystal polymers have unique applications in various fields Compared to single crystals of small molecules. Because of the long chain and various conformation, it is hard for the polymers to be crystallized directly from solution. Few examples of polymer single crystals prepared in this way suffered from low quality and small size. [ 26 ] Topochemical polymerization provides a potential solution to yield high-quality polymer single crystals.
If the polymer is still mono crystalline, the transformation from single-crystal monomer to polymer is called single-crystal-to-single-crystal (SCSC) transformation, [ 14 ] which required a more sophisticated design than normal topochemical polymerization. In order to prevent the polymer from breaking into polycrystalline powder, the stress-releasing strategies should be carefully considered. However, the study on general criteria of SCSC transition is still in its infancy and requires further study.
In addition to organic polymers, coordination polymers can also be prepared with topochemical polymerization. The various conformations of metal-organic complexes provide large libraries of monomer geometry. [ 27 ] In addition, the length and angle of metal-ligand bonds are relatively flexible so that stress generated by polymerization is able to be released.
The Two-dimensional (2-D) polymers formed by topochemical polymerization are popular topics in material chemistry. By synthesizing and polymerizing monomers with functionality greater than 2, the 2-D networks instead of linear polymers can be obtained. [ 28 ] [4+4] and [4+2] involving anthracene units are popular choices for 2D-polymer synthesis. 2-D covalent networks with high crystallinity can be produced in this way in high conversion. [ 5 ] [ 29 ] Recently, schluter et al. synthesized a 2D polymer via 2+2 topochemical cycloaddition reaction. [1] | https://en.wikipedia.org/wiki/Topochemical_polymerization |
A topogenic sequence is a collective term used for a peptide sequence present at nascent proteins essential for their insertion and orienting in cellular membranes . The sequences are also used to translocate proteins across various intracellular membranes, [ 1 ] and ensure they are transported to the correct organelle after synthesis. [ 2 ] The position of the sequence may be at the end, e.g. N-terminal signal sequence, or in mid parts of the nascent protein, e.g. stop-transfer anchor sequences and signal-anchor sequences. [ 3 ] If the sequence is at the end of the polypeptide, it is cleaved off after entering the ER-lumen (via a translocon) by a signal peptidase, and subsequently degraded.
As an example, the vast majority of all known complex plastid preproteins (an 'unactivated' protein) encoded in the nucleus possess a topogenic sequence. [ 2 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topogenic_sequence |
The topographic wetness index ( TWI ), also known as the compound topographic index (CTI), is a steady state wetness index. It is commonly used to quantify topographic control on hydrological processes. [ 1 ] The index is a function of both the slope and the upstream contributing area per unit width orthogonal to the flow direction. The index was designed for hillslope catenas . Accumulation numbers in flat areas will be very large, so TWI will not be a relevant variable. The index is highly correlated with several soil attributes such as horizon depth, silt percentage, organic matter content, and phosphorus . [ 2 ] Methods of computing this index differ primarily in the way the upslope contributing area is calculated.
The topographic wetness index is defined as:
ln a tan b {\displaystyle \ln {a \over \tan b}}
where a {\displaystyle a} is the local upslope area draining through a certain point per unit contour length and tan b {\displaystyle \tan b} is the local slope in radians . The TWI has been used to study spatial scale effects on hydrological processes. The topographic wetness index (TWI) was developed by Beven and Kirkby [ 3 ] within the runoff model TOPMODEL. Although the topographic wetness index is not a unitless number, it is sufficiently approximate that its interpretation doesn't rely on its physical units. Rather, it should be interpreted that areas with similar TWIs become saturated under similar moisture conditions, as described by Dr John Lindsay of the University of Guelph. [ 4 ]
The TWI has been used to study spatial scale effects on hydrological processes and to identify hydrological flow paths for geochemical modelling, as well as to characterize biological processes such as annual net primary production , vegetation patterns, and forest site quality. | https://en.wikipedia.org/wiki/Topographic_wetness_index |
In medicine , " topographical codes " (or "topography codes") are codes that indicate a specific location in the body. [ 1 ]
Only the first of these is a system dedicated only to topography. The others are more generalized systems that contain topographic axes.
This anatomy article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topographical_code |
Topoisomers or topological isomers are molecules with the same chemical formula and stereochemical bond connectivities but different topologies . Examples of molecules for which there exist topoisomers include DNA , which can form knots , and catenanes . Each topoisomer of a given DNA molecule possesses a different linking number associated with it. DNA topoisomers can be interchanged by enzymes called topoisomerases . [1] Using a topoisomerase along with an intercalator , topoisomers with different linking number may be separated on an agarose gel via gel electrophoresis .
This nanotechnology-related article is a stub . You can help Wikipedia by expanding it .
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topoisomer |
In mathematics , topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by Vladimir Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory . It connects many ideas from algebra to ideas in topology . As described in Askold Khovanskii 's book: "According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way."
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_Galois_theory |
In mathematics , a topological abelian group , or TAG , is a topological group that is also an abelian group .
That is, a TAG is both a group and a topological space , the group operations are continuous , and the group's binary operation is commutative .
The theory of topological groups applies also to TAGs, but more can be done with TAGs. Locally compact TAGs, in particular, are used heavily in harmonic analysis .
This topology-related article is a stub . You can help Wikipedia by expanding it .
This group theory -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_abelian_group |
In mathematics, topological complexity of a topological space X (also denoted by TC( X )) is a topological invariant closely connected to the motion planning problem [ further explanation needed ] , introduced by Michael Farber in 2003.
Let X be a topological space and P X = { γ : [ 0 , 1 ] → X } {\displaystyle PX=\{\gamma :[0,1]\,\to \,X\}} be the space of all continuous paths in X . Define the projection π : P X → X × X {\displaystyle \pi :PX\to \,X\times X} by π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) {\displaystyle \pi (\gamma )=(\gamma (0),\gamma (1))} . The topological complexity is the minimal number k such that
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_complexity |
In quantum many-body physics , topological degeneracy is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations . [ 1 ]
Topological degeneracy can be used to protect qubits which allows topological quantum computation . [ 2 ] It is believed that topological degeneracy implies topological order (or long-range entanglement [ 3 ] ) in the ground state. [ 4 ] Many-body states with topological degeneracy are described by topological quantum field theory at low energies.
Topological degeneracy was first introduced to physically define topological order. [ 5 ] In two-dimensional space, the topological degeneracy depends on the topology of space, and the topological degeneracy on high genus Riemann surfaces encode all information on the quantum dimensions and the fusion algebra of the quasiparticles. In particular, the topological degeneracy on torus is equal to the number of quasiparticles types.
The topological degeneracy also appears in the situation with topological defects (such as vortices, dislocations, holes in 2D sample, ends of a 1D sample, etc.), where the topological degeneracy depends on the number of defects. Braiding those topological defect leads to topologically protected non-Abelian geometric phase , which can be used to perform topologically protected quantum computation .
Topological degeneracy of topological order can be defined on a closed space or an open space with gapped boundaries or gapped domain
walls, [ 6 ] including both Abelian topological orders [ 7 ] [ 8 ] and non-Abelian topological orders. [ 9 ] [ 10 ] The application of these types of systems for quantum computation has been proposed. [ 11 ] In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global or gauge symmetries. [ 12 ]
The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors [ 13 ] ) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy
where number of the degenerate states is given by 2 N d / 2 / 2 {\displaystyle 2^{N_{d}/2}/2} , where N d {\displaystyle N_{d}} is the number of the defects (such as the number of vortices).
Such topological degeneracy is referred as "Majorana zero-mode" on the defects. [ 14 ] [ 15 ] In contrast, there are many types of topological degeneracy for interacting systems. [ 16 ] [ 17 ] [ 18 ] A systematic description of topological degeneracy is given by tensor category (or monoidal category ) theory. | https://en.wikipedia.org/wiki/Topological_degeneracy |
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane . It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory . When one solution of an equation is easily found, degree theory can often be used to prove existence of a second, nontrivial, solution. There are different types of degree for different types of maps: e.g. for maps between Banach spaces there is the Brouwer degree in R n , the Leray-Schauder degree for compact mappings in normed spaces , the coincidence degree and various other types. There is also a degree for continuous maps between manifolds .
Topological degree theory has applications in complementarity problems , differential equations , differential inclusions and dynamical systems .
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_degree_theory |
In mathematics , the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler , Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai , or metric entropy . Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension . The second definition clarified the meaning of the topological entropy: for a system given by an iterated function , the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy.
A topological dynamical system consists of a Hausdorff topological space X (usually assumed to be compact ) and a continuous self-map f : X → X . Its topological entropy is a nonnegative extended real number that can be defined in various ways, which are known to be equivalent.
Let X be a compact Hausdorff topological space. For any finite open cover C of X , let H ( C ) be the logarithm (usually to base 2) of the smallest number of elements of C that cover X . [ 1 ] For two covers C and D , let C ∨ D {\displaystyle C\vee D} be their (minimal) common refinement, which consists of all the non-empty intersections of a set from C with a set from D , and similarly for multiple covers.
For any continuous map f : X → X , the following limit exists:
Then the topological entropy of f , denoted h ( f ), is defined to be the supremum of H ( f , C ) over all possible finite covers C of X .
The parts of C may be viewed as symbols that (partially) describe the position of a point x in X : all points x ∈ C i are assigned the symbol C i . Imagine that the position of x is (imperfectly) measured by a certain device and that each part of C corresponds to one possible outcome of the measurement. H ( C ∨ f − 1 C ∨ … ∨ f − n + 1 C ) {\displaystyle H(C\vee f^{-1}C\vee \ldots \vee f^{-n+1}C)} then represents the logarithm of the minimal number of "words" of length n needed to encode the points of X according to the behavior of their first n − 1 iterates under f , or, put differently, the total number of "scenarios" of the behavior of these iterates, as "seen" by the partition C . Thus the topological entropy is the average (per iteration) amount of information needed to describe long iterations of the map f .
This definition [ 2 ] [ 3 ] [ 4 ] uses a metric on X (actually, a uniform structure would suffice). This is a narrower definition than that of Adler, Konheim, and McAndrew, [ 5 ] as it requires the additional metric structure on the topological space (but is independent of the choice of metrics generating the given topology). However, in practice, the Bowen-Dinaburg topological entropy is usually much easier to calculate.
Let ( X , d ) be a compact metric space and f : X → X be a continuous map . For each natural number n , a new metric d n is defined on X by the formula
Given any ε > 0 and n ≥ 1, two points of X are ε -close with respect to this metric if their first n iterates are ε -close. This metric allows one to distinguish in a neighborhood of an orbit the points that move away from each other during the iteration from the points that travel together. A subset E of X is said to be ( n , ε )-separated if each pair of distinct points of E is at least ε apart in the metric d n .
Denote by N ( n , ε ) the maximum cardinality of an ( n , ε )-separated set. The topological entropy of the map f is defined by
Since X is compact, N ( n , ε ) is finite and represents the number of distinguishable orbit segments of length n , assuming that we cannot distinguish points within ε of one another. A straightforward argument shows that the limit defining h ( f ) always exists in the extended real line (but could be infinite). This limit may be interpreted as the measure of the average exponential growth of the number of distinguishable orbit segments. In this sense, it measures complexity of the topological dynamical system ( X , f ). Rufus Bowen extended this definition of topological entropy in a way which permits X to be non-compact under the assumption that the map f is uniformly continuous .
Recent studies have extended topological entropy to symbolic conditional entropy in layered dynamical systems, generalizing classical entropy measures to more abstract symbolic and informational structures. [ 1 ]
This article incorporates material from Topological Entropy on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Topological_entropy |
The topological entanglement entropy [ 1 ] [ 2 ] [ 3 ] or topological entropy , usually denoted by γ {\displaystyle \gamma } , is a number characterizing many-body states that possess topological order .
A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state. So the topological entanglement entropy links topological order with pattern of long range quantum entanglements.
Given a topologically ordered state, the topological entropy can be extracted from the asymptotic behavior of the Von Neumann entropy measuring the quantum entanglement between a spatial block and the rest of the system. The entanglement entropy of a simply connected region of boundary length L , within an infinite two-dimensional topologically ordered state, has the following form for large L :
where − γ {\displaystyle -\gamma } is the topological entanglement entropy.
The topological entanglement entropy is equal to the logarithm of the total quantum dimension of the quasiparticle excitations of the state.
For example, the simplest fractional quantum Hall states, the Laughlin states at filling fraction 1/ m , have γ = ½log( m ). The Z 2 fractionalized states, such as topologically ordered states of Z 2 spin-liquid, quantum dimer models on non-bipartite lattices, and Kitaev's toric code state, are characterized γ = log(2).
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it .
This article about statistical mechanics is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_entropy_in_physics |
Topological ideas are relevant to fluid dynamics (including magnetohydrodynamics ) at the kinematic level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particularly susceptible to topological techniques. Thus, for example, the Thurston–Nielsen classification has been fruitfully applied to the problem of stirring in two-dimensions by any number of stirrers following a time-periodic 'stirring protocol' (Boyland, Aref & Stremler 2000). Other studies are concerned with flows having chaotic particle paths, and associated exponential rates of mixing (Ottino 1989).
At the dynamic level, the fact that vortex lines are transported by any flow governed by the classical Euler equations implies conservation of any vortical structure within the flow. Such structures are characterised at least in part by the helicity of certain sub-regions of the flow field, a topological invariant of the equations. Helicity plays a central role in dynamo theory , the theory of spontaneous generation of magnetic fields in stars and planets (Moffatt 1978, Parker 1979, Krause & Rädler 1980). It is known that, with few exceptions, any statistically homogeneous turbulent flow having nonzero mean helicity in a sufficiently large expanse of conducting fluid will generate a large-scale magnetic field through dynamo action. Such fields themselves exhibit magnetic helicity , reflecting their own topologically nontrivial structure.
Much interest attaches to the determination of states of minimum energy, subject to prescribed topology. Many problems of fluid dynamics and magnetohydrodynamics fall within this category. Recent developments in topological fluid dynamics include also applications to magnetic braids in the solar corona , DNA knotting by topoisomerases , polymer entanglement in chemical physics and chaotic behavior in dynamical systems. A mathematical introduction to this subject is given by Arnold & Khesin (1998) and recent survey articles and contributions may be found in Ricca (2009), and Moffatt, Bajer & Kimura (2013).
Topology is also crucial to the structure of neutral surfaces in a fluid (such as the ocean) where the equation of state nonlinearly depends on multiple components (e.g. salinity and heat). Fluid parcels remain neutrally buoyant as they move along neutral surfaces, despite variations in salinity or heat. On such surfaces, the salinity and heat are functionally related, but this function is multivalued . The spatial regions within which this function becomes single-valued are those where there is at most one contour of salinity (or heat) per isovalue, which are precisely the regions associated with each edge of the Reeb graph of the salinity (or heat) on the surface (Stanley 2019). | https://en.wikipedia.org/wiki/Topological_fluid_dynamics |
Topological geometry deals with incidence structures consisting of a point set P {\displaystyle P} and a family L {\displaystyle {\mathfrak {L}}} of subsets of P {\displaystyle P} called lines or circles etc. such that both P {\displaystyle P} and L {\displaystyle {\mathfrak {L}}} carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups , many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines.
Linear geometries are incidence structures in which any two distinct points x {\displaystyle x} and y {\displaystyle y} are joined by a unique line x y {\displaystyle xy} . Such geometries are called topological if x y {\displaystyle xy} depends continuously on the pair ( x , y ) {\displaystyle (x,y)} with respect to given topologies on the point set and the line set. The dual of a linear geometry is obtained by interchanging the roles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the Handbook of incidence geometry . [ 1 ] The most extensively investigated topological linear geometries are those which are also dual topological linear geometries. Such geometries are known as topological projective planes .
A systematic study of these planes began in 1954 with a paper by Skornyakov. [ 2 ] Earlier, the topological properties of the real plane had been introduced via ordering relations on the affine lines, see, e.g., Hilbert , [ 3 ] Coxeter , [ 4 ] and O. Wyler. [ 5 ] The completeness of the ordering is equivalent to local compactness and implies that the affine lines are homeomorphic to R {\displaystyle \mathbb {R} } and that the point space is connected . Note that the rational numbers do not suffice to describe our intuitive notions of plane geometry and that some extension of the rational field is necessary. In fact, the equation x 2 + y 2 = 3 {\displaystyle x^{2}+y^{2}=3} for a circle has no rational solution.
The approach to the topological properties of projective planes via ordering relations is not possible, however, for the planes coordinatized by the complex numbers , the quaternions or the octonion algebra. [ 6 ] The point spaces as well as the line spaces of these classical planes (over the real numbers, the complex numbers, the quaternions, and the octonions) are compact manifolds of dimension 2 m , 1 ≤ m ≤ 4 {\displaystyle 2^{m},\,1\leq m\leq 4} .
The notion of the dimension of a topological space plays a prominent rôle in the study of topological, in particular of compact connected planes. For a normal space X {\displaystyle X} , the dimension dim X {\displaystyle \dim X} can be characterized as follows:
If S n {\displaystyle \mathbb {S} _{n}} denotes the n {\displaystyle n} -sphere, then dim X ≤ n {\displaystyle \dim X\leq n} if, and only if, for every closed subspace A ⊂ X {\displaystyle A\subset X} each continuous map φ : A → S n {\displaystyle \varphi :A\to \mathbb {S} _{n}} has a continuous extension ψ : X → S n {\displaystyle \psi :X\to \mathbb {S} _{n}} .
For details and other definitions of a dimension see [ 7 ] and the references given there, in particular Engelking [ 8 ] or Fedorchuk. [ 9 ]
The lines of a compact topological plane with a 2-dimensional point space form a family of curves homeomorphic to a circle, and this fact characterizes these planes among the topological projective planes. [ 10 ] Equivalently, the point space is a surface . Early examples not isomorphic to the classical real plane E {\displaystyle {\mathcal {E}}} have been given by Hilbert [ 3 ] [ 11 ] and Moulton . [ 12 ] The continuity properties of these examples have not been considered explicitly at that time, they may have been taken for granted. Hilbert’s construction can be modified to obtain uncountably many pairwise non-isomorphic 2 {\displaystyle 2} -dimensional compact planes. The traditional way to distinguish E {\displaystyle {\mathcal {E}}} from the other 2 {\displaystyle 2} -dimensional planes is by the validity of Desargues’s theorem or the theorem of Pappos (see, e.g., Pickert [ 13 ] for a discussion of these two configuration theorems). The latter is known to imply the former ( Hessenberg [ 14 ] ). The theorem of Desargues expresses a kind of homogeneity of the plane. In general, it holds in a projective plane if, and only if, the plane can be coordinatized by a (not necessarily commutative) field, [ 3 ] [ 15 ] [ 13 ] hence it implies that the group of automorphisms is transitive on the set of quadrangles ( 4 {\displaystyle 4} points no 3 {\displaystyle 3} of which are collinear). In the present setting, a much weaker homogeneity condition characterizes E {\displaystyle {\mathcal {E}}} :
Theorem. If the automorphism group Σ {\displaystyle \Sigma } of a 2 {\displaystyle 2} -dimensional compact plane P {\displaystyle {\mathcal {P}}} is transitive on the point set (or the line set), then Σ {\displaystyle \Sigma } has a compact subgroup Φ {\displaystyle \Phi } which is even transitive on the set of flags (=incident point-line pairs), and P {\displaystyle {\mathcal {P}}} is classical . [ 10 ]
The automorphism group Σ = Aut P {\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}} of a 2 {\displaystyle 2} -dimensional compact plane P {\displaystyle {\mathcal {P}}} , taken with the topology of uniform convergence on the point space, is a locally compact group of dimension at most 8 {\displaystyle 8} , in fact even a Lie group . All 2 {\displaystyle 2} -dimensional planes such that dim Σ ≥ 3 {\displaystyle \dim \Sigma \geq 3} can be described explicitly; [ 10 ] those with dim Σ = 4 {\displaystyle \dim \Sigma =4} are exactly the Moulton planes, the classical plane E {\displaystyle {\mathcal {E}}} is the only 2 {\displaystyle 2} -dimensional plane with dim Σ > 4 {\displaystyle \dim \Sigma >4} ; see also. [ 16 ]
The results on 2 {\displaystyle 2} -dimensional planes have been extended to compact planes of dimension > 2 {\displaystyle >2} . This is possible due to the following basic theorem:
Topology of compact planes. If the dimension of the point space P {\displaystyle P} of a compact connected projective plane is finite, then dim P = 2 m {\displaystyle \dim P=2^{m}} with m ∈ { 1 , 2 , 3 , 4 } {\displaystyle m\in \{1,2,3,4\}} . Moreover, each line is a homotopy sphere of dimension 2 m − 1 {\displaystyle 2^{m-1}} , see [ 17 ] or. [ 18 ]
Special aspects of 4-dimensional planes are treated in, [ 19 ] more recent results can be found in. [ 20 ] The lines of a 4 {\displaystyle 4} -dimensional compact plane are homeomorphic to the 2 {\displaystyle 2} -sphere; [ 21 ] in the cases m > 2 {\displaystyle m>2} the lines are not known to be manifolds, but in all examples which have been found so far the lines are spheres. A subplane B {\displaystyle {\mathcal {B}}} of a projective plane P {\displaystyle {\mathcal {P}}} is said to be a Baer subplane, [ 22 ] if each point of P {\displaystyle {\mathcal {P}}} is incident with a line of B {\displaystyle {\mathcal {B}}} and each line of P {\displaystyle {\mathcal {P}}} contains a point of B {\displaystyle {\mathcal {B}}} . A closed subplane B {\displaystyle {\mathcal {B}}} is a Baer subplane of a compact connected plane P {\displaystyle {\mathcal {P}}} if, and only if, the point space of B {\displaystyle {\mathcal {B}}} and a line of P {\displaystyle {\mathcal {P}}} have the same dimension. Hence the lines of an 8-dimensional plane P {\displaystyle {\mathcal {P}}} are homeomorphic to a sphere S 4 {\displaystyle \mathbb {S} _{4}} if P {\displaystyle {\mathcal {P}}} has a closed Baer subplane. [ 23 ]
Homogeneous planes. If P {\displaystyle {\mathcal {P}}} is a compact connected projective plane and if Σ = Aut P {\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}} is transitive on the point set of P {\displaystyle {\mathcal {P}}} , then Σ {\displaystyle \Sigma } has a flag-transitive compact subgroup Φ {\displaystyle \Phi } and P {\displaystyle {\mathcal {P}}} is classical , see [ 24 ] or. [ 25 ] In fact, Φ {\displaystyle \Phi } is an elliptic motion group. [ 26 ]
Let P {\displaystyle {\mathcal {P}}} be a compact plane of dimension 2 m , m = 2 , 3 , 4 {\displaystyle 2^{m},\;m=2,3,4} , and write Σ = Aut P {\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}} . If dim Σ > 8 , 18 , 40 {\displaystyle \dim \Sigma >8,18,40} , then P {\displaystyle {\mathcal {P}}} is classical, [ 27 ] and Aut P {\displaystyle \operatorname {Aut} {\mathcal {P}}} is a simple Lie group of dimension 16 , 35 , 78 {\displaystyle 16,35,78} respectively. All planes P {\displaystyle {\mathcal {P}}} with dim Σ = 8 , 18 , 40 {\displaystyle \dim \Sigma =8,18,40} are known explicitly. [ 28 ] The planes with dim Σ = 40 {\displaystyle \dim \Sigma =40} are exactly the projective closures of the affine planes coordinatized by a so-called mutation ( O , + , ∘ ) {\displaystyle (\mathbb {O} ,+,\circ )} of the octonion algebra ( O , + , ) {\displaystyle (\mathbb {O} ,+,\ \,)} , where the new multiplication ∘ {\displaystyle \circ } is defined as follows: choose a real number t {\displaystyle t} with 1 / 2 < t ≠ 1 {\displaystyle 1/2<t\neq 1} and put a ∘ b = t ⋅ a b + ( 1 − t ) ⋅ b a {\displaystyle a\circ b=t\cdot ab+(1-t)\cdot ba} . Vast families of planes with a group of large dimension have been discovered systematically starting from assumptions about their automorphism groups, see, e.g.,. [ 20 ] [ 29 ] [ 30 ] [ 31 ] [ 32 ] Many of them are projective closures of translation planes (affine planes admitting a sharply transitive group of automorphisms mapping each line to a parallel), cf.; [ 33 ] see also [ 34 ] for more recent results in the case m = 3 {\displaystyle m=3} and [ 30 ] for m = 4 {\displaystyle m=4} .
Subplanes of projective spaces of geometrical dimension at least 3 are necessarily Desarguesian, see [ 35 ] §1 or [ 4 ] §16 or. [ 36 ] Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field. [ 37 ]
The classical non-euclidean hyperbolic plane can be represented by the intersections of the straight lines in the real plane with an open circular disk. More generally, open (convex) parts of the classical affine planes are typical stable planes. A survey of these geometries can be found in, [ 38 ] for the 2 {\displaystyle 2} -dimensional case see also. [ 39 ]
Precisely, a stable plane S {\displaystyle {\mathcal {S}}} is a topological linear geometry ( P , L ) {\displaystyle (P,{\mathfrak {L}})} such that
Note that stability excludes geometries like the 3 {\displaystyle 3} -dimensional affine space over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } .
A stable plane S {\displaystyle {\mathcal {S}}} is a projective plane if, and only if, P {\displaystyle P} is compact. [ 40 ]
As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension 2 m − 1 {\displaystyle 2^{m-1}} , and dim P = 2 m {\displaystyle \dim P=2^{m}} with m ∈ { 1 , 2 , 3 , 4 } {\displaystyle m\in \{1,2,3,4\}} , see [ 17 ] or. [ 41 ] Moreover, the point space P {\displaystyle P} is locally contractible. [ 17 ] [ 42 ]
' Compact groups of (proper) stable planes are rather small. Let Φ d {\displaystyle \Phi _{d}} denote a maximal compact subgroup of the automorphism group of the classical d {\displaystyle d} -dimensional projective plane P d {\displaystyle {\mathcal {P}}_{d}} . Then the following theorem holds: If a d {\displaystyle d} -dimensional stable plane S {\displaystyle {\mathcal {S}}} admits a compact group Γ {\displaystyle \Gamma } of automorphisms such that dim Γ > dim Φ d − d {\displaystyle \dim \Gamma >\dim \Phi _{d}-d} , then S ≅ P d {\displaystyle {\mathcal {S}}\cong {\mathcal {P}}_{d}} , see. [ 43 ]
Flag-homogeneous stable planes. Let S = ( P , L ) {\displaystyle {\mathcal {S}}=(P,{\mathfrak {L}})} be a stable plane. If the automorphism group Aut S {\displaystyle \operatorname {Aut} {\mathcal {S}}} is flag-transitive, then S {\displaystyle {\mathcal {S}}} is a classical projective or affine plane, or S {\displaystyle {\mathcal {S}}} is isomorphic to the interior of the absolute sphere of the hyperbolic polarity of a classical plane ; see. [ 44 ] [ 45 ] [ 46 ]
In contrast to the projective case, there is an abundance of point-homogeneous stable planes, among them vast classes of translation planes, see [ 33 ] and. [ 47 ]
Affine translation planes have the following property:
More generally, a symmetric plane is a stable plane S = ( P , L ) {\displaystyle {\mathcal {S}}=(P,{\mathfrak {L}})} satisfying the aforementioned condition; see, [ 48 ] cf. [ 49 ] for a survey of these geometries. By [ 50 ] Corollary 5.5, the group Δ {\displaystyle \Delta } is a Lie group and the point space P {\displaystyle P} is a manifold. It follows that S {\displaystyle {\mathcal {S}}} is a symmetric space . By means of the Lie theory of symmetric spaces, all symmetric planes with a point set of dimension 2 {\displaystyle 2} or 4 {\displaystyle 4} have been classified. [ 48 ] [ 51 ] They are either translation planes or they are determined by a Hermitian form . An easy example is the real hyperbolic plane.
Classical models [ 52 ] are given by the plane sections of a quadratic surface S {\displaystyle S} in real projective 3 {\displaystyle 3} -space; if S {\displaystyle S} is a sphere, the geometry is called a Möbius plane . [ 39 ] The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical Minkowski plane , cf. [ 53 ] for generalizations. If S {\displaystyle S} is an elliptic cone without its vertex, the geometry is called a Laguerre plane . Collectively these planes are sometimes referred to as Benz planes . A topological Benz plane is classical, if each point has a neighbourhood which is isomorphic to some open piece of the corresponding classical Benz plane . [ 54 ]
Möbius planes consist of a family C {\displaystyle {\mathfrak {C}}} of circles, which are topological 1-spheres, on the 2 {\displaystyle 2} -sphere S {\displaystyle S} such that for each point p {\displaystyle p} the derived structure ( S ∖ { p } , { C ∖ { p } ∣ p ∈ C ∈ C } ) {\displaystyle (S\setminus \{p\},\{C\setminus \{p\}\mid p\in C\in {\mathfrak {C}}\})} is a topological affine plane. [ 55 ] In particular, any 3 {\displaystyle 3} distinct points are joined by a unique circle. The circle space C {\displaystyle {\mathfrak {C}}} is then homeomorphic to real projective 3 {\displaystyle 3} -space with one point deleted. [ 56 ] A large class of examples is given by the plane sections of an egg-like surface in real 3 {\displaystyle 3} -space.
If the automorphism group Σ {\displaystyle \Sigma } of a Möbius plane is transitive on the point set S {\displaystyle S} or on the set C {\displaystyle {\mathfrak {C}}} of circles, or if dim Σ ≥ 4 {\displaystyle \dim \Sigma \geq 4} , then ( S , C ) {\displaystyle (S,{\mathfrak {C}})} is classical and dim Σ = 6 {\displaystyle \dim \Sigma =6} , see. [ 57 ] [ 58 ]
In contrast to compact projective planes there are no topological Möbius planes with circles of dimension > 1 {\displaystyle >1} , in particular no compact Möbius planes with a 4 {\displaystyle 4} -dimensional point space. [ 59 ] All 2-dimensional Möbius planes such that dim Σ ≥ 3 {\displaystyle \dim \Sigma \geq 3} can be described explicitly. [ 60 ] [ 61 ]
The classical model of a Laguerre plane consists of a circular cylindrical surface C {\displaystyle C} in real 3 {\displaystyle 3} -space R 3 {\displaystyle \mathbb {R} ^{3}} as point set and the compact plane sections of C {\displaystyle C} as circles. Pairs of points which are not joined by a circle are called parallel . Let P {\displaystyle P} denote a class of parallel points. Then C ∖ P {\displaystyle C\setminus P} is a plane R 2 {\displaystyle \mathbb {R} ^{2}} , the circles can be represented in this plane by parabolas of the form y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} .
In an analogous way, the classical 4 {\displaystyle 4} -dimensional Laguerre plane is related to the geometry of complex quadratic polynomials. In general, the axioms of a locally compact connected Laguerre plane require that the derived planes embed into compact projective planes of finite dimension. A circle not passing through the point of derivation induces an oval in the derived projective plane. By [ 62 ] or, [ 63 ] circles are homeomorphic to spheres of dimension 1 {\displaystyle 1} or 2 {\displaystyle 2} . Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder C {\displaystyle C} or it is a 4 {\displaystyle 4} -dimensional manifold, cf. [ 64 ] A large class of 2 {\displaystyle 2} -dimensional examples, called ovoidal Laguerre planes, is given by the plane sections of a cylinder in real 3-space whose base is an oval in R 2 {\displaystyle \mathbb {R} ^{2}} .
The automorphism group of a 2 d {\displaystyle 2d} -dimensional Laguerre plane ( d = 1 , 2 {\displaystyle d=1,2} ) is a Lie group with respect to the topology of uniform convergence on compact subsets of the point space; furthermore, this group has dimension at most 7 d {\displaystyle 7d} . All automorphisms of a Laguerre plane which fix each parallel class form a normal subgroup, the kernel of the full automorphism group. The 2 {\displaystyle 2} -dimensional Laguerre planes with dim Σ = 5 {\displaystyle \dim \Sigma =5} are exactly the ovoidal planes over proper skew parabolae. [ 65 ] The classical 2 d {\displaystyle 2d} -dimensional Laguerre planes are the only ones such that dim Σ > 5 d {\displaystyle \dim \Sigma >5d} , see, [ 66 ] cf. also. [ 67 ]
If the automorphism group Σ {\displaystyle \Sigma } of a 2 d {\displaystyle 2d} -dimensional Laguerre plane L {\displaystyle {\mathcal {L}}} is transitive on the set of parallel classes, and if the kernel T ◃ Σ {\displaystyle T\triangleleft \Sigma } is transitive on the set of circles, then L {\displaystyle {\mathcal {L}}} is classical , see [ 68 ] [ 67 ] 2.1,2.
However, transitivity of the automorphism group on the set of circles does not suffice to characterize the classical model among the 2 d {\displaystyle 2d} -dimensional Laguerre planes.
The classical model of a Minkowski plane has the torus S 1 × S 1 {\displaystyle \mathbb {S} _{1}\times \mathbb {S} _{1}} as point space, circles are the graphs of real fractional linear maps on S 1 = R ∪ { ∞ } {\displaystyle \mathbb {S} _{1}=\mathbb {R} \cup \{\infty \}} . As with Laguerre planes, the point space of a locally compact connected Minkowski plane is 1 {\displaystyle 1} - or 2 {\displaystyle 2} -dimensional; the point space is then homeomorphic to a torus or to S 2 × S 2 {\displaystyle \mathbb {S} _{2}\times \mathbb {S} _{2}} , see. [ 69 ]
If the automorphism group Σ {\displaystyle \Sigma } of a Minkowski plane M {\displaystyle {\mathcal {M}}} of dimension 2 d {\displaystyle 2d} is flag-transitive, then M {\displaystyle {\mathcal {M}}} is classical . [ 70 ]
The automorphism group of a 2 d {\displaystyle 2d} -dimensional Minkowski plane is a Lie group of dimension at most 6 d {\displaystyle 6d} . All 2 {\displaystyle 2} -dimensional Minkowski planes such that dim Σ ≥ 4 {\displaystyle \dim \Sigma \geq 4} can be described explicitly. [ 71 ] The classical 2 d {\displaystyle 2d} -dimensional Minkowski plane is the only one with dim Σ > 4 d {\displaystyle \dim \Sigma >4d} , see. [ 72 ] | https://en.wikipedia.org/wiki/Topological_geometry |
In the fields of chemical graph theory , molecular topology , and mathematical chemistry , a topological index , also known as a connectivity index , is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. [ 1 ] Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant . Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure . [ 2 ]
Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index. [ 3 ] [ 4 ] As a result, the topological index can be defined as two-dimensional descriptors that can be easily calculated from the molecular graphs, and do not depend on the way the graph is depicted or labeled and no need of energy minimization of the chemical structure.
The simplest topological indices do not recognize double bonds and atom types (C, N, O etc.) and ignore hydrogen atoms ("hydrogen suppressed") and defined for connected undirected molecular graphs only. [ 5 ] More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index. [ 6 ] Other examples include the Wiener index , Randić's molecular connectivity index , Balaban’s J index , [ 7 ] and the TAU descriptors. [ 8 ] [ 9 ] The extended topochemical atom (ETA) [ 10 ] indices have been developed based on refinement of TAU descriptors.
Hosoya index and Wiener index are global (integral) indices to describe entire molecule, Bonchev and Polansky introduced local (differential) index for every atom in a molecule. [ 5 ] Another examples of local indices are modifications of Hosoya index. [ 11 ]
A topological index may have the same value for a subset of different molecular graphs, i.e. the index is unable to discriminate the graphs from this subset. The discrimination capability is very important characteristic of topological index. To increase the discrimination capability a few topological indices may be combined to superindex . [ 12 ]
Computational complexity is another important characteristic of topological index. The Wiener index, Randic's molecular connectivity index, Balaban's J index may be calculated by fast algorithms, in contrast to Hosoya index and its modifications for which non-exponential algorithms are unknown. [ 11 ]
QSARs represent predictive models derived from application of statistical tools correlating biological activity (including desirable therapeutic effect and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties . QSARs are being applied in many disciplines for example risk assessment , toxicity prediction, and regulatory decisions [ 13 ] in addition to drug discovery and lead optimization . [ 14 ]
For example, ETA indices have been applied in the development of predictive QSAR/QSPR/QSTR models. [ 15 ] | https://en.wikipedia.org/wiki/Topological_index |
A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor , [ 3 ] meaning that electrons can only move along the surface of the material.
A topological insulator is an insulator for the same reason a " trivial " (ordinary) insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. [ 4 ] The topological insulator cannot be continuously transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state. Thus, due to the continuity of the underlying field, the border of a topological insulator with a trivial insulator (including vacuum , which is topologically trivial) is forced to support conducting edge states . [ 5 ]
Since this results from a global property of the topological insulator's band structure , local (symmetry-preserving) perturbations cannot damage this surface state. [ 6 ] This is unique to topological insulators: while ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property.
This leads to a more formal definition of a topological insulator: an insulator which cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state. [ 5 ] In other words, topological insulators and trivial insulators are separate regions in the phase diagram , connected only by conducting phases. In this way, topological insulators provide an example of a state of matter not described by the Landau symmetry-breaking theory that defines ordinary states of matter. [ 6 ]
The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries , and can be classified using the so-called periodic table of topological insulators . Some combinations of dimension and symmetries forbid topological insulators completely. [ 7 ] All topological insulators have at least U(1) symmetry from particle number conservation, and often have time-reversal symmetry from the absence of a magnetic field. In this way, topological insulators are an example of symmetry-protected topological order . [ 8 ] So-called "topological invariants", taking values in Z 2 {\displaystyle \mathbb {Z} _{2}} or Z {\displaystyle \mathbb {Z} } , allow classification of insulators as trivial or topological, and can be computed by various methods. [ 7 ]
The surface states of topological insulators can have exotic properties. For example, in time-reversal symmetric 3D topological insulators, surface states have their spin locked at a right-angle to their momentum (spin-momentum locking). At a given energy the only other available electronic states have different spin, so "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic.
Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media. There exist photonic , [ 9 ] magnetic , [ 10 ] and acoustic [ 11 ] topological insulators, among others.
The first models of 3D topological insulators were proposed by B. A. Volkov and O. A. Pankratov in 1985, [ 12 ] and subsequently by Pankratov, S. V. Pakhomov, and Volkov in 1987. [ 13 ] Gapless 2D Dirac states were shown to exist at the band inversion contact in PbTe / SnTe [ 12 ] and HgTe / CdTe [ 13 ] heterostructures.
Existence of interface Dirac states in HgTe/CdTe was experimentally verified by Laurens W. Molenkamp's group in 2D topological insulators in 2007. [ 14 ]
Later sets of theoretical models for the 2D topological insulator (also known as the quantum spin Hall insulators) were proposed by Charles L. Kane and Eugene J. Mele in 2005, [ 15 ] and also by B. Andrei Bernevig and Shoucheng Zhang in 2006. [ 16 ] The Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariant was constructed and the importance of the time reversal symmetry was clarified in the work by Kane and Mele. [ 17 ] Subsequently, Bernevig, Taylor L. Hughes and Zhang made a theoretical prediction that 2D topological insulator with one-dimensional (1D) helical edge states would be realized in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride. [ 18 ] The transport due to 1D helical edge states was indeed observed in the experiments by Molenkamp's group in 2007. [ 14 ]
Although the topological classification and the importance of time-reversal symmetry was pointed in the 2000s, all the necessary ingredients and physics of topological insulators were already understood in the works from the 1980s.
In 2007, it was predicted that 3D topological insulators might be found in binary compounds involving bismuth , [ 19 ] [ 20 ] [ 21 ] [ 22 ] and in particular "strong topological insulators" exist that cannot be reduced to multiple copies of the quantum spin Hall state . [ 23 ]
2D Topological insulators were first realized in system containing HgTe quantum wells sandwiched between cadmium telluride in 2007.
The first 3D topological insulator to be realized experimentally was Bi 1 − x Sb x . [ 24 ] [ 25 ] [ 26 ] Bismuth in its pure state, is a semimetal with a small electronic band gap. Using angle-resolved photoemission spectroscopy , and many other measurements, it was observed that Bi 1 − x Sb x alloy exhibits an odd surface state (SS) crossing between any pair of Kramers points and the bulk features massive Dirac fermions. [ 25 ] Additionally, bulk Bi 1 − x Sb x has been predicted to have 3D Dirac particles . [ 27 ] This prediction is of particular interest due to the observation of charge quantum Hall fractionalization in 2D graphene [ 28 ] and pure bismuth. [ 29 ]
Shortly thereafter symmetry-protected surface states were also observed in pure antimony , bismuth selenide , bismuth telluride and antimony telluride using angle-resolved photoemission spectroscopy (ARPES). [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ] and bismuth selenide . [ 34 ] [ 35 ] Many semiconductors within the large family of Heusler materials are now believed to exhibit topological surface states. [ 36 ] [ 37 ] In some of these materials, the Fermi level actually falls in either the conduction or valence bands due to naturally-occurring defects, and must be pushed into the bulk gap by doping or gating. [ 38 ] [ 39 ] The surface states of a 3D topological insulator is a new type of two-dimensional electron gas (2DEG) where the electron's spin is locked to its linear momentum. [ 31 ]
Fully bulk-insulating or intrinsic 3D topological insulator states exist in Bi-based materials as demonstrated in surface transport measurements. [ 40 ] In a new Bi based chalcogenide (Bi 1.1 Sb 0.9 Te 2 S) with slightly Sn - doping, exhibits an intrinsic semiconductor behavior with Fermi energy and Dirac point lie in the bulk gap and the surface states were probed by the charge transport experiments. [ 41 ]
It was proposed in 2008 and 2009 that topological insulators are best understood not as surface conductors per se, but as bulk 3D magnetoelectrics with a quantized magnetoelectric effect. [ 42 ] [ 43 ] This can be revealed by placing topological insulators in magnetic field. The effect can be described in language similar to that of the hypothetical axion particle of particle physics. [ 44 ] The effect was reported by researchers at Johns Hopkins University and Rutgers University using THz spectroscopy who showed that the Faraday rotation was quantized by the fine structure constant. [ 45 ]
In 2012, topological Kondo insulators were identified in samarium hexaboride , which is a bulk insulator at low temperatures. [ 46 ] [ 47 ]
In 2014, it was shown that magnetic components, like the ones in spin-torque computer memory , can be manipulated by topological insulators. [ 48 ] [ 49 ] The effect is related to metal–insulator transitions ( Bose–Hubbard model ). [ citation needed ]
Topological insulators are challenging to synthesize, and limited in topological phases accessible with solid-state materials. [ 50 ] This has motivated the search for topological phases on the systems that simulate the same principles underlying topological insulators. Discrete time quantum walks (DTQW) have been proposed for making Floquet topological insulators (FTI). This periodically driven system simulates an effective ( Floquet ) Hamiltonian that is topologically nontrivial. [ 51 ] This system replicates the effective Hamiltonians from all universal classes of 1- to 3-D topological insulators. [ 52 ] [ 53 ] [ 54 ] [ 55 ] Interestingly, topological properties of Floquet topological insulators could be controlled via an external periodic drive rather than an external magnetic field. An atomic lattice empowered by distance selective Rydberg interaction could simulate different classes of FTI over a couple of hundred sites and steps in 1, 2 or 3 dimensions. [ 55 ] The long-range interaction allows designing topologically ordered periodic boundary conditions, further enriching the realizable topological phases. [ 55 ]
Spin-momentum locking [ 31 ] in the topological insulator allows symmetry-protected surface states to host Majorana particles if superconductivity is induced on the surface of 3D topological insulators via proximity effects. [ 56 ] (Note that Majorana zero-mode can also appear without topological insulators. [ 57 ] ) The non-trivialness of topological insulators is encoded in the existence of a gas of helical Dirac fermions . Dirac particles which behave like massless relativistic fermions have been observed in 3D topological insulators. Note that the gapless surface states of topological insulators differ from those in the quantum Hall effect : the gapless surface states of topological insulators are symmetry-protected (i.e., not topological), while the gapless surface states in quantum Hall effect are topological (i.e., robust against any local perturbations that can break all the symmetries). The Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the Z 2 {\displaystyle \mathbb {Z} _{2}} invariants. An experimental method to measure Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants was demonstrated which provide a measure of the Z 2 {\displaystyle \mathbb {Z} _{2}} topological order. [ 58 ] (Note that the term Z 2 {\displaystyle \mathbb {Z} _{2}} topological order has also been used to describe the topological order with emergent Z 2 {\displaystyle \mathbb {Z} _{2}} gauge theory discovered in 1991. [ 59 ] [ 60 ] ) More generally (in what is known as the ten-fold way ) for each spatial dimensionality, each of the ten Altland—Zirnbauer symmetry classes of random Hamiltonians labelled by the type of discrete symmetry (time-reversal symmetry, particle-hole symmetry, and chiral symmetry) has a corresponding group of topological invariants (either Z {\displaystyle \mathbb {Z} } , Z 2 {\displaystyle \mathbb {Z} _{2}} or trivial) as described by the periodic table of topological invariants . [ 61 ]
The most promising applications of topological insulators are spintronic devices and dissipationless transistors for quantum computers based on the quantum Hall effect [ 14 ] and quantum anomalous Hall effect . [ 62 ] In addition, topological insulator materials have also found practical applications in advanced magnetoelectronic and optoelectronic devices. [ 63 ] [ 64 ]
Some of the most well-known topological insulators are also thermoelectric materials , such as Bi 2 Te 3 and its alloys with Bi 2 Se 3 (n-type thermoelectrics) and Sb 2 Te 3 (p-type thermoelectrics). [ 65 ] High thermoelectric power conversion efficiency is realized in materials with low thermal conductivity, high electrical conductivity, and high Seebeck coefficient (i.e., the incremental change in voltage due to an incremental change in temperature). Topological insulators are often composed of heavy atoms, which tends to lower thermal conductivity and are therefore beneficial for thermoelectrics. A recent study also showed that good electrical characteristics (i.e., high electrical conductivity and Seebeck coefficient) can arise in topological insulators due to warping of the bulk band structure, which is driven by band inversion. [ 66 ] Often, the electrical conductivity and Seebeck coefficient are conflicting properties of thermoelectrics and difficult to optimize simultaneously. Band warping, induced by band inversion in a topological insulator, can mediate the two properties by reducing the effective mass of electrons/holes and increasing the valley degeneracy (i.e., the number of electronic bands that are contributing to charge transport). As a result, topological insulators are generally interesting candidates for thermoelectric applications.
Topological insulators can be grown using different methods such as metal-organic chemical vapor deposition (MOCVD), [ 67 ]
physical vapor deposition (PVD), [ 68 ] solvothermal synthesis, [ 69 ] sonochemical technique [ 70 ] and molecular beam epitaxy
(MBE). [ 34 ] MBE has so far been the most common experimental technique. The growth of thin film topological insulators is governed by weak van der Waals interactions . [ 71 ] The weak interaction allows to exfoliate the thin film from bulk crystal with a clean and perfect surface. The van der Waals interactions in epitaxy also known as van der Waals epitaxy (VDWE), is a phenomenon governed by weak van der Waals interactions between layered materials of different or same elements [ 72 ] in which the materials are stacked on top of each other. This approach allows the growth of layered topological insulators on other substrates for heterostructure and integrated circuits . [ 72 ]
MBE growth of topological insulators
Molecular beam epitaxy (MBE) is an epitaxy method for the growth of a crystalline material on a crystalline substrate to form an ordered layer. MBE is performed in high vacuum or ultra-high vacuum , the elements are heated in different electron beam evaporators until they sublime . The gaseous elements then condense on the wafer where they react with each other to form single crystals .
MBE is an appropriate technique for the growth of high quality single-crystal films. In order to avoid a huge lattice mismatch and defects at the interface, the substrate and thin film are expected to have similar lattice constants. MBE has an advantage over other methods due to the fact that the synthesis is performed in high vacuum hence resulting in less contamination. Additionally, lattice defect is reduced due to the ability to influence the growth rate and the ratio of species of source materials present at the substrate interface. [ 73 ] Furthermore, in MBE, samples can be grown layer by layer which results in flat surfaces with smooth interface for engineered heterostructures. Moreover, MBE synthesis technique benefits from the ease of moving a topological insulator sample from the growth chamber to a characterization chamber such as angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) studies. [ 74 ]
Due to the weak van der Waals bonding, which relaxes the lattice-matching condition, TI can be grown on a wide variety of substrates [ 75 ] such as Si(111), [ 76 ] [ 77 ] Al 2 O 3 , GaAs (111), [ 78 ]
InP (111), CdS (0001) and Y 3 Fe 5 O 12 .
The physical vapor deposition (PVD) technique does not suffer from the disadvantages of the exfoliation method and, at the same time, it is much simpler and cheaper than the fully controlled growth by molecular-beam epitaxy. The PVD method enables a reproducible synthesis of single crystals of various layered quasi-two-dimensional materials including topological insulators (i.e., Bi 2 Se 3 , Bi 2 Te 3 ). [ 79 ] The resulted single crystals have a well-defined crystallographic orientation; their composition, thickness, size, and the surface density on the desired substrate can be controlled.
The thickness control is particularly important for 3D TIs in which the trivial (bulky) electronic channels usually dominate the transport properties and mask the response of the topological (surface) modes. By reducing the thickness, one lowers the contribution of trivial bulk channels into the total conduction, thus forcing the topological modes to carry the electric current. [ 80 ]
Thus far, the field of topological insulators has been focused on bismuth and antimony chalcogenide based materials such as Bi 2 Se 3 , Bi 2 Te 3 , Sb 2 Te 3 or Bi 1 − x Sb x , Bi 1.1 Sb 0.9 Te 2 S. [ 41 ] The choice of chalcogenides is related to the van der Waals relaxation of the lattice matching strength which restricts the number of materials and substrates. [ 73 ] Bismuth chalcogenides have been studied extensively for TIs and their applications in thermoelectric materials . The van der Waals interaction in TIs exhibit important features due to low surface energy. For instance, the surface of Bi 2 Te 3 is usually terminated by Te due to its low surface energy. [ 34 ]
Bismuth chalcogenides have been successfully grown on different substrates. In particular, Si has been a good substrate for the successful growth of Bi 2 Te 3 . However, the use of sapphire as substrate has not been so encouraging due to a large mismatch of about 15%. [ 81 ] The selection of appropriate substrate can improve the overall properties of TI. The use of buffer layer can reduce the lattice match hence improving the electrical properties of TI. [ 81 ] Bi 2 Se 3 can be grown on top of various Bi 2 − x In x Se 3 buffers. Table 1 shows Bi 2 Se 3 , Bi 2 Te 3 , Sb 2 Te 3 on different substrates and the resulting lattice mismatch. Generally, regardless of the substrate used, the resulting films have a textured surface that is characterized by pyramidal single-crystal domains with quintuple-layer steps. The size and relative proportion of these pyramidal domains vary with factors that include film thickness, lattice mismatch with the substrate and interfacial chemistry-dependent film nucleation. The synthesis of thin films have the stoichiometry problem due to the high vapor pressures of the elements. Thus, binary tetradymites are extrinsically doped as n-type ( Bi 2 Se 3 , Bi 2 Te 3 ) or p-type ( Sb 2 Te 3 ). [ 73 ] Due to the weak van der Waals bonding, graphene is one of the preferred substrates for TI growth despite the large lattice mismatch.
The first step of topological insulators identification takes place right after synthesis, meaning without breaking the vacuum and moving the sample to an atmosphere. That could be done by using angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) techniques. [ 74 ] Further measurements includes structural and chemical probes such as X-ray diffraction and energy-dispersive spectroscopy but depending on the sample quality, the lack of sensitivity could remain. Transport measurements cannot uniquely pinpoint the Z 2 {\displaystyle \mathbb {Z} _{2}} topology by definition of the state.
Bloch's theorem allows a full characterization of the wave propagation properties of a material by assigning a matrix to each wave vector in the Brillouin zone .
Mathematically, this assignment creates a vector bundle . Different materials will have different wave propagation properties, and thus different vector bundles. If we consider all insulators (materials with a band gap), this creates a space of vector bundles. It is the topology of this space (modulo trivial bands) from which the "topology" in topological insulators arises. [ 7 ]
Specifically, the number of connected components of the space indicates how many different "islands" of insulators exist amongst the metallic states. Insulators in the connected component containing the vacuum state are identified as "trivial", and all other insulators as "topological". The connected component in which an insulator lies can be identified with a number, referred to as a "topological invariant". [ 7 ]
This space can be restricted under the presence of symmetries, changing the resulting topology. Although unitary symmetries are usually significant in quantum mechanics, they have no effect on the topology here. [ 82 ] Instead, the three symmetries typically considered are time-reversal symmetry, particle-hole symmetry, and chiral symmetry (also called sublattice symmetry). Mathematically, these are represented as, respectively: an anti-unitary operator which commutes with the Hamiltonian ; an anti-unitary operator which anti-commutes with the Hamiltonian; and a unitary operator which anti-commutes with the Hamiltonian. All combinations of the three together with each spatial dimension result in the so-called periodic table of topological insulators . [ 7 ]
The field of topological insulators still needs to be developed. The best bismuth chalcogenide topological insulators have about 10 meV bandgap variation due to the charge. Further development should focus on the examination of both: the presence of high-symmetry electronic bands and simply synthesized materials. One of the candidates is half-Heusler compounds . [ 74 ] These crystal structures can consist of a large number of elements. Band structures and energy gaps are very sensitive to the valence configuration; because of the increased likelihood of intersite exchange and disorder, they are also very sensitive to specific crystalline configurations. A nontrivial band structure that exhibits band ordering analogous to that of the known 2D and 3D TI materials was predicted in a variety of 18-electron half-Heusler compounds using first-principles calculations. [ 83 ] These materials have not yet shown any sign of intrinsic topological insulator behavior in actual experiments. | https://en.wikipedia.org/wiki/Topological_insulator |
A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor , [ 3 ] meaning that electrons can only move along the surface of the material.
A topological insulator is an insulator for the same reason a " trivial " (ordinary) insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. [ 4 ] The topological insulator cannot be continuously transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state. Thus, due to the continuity of the underlying field, the border of a topological insulator with a trivial insulator (including vacuum , which is topologically trivial) is forced to support conducting edge states . [ 5 ]
Since this results from a global property of the topological insulator's band structure , local (symmetry-preserving) perturbations cannot damage this surface state. [ 6 ] This is unique to topological insulators: while ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property.
This leads to a more formal definition of a topological insulator: an insulator which cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state. [ 5 ] In other words, topological insulators and trivial insulators are separate regions in the phase diagram , connected only by conducting phases. In this way, topological insulators provide an example of a state of matter not described by the Landau symmetry-breaking theory that defines ordinary states of matter. [ 6 ]
The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries , and can be classified using the so-called periodic table of topological insulators . Some combinations of dimension and symmetries forbid topological insulators completely. [ 7 ] All topological insulators have at least U(1) symmetry from particle number conservation, and often have time-reversal symmetry from the absence of a magnetic field. In this way, topological insulators are an example of symmetry-protected topological order . [ 8 ] So-called "topological invariants", taking values in Z 2 {\displaystyle \mathbb {Z} _{2}} or Z {\displaystyle \mathbb {Z} } , allow classification of insulators as trivial or topological, and can be computed by various methods. [ 7 ]
The surface states of topological insulators can have exotic properties. For example, in time-reversal symmetric 3D topological insulators, surface states have their spin locked at a right-angle to their momentum (spin-momentum locking). At a given energy the only other available electronic states have different spin, so "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic.
Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media. There exist photonic , [ 9 ] magnetic , [ 10 ] and acoustic [ 11 ] topological insulators, among others.
The first models of 3D topological insulators were proposed by B. A. Volkov and O. A. Pankratov in 1985, [ 12 ] and subsequently by Pankratov, S. V. Pakhomov, and Volkov in 1987. [ 13 ] Gapless 2D Dirac states were shown to exist at the band inversion contact in PbTe / SnTe [ 12 ] and HgTe / CdTe [ 13 ] heterostructures.
Existence of interface Dirac states in HgTe/CdTe was experimentally verified by Laurens W. Molenkamp's group in 2D topological insulators in 2007. [ 14 ]
Later sets of theoretical models for the 2D topological insulator (also known as the quantum spin Hall insulators) were proposed by Charles L. Kane and Eugene J. Mele in 2005, [ 15 ] and also by B. Andrei Bernevig and Shoucheng Zhang in 2006. [ 16 ] The Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariant was constructed and the importance of the time reversal symmetry was clarified in the work by Kane and Mele. [ 17 ] Subsequently, Bernevig, Taylor L. Hughes and Zhang made a theoretical prediction that 2D topological insulator with one-dimensional (1D) helical edge states would be realized in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride. [ 18 ] The transport due to 1D helical edge states was indeed observed in the experiments by Molenkamp's group in 2007. [ 14 ]
Although the topological classification and the importance of time-reversal symmetry was pointed in the 2000s, all the necessary ingredients and physics of topological insulators were already understood in the works from the 1980s.
In 2007, it was predicted that 3D topological insulators might be found in binary compounds involving bismuth , [ 19 ] [ 20 ] [ 21 ] [ 22 ] and in particular "strong topological insulators" exist that cannot be reduced to multiple copies of the quantum spin Hall state . [ 23 ]
2D Topological insulators were first realized in system containing HgTe quantum wells sandwiched between cadmium telluride in 2007.
The first 3D topological insulator to be realized experimentally was Bi 1 − x Sb x . [ 24 ] [ 25 ] [ 26 ] Bismuth in its pure state, is a semimetal with a small electronic band gap. Using angle-resolved photoemission spectroscopy , and many other measurements, it was observed that Bi 1 − x Sb x alloy exhibits an odd surface state (SS) crossing between any pair of Kramers points and the bulk features massive Dirac fermions. [ 25 ] Additionally, bulk Bi 1 − x Sb x has been predicted to have 3D Dirac particles . [ 27 ] This prediction is of particular interest due to the observation of charge quantum Hall fractionalization in 2D graphene [ 28 ] and pure bismuth. [ 29 ]
Shortly thereafter symmetry-protected surface states were also observed in pure antimony , bismuth selenide , bismuth telluride and antimony telluride using angle-resolved photoemission spectroscopy (ARPES). [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ] and bismuth selenide . [ 34 ] [ 35 ] Many semiconductors within the large family of Heusler materials are now believed to exhibit topological surface states. [ 36 ] [ 37 ] In some of these materials, the Fermi level actually falls in either the conduction or valence bands due to naturally-occurring defects, and must be pushed into the bulk gap by doping or gating. [ 38 ] [ 39 ] The surface states of a 3D topological insulator is a new type of two-dimensional electron gas (2DEG) where the electron's spin is locked to its linear momentum. [ 31 ]
Fully bulk-insulating or intrinsic 3D topological insulator states exist in Bi-based materials as demonstrated in surface transport measurements. [ 40 ] In a new Bi based chalcogenide (Bi 1.1 Sb 0.9 Te 2 S) with slightly Sn - doping, exhibits an intrinsic semiconductor behavior with Fermi energy and Dirac point lie in the bulk gap and the surface states were probed by the charge transport experiments. [ 41 ]
It was proposed in 2008 and 2009 that topological insulators are best understood not as surface conductors per se, but as bulk 3D magnetoelectrics with a quantized magnetoelectric effect. [ 42 ] [ 43 ] This can be revealed by placing topological insulators in magnetic field. The effect can be described in language similar to that of the hypothetical axion particle of particle physics. [ 44 ] The effect was reported by researchers at Johns Hopkins University and Rutgers University using THz spectroscopy who showed that the Faraday rotation was quantized by the fine structure constant. [ 45 ]
In 2012, topological Kondo insulators were identified in samarium hexaboride , which is a bulk insulator at low temperatures. [ 46 ] [ 47 ]
In 2014, it was shown that magnetic components, like the ones in spin-torque computer memory , can be manipulated by topological insulators. [ 48 ] [ 49 ] The effect is related to metal–insulator transitions ( Bose–Hubbard model ). [ citation needed ]
Topological insulators are challenging to synthesize, and limited in topological phases accessible with solid-state materials. [ 50 ] This has motivated the search for topological phases on the systems that simulate the same principles underlying topological insulators. Discrete time quantum walks (DTQW) have been proposed for making Floquet topological insulators (FTI). This periodically driven system simulates an effective ( Floquet ) Hamiltonian that is topologically nontrivial. [ 51 ] This system replicates the effective Hamiltonians from all universal classes of 1- to 3-D topological insulators. [ 52 ] [ 53 ] [ 54 ] [ 55 ] Interestingly, topological properties of Floquet topological insulators could be controlled via an external periodic drive rather than an external magnetic field. An atomic lattice empowered by distance selective Rydberg interaction could simulate different classes of FTI over a couple of hundred sites and steps in 1, 2 or 3 dimensions. [ 55 ] The long-range interaction allows designing topologically ordered periodic boundary conditions, further enriching the realizable topological phases. [ 55 ]
Spin-momentum locking [ 31 ] in the topological insulator allows symmetry-protected surface states to host Majorana particles if superconductivity is induced on the surface of 3D topological insulators via proximity effects. [ 56 ] (Note that Majorana zero-mode can also appear without topological insulators. [ 57 ] ) The non-trivialness of topological insulators is encoded in the existence of a gas of helical Dirac fermions . Dirac particles which behave like massless relativistic fermions have been observed in 3D topological insulators. Note that the gapless surface states of topological insulators differ from those in the quantum Hall effect : the gapless surface states of topological insulators are symmetry-protected (i.e., not topological), while the gapless surface states in quantum Hall effect are topological (i.e., robust against any local perturbations that can break all the symmetries). The Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the Z 2 {\displaystyle \mathbb {Z} _{2}} invariants. An experimental method to measure Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants was demonstrated which provide a measure of the Z 2 {\displaystyle \mathbb {Z} _{2}} topological order. [ 58 ] (Note that the term Z 2 {\displaystyle \mathbb {Z} _{2}} topological order has also been used to describe the topological order with emergent Z 2 {\displaystyle \mathbb {Z} _{2}} gauge theory discovered in 1991. [ 59 ] [ 60 ] ) More generally (in what is known as the ten-fold way ) for each spatial dimensionality, each of the ten Altland—Zirnbauer symmetry classes of random Hamiltonians labelled by the type of discrete symmetry (time-reversal symmetry, particle-hole symmetry, and chiral symmetry) has a corresponding group of topological invariants (either Z {\displaystyle \mathbb {Z} } , Z 2 {\displaystyle \mathbb {Z} _{2}} or trivial) as described by the periodic table of topological invariants . [ 61 ]
The most promising applications of topological insulators are spintronic devices and dissipationless transistors for quantum computers based on the quantum Hall effect [ 14 ] and quantum anomalous Hall effect . [ 62 ] In addition, topological insulator materials have also found practical applications in advanced magnetoelectronic and optoelectronic devices. [ 63 ] [ 64 ]
Some of the most well-known topological insulators are also thermoelectric materials , such as Bi 2 Te 3 and its alloys with Bi 2 Se 3 (n-type thermoelectrics) and Sb 2 Te 3 (p-type thermoelectrics). [ 65 ] High thermoelectric power conversion efficiency is realized in materials with low thermal conductivity, high electrical conductivity, and high Seebeck coefficient (i.e., the incremental change in voltage due to an incremental change in temperature). Topological insulators are often composed of heavy atoms, which tends to lower thermal conductivity and are therefore beneficial for thermoelectrics. A recent study also showed that good electrical characteristics (i.e., high electrical conductivity and Seebeck coefficient) can arise in topological insulators due to warping of the bulk band structure, which is driven by band inversion. [ 66 ] Often, the electrical conductivity and Seebeck coefficient are conflicting properties of thermoelectrics and difficult to optimize simultaneously. Band warping, induced by band inversion in a topological insulator, can mediate the two properties by reducing the effective mass of electrons/holes and increasing the valley degeneracy (i.e., the number of electronic bands that are contributing to charge transport). As a result, topological insulators are generally interesting candidates for thermoelectric applications.
Topological insulators can be grown using different methods such as metal-organic chemical vapor deposition (MOCVD), [ 67 ]
physical vapor deposition (PVD), [ 68 ] solvothermal synthesis, [ 69 ] sonochemical technique [ 70 ] and molecular beam epitaxy
(MBE). [ 34 ] MBE has so far been the most common experimental technique. The growth of thin film topological insulators is governed by weak van der Waals interactions . [ 71 ] The weak interaction allows to exfoliate the thin film from bulk crystal with a clean and perfect surface. The van der Waals interactions in epitaxy also known as van der Waals epitaxy (VDWE), is a phenomenon governed by weak van der Waals interactions between layered materials of different or same elements [ 72 ] in which the materials are stacked on top of each other. This approach allows the growth of layered topological insulators on other substrates for heterostructure and integrated circuits . [ 72 ]
MBE growth of topological insulators
Molecular beam epitaxy (MBE) is an epitaxy method for the growth of a crystalline material on a crystalline substrate to form an ordered layer. MBE is performed in high vacuum or ultra-high vacuum , the elements are heated in different electron beam evaporators until they sublime . The gaseous elements then condense on the wafer where they react with each other to form single crystals .
MBE is an appropriate technique for the growth of high quality single-crystal films. In order to avoid a huge lattice mismatch and defects at the interface, the substrate and thin film are expected to have similar lattice constants. MBE has an advantage over other methods due to the fact that the synthesis is performed in high vacuum hence resulting in less contamination. Additionally, lattice defect is reduced due to the ability to influence the growth rate and the ratio of species of source materials present at the substrate interface. [ 73 ] Furthermore, in MBE, samples can be grown layer by layer which results in flat surfaces with smooth interface for engineered heterostructures. Moreover, MBE synthesis technique benefits from the ease of moving a topological insulator sample from the growth chamber to a characterization chamber such as angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) studies. [ 74 ]
Due to the weak van der Waals bonding, which relaxes the lattice-matching condition, TI can be grown on a wide variety of substrates [ 75 ] such as Si(111), [ 76 ] [ 77 ] Al 2 O 3 , GaAs (111), [ 78 ]
InP (111), CdS (0001) and Y 3 Fe 5 O 12 .
The physical vapor deposition (PVD) technique does not suffer from the disadvantages of the exfoliation method and, at the same time, it is much simpler and cheaper than the fully controlled growth by molecular-beam epitaxy. The PVD method enables a reproducible synthesis of single crystals of various layered quasi-two-dimensional materials including topological insulators (i.e., Bi 2 Se 3 , Bi 2 Te 3 ). [ 79 ] The resulted single crystals have a well-defined crystallographic orientation; their composition, thickness, size, and the surface density on the desired substrate can be controlled.
The thickness control is particularly important for 3D TIs in which the trivial (bulky) electronic channels usually dominate the transport properties and mask the response of the topological (surface) modes. By reducing the thickness, one lowers the contribution of trivial bulk channels into the total conduction, thus forcing the topological modes to carry the electric current. [ 80 ]
Thus far, the field of topological insulators has been focused on bismuth and antimony chalcogenide based materials such as Bi 2 Se 3 , Bi 2 Te 3 , Sb 2 Te 3 or Bi 1 − x Sb x , Bi 1.1 Sb 0.9 Te 2 S. [ 41 ] The choice of chalcogenides is related to the van der Waals relaxation of the lattice matching strength which restricts the number of materials and substrates. [ 73 ] Bismuth chalcogenides have been studied extensively for TIs and their applications in thermoelectric materials . The van der Waals interaction in TIs exhibit important features due to low surface energy. For instance, the surface of Bi 2 Te 3 is usually terminated by Te due to its low surface energy. [ 34 ]
Bismuth chalcogenides have been successfully grown on different substrates. In particular, Si has been a good substrate for the successful growth of Bi 2 Te 3 . However, the use of sapphire as substrate has not been so encouraging due to a large mismatch of about 15%. [ 81 ] The selection of appropriate substrate can improve the overall properties of TI. The use of buffer layer can reduce the lattice match hence improving the electrical properties of TI. [ 81 ] Bi 2 Se 3 can be grown on top of various Bi 2 − x In x Se 3 buffers. Table 1 shows Bi 2 Se 3 , Bi 2 Te 3 , Sb 2 Te 3 on different substrates and the resulting lattice mismatch. Generally, regardless of the substrate used, the resulting films have a textured surface that is characterized by pyramidal single-crystal domains with quintuple-layer steps. The size and relative proportion of these pyramidal domains vary with factors that include film thickness, lattice mismatch with the substrate and interfacial chemistry-dependent film nucleation. The synthesis of thin films have the stoichiometry problem due to the high vapor pressures of the elements. Thus, binary tetradymites are extrinsically doped as n-type ( Bi 2 Se 3 , Bi 2 Te 3 ) or p-type ( Sb 2 Te 3 ). [ 73 ] Due to the weak van der Waals bonding, graphene is one of the preferred substrates for TI growth despite the large lattice mismatch.
The first step of topological insulators identification takes place right after synthesis, meaning without breaking the vacuum and moving the sample to an atmosphere. That could be done by using angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) techniques. [ 74 ] Further measurements includes structural and chemical probes such as X-ray diffraction and energy-dispersive spectroscopy but depending on the sample quality, the lack of sensitivity could remain. Transport measurements cannot uniquely pinpoint the Z 2 {\displaystyle \mathbb {Z} _{2}} topology by definition of the state.
Bloch's theorem allows a full characterization of the wave propagation properties of a material by assigning a matrix to each wave vector in the Brillouin zone .
Mathematically, this assignment creates a vector bundle . Different materials will have different wave propagation properties, and thus different vector bundles. If we consider all insulators (materials with a band gap), this creates a space of vector bundles. It is the topology of this space (modulo trivial bands) from which the "topology" in topological insulators arises. [ 7 ]
Specifically, the number of connected components of the space indicates how many different "islands" of insulators exist amongst the metallic states. Insulators in the connected component containing the vacuum state are identified as "trivial", and all other insulators as "topological". The connected component in which an insulator lies can be identified with a number, referred to as a "topological invariant". [ 7 ]
This space can be restricted under the presence of symmetries, changing the resulting topology. Although unitary symmetries are usually significant in quantum mechanics, they have no effect on the topology here. [ 82 ] Instead, the three symmetries typically considered are time-reversal symmetry, particle-hole symmetry, and chiral symmetry (also called sublattice symmetry). Mathematically, these are represented as, respectively: an anti-unitary operator which commutes with the Hamiltonian ; an anti-unitary operator which anti-commutes with the Hamiltonian; and a unitary operator which anti-commutes with the Hamiltonian. All combinations of the three together with each spatial dimension result in the so-called periodic table of topological insulators . [ 7 ]
The field of topological insulators still needs to be developed. The best bismuth chalcogenide topological insulators have about 10 meV bandgap variation due to the charge. Further development should focus on the examination of both: the presence of high-symmetry electronic bands and simply synthesized materials. One of the candidates is half-Heusler compounds . [ 74 ] These crystal structures can consist of a large number of elements. Band structures and energy gaps are very sensitive to the valence configuration; because of the increased likelihood of intersite exchange and disorder, they are also very sensitive to specific crystalline configurations. A nontrivial band structure that exhibits band ordering analogous to that of the known 2D and 3D TI materials was predicted in a variety of 18-electron half-Heusler compounds using first-principles calculations. [ 83 ] These materials have not yet shown any sign of intrinsic topological insulator behavior in actual experiments. | https://en.wikipedia.org/wiki/Topological_insulator_growth |
In mathematics , a topological module is a module over a topological ring such that scalar multiplication and addition are continuous .
A topological vector space is a topological module over a topological field .
An abelian topological group can be considered as a topological module over Z , {\displaystyle \mathbb {Z} ,} where Z {\displaystyle \mathbb {Z} } is the ring of integers with the discrete topology .
A topological ring is a topological module over each of its subrings .
A more complicated example is the I {\displaystyle I} - adic topology on a ring and its modules. Let I {\displaystyle I} be an ideal of a ring R . {\displaystyle R.} The sets of the form x + I n {\displaystyle x+I^{n}} for all x ∈ R {\displaystyle x\in R} and all positive integers n , {\displaystyle n,} form a base for a topology on R {\displaystyle R} that makes R {\displaystyle R} into a topological ring. Then for any left R {\displaystyle R} -module M , {\displaystyle M,} the sets of the form x + I n M , {\displaystyle x+I^{n}M,} for all x ∈ M {\displaystyle x\in M} and all positive integers n , {\displaystyle n,} form a base for a topology on M {\displaystyle M} that makes M {\displaystyle M} into a topological module over the topological ring R . {\displaystyle R.}
This topology-related article is a stub . You can help Wikipedia by expanding it .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_module |
In physics , topological order [ 1 ] describes a state or phase of matter that arises system with non-local interactions, such as entanglement in quantum mechanics, and floppy modes in elastic systems. [ 2 ] Whereas classical phases of matter such as gases and solids correspond to microscopic patterns in the spatial arrangement of particles arising from short range interactions, topological orders correspond to patterns of long-range quantum entanglement . [ 3 ] States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Technically, topological order occurs at zero temperature. Various topologically ordered states have interesting properties, such as (1) ground state degeneracy [ 4 ] and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer ; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles ; [ 5 ] (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids , [ 6 ] [ 7 ] [ 8 ] [ 9 ] and the quantum Hall effect , [ 10 ] [ 11 ] along with potential applications to fault-tolerant quantum computation . [ 12 ]
Topological insulators [ 13 ] and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged, but are examples of symmetry-protected topological order .
Matter composed of atoms can have different properties and appear in different forms, such as solid , liquid , superfluid , etc. These various forms of matter are often called states of matter or phases . According to condensed matter physics and the principle of emergence , the different properties of materials generally arise from the different ways in which the atoms are organized in the materials. Those different organizations of the atoms (or other particles) are formally called the orders in the materials. [ 14 ]
Atoms can organize in many ways which lead to many different orders and many different types of materials. Landau symmetry-breaking theory provides a general understanding of these different orders. It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition ), what happens is that the symmetry of the organization of the atoms changes.
For example, atoms have a random distribution in a liquid , so a liquid remains the same as we displace atoms by an arbitrary distance. We say that a liquid has a continuous translation symmetry . After a phase transition, a liquid can turn into a crystal . In a crystal, atoms organize into a regular array (a lattice ). A lattice remains unchanged only when we displace it by a particular distance (integer times a lattice constant ), so a crystal has only discrete translation symmetry . The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Similarly this holds for rotational symmetry. Such a change in symmetry is called symmetry breaking . The essence of the difference between liquids and crystals is therefore that the organizations of atoms have different symmetries in the two phases.
Landau symmetry-breaking theory has been a very successful theory. For a long time, physicists believed that Landau Theory described all possible orders in materials, and all possible (continuous) phase transitions.
However, since the late 1980s, it has become gradually apparent that Landau symmetry-breaking theory may not describe all possible orders. In an attempt to explain high temperature superconductivity [ 15 ] the chiral spin state was introduced. [ 6 ] [ 7 ] At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. This should be the end of the story according to Landau's symmetry breaking description of orders. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description. [ 16 ] The proposed, new kind of order was named "topological order". [ 1 ] The name "topological order" is motivated by the low energy effective theory of the chiral spin states which is a topological quantum field theory (TQFT). [ 17 ] [ 18 ] [ 19 ] New quantum numbers, such as ground state degeneracy [ 16 ] (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders [ 20 ] [ 21 ] and non-Abelian topological orders [ 22 ] [ 23 ] ) and the non-Abelian geometric phase of degenerate ground states, [ 1 ] were introduced to characterize and define the different topological orders in chiral spin states. More recently, it was shown that topological orders can also be characterized by topological entropy . [ 24 ] [ 25 ]
But experiments [ which? ] soon indicated [ how? ] that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states. [ 4 ] Just like chiral spin states, different quantum Hall states all have the same symmetry and are outside the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations.
The fractional quantum Hall (FQH) state was discovered in 1982 [ 10 ] [ 11 ] before the introduction of the concept of topological order in 1989. But the FQH state is not the first experimentally discovered topologically ordered state. The superconductor , discovered in 1911, is the first experimentally discovered topologically ordered state; it has Z 2 topological order. [ note 1 ]
Although topologically ordered states usually appear in strongly interacting boson/fermion systems, a simple kind of topological order can also appear in free fermion systems. This kind of topological order corresponds to integral quantum Hall state, which can be characterized by the Chern number of the filled energy band if we consider integer quantum Hall state on a lattice. Theoretical calculations have proposed that such Chern numbers can be measured for a free fermion system experimentally. [ 29 ] [ 30 ] It is also well known that such a Chern number can be measured (maybe indirectly) by edge states .
The most important characterization of topological orders would be the underlying fractionalized excitations (such as anyons ) and their fusion statistics and braiding statistics (which can go beyond the quantum statistics of bosons or fermions ). Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the crucial signatures for identifying 3+1 dimensional topological orders. [ 31 ] [ 32 ] [ 33 ] The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular topological quantum field theory in 4 spacetime dimensions. [ 33 ]
A large class of 2+1D topological orders is realized through a mechanism called string-net condensation . [ 34 ] This class of topological orders can have a gapped edge and are classified by unitary fusion category (or monoidal category ) theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered.
The collective motions of condensed strings give rise to excitations above the string-net condensed states. Those excitations turn out to be gauge bosons . The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi or fractional statistics . [ 35 ]
The condensations of other extended objects such as " membranes ", [ 36 ] "brane-nets", [ 37 ] and fractals also lead to topologically ordered phases [ 38 ] and "quantum glassiness". [ 39 ] [ 40 ]
We know that group theory is the mathematical foundation of symmetry-breaking orders. What is the mathematical foundation of topological order?
It was found that a subclass of 2+1D topological orders—Abelian topological orders—can be classified by a K-matrix approach. [ 41 ] [ 42 ] [ 43 ] [ 44 ] The string-net condensation suggests that tensor category (such as fusion category or monoidal category ) is part of the mathematical foundation of topological order in 2+1D. The more recent researches suggest that
(up to invertible topological orders that have no fractionalized excitations):
Topological order in higher dimensions may be related to n-Category theory. Quantum operator algebra is a very important mathematical tool in studying topological orders.
Some also suggest that topological order is mathematically described by extended quantum symmetry . [ 45 ]
The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabytes of information. Liquid crystals that break the rotational symmetry of molecules find wide application in display technology. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors . Different types of topological orders are even richer than different types of symmetry-breaking orders. This suggests their potential for exciting, novel applications.
One theorized application would be to use topologically ordered states as media for quantum computing in a technique known as topological quantum computing . A topologically ordered state is a state with complicated non-local quantum entanglement . The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence . This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. [ 46 ] The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing quantum computations . [ 47 ] Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant . [ 12 ]
Topologically ordered states in general have a special property that they contain non-trivial boundary states. In many cases, those boundary states become perfect conducting channel that can conduct electricity without generating heat. [ 48 ] This can be another potential application of topological order in electronic devices.
Similarly to topological order, topological insulators [ 49 ] [ 50 ] also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and the application of topological insulators.
This observation naturally leads to a question:
are topological insulators examples of topologically ordered states?
In fact topological insulators are different from topologically ordered states defined in this article.
Topological insulators only have short-ranged entanglements and have no topological order, while the topological order defined in this article is a pattern of long-range entanglement. Topological order is robust against any perturbations. It has emergent gauge theory, emergent fractional charge and fractional statistics. In contrast, topological insulators are robust only against perturbations that respect time-reversal and U(1) symmetries. Their quasi-particle excitations have no fractional charge and fractional statistics. Strictly speaking, topological insulator is an example of symmetry-protected topological (SPT) order , [ 51 ] where the first example of SPT order is the Haldane phase of spin-1 chain. [ 52 ] [ 53 ] [ 54 ] [ 55 ] But the Haldane phase of spin-2 chain has no SPT order.
Landau symmetry-breaking theory is a cornerstone of condensed matter physics . It is used to define the territory of condensed matter research. The existence of topological order appears to indicate that nature is much richer than Landau symmetry-breaking theory has so far indicated. So topological order opens up a new direction in condensed matter physics—a new direction of highly entangled quantum matter. We realize that quantum phases of matter (i.e. the zero-temperature phases of matter) can be divided into two classes: long range entangled states and short range entangled states. [ 3 ] Topological order is the notion that describes the long range entangled states: topological order = pattern of long range entanglements. Short range entangled states are trivial in the sense that they all belong to one phase. However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases. Those phases are said to contain SPT order . [ 51 ] SPT order generalizes the notion of topological insulator to interacting systems.
Some suggest that topological order (or more precisely, string-net condensation ) in local bosonic (spin) models has the potential to provide a unified origin for photons , electrons and other elementary particles in our universe. [ 5 ] | https://en.wikipedia.org/wiki/Topological_order |
Topological polymers may refer to a polymeric molecule that possesses unique spatial features, such as linear, branched, or cyclic architectures. [ 1 ] It could also refer to polymer networks that exhibit distinct topologies owing to special crosslinkers. [ 2 ] When self-assembling or crosslinking in a certain way, polymeric species with simple topological identity could also demonstrate complicated topological structures in a larger spatial scale. Topological structures, along with the chemical composition, determine the macroscopic physical properties of polymeric materials. [ 3 ] [ 4 ]
Topological polymers, or polymer topology, could refer to a single polymeric chain with topological information or a polymer network with special junctions or connections. When the topology of a polymeric chain or network is investigated, the exact chemical composition is usually neglected, but the way of junctions and connections is more considered. Various topological structures, on one hand, could potentially change the interactions ( van der Waals interaction , hydrogen bonding , etc.) between each of the polymer chain. On the other hand, topology also determines the hierarchical structures within a polymer network, from a microscopic level (<1 nm) to a macroscopic level (10-100 nm), which eventually affords polymeric materials with completely different physical properties, [ 2 ] such as mechanical property, [ 5 ] glass transition temperature , [ 6 ] gelation concentration. [ 7 ]
In early 1950s, Paul J. Flory was the pioneer who developed theories to explain topology within a polymer network, and the structure-property relationships between the topology and the mechanical property, like elasticity, was initially established afterwards. [ 3 ] Later in 1980s, Bertrand Duplantier developed theories to describe any polymer network topologies using statistical mechanics, which could help to derive topology-dependent critical exponents in a polymer network. [ 8 ] In early 2000s, Yasuyuki Tezuka and coworkers were the first ones that systematically described a single molecular chain with topological information. [ 9 ]
Adapted from Y. Tezuka and coworker's description of a topological polymer chain with more generalized rules, [ 9 ] the topology notation rules are to be introduced first, followed by three classical classifications, including linear, branched and cyclic polymer topologies, and they are classified in a table reorganized and redrawn from Y. Tezuka and coworker [ 9 ] (Copyright, 2001 by American Chemical Society).
A general polymer chain could be generalized into an undirected graph with nodes (vertices or points) and edges (lines or links) based on graph theory . In a graph theory topology, two sets of nodes are present, termini and junctions. The quantity ‘ degree ’ represents the number of edges linked to each node, if the degree of a certain node is larger than 3 (including 3), the node is a junction, while the degree of a node is 1, the node is a terminus. There are no nodes with a degree of 2 since they could be generalized into their adjacent nodes.
As for a certain polymer, as long as the topology is fixed, a specific topology notation could be generated using the following rules:
A general polymer chain notation could be expressed as:
P m ( x , y ) [ s 1 ( s 11 , s 12 , . . ) , s 2 ( s 21 , s 22 , . . . ) , . . . ] {\displaystyle P_{m}(x,y)[s_{1}(s_{11},s_{12},..),s_{2}(s_{21},s_{22},...),...]}
i. For branched topology, a main chain is first selected, and the degree of each junction nodes along the chain should be noted as s i {\displaystyle s_{i}} connected by a hyphen. If there is a side chain on any of the main chain node, s i j {\displaystyle s_{ij}} should be noted with a bracket following the main chain notation.
ii. For monocyclic topology, the outward branch should be firstly identified with the number of branches at each of the junctions as s i {\displaystyle s_{i}} connected by a hyphen. Then the topology of each branch should be identified using the rule in i as s i j {\displaystyle s_{ij}} using a bracket following the s i {\displaystyle s_{i}} notations.
iii. For multicyclic topology, superscript letter ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} and so on) is used to describe internal connections within an existing ring.
Linear topology is a special topological structure that exclusively has two nodes as the termini without any junction nodes. High-density polyethylene (HDPE) could be regarded as a linear polymer chain with very small amount of branching, the linear topology has been listed below: [ 9 ] Linear chains capable of forming intra-chain interactions can fold into a wide range of circuit topologies . Examples include biopolymers such as proteins and nucleic acids .
When side chains are introduced into a linear polymer chain, a branched topology forms. Linear polymers are special types of branched polymers with zero junction nodes, but they are cataloged into two classifications to distinguish their special macroscopic properties. [ 2 ] Branched polymers with the same molecular weight usually demonstrate different physical properties due to that branching could generally decrease the van der Waals interactions between each of the polymer chain. Several well-known branched polymers have been synthesized, such as star-shape polymer , comb polymer and dendrimer . Selected branched topologies have been listed below: [ 9 ]
Cyclic structures are of interest topologically because there are no termini in this topology and the physical property could be dramatically different as a result of the restriction of the termini.
Monocyclic topology is a topological structure with only one cycle in the polymer chain, and it could be coupled with outward branching structures. Selected monocyclic topologies have been listed below: [ 9 ]
Bicyclic topology refers to a structure that two cycles connected internally or externally are present in a polymer chain. Selected bicyclic topologies are listed below: [ 9 ]
Similar to monocyclic and bicyclic topologies, polycyclic topologies possess more cycles in a polymer chain and are more synthetically challenging. Selected polycyclic (tricyclic) topologies are listed below: [ 9 ]
Unlike single chain polymeric species, polymer network topology is more complicated as a result of the amorphous feature so that a simple notation is usually not feasible. To analyze the topology of a network, the crosslinkers, including the branched crosslinker and cyclic crosslinker, are considered.
Branched crosslinkers are entities that do not form cyclic topologies, which could be simply understood by branched topological polymer chain above. The ‘degree’ of branched demonstrates the theoretical number of polymer strands at the junctions of the crosslinker, also known as branch functionality ( f ). [ 2 ]
Combining monomers with different degree of branch functionality could generate various topological network with distinct elastic property. Meanwhile, amphiphilic polymers , such as block copolymers , when forming micelle structures, could also be treated as a branched crosslinker with high degree of branch functionality.
Branched crosslinkers should in principle form branched polymer network, but in practice, they could also generate loops and cycles.
Cyclic crosslinkers are more sophisticated and show multiple possibilities. Loops or cycles could form in a smaller scale between two polymer chains or in a larger scale among multiple polymer strands. Besides, bicyclic topology is likely to form if two loops are catenated or linked internally or externally. Special cyclic crosslinking is more attractive within rotaxanes or catenanes since cycles are already present in those molecules.
The characterization of cyclic topologies within a polymer network, compared to branched crosslinker, is relatively harder to perform. Conventional techniques such as rheology and tensile strength analysis are used to offer semiquantitative insights into the polymer topologies. [ 10 ] Recently, the development of multiple quantum nuclear magnetic resonance (NMR) [ 11 ] and network disassembly spectrometry (NDS) [ 12 ] techniques provides quantitative characterizations of loops or cycles in a polymer network. [ 2 ]
The synthesis of branched polymers (grafted polymer, comb polymer, star-shape polymer and dendrimer) has been well developed using well-known polymerization methodology such as cationic / anionic polymerization. [ 14 ]
Unlike branched polymer chain synthesis, the synthesis of cyclic polymer is more challenging. General cyclic species involve the combination between two fragments or among several fragments. Electrostatic self-assembly and covalent fixation is one of the most effective strategies to synthesize cyclic topological polymer. The reaction is driven by the electrostatic interactions between telechelic polytetrahydrofurans with cyclic ammonium salt and pluricaboxylate counterions. Upon dilution, the anions and cations could self-assemble into a cyclic structure, followed by a covalent fixation by heat or other external stimuli to undergo ring-opening reaction and close the chain into a cycle. [ 13 ]
Polymer networks intrinsically have various spatial features due to their amorphous property within a three-dimensional network. There are generally two ways to introduce spatially unique entities into a polymer network:
The topology of a polymer chain or a polymer network is crucial in determining the macroscopic properties of a polymeric material, especially mechanical properties like elasticity and physical properties involving phase transitions . To date, several polymers with topological interest have been developed, which have been used for many applications, such as mechanical elastomer , [ 16 ] energy, [ 17 ] and so on. Below are some of the representative topological polymers or polymer networks.
Interpenetration polymers are polymer networks involving two and more polymer strands which are spatially intertwining with each other to form unique spatial topologies. [ 18 ]
Dendrimer is a special branched polymers with a larger fraction of terminal nodes compared to the junction nodes and could be used for applications in drug delivery [ 19 ] or catalysis. [ 20 ]
Polyrotaxane is a polymer chain or a polymer network with mechanical interlock structures between ring-like molecules and polymer chain, where both the rings and the linear polymer chain could serve as the crosslinker to form a polymer network. [ 21 ] | https://en.wikipedia.org/wiki/Topological_polymers |
A topological quantum computer is a type of quantum computer . It utilizes anyons , a type of quasiparticle that occurs in two-dimensional systems. The anyons' world lines intertwine to form braids in a three-dimensional spacetime (one temporal and two spatial dimensions). The braids act as the logic gates of the computer. The primary advantage of using quantum braids over trapped quantum particles is in their stability. While small but cumulative perturbations can cause quantum states to decohere and introduce errors in traditional quantum computations, such perturbations do not alter the topological properties of the braids. This stability is akin to the difference between cutting and reattaching a string to form a different braid versus a ball (representing an ordinary quantum particle in four-dimensional spacetime) colliding with a wall. It was proposed by Russian-American physicist Alexei Kitaev in 1997. [ 2 ]
While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate that these elements may be created in the real world by using semiconductors made of gallium arsenide at a temperature of nearly absolute zero and subject to strong magnetic fields .
Anyons are quasiparticles in a two-dimensional space. Anyons are neither fermions nor bosons , but like fermions, they cannot occupy the same state. Thus, the world lines of two anyons cannot intersect or merge, which allows their paths to form stable braids in space-time. Anyons can form from excitations in a cold two-dimensional electron gas in a very strong magnetic field and carry fractional units of magnetic flux. This phenomenon is called the fractional quantum Hall effect . In typical laboratory systems, the electron gas occupies a thin semiconducting layer sandwiched between layers of aluminum gallium arsenide. [ 3 ] [ 4 ]
When anyons are braided, the transformation of the quantum state of the system depends only on the topological class of the anyons' trajectories (which are classified according to the braid group ). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories. [ 5 ] In 2005, Sankar Das Sarma , Michael Freedman , and Chetan Nayak proposed a quantum Hall device that would realize a topological qubit. In 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou [ 6 ] claimed to have created and observed the first experimental evidence for using a fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. Non-abelian anyons, a species required for topological quantum computers, have yet to be experimentally confirmed. Possible experimental evidence has been found, [ 7 ] but the conclusions remain contested. [ 8 ] In 2018, scientists again claimed to have isolated the required Majorana particles, but the finding was retracted in 2021. Quanta Magazine stated in 2021 that "no one has convincingly shown the existence of even a single (Majorana zero-mode) quasiparticle", [ 9 ] although in 2023 a new article [ 10 ] by the magazine has covered some preprints by Google [ 11 ] and Quantinuum [ 12 ] claiming the realization of non-abelian anyons on quantum processors, the first used a toric code with twist defects as a topological degeneracy (or topological defect ) while the second used a different but related protocol both of which can be understood as Majorana bound states in quantum error correction .
Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum Turing machine model. [ 13 ] That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.
To live up to its name, a topological quantum computer must provide the unique computation properties promised by a conventional quantum computer design, which uses trapped quantum particles. In 2000, Michael H. Freedman , Alexei Kitaev , Michael J. Larsen , and Zhenghan Wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do, and vice versa. [ 13 ] [ 14 ] [ 15 ]
They found that a conventional quantum computer device, given an error-free operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.
Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. Simulating the dynamics of a topological quantum computer may be a promising method of implementing fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf, Harrington, and Goyal have studied one model, with promising simulation results. [ 16 ]
One of the prominent examples in topological quantum computing is with a system of Fibonacci anyons. A Fibonacci anyon has been described as "an emergent particle with the property that as you add more particles to the system, the number of quantum states grows like the Fibonacci sequence, 1, 2, 3, 5, 8, etc." [ 17 ] In the context of conformal field theory, fibonacci anyons are described by the Yang–Lee model, the SU(2) special case of the Chern–Simons theory and Wess–Zumino–Witten models . [ 18 ] These anyons can be used to create generic gates for topological quantum computing. There are three main steps for creating a model:
Fibonacci anyons are defined by three qualities:
The last ‘fusion’ rule can be extended this to a system of three anyons:
Thus, fusing three anyons will yield a final state of total charge τ {\displaystyle \tau } in 2 ways, or a charge of 1 {\displaystyle 1} in exactly one way. We use three states to define our basis. [ 19 ] However, because we wish to encode these three anyon states as superpositions of 0 and 1, we need to limit the basis to a two-dimensional Hilbert space. Thus, we consider only two states with a total charge of τ {\displaystyle \tau } . This choice is purely phenomenological. In these states, we group the two leftmost anyons into a 'control group', and leave the rightmost as a 'non-computational anyon'. We classify a | 0 ⟩ {\displaystyle |0\rangle } state as one where the control group has total 'fused' charge of 1 {\displaystyle 1} , and a state of | 1 ⟩ {\displaystyle |1\rangle } has a control group with a total 'fused' charge of τ {\displaystyle \tau } . For a more complete description, see Nayak. [ 19 ]
Following the ideas above, adiabatically braiding these anyons around each-other will result in a unitary transformation. These braid operators are a result of two subclasses of operators:
The R matrix can be conceptually thought of as the topological phase that is imparted onto the anyons during the braid. As the anyons wind around each-other, they pick up some phase due to the Aharonov–Bohm effect.
The F matrix is a result of the physical rotations of the anyons. As they braid between each-other, it is important to realize that the bottom two anyons—the control group—will still distinguish the state of the qubit. Thus, braiding the anyons will change which anyons are in the control group, and therefore change the basis. We evaluate the anyons by always fusing the control group (the bottom anyons) together first, so exchanging which anyons these are will rotate the system. Because these anyons are non-abelian , the order of the anyons (which ones are within the control group) will matter, and as such they will transform the system.
The complete braid operator can be derived as:
In order to mathematically construct the F and R operators, we can consider permutations of these F and R operators. We know that if we sequentially change the basis that we are operating on, this will eventually lead us back to the same basis. Similarly, we know that if we braid anyons around each-other a certain number of times, this will lead back to the same state. These axioms are called the pentagonal and hexagonal axioms respectively as performing the operation can be visualized with a pentagon/hexagon of state transformations. Although mathematically difficult, [ 20 ] these can be approached much more successfully visually.
With these braid operators, we can finally formalize the notion of braids in terms of how they act on our Hilbert space and construct arbitrary universal quantum gates. [ 21 ]
In 2018, Leo Kouwenhoven working for Microsoft published a paper in Nature indicating to have found firm evidence of "zero-bias peaks" indicating Majorana quasiparticles. In 2020, the paper got an editorial note of concern. In 2021, in a follow-up paper it was indicated that the data in the 2018 paper was incomplete and misrepresented the results. [ 22 ]
In 2023, Microsoft Quantum researchers published a paper in Physical Review that described a new device that can represent a logical qubit with hardware stability, measuring a phase of matter consistent with the observation of topological superconductivity and Majorana zero modes. [ 23 ] The scientists reported that "such devices have demonstrated low enough disorder to pass the topological gap protocol, proving the technology is viable". [ 24 ] This publication has been criticized by other scientists for not providing sufficient evidence for Majorana modes as in previous papers. [ 25 ] In a 2025 paper, Microsoft determined fermion parity in Majorana zero modes in a single shot, claiming to develop the first topological chip using a new state of matter called topoconductors . [ 26 ] [ 27 ] [ 28 ] | https://en.wikipedia.org/wiki/Topological_quantum_computer |
In gauge theory and mathematical physics , a topological quantum field theory (or topological field theory or TQFT ) is a quantum field theory that computes topological invariants .
While TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology , and to the theory of moduli spaces in algebraic geometry . Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for mathematical work related to topological field theory.
In condensed matter physics , topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.
In a topological field theory, correlation functions do not depend on the metric of spacetime . This means that the theory is not sensitive to changes in the shape of spacetime; if spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants.
Topological field theories are not very interesting on flat Minkowski spacetime used in particle physics. Minkowski space can be contracted to a point , so a TQFT applied to Minkowski space results in trivial topological invariants. Consequently, TQFTs are usually applied to curved spacetimes, such as, for example, Riemann surfaces . Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher-dimensional theories exist, but they are not very well understood [ citation needed ] .
Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigations into this class of models.
(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model targets infinite-dimensional projective space, and if such a thing could be defined it would have countably infinitely many degrees of freedom.)
The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also sometimes referred to as cohomological field theories. See ( Schwarz 2000 ).
In Schwarz-type TQFTs , the correlation functions or partition functions of the system are computed by the path integral of metric-independent action functionals. For instance, in the BF model , the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is
The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. The first example appeared in 1977 and is due to A. Schwarz ; its action functional is:
Another more famous example is Chern–Simons theory , which can be applied to knot invariants . In general, partition functions depend on a metric but the above examples are metric-independent.
The first example of Witten-type TQFTs appeared in Witten's paper in 1988 ( Witten 1988a ), i.e. topological Yang–Mills theory in four dimensions. Though its action functional contains the spacetime metric g αβ , after a topological twist it turns out to be metric independent. The independence of the stress-energy tensor T αβ of the system from the metric depends on whether the BRST-operator is closed. Following Witten's example many other examples can be found in string theory .
Witten-type TQFTs arise if the following conditions are satisfied:
As an example ( Linker 2015 ): Given a 2-form field B {\displaystyle B} with the differential operator δ {\displaystyle \delta } which satisfies δ 2 = 0 {\displaystyle \delta ^{2}=0} , then the action S = ∫ M B ∧ δ B {\displaystyle S=\int \limits _{M}B\wedge \delta B} has a symmetry if δ B ∧ δ B = 0 {\displaystyle \delta B\wedge \delta B=0} since
Further, the following holds (under the condition that δ {\displaystyle \delta } is independent on B {\displaystyle B} and acts similarly to a functional derivative ):
The expression δ δ B α β S {\displaystyle {\frac {\delta }{\delta B^{\alpha \beta }}}S} is proportional to δ G {\displaystyle \delta G} with another 2-form G {\displaystyle G} .
Now any averages of observables ⟨ O i ⟩ := ∫ d μ O i e i S {\displaystyle \left\langle O_{i}\right\rangle :=\int d\mu O_{i}e^{iS}} for the corresponding Haar measure μ {\displaystyle \mu } are independent on the "geometric" field B {\displaystyle B} and are therefore topological:
The third equality uses the fact that δ O i = δ S = 0 {\displaystyle \delta O_{i}=\delta S=0} and the invariance of the Haar measure under symmetry transformations. Since ∫ d μ O i G e i S {\displaystyle \int d\mu O_{i}Ge^{iS}} is only a number, its Lie derivative vanishes.
Atiyah suggested a set of axioms for topological quantum field theory, inspired by Segal 's proposed axioms for conformal field theory (subsequently, Segal's idea was summarized in Segal (2001) ), and Witten's geometric meaning of supersymmetry in Witten (1982) . Atiyah's axioms are constructed by gluing the boundary with a differentiable (topological or continuous) transformation, while Segal's axioms are for conformal transformations. These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs. The basic idea is that a TQFT is a functor from a certain category of cobordisms to the category of vector spaces .
There are in fact two different sets of axioms which could reasonably be called the Atiyah axioms. These axioms differ basically in whether or not they apply to a TQFT defined on a single fixed n -dimensional Riemannian / Lorentzian spacetime M or a TQFT defined on all n -dimensional spacetimes at once.
Let Λ be a commutative ring with 1 (for almost all real-world purposes we will have Λ = Z , R or C ). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension d defined over a ground ring Λ as following:
These data are subject to the following axioms (4 and 5 were added by Atiyah):
Remark . If for a closed manifold M we view Z ( M ) as a numerical invariant, then for a manifold with a boundary we should think of Z ( M ) ∈ Z (∂ M ) as a "relative" invariant. Let f : Σ → Σ be an orientation-preserving diffeomorphism, and identify opposite ends of Σ × I by f . This gives a manifold Σ f and our axioms imply
where Σ( f ) is the induced automorphism of Z (Σ).
Remark . For a manifold M with boundary Σ we can always form the double M ∪ Σ M ∗ {\displaystyle M\cup _{\Sigma }M^{*}} which is a closed manifold. The fifth axiom shows that
where on the right we compute the norm in the hermitian (possibly indefinite) metric.
Physically (2) + (4) are related to relativistic invariance while (3) + (5) are indicative of the quantum nature of the theory.
Σ is meant to indicate the physical space (usually, d = 3 for standard physics) and the extra dimension in Σ × I is "imaginary" time. The space Z (Σ) is the Hilbert space of the quantum theory and a physical theory, with a Hamiltonian H , will have a time evolution operator e itH or an "imaginary time" operator e −tH . The main feature of topological QFTs is that H = 0, which implies that there is no real dynamics or propagation along the cylinder Σ × I . However, there can be non-trivial "propagation" (or tunneling amplitudes) from Σ 0 to Σ 1 through an intervening manifold M with ∂ M = Σ 0 ∗ ∪ Σ 1 {\displaystyle \partial M=\Sigma _{0}^{*}\cup \Sigma _{1}} ; this reflects the topology of M .
If ∂ M = Σ, then the distinguished vector Z ( M ) in the Hilbert space Z (Σ) is thought of as the vacuum state defined by M . For a closed manifold M the number Z ( M ) is the vacuum expectation value . In analogy with statistical mechanics it is also called the partition function .
The reason why a theory with a zero Hamiltonian can be sensibly formulated resides in the Feynman path integral approach to QFT. This incorporates relativistic invariance (which applies to general ( d + 1)-dimensional "spacetimes") and the theory is formally defined by a suitable Lagrangian —a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate to the topology of M .
In 1988, M. Atiyah published a paper in which he described many new examples of topological quantum field theory that were considered at that time ( Atiyah 1988a )( Atiyah 1988b ). It contains some new topological invariants along with some new ideas: Casson invariant , Donaldson invariant , Gromov's theory , Floer homology and Jones–Witten theory .
In this case Σ consists of finitely many points. To a single point we associate a vector space V = Z (point) and to n -points the n -fold tensor product: V ⊗ n = V ⊗ … ⊗ V . The symmetric group S n acts on V ⊗ n . A standard way to get the quantum Hilbert space is to start with a classical symplectic manifold (or phase space ) and then quantize it. Let us extend S n to a compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle , then quantization leads to the irreducible representations V of G . This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem . The Lagrangian of these theories is the classical action ( holonomy of the line bundle). Thus topological QFT's with d = 0 relate naturally to the classical representation theory of Lie groups and the symmetric group .
We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold X . Along with Witten (1982) holonomy such loops as used in the case of d = 0 as a Lagrangian are then used to modify the Hamiltonian. For a closed surface M the invariant Z ( M ) of the theory is the number of pseudo holomorphic maps f : M → X in the sense of Gromov (they are ordinary holomorphic maps if X is a Kähler manifold ). If this number becomes infinite i.e. if there are "moduli", then we must fix further data on M . This can be done by picking some points P i and then looking at holomorphic maps f : M → X with f ( P i ) constrained to lie on a fixed hyperplane. Witten (1988b) has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e. Floer homology , based on Witten's Morse theory ideas; for the case when the boundary conditions are over the interval instead of being periodic, the path initial and end-points lie on two fixed Lagrangian submanifolds . This theory has been developed as Gromov–Witten invariant theory.
Another example is Holomorphic Conformal Field Theory . This might not have been considered strictly topological quantum field theory at the time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to the compact Lie group G in which the classical phase consists of a central extension of the loop group (LG) . Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of LG . The group Diff + ( S 1 ) now substitutes for the symmetric group and plays an important role. As a result, the partition function in such theories depends on complex structure , thus it is not purely topological.
Jones–Witten theory is the most important theory in this case. Here the classical phase space, associated with a closed surface Σ is the moduli space of a flat G -bundle over Σ. The Lagrangian is an integer multiple of the Chern–Simons function of a G -connection on a 3-manifold (which has to be "framed"). The integer multiple k , called the level, is a parameter of the theory and k → ∞ gives the classical limit. This theory can be naturally coupled with the d = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that the partition function for a (framed) link in the 3-sphere is just the value of the Jones polynomial for a suitable root of unity. The theory can be defined over the relevant cyclotomic field , see Atiyah (1988b) . By considering a Riemann surface with boundary, we can couple it to the d = 1 conformal theory instead of coupling d = 2 theory to d = 0. This has developed into Jones–Witten theory and has led to the discovery of deep connections between knot theory and quantum field theory.
Donaldson has defined the integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons. These invariants are polynomials on the second homology. Thus 4-manifolds should have extra data consisting of the symmetric algebra of H 2 . Witten (1988a) has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory. Witten's formula might be understood as an infinite-dimensional analogue of the Gauss–Bonnet theorem . At a later date, this theory was further developed and became the Seiberg–Witten gauge theory which reduces SU(2) to U(1) in N = 2, d = 4 gauge theory. The Hamiltonian version of the theory has been developed by Andreas Floer in terms of the space of connections on a 3-manifold. Floer uses the Chern–Simons function , which is the Lagrangian of Jones–Witten theory to modify the Hamiltonian. For details, see Atiyah (1988b) . Witten (1988a) has also shown how one can couple the d = 3 and d = 1 theories together: this is quite analogous to the coupling between d = 2 and d = 0 in Jones–Witten theory.
Now, topological field theory is viewed as a functor , not on a fixed dimension but on all dimensions at the same time.
Let Bord M be the category whose morphisms are n -dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M , and so form the quotient category hBord M : The objects in hBord M are the objects of Bord M , and the morphisms of hBord M are homotopy equivalence classes of morphisms in Bord M . A TQFT on M is a symmetric monoidal functor from hBord M to the category of vector spaces.
Note that cobordisms can, if their boundaries match, be sewn together to form a new bordism. This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.
There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras .
To consider all spacetimes at once, it is necessary to replace hBord M by a larger category. So let Bord n be the category of bordisms, i.e. the category whose morphisms are n -dimensional manifolds with boundary, and whose objects are the connected components of the boundaries of n-dimensional manifolds. (Note that any ( n −1)-dimensional manifold may appear as an object in Bord n .) As above, regard two morphisms in Bord n as equivalent if they are homotopic, and form the quotient category hBord n . Bord n is a monoidal category under the operation which maps two bordisms to the bordism made from their disjoint union. A TQFT on n -dimensional manifolds is then a functor from hBord n to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.
For example, for (1 + 1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on the grouping of boundary components, and thus (1+1)-dimension TQFTs correspond to Frobenius algebras .
Furthermore, we can consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds related by the above bordisms, and from them we can obtain ample and important examples.
Looking at the development of topological quantum field theory, we should consider its many applications to Seiberg–Witten gauge theory , topological string theory , the relationship between knot theory and quantum field theory, and quantum knot invariants . Furthermore, it has generated topics of great interest in both mathematics and physics. Also of important recent interest are non-local operators in TQFT ( Gukov & Kapustin (2013) ). If string theory is viewed as the fundamental, then non-local TQFTs can be viewed as non-physical models that provide a computationally efficient approximation to local string theory.
Stochastic (partial) differential equations (SDEs) are the foundation for models of everything in nature above the scale of quantum degeneracy and coherence and are essentially Witten-type TQFTs. All SDEs possess topological or BRST supersymmetry , δ {\displaystyle \delta } , and in the operator representation of stochastic dynamics is the exterior derivative , which is commutative with the stochastic evolution operator. This supersymmetry preserves the continuity of phase space by continuous flows, and the phenomenon of supersymmetric spontaneous breakdown by a global non-supersymmetric ground state encompasses such well-established physical concepts as chaos , turbulence , 1/f and crackling noises, self-organized criticality etc. The topological sector of the theory for any SDE can be recognized as a Witten-type TQFT. | https://en.wikipedia.org/wiki/Topological_quantum_field_theory |
In mathematics , topological recursion is a recursive definition of invariants of spectral curves.
It has applications in enumerative geometry , random matrix theory , mathematical physics , string theory , knot theory .
The topological recursion is a construction in algebraic geometry . [ 1 ] It takes as initial data a spectral curve : the data of ( Σ , Σ 0 , x , ω 0 , 1 , ω 0 , 2 ) {\displaystyle \left(\Sigma ,\Sigma _{0},x,\omega _{0,1},\omega _{0,2}\right)} , where: x : Σ → Σ 0 {\displaystyle x:\Sigma \to \Sigma _{0}} is a covering of Riemann surfaces with ramification points; ω 0 , 1 {\displaystyle \omega _{0,1}} is a meromorphic differential 1-form on Σ {\displaystyle \Sigma } , regular at the ramification points ; ω 0 , 2 {\displaystyle \omega _{0,2}} is a symmetric meromorphic bilinear differential form on Σ 2 {\displaystyle \Sigma ^{2}} having a double pole on the diagonal and no residue.
The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms ω g , n {\displaystyle \omega _{g,n}} on Σ n {\displaystyle \Sigma ^{n}} , with poles at ramification points only, for integers g≥0 such that 2g-2+n>0. The definition is a recursion on the integer 2g-2+n.
In many applications, the n-form ω g , n {\displaystyle \omega _{g,n}} is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2g-2+n the Euler characteristics , whence the name "topological recursion".
The topological recursion was first discovered in random matrices . One main goal of random matrix theory, is to find the large size asymptotic expansion of n-point correlation functions, and in some suitable cases, the asymptotic expansion takes the form of a power series . The n-form ω g , n {\displaystyle \omega _{g,n}} is then the g th coefficient in the asymptotic expansion of the n-point correlation function. It was found [ 2 ] [ 3 ] [ 4 ] that the coefficients ω g , n {\displaystyle \omega _{g,n}} always obey a same recursion on 2g-2+n. The idea to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007 [ 1 ] who studied the main properties of those invariants.
An important application of topological recursion was to Gromov–Witten invariants . Marino and BKMP [ 5 ] conjectured that Gromov–Witten invariants of a toric Calabi–Yau 3-fold X {\displaystyle {\mathfrak {X}}} are the TR invariants of a spectral curve that is the mirror of X {\displaystyle {\mathfrak {X}}} .
Since then, topological recursion has generated a lot of activity in particular in enumerative geometry .
The link to Givental formalism and Frobenius manifolds has been established. [ 6 ]
(Case of simple branch points. For higher order branchpoints, see the section Higher order ramifications below)
ω g , n ( z 1 , z 2 , … , z n ) = ∑ a = branchpoints Res z → a K ( z 1 , z , σ a ( z ) ) ( ω g − 1 , n + 1 ( z , σ a ( z ) , z 2 , … , z n ) + ∑ ′ I 1 ⊎ I 2 = { z 2 , … , z n } g 1 + g 2 = g ω g 1 , 1 + # I 1 ( z , I 1 ) ω g 2 , 1 + # I 2 ( σ a ( z ) , I 2 ) ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})&=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g-1,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})\\&\qquad \qquad \qquad +\mathop {{\sum }'} _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(\sigma _{a}(z),I_{2}){\Big )}\end{aligned}}} where K ( z 1 , z 2 , z 3 ) {\displaystyle K(z_{1},z_{2},z_{3})} is called the recursion kernel: K ( z 1 , z 2 , z 3 ) = 1 2 ∫ z ′ = z 3 z 2 ω 0 , 2 ( z 1 , z ′ ) ω 0 , 1 ( z 2 ) − ω 0 , 1 ( z 3 ) {\displaystyle K(z_{1},z_{2},z_{3})={\frac {{\frac {1}{2}}\int _{z'=z_{3}}^{z_{2}}\omega _{0,2}(z_{1},z')}{\omega _{0,1}(z_{2})-\omega _{0,1}(z_{3})}}} and σ a {\displaystyle \sigma _{a}} is the local Galois involution near a branch point a {\displaystyle a} , it is such that x ( σ a ( z ) ) = x ( z ) {\displaystyle x(\sigma _{a}(z))=x(z)} .
The primed sum ∑ ′ {\displaystyle {\sum }'} means excluding the two terms ( g 1 , I 1 ) = ( 0 , ∅ ) {\displaystyle (g_{1},I_{1})=(0,\emptyset )} and ( g 2 , I 2 ) = ( 0 , ∅ ) {\displaystyle (g_{2},I_{2})=(0,\emptyset )} .
F g = ω g , 0 = 1 2 − 2 g ∑ a = branchpoints Res z → a F 0 , 1 ( z ) ω g , 1 ( z ) {\displaystyle F_{g}=\omega _{g,0}={\frac {1}{2-2g}}\ \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\omega _{g,1}(z)} with d F 0 , 1 = ω 0 , 1 {\displaystyle dF_{0,1}=\omega _{0,1}} any antiderivative of ω 0 , 1 {\displaystyle \omega _{0,1}} .
∑ a = branchpoints Res z → a F 0 , 1 ( z ) ω g , n + 1 ( z 1 , … , z n , z ) = ( 2 g − 2 + n ) ω g , n ( z 1 , … , z n ) {\displaystyle \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\ \omega _{g,n+1}(z_{1},\dots ,z_{n},z)=(2g-2+n)\omega _{g,n}(z_{1},\dots ,z_{n})} where d F 0 , 1 = ω 0 , 1 {\displaystyle dF_{0,1}=\omega _{0,1}} .
∑ z ∈ x − 1 ( x ) ω g , n + 1 ( z , z 1 , … , z n ) {\displaystyle \sum _{z\in x^{-1}(x)}\omega _{g,n+1}(z,z_{1},\dots ,z_{n})} ∑ { z ≠ z ′ } ⊂ x − 1 ( x ) ( ω g , n + 1 ( z , z ′ , z 2 , … , z n ) + ∑ I 1 ⊎ I 2 = { z 2 , … , z n } g 1 + g 2 = g ω g 1 , 1 + # I 1 ( z , I 1 ) ω g 2 , 1 + # I 2 ( z ′ , I 2 ) ) {\displaystyle \sum _{\{z\neq z'\}\subset x^{-1}(x)}{\Big (}\omega _{g,n+1}(z,z',z_{2},\dots ,z_{n})+\sum _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(z',I_{2}){\Big )}} where the sum has no prime, i.e. no term excluded.
In case the branchpoints are not simple, the definition is amended as follows [ 7 ] (simple branchpoints correspond to k=2):
ω g , n ( z 1 , z 2 , … , z n ) = ∑ a = branchpoints Res z → a ∑ k = 2 o r d e r x ( a ) ∑ J ⊂ x − 1 ( x ( z ) ) ∖ { z } , # J = k − 1 K k ( z 1 , z , J ) ⋅ ∑ J 1 , … , J ℓ ⊢ J ∪ { z } ∑ I 1 ⊎ … I ℓ = { z 2 , … , z n } g 1 + ⋯ + g ℓ = g + ℓ − k ′ ∏ i = 1 l ω g i , # J i + # I i ( J i , I i ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=&\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}\sum _{k=2}^{{\rm {order}}_{x}(a)}\sum _{J\subset x^{-1}(x(z))\setminus \{z\},\,\#J=k-1}K_{k}(z_{1},z,J)\\&\qquad \cdot \sum _{J_{1},\dots ,J_{\ell }\vdash J\cup \{z\}}\sum '_{\overset {g_{1}+\dots +g_{\ell }=g+\ell -k}{I_{1}\uplus \dots I_{\ell }=\{z_{2},\dots ,z_{n}\}}}\prod _{i=1}^{l}\omega _{g_{i},\#J_{i}+\#I_{i}}(J_{i},I_{i})\end{aligned}}}
The first sum is over partitions J 1 , … , J ℓ {\displaystyle J_{1},\dots ,J_{\ell }} of J ∪ { z } {\displaystyle J\cup \{z\}} with non empty parts J i ≠ ∅ {\displaystyle J_{i}\neq \emptyset } , and in the second sum, the prime means excluding all terms such that ( g i , # J i + # I i ) = ( 0 , 1 ) {\displaystyle (g_{i},\#J_{i}+\#I_{i})=(0,1)} .
K k {\displaystyle K_{k}} is called the recursion kernel: K k ( z 0 , z 1 , … , z k ) = ∫ z ′ = ∗ z 1 ω 0 , 2 ( z 0 , z ′ ) ∏ i = 2 k ( ω 0 , 1 ( z 1 ) − ω 0 , 1 ( z i ) ) {\displaystyle K_{k}(z_{0},z_{1},\dots ,z_{k})={\frac {\int _{z'=*}^{z_{1}}\omega _{0,2}(z_{0},z')}{\prod _{i=2}^{k}(\omega _{0,1}(z_{1})-\omega _{0,1}(z_{i}))}}} The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants ω g , n {\displaystyle \omega _{g,n}} will not depend on it.
The invariants ω g , n {\displaystyle \omega _{g,n}} can be written in terms of intersection numbers of tautological classes: [ 8 ] (*) ω g , n ( z 1 , … , z n ) = 2 3 g − 3 + n ∑ G = Graphs 1 # Aut ( G ) ∫ ( ∏ v = vertices M ¯ g v , n v ) ∏ v = vertices e ∑ k t ^ σ ( v ) , k κ k ∏ ( p , p ′ ) = nodal points ( ∑ d , d ′ B σ ( p ) , 2 d ; σ ( p ′ ) , 2 d ′ ψ p d ψ p ′ d ′ ) ∏ p i = marked points i = 1 , … , n ( ∑ d i ψ p i d i d ξ σ ( p i ) , d i ( z i ) ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},\dots ,z_{n})=2^{3g-3+n}&\sum _{G={\text{Graphs}}}{\frac {1}{\#{\text{Aut}}(G)}}\int _{\left(\prod _{v={\text{vertices}}}{\overline {\mathcal {M}}}_{g_{v},n_{v}}\right)}\,\,\prod _{v={\text{vertices}}}e^{\sum _{k}{\hat {t}}_{\sigma (v),k}\kappa _{k}}\\&\prod _{(p,p')={\text{nodal points}}}\left(\sum _{d,d'}B_{\sigma (p),2d;\sigma (p'),2d'}\psi _{p}^{d}\psi _{p'}^{d'}\right)\prod _{p_{i}={\text{marked points}}\,i=1,\dots ,n}\left(\sum _{d_{i}}\psi _{p_{i}}^{d_{i}}d\xi _{\sigma (p_{i}),d_{i}}(z_{i})\right)\end{aligned}}} where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus g {\displaystyle g} , and n {\displaystyle n} smooth labeled marked points p 1 , … , p n {\displaystyle p_{1},\dots ,p_{n}} , and equipped with a map σ : { vertices } → { branchpoints } {\displaystyle \sigma :\{{\text{vertices}}\}\to \{{\text{branchpoints}}\}} . ψ p = c 1 ( L p ) {\displaystyle \psi _{p}=c_{1}({\mathcal {L}}_{p})} is the Chern class of the cotangent line bundle L p {\displaystyle {\mathcal {L}}_{p}} whose fiber is the cotangent plane at p {\displaystyle p} . κ k {\displaystyle \kappa _{k}} is the k {\displaystyle k} th Mumford's kappa class.
The coefficients t ^ a , k {\displaystyle {\hat {t}}_{a,k}} , B a , k ; a ′ , k ′ {\displaystyle B_{a,k;a',k'}} , d ξ a , k ( z ) {\displaystyle d\xi _{a,k}(z)} , are the Taylor expansion coefficients of ω 0 , 1 {\displaystyle \omega _{0,1}} and ω 0 , 2 {\displaystyle \omega _{0,2}} in the vicinity of branchpoints as follows:
in the vicinity of a branchpoint a {\displaystyle a} (assumed simple), a local coordinate is ζ a ( z ) = x ( z ) − a {\displaystyle \zeta _{a}(z)={\sqrt {x(z)-a}}} . The Taylor expansion of ω 0 , 2 ( z , z ′ ) {\displaystyle \omega _{0,2}(z,z')} near branchpoints z → a {\displaystyle z\to a} , z ′ → a ′ {\displaystyle z'\to a'} defines the coefficients B a , d ; a ′ , d ′ {\displaystyle B_{a,d;a',d'}} ω 0 , 2 ( z , z ′ ) ∼ z → a , z ′ → a ′ ( δ a , a ′ ( ζ a ( z ) − ζ a ′ ( z ′ ) ) 2 + 2 π ∑ d , d ′ = 0 ∞ B a , d ; a ′ , d ′ Γ ( d + 1 2 ) Γ ( d ′ + 1 2 ) ζ a ( z ) d ζ a ′ ( z ′ ) d ′ ) d ζ a ( z ) d ζ a ′ ( z ′ ) {\displaystyle \omega _{0,2}(z,z')\mathop {\sim } _{z\to a,\ z'\to a'}\left({\frac {\delta _{a,a'}}{(\zeta _{a}(z)-\zeta _{a'}(z'))^{2}}}+2\pi \sum _{d,d'=0}^{\infty }{\frac {B_{a,d;a',d'}}{\Gamma ({\frac {d+1}{2}})\Gamma ({\frac {d'+1}{2}})}}\,\zeta _{a}(z)^{d}\zeta _{a'}(z')^{d'}\right)d\zeta _{a}(z)d\zeta _{a'}(z')} . The Taylor expansion at z ′ → a {\displaystyle z'\to a} , defines the 1-forms coefficients d ξ a , d ( z ) {\displaystyle d\xi _{a,d}(z)} d ξ a , d ( z ) = − Γ ( d + 1 2 ) Γ ( 1 2 ) Res z ′ → a ( x ( z ′ ) − a ) − d − 1 2 ω 0 , 2 ( z , z ′ ) {\displaystyle d\xi _{a,d}(z)={\frac {-\Gamma (d+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}\operatorname {Res} _{z'\to a}(x(z')-a)^{-d-{\frac {1}{2}}}\omega _{0,2}(z,z')} whose Taylor expansion near a branchpoint a ′ {\displaystyle a'} is d ξ a , d ( z ) ∼ z → a ′ − δ a , a ′ ( 2 d + 1 ) ! ! d ζ a ( z ) 2 d ζ a ( z ) 2 d + 2 + ∑ k = 0 ∞ B a , 2 d ; a ′ , 2 k 2 k + 1 ( 2 k − 1 ) ! ! ζ a ′ ( z ) 2 k d ζ a ′ ( z ) {\displaystyle d\xi _{a,d}(z)\mathop {\sim } _{z\to a'}{\frac {-\delta _{a,a'}(2d+1)!!d\zeta _{a}(z)}{2^{d}\zeta _{a}(z)^{2d+2}}}+\sum _{k=0}^{\infty }{\frac {B_{a,2d;a',2k}2^{k+1}}{(2k-1)!!}}\zeta _{a'}(z)^{2k}d\zeta _{a'}(z)} . Write also the Taylor expansion of ω 0 , 1 {\displaystyle \omega _{0,1}} ω 0 , 1 ( z ) ∼ z → a ∑ k = 0 ∞ t a , k Γ ( 1 2 ) ( k + 1 ) Γ ( k + 1 2 ) ζ a ( z ) k d ζ a ( z ) {\displaystyle \omega _{0,1}(z)\mathop {\sim } _{z\to a}\sum _{k=0}^{\infty }t_{a,k}\ {\frac {\Gamma ({\frac {1}{2}})}{(k+1)\Gamma ({\frac {k+1}{2}})}}\ \zeta _{a}(z)^{k}d\zeta _{a}(z)} . Equivalently, the coefficients t a , k {\displaystyle t_{a,k}} can be found from expansion coefficients of the Laplace transform, and the coefficients t ^ a , k {\displaystyle {\hat {t}}_{a,k}} are the expansion coefficients of the log of the Laplace transform ∫ x ( z ) − x ( a ) ∈ R + ω 0 , 1 ( z ) e − u x ( z ) = e − u x ( a ) π 2 u 3 / 2 ∑ k = 0 ∞ t a , k u − k = e − u x ( a ) π 2 u 3 / 2 e − ∑ k = 0 ∞ t ^ a , k u − k {\displaystyle \int _{x(z)-x(a)\in \mathbb {R} _{+}}\omega _{0,1}(z)e^{-ux(z)}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}\sum _{k=0}^{\infty }t_{a,k}u^{-k}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}e^{-\sum _{k=0}^{\infty }{\hat {t}}_{a,k}u^{-k}}} .
For example, we have ω 0 , 3 ( z 1 , z 2 , z 3 ) = ∑ a e t ^ a , 0 d ξ a , 0 ( z 1 ) d ξ a , 0 ( z 2 ) d ξ a , 0 ( z 3 ) . {\displaystyle \omega _{0,3}(z_{1},z_{2},z_{3})=\sum _{a}e^{{\hat {t}}_{a,0}}d\xi _{a,0}(z_{1})d\xi _{a,0}(z_{2})d\xi _{a,0}(z_{3}).}
ω 1 , 1 ( z ) = 2 ∑ a e t ^ a , 0 ( 1 24 d ξ a , 1 ( z ) + t ^ a , 1 24 d ξ a , 0 ( z ) + 1 2 B a , 0 ; a , 0 d ξ a , 0 ( z ) ) . {\displaystyle \omega _{1,1}(z)=2\sum _{a}e^{{\hat {t}}_{a,0}}\left({\frac {1}{24}}d\xi _{a,1}(z)+{\frac {{\hat {t}}_{a,1}}{24}}d\xi _{a,0}(z)+{\frac {1}{2}}B_{a,0;a,0}d\xi _{a,0}(z)\right).}
The formula (*) generalizes ELSV formula as well as Mumford 's formula and Mariño - Vafa formula.
M. Mirzakhani's recursion for hyperbolic volumes of moduli spaces is an instance of topological recursion.
For the choice of spectral curve ( C ; C ; x : z ↦ z 2 ; ω 0 , 1 ( z ) = 4 π z sin ( π z ) d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)={\frac {4}{\pi }}z\sin {(\pi z)}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)} the n-form ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is the Laplace transform of the Weil-Petersson volume F g , n ( z 1 , … , z n ) = ∫ 0 ∞ e − z 1 L 1 d L 1 … ∫ 0 ∞ e − z n L n d L n ∫ M g , n ( L 1 , … , L n ) w {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\int _{0}^{\infty }e^{-z_{1}L_{1}}dL_{1}\dots \int _{0}^{\infty }e^{-z_{n}L_{n}}dL_{n}\quad \int _{{\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})}w} where M g , n ( L 1 , … , L n ) {\displaystyle {\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})} is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths L 1 , … , L n {\displaystyle L_{1},\dots ,L_{n}} , and w {\displaystyle w} is the Weil-Petersson volume form. The topological recursion for the n-forms ω g , n ( z 1 , … , z n ) {\displaystyle \omega _{g,n}(z_{1},\dots ,z_{n})} , is then equivalent to Mirzakhani's recursion.
For the choice of spectral curve ( C ; C ; x : z ↦ z 2 ; ω 0 , 1 ( z ) = 2 z 2 d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)=2z^{2}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)} the n-form ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is F g , n ( z 1 , … , z n ) = 2 2 − 2 g − n ∑ d 1 + ⋯ + d n = 3 g − 3 + n ∏ i = 1 n ( 2 d i − 1 ) ! ! z i 2 d i + 1 ⟨ τ d 1 … τ d n ⟩ g {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=2^{2-2g-n}\sum _{d_{1}+\dots +d_{n}=3g-3+n}\prod _{i=1}^{n}{\frac {(2d_{i}-1)!!}{z_{i}^{2d_{i}+1}}}\quad \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}} where ⟨ τ d 1 … τ d n ⟩ g {\displaystyle \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}} is the Witten-Kontsevich intersection number of Chern classes of cotangent line bundles in the compactified moduli space of Riemann surfaces of genus g with n smooth marked points.
For the choice of spectral curve ( C ; C ; x : z ↦ − z + ln z ; ω 0 , 1 ( z ) = ( 1 − z ) d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto -z+\ln {z};\ \omega _{0,1}(z)=(1-z)dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)} the n-form ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is F g , n ( z 1 , … , z n ) = ∑ ℓ ( μ ) ≤ n m μ ( e x ( z 1 ) , … , e x ( z n ) ) h g , μ 1 , … , μ n {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\ell (\mu )\leq n}m_{\mu }(e^{x(z_{1})},\dots ,e^{x(z_{n})})\quad h_{g,\mu _{1},\dots ,\mu _{n}}} where h g , μ {\displaystyle h_{g,\mu }} is the connected simple Hurwitz number of genus g with ramification μ = ( μ 1 , … , μ n ) {\displaystyle \mu =(\mu _{1},\dots ,\mu _{n})} : the number of branch covers of the Riemann sphere by a genus g connected surface, with 2g-2+n simple ramification points, and one point with ramification profile given by the partition μ {\displaystyle \mu } .
Let X {\displaystyle {\mathfrak {X}}} a toric Calabi–Yau 3-fold, with Kähler moduli t 1 , … , t b 2 ( X ) {\displaystyle t_{1},\dots ,t_{b_{2}({\mathfrak {X}})}} .
Its mirror manifold is singular over a complex plane curve Σ {\displaystyle \Sigma } given by a polynomial equation P ( e x , e y ) = 0 {\displaystyle P(e^{x},e^{y})=0} , whose coefficients are functions of the Kähler moduli.
For the choice of spectral curve ( Σ ; C ∗ ; x ; ω 0 , 1 = y d x ; ω 0 , 2 ) {\displaystyle \left(\Sigma ;\ \mathbb {C} ^{*};\ x;\ \omega _{0,1}=ydx;\,\omega _{0,2}\right)} with ω 0 , 2 {\displaystyle \omega _{0,2}} the fundamental second kind differential on Σ {\displaystyle \Sigma } , According to the BKMP [ 5 ] conjecture, the n-form ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is F g , n ( z 1 , … , z n ) = ∑ d ∈ H 2 ( X , Z ) ∑ μ 1 , … , μ n ∈ H 1 ( L , Z ) t d ∏ i = 1 n e x ( z i ) N g ( X , L ; d , μ 1 , … , μ n ) {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\mathbf {d} \in H_{2}({\mathfrak {X}},\mathbb {Z} )}\sum _{\mu _{1},\dots ,\mu _{n}\in H_{1}({\mathcal {L}},\mathbb {Z} )}t^{d}\prod _{i=1}^{n}e^{x(z_{i})}{\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})} where N g ( X , L ; d , μ 1 , … , μ n ) = ∫ [ M ¯ g , n ( X , L , d , μ 1 , … , μ n ) ] v i r 1 {\displaystyle {\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})=\int _{[{\overline {\mathcal {M}}}_{g,n}({\mathfrak {X}},{\mathcal {L}},\mathbf {d} ,\mu _{1},\dots ,\mu _{n})]^{\rm {vir}}}1} is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into X {\displaystyle {\mathfrak {X}}} , with n boundaries mapped to a special Lagrangian submanifold L {\displaystyle {\mathcal {L}}} . d = ( d 1 , … , d b 2 ( X ) ) {\displaystyle \mathbf {d} =(d_{1},\dots ,d_{b_{2}({\mathfrak {X}})})} is the 2nd relative homology class of the surface's image, and μ i ∈ H 1 ( L , Z ) {\displaystyle \mu _{i}\in H_{1}({\mathcal {L}},\mathbb {Z} )} are homology classes (winding number) of the boundary images. The BKMP [ 5 ] conjecture has since then been proven.
[ 1 ] | https://en.wikipedia.org/wiki/Topological_recursion |
In the mathematical field of topology , a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M . [ 1 ]
A central problem in topology is determining when two spaces are the same i.e. homeomorphic or diffeomorphic. Constructing a morphism explicitly is almost always impractical. If we put further condition on one or both spaces (manifolds) we can exploit this additional structure in order to show that the desired morphism must exist.
Rigidity theorem is about when a fairly weak equivalence between two manifolds (usually a homotopy equivalence ) implies the existence of stronger equivalence homeomorphism, diffeomorphism or isometry .
A closed topological manifold M is called topological rigid if any homotopy equivalence f : N → M with some manifold N as source and M as target is homotopic to a homeomorphism.
Example 1. If closed 2-manifolds M and N are homotopically equivalent then they are homeomorphic. Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism.
Example 2. If a closed manifold M n ( n ≠ 3) is homotopy-equivalent to S n then M n is homeomorphic to S n .
A diffeomorphism of flat-Riemannian manifolds is said to be affine iff it carries geodesics to geodesic.
If f : M → N is a homotopy equivalence between flat closed connected Riemannian manifolds then f is homotopic to an affine homeomorphism.
Theorem: Let M and N be compact , locally symmetric Riemannian manifolds with everywhere non-positive curvature having no closed one or two dimensional geodesic subspace which are direct factor locally. If f : M → N is a homotopy equivalence then f is homotopic to an isometry.
Theorem (Mostow's theorem for hyperbolic n - manifolds, n ≥ 3): If M and N are complete hyperbolic n -manifolds, n ≥ 3 with finite volume and f : M → N is a homotopy equivalence then f is homotopic to an isometry.
These results are named after George Mostow .
Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n -space H , where n ≥ 3, whose quotients H /Γ and H /Δ have finite volume. If Γ and Δ are isomorphic as discrete groups then they are conjugate.
(1) In the 2-dimensional case any manifold of genus at least two has a hyperbolic structure. Mostow's rigidity theorem does not apply in this case. In fact, there are many hyperbolic structures on any such manifold; each such structure corresponds to a point in Teichmuller space.
(2) On the other hand, if M and N are 2-manifolds of finite volume then it is easy to show that they are homeomorphic exactly when their fundamental groups are the same.
The group of isometries of a finite-volume hyperbolic n -manifold M (for n ≥ 3) is finitely generated [ 2 ] and isomorphic to π 1 ( M ). | https://en.wikipedia.org/wiki/Topological_rigidity |
In mathematics , a topological ring is a ring R {\displaystyle R} that is also a topological space such that both the addition and the multiplication are continuous as maps: [ 1 ] R × R → R {\displaystyle R\times R\to R} where R × R {\displaystyle R\times R} carries the product topology . That means R {\displaystyle R} is an additive topological group and a multiplicative topological semigroup .
Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field . [ 2 ]
The group of units R × {\displaystyle R^{\times }} of a topological ring R {\displaystyle R} is a topological group when endowed with the topology coming from the embedding of R × {\displaystyle R^{\times }} into the product R × R {\displaystyle R\times R} as ( x , x − 1 ) . {\displaystyle \left(x,x^{-1}\right).} However, if the unit group is endowed with the subspace topology as a subspace of R , {\displaystyle R,} it may not be a topological group, because inversion on R × {\displaystyle R^{\times }} need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field ; its unit group, called the idele group , is not a topological group in the subspace topology. If inversion on R × {\displaystyle R^{\times }} is continuous in the subspace topology of R {\displaystyle R} then these two topologies on R × {\displaystyle R^{\times }} are the same.
If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a topological group (for + {\displaystyle +} ) in which multiplication is continuous, too.
Topological rings occur in mathematical analysis , for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space ; all Banach algebras are topological rings. The rational , real , complex and p {\displaystyle p} -adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.
In commutative algebra , the following construction is common: given an ideal I {\displaystyle I} in a commutative ring R , {\displaystyle R,} the I -adic topology on R {\displaystyle R} is defined as follows: a subset U {\displaystyle U} of R {\displaystyle R} is open if and only if for every x ∈ U {\displaystyle x\in U} there exists a natural number n {\displaystyle n} such that x + I n ⊆ U . {\displaystyle x+I^{n}\subseteq U.} This turns R {\displaystyle R} into a topological ring. The I {\displaystyle I} -adic topology is Hausdorff if and only if the intersection of all powers of I {\displaystyle I} is the zero ideal ( 0 ) . {\displaystyle (0).}
The p {\displaystyle p} -adic topology on the integers is an example of an I {\displaystyle I} -adic topology (with I = p Z {\displaystyle I=p\mathbb {Z} } ).
Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring R {\displaystyle R} is complete . If it is not, then it can be completed : one can find an essentially unique complete topological ring S {\displaystyle S} that contains R {\displaystyle R} as a dense subring such that the given topology on R {\displaystyle R} equals the subspace topology arising from S . {\displaystyle S.} If the starting ring R {\displaystyle R} is metric, the ring S {\displaystyle S} can be constructed as a set of equivalence classes of Cauchy sequences in R , {\displaystyle R,} this equivalence relation makes the ring S {\displaystyle S} Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) c : R → S {\displaystyle c:R\to S} such that, for all CM f : R → T {\displaystyle f:R\to T} where T {\displaystyle T} is Hausdorff and complete, there exists a unique CM g : S → T {\displaystyle g:S\to T} such that f = g ∘ c . {\displaystyle f=g\circ c.} If R {\displaystyle R} is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions f : R → Q {\displaystyle f:\mathbb {R} \to \mathbb {Q} } endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see Bourbaki , General Topology, III.6.5).
The rings of formal power series and the p {\displaystyle p} -adic integers are most naturally defined as completions of certain topological rings carrying I {\displaystyle I} -adic topologies .
Some of the most important examples are topological fields . A topological field is a topological ring that is also a field , and such that inversion of non zero elements is a continuous function. The most common examples are the complex numbers and all its subfields , and the valued fields , which include the p {\displaystyle p} -adic fields . | https://en.wikipedia.org/wiki/Topological_ring |
In mathematics , a topological semigroup is a semigroup that is simultaneously a topological space , and whose semigroup operation is continuous . [ 1 ]
Every topological group is a topological semigroup.
This algebra -related article is a stub . You can help Wikipedia by expanding it .
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_semigroup |
In condensed matter physics and materials chemistry , a topological superconductor is a material that conducts electricity with zero electrical resistivity , and has non-trivial topology which gives it certain unique properties. These materials behave as superconductors that feature exotic edge states , known as Majorana zero modes . [ 1 ] [ 2 ]
Topological superconductors are characterized by the topological order related to their electronic band structure . [ 2 ] These materials can be classified using the periodic table of topological superconductors , which categorizes topological phases based on time-reversal symmetry , particle-hole symmetry , and chiral symmetry . [ 2 ]
An example of a simple topological superconductor in one-dimension is the Kitaev chain . [ 2 ]
In 2015, uranium ditelluride (UTe 2 ) was found to behave as a topological superconductor. [ 2 ]
A notable application of topological superconductors is in the realm of topological quantum computing , where Majorana zero modes can be used to implement fault-tolerant quantum gates via braiding operations. This approach leverages the non-Abelian statistics of Majorana modes to perform computations that are protected from local sources of decoherence. [ 1 ] [ 2 ]
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topological_superconductor |
In mathematics , there are usually many different ways to construct a topological tensor product of two topological vector spaces . For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces ), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
One of the original motivations for topological tensor products ⊗ ^ {\displaystyle {\hat {\otimes }}} is the fact that tensor products of the spaces of smooth real-valued functions on R n {\displaystyle \mathbb {R} ^{n}} do not behave as expected. There is an injection
but this is not an isomorphism . For example, the function f ( x , y ) = e x y {\displaystyle f(x,y)=e^{xy}} cannot be expressed as a finite linear combination of smooth functions in C ∞ ( R x ) ⊗ C ∞ ( R y ) . {\displaystyle C^{\infty }(\mathbb {R} _{x})\otimes C^{\infty }(\mathbb {R} _{y}).} [ 1 ] We only get an isomorphism after constructing the topological tensor product; i.e.,
This article first details the construction in the Banach space case. The space C ∞ ( R n ) {\displaystyle C^{\infty }(\mathbb {R} ^{n})} is not a Banach space and further cases are discussed at the end.
The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B . So in particular it has a natural positive definite quadratic form , and the corresponding completion is a Hilbert space A ⊗ B , called the (Hilbert space) tensor product of A and B .
If the vectors a i and b j run through orthonormal bases of A and B , then the vectors a i ⊗ b j form an orthonormal basis of A ⊗ B .
We shall use the notation from ( Ryan 2002 ) in this section. The obvious way to define the tensor product of two Banach spaces A {\displaystyle A} and B {\displaystyle B} is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.
If A {\displaystyle A} and B {\displaystyle B} are Banach spaces the algebraic tensor product of A {\displaystyle A} and B {\displaystyle B} means the tensor product of A {\displaystyle A} and B {\displaystyle B} as vector spaces and is denoted by A ⊗ B . {\displaystyle A\otimes B.} The algebraic tensor product A ⊗ B {\displaystyle A\otimes B} consists of all finite sums x = ∑ i = 1 n a i ⊗ b i , {\displaystyle x=\sum _{i=1}^{n}a_{i}\otimes b_{i},} where n {\displaystyle n} is a natural number depending on x {\displaystyle x} and a i ∈ A {\displaystyle a_{i}\in A} and b i ∈ B {\displaystyle b_{i}\in B} for i = 1 , … , n . {\displaystyle i=1,\ldots ,n.}
When A {\displaystyle A} and B {\displaystyle B} are Banach spaces, a crossnorm (or cross norm ) p {\displaystyle p} on the algebraic tensor product A ⊗ B {\displaystyle A\otimes B} is a norm satisfying the conditions p ( a ⊗ b ) = ‖ a ‖ ‖ b ‖ , {\displaystyle p(a\otimes b)=\|a\|\|b\|,} p ′ ( a ′ ⊗ b ′ ) = ‖ a ′ ‖ ‖ b ′ ‖ . {\displaystyle p'(a'\otimes b')=\|a'\|\|b'\|.} Here a ′ {\displaystyle a^{\prime }} and b ′ {\displaystyle b^{\prime }} are elements of the topological dual spaces of A {\displaystyle A} and B , {\displaystyle B,} respectively, and p ′ {\displaystyle p^{\prime }} is the dual norm of p . {\displaystyle p.} The term reasonable crossnorm is also used for the definition above.
There is a cross norm π {\displaystyle \pi } called the projective cross norm, given by π ( x ) = inf { ∑ i = 1 n ‖ a i ‖ ‖ b i ‖ : x = ∑ i = 1 n a i ⊗ b i } , {\displaystyle \pi (x)=\inf \left\{\sum _{i=1}^{n}\|a_{i}\|\|b_{i}\|:x=\sum _{i=1}^{n}a_{i}\otimes b_{i}\right\},} where x ∈ A ⊗ B . {\displaystyle x\in A\otimes B.}
It turns out that the projective cross norm agrees with the largest cross norm ( Ryan 2002 , pp. 15–16).
There is a cross norm ε {\displaystyle \varepsilon } called the injective cross norm, given by ε ( x ) = sup { | ( a ′ ⊗ b ′ ) ( x ) | : a ′ ∈ A ′ , b ′ ∈ B ′ , ‖ a ′ ‖ = ‖ b ′ ‖ = 1 } {\displaystyle \varepsilon (x)=\sup \left\{\left|(a'\otimes b')(x)\right|:a'\in A',b'\in B',\|a'\|=\|b'\|=1\right\}} where x ∈ A ⊗ B . {\displaystyle x\in A\otimes B.} Here A ′ {\displaystyle A^{\prime }} and B ′ {\displaystyle B^{\prime }} denote the topological duals of A {\displaystyle A} and B , {\displaystyle B,} respectively.
Note hereby that the injective cross norm is only in some reasonable sense the "smallest".
The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by A ⊗ ^ π B {\displaystyle A\operatorname {\hat {\otimes }} _{\pi }B} and A ⊗ ^ ε B . {\displaystyle A\operatorname {\hat {\otimes }} _{\varepsilon }B.}
When A {\displaystyle A} and B {\displaystyle B} are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by σ , {\displaystyle \sigma ,} so the Hilbert space tensor product in the section above would be A ⊗ ^ σ B . {\displaystyle A\operatorname {\hat {\otimes }} _{\sigma }B.}
A uniform crossnorm α {\displaystyle \alpha } is an assignment to each pair ( X , Y ) {\displaystyle (X,Y)} of Banach spaces of a reasonable crossnorm on X ⊗ Y {\displaystyle X\otimes Y} so that if X , W , Y , Z {\displaystyle X,W,Y,Z} are arbitrary Banach spaces then for all (continuous linear) operators S : X → W {\displaystyle S:X\to W} and T : Y → Z {\displaystyle T:Y\to Z} the operator S ⊗ T : X ⊗ α Y → W ⊗ α Z {\displaystyle S\otimes T:X\otimes _{\alpha }Y\to W\otimes _{\alpha }Z} is continuous and ‖ S ⊗ T ‖ ≤ ‖ S ‖ ‖ T ‖ . {\displaystyle \|S\otimes T\|\leq \|S\|\|T\|.} If A {\displaystyle A} and B {\displaystyle B} are two Banach spaces and α {\displaystyle \alpha } is a uniform cross norm then α {\displaystyle \alpha } defines a reasonable cross norm on the algebraic tensor product A ⊗ B . {\displaystyle A\otimes B.} The normed linear space obtained by equipping A ⊗ B {\displaystyle A\otimes B} with that norm is denoted by A ⊗ α B . {\displaystyle A\otimes _{\alpha }B.} The completion of A ⊗ α B , {\displaystyle A\otimes _{\alpha }B,} which is a Banach space, is denoted by A ⊗ ^ α B . {\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B.} The value of the norm given by α {\displaystyle \alpha } on A ⊗ B {\displaystyle A\otimes B} and on the completed tensor product A ⊗ ^ α B {\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B} for an element x {\displaystyle x} in A ⊗ ^ α B {\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B} (or A ⊗ α B {\displaystyle A\otimes _{\alpha }B} ) is denoted by α A , B ( x ) or α ( x ) . {\displaystyle \alpha _{A,B}(x){\text{ or }}\alpha (x).}
A uniform crossnorm α {\displaystyle \alpha } is said to be finitely generated if, for every pair ( X , Y ) {\displaystyle (X,Y)} of Banach spaces and every u ∈ X ⊗ Y , {\displaystyle u\in X\otimes Y,} α ( u ; X ⊗ Y ) = inf { α ( u ; M ⊗ N ) : dim M , dim N < ∞ } . {\displaystyle \alpha (u;X\otimes Y)=\inf\{\alpha (u;M\otimes N):\dim M,\dim N<\infty \}.}
A uniform crossnorm α {\displaystyle \alpha } is cofinitely generated if, for every pair ( X , Y ) {\displaystyle (X,Y)} of Banach spaces and every u ∈ X ⊗ Y , {\displaystyle u\in X\otimes Y,} α ( u ) = sup { α ( ( Q E ⊗ Q F ) u ; ( X / E ) ⊗ ( Y / F ) ) : dim X / E , dim Y / F < ∞ } . {\displaystyle \alpha (u)=\sup\{\alpha ((Q_{E}\otimes Q_{F})u;(X/E)\otimes (Y/F)):\dim X/E,\dim Y/F<\infty \}.}
A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm π {\displaystyle \pi } and the injective cross norm ε {\displaystyle \varepsilon } defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.
If A {\displaystyle A} and B {\displaystyle B} are arbitrary Banach spaces and α {\displaystyle \alpha } is an arbitrary uniform cross norm then ε A , B ( x ) ≤ α A , B ( x ) ≤ π A , B ( x ) . {\displaystyle \varepsilon _{A,B}(x)\leq \alpha _{A,B}(x)\leq \pi _{A,B}(x).}
The topologies of locally convex topological vector spaces A {\displaystyle A} and B {\displaystyle B} are given by families of seminorms . For each choice of seminorm on A {\displaystyle A} and on B {\displaystyle B} we can define the corresponding family of cross norms on the algebraic tensor product A ⊗ B , {\displaystyle A\otimes B,} and by choosing one cross norm from each family we get some cross norms on A ⊗ B , {\displaystyle A\otimes B,} defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on A ⊗ B {\displaystyle A\otimes B} are called the projective and injective tensor products, and denoted by A ⊗ γ B {\displaystyle A\otimes _{\gamma }B} and A ⊗ λ B . {\displaystyle A\otimes _{\lambda }B.} There is a natural map from A ⊗ γ B {\displaystyle A\otimes _{\gamma }B} to A ⊗ λ B . {\displaystyle A\otimes _{\lambda }B.}
If A {\displaystyle A} or B {\displaystyle B} is a nuclear space then the natural map from A ⊗ γ B {\displaystyle A\otimes _{\gamma }B} to A ⊗ λ B {\displaystyle A\otimes _{\lambda }B} is an isomorphism. Roughly speaking, this means that if A {\displaystyle A} or B {\displaystyle B} is nuclear, then there is only one sensible tensor product of A {\displaystyle A} and B {\displaystyle B} .
This property characterizes nuclear spaces. | https://en.wikipedia.org/wiki/Topological_tensor_product |
A topologically associating domain (TAD) is a self-interacting genomic region, meaning that DNA sequences within a TAD physically interact with each other more frequently than with sequences outside the TAD. [ 1 ] The average size of a topologically associating domain (TAD) is 1000 kb in humans, 880 kb in mouse cells, and 140 kb in fruit flies. [ 2 ] [ 3 ] Boundaries at both side of these domains are conserved between different mammalian cell types and even across species [ 2 ] and are highly enriched with CCCTC-binding factor (CTCF) and cohesin . [ 1 ] In addition, some types of genes (such as transfer RNA genes and housekeeping genes ) appear near TAD boundaries more often than would be expected by chance. [ 4 ] [ 5 ]
The functions of TADs are not fully understood and are still a matter of debate. Most of the studies indicate TADs regulate gene expression by limiting the enhancer - promoter interaction to each TAD; [ 6 ] however, a recent study uncouples TAD organization and gene expression. [ 7 ] Disruption of TAD boundaries are found to be associated with wide range of diseases such as cancer , [ 8 ] [ 9 ] [ 10 ] variety of limb malformations such as synpolydactyly , Cooks syndrome , and F-syndrome, [ 11 ] and number of brain disorders like Hypoplastic corpus callosum and Adult-onset demyelinating leukodystrophy. [ 11 ] Furthermore, studies have revealed that interactions between promoters and enhancers spanning single or multiple TADs, are fundamental to the exact dynamics of gene expression. [ 12 ] The genomic elements underlying these interactions are named distal tethering elements (DTEs) and it has been shown that these elements are important for precise gene activation of Hox genes in early embryogenesis of D. melanogaster . [ 12 ]
The mechanisms underlying TAD formation are also complex and not yet fully elucidated, though a number of protein complexes and DNA elements are associated with TAD boundaries. However, the handcuff model and the loop extrusion model describe the TAD formation by the aid of CTCF and cohesin proteins. [ 13 ] Furthermore, it has been proposed that the stiffness of TAD boundaries itself could cause the domain insulation and TAD formation. [ 13 ]
TADs are defined as regions whose DNA sequences preferentially contact each other. They were discovered in 2012 using chromosome conformation capture techniques including Hi-C . [ 4 ] [ 14 ] [ 5 ] They have been shown to be present in multiple species, [ 15 ] including fruit flies ( Drosophila ), [ 16 ] mouse , [ 4 ] plants, fungi and human [ 5 ] genomes. In bacteria, they are referred to as Chromosomal Interacting Domains (CIDs). [ 15 ]
TAD locations are defined by applying an algorithm to Hi-C data. For example, TADs are often called according to the so-called "directionality index". [ 5 ] The directionality index is calculated for individual 40kb bins, by collecting the reads that fall in the bin, and observing whether their paired reads map upstream or downstream of the bin (read pairs are required to span no more than 2Mb). A positive value indicates that more read pairs lie downstream than upstream, and a negative value indicates the reverse. Mathematically, the directionality index is a signed chi-square statistic.
The development of specialized genome browsers and visualization tools [ 17 ] such as Juicebox, [ 18 ] HiGlass [ 19 ] /HiPiler, [ 20 ] The 3D Genome Browser, [ 21 ] 3DIV, [ 22 ] 3D-GNOME, [ 23 ] and TADKB [ 24 ] have enabled us to visualize the TAD organization of regions of interest in different cell types.
A number of proteins are known to be associated with TAD formation including the protein CTCF and the protein complex cohesin . [ 1 ] It is also unknown what components are required at TAD boundaries; however, in mammalian cells, it has been shown that these boundary regions have comparatively high levels of CTCF binding. In addition, some types of genes (such as transfer RNA genes and housekeeping genes ) appear near TAD boundaries more often than would be expected by chance. [ 4 ] [ 5 ]
Computer simulations have shown that chromatin loop extrusion driven by cohesin motors can generate TADs. [ 25 ] [ 26 ] In the loop extrusion model, cohesin binds chromatin, pulls it in, and extrudes chromatin to progressively grow a loop. Chromatin on both sides of the cohesin complex is extruded until cohesin encounters a chromatin-bound CTCF protein, typically located at the boundary of a TAD. In this way, TAD boundaries can be brought together as the anchors of a chromatin loop. [ 27 ] Indeed, in vitro, cohesin has been observed to processively extrude DNA loops in an ATP-dependent manner [ 28 ] [ 29 ] [ 30 ] and stall at CTCF. [ 31 ] [ 32 ] However, some in vitro data indicates that the observed loops may be artifacts. [ 33 ] [ 34 ] Importantly, since cohesins can dynamically unbind from chromatin, this model suggests that TADs (and associated chromatin loops) are dynamic, transient structures, [ 25 ] in agreement with in vivo observations. [ 35 ] [ 36 ] [ 37 ] [ 38 ]
Other mechanisms for TAD formation have been suggested. For example, some simulations suggest that transcription-generated supercoiling can relocalize cohesin to TAD boundaries [ 39 ] [ 40 ] or that passively diffusing cohesin “slip links” [ 41 ] [ 42 ] can generate TADs.
TADs have been reported to be relatively constant between different cell types (in stem cells and blood cells, for example), and even between species in specific cases. [ 5 ] [ 43 ] [ 44 ] [ 45 ] Comparative TAD analysis between Drosophila melanogaster and Drosophila subobscura , with a divergence time of approximately 49 million years, has revealed a conservation in range of 30-40%. [ 46 ]
The majority of observed interactions between promoters and enhancers do not cross TAD boundaries. Removing a TAD boundary (for example, using CRISPR to delete the relevant region of the genome) can allow new promoter-enhancer contacts to form. This can affect gene expression nearby - such misregulation has been shown to cause limb malformations (e.g. polydactyly ) in humans and mice. [ 44 ]
Computer simulations have shown that transcription-induced supercoiling of chromatin fibres can explain how TADs are formed and how they can assure very efficient interactions between enhancers and their cognate promoters located in the same TAD. [ 39 ]
Replication timing domains have been shown to be associated with TADs as their boundary is co localized with the boundaries of TADs that are located at either sides of compartments. [ 47 ] Insulated neighborhoods , DNA loops formed by CTCF/cohesin-bound regions, are proposed to functionally underlie TADs. [ 48 ]
Genome rearrangement breakpoint have shown to be enriched at the TAD boundaries in D. melanogaster . [ 49 ]
Disruption of TAD boundaries can affect the expression of nearby genes, and this can cause disease. [ 50 ]
For example, genomic structural variants that disrupt TAD boundaries have been reported to cause developmental disorders such as human limb malformations. [ 51 ] [ 52 ] [ 53 ] Additionally, several studies have provided evidence that the disruption or rearrangement of TAD boundaries can provide growth advantages to certain cancers, such as T-cell acute lymphoblastic leukemia (T-ALL), [ 54 ] gliomas, [ 55 ] and lung cancer. [ 56 ]
Lamina-associated domains (LADs) are parts of the chromatin that heavily interact with the lamina, a network-like structure at the inner membrane of the nucleus . [ 57 ] LADs consist mostly of transcriptionally silent chromatin, being enriched with trimethylated Lys27 on histone H3 , (i.e. H3K27me3 ); which is a common posttranslational histone modification of heterochromatin . [ 58 ] LADs have CTCF-binding sites at their periphery. [ 57 ] | https://en.wikipedia.org/wiki/Topologically_associating_domain |
Topology (from the Greek words τόπος , 'place, location', and λόγος , 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations , such as stretching , twisting , crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity . Euclidean spaces , and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that is invariant under such deformations is a topological property . The following are basic examples of topological properties: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle; connectedness , which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Wilhelm Leibniz , who in the 17th century envisioned the geometria situs and analysis situs . Leonhard Euler 's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed.
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one-dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory .
Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick ." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg , the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.
To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism . The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous example, known as the "Topologist's Breakfast", is that a topologist cannot distinguish a coffee mug from a doughnut; a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it while shrinking the hole into a handle. [ 1 ]
Homeomorphism can be considered the most basic topological equivalence . Another is homotopy equivalence . This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. [ 2 ] Among these are certain questions in geometry investigated by Leonhard Euler . His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. [ 2 ] On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron . This led to his polyhedron formula , V − E + F = 2 (where V , E , and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. [ 3 ]
Further contributions were made by Augustin-Louis Cauchy , Ludwig Schläfli , Johann Benedict Listing , Bernhard Riemann and Enrico Betti . [ 4 ] Listing introduced the term "Topologie" in Vorstudien zur Topologie , written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print. [ 5 ] The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". [ 6 ]
Their work was corrected, consolidated and greatly extended by Henri Poincaré . In 1895, he published his ground-breaking paper on Analysis Situs , which introduced the concepts now known as homotopy and homology , which are now considered part of algebraic topology . [ 4 ]
The development of topology in the 20th century was marked by significant advances in both foundational theory and its application to other fields of mathematics. Unifying the work on function spaces of Georg Cantor , Vito Volterra , Cesare Arzelà , Jacques Hadamard , Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. [ 7 ] A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and defined what is now called a Hausdorff space . [ 8 ] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski . [ 9 ]
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series . For further developments, see point-set topology and algebraic topology.
The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects". [ 10 ]
The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line , the complex plane , and the Cantor set can be thought of as the same set with different topologies.
Formally, let X be a set and let τ be a family of subsets of X . Then τ is called a topology on X if:
If τ is a topology on X , then the pair ( X , τ ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ . By definition, every topology is a π -system .
The members of τ are called open sets in X . A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set ), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x .
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus . If a continuous function is one-to-one and onto , and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.
While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n . Lines and circles , but not figure eights , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. Examples include the plane , the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane , which cannot (that is, all their realizations are surfaces that are not manifolds).
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. [ 11 ] [ 12 ] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets , which is closed under finite intersections and (finite or infinite) unions . The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby , arbitrarily small , and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.
Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric . In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces . Having a metric simplifies many proofs.
Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. [ 13 ] The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
The most important of these invariants are homotopy groups , homology, and cohomology .
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
Differential topology is the field dealing with differentiable functions on differentiable manifolds . [ 14 ] It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.
Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. [ 15 ] Some examples of topics in geometric topology are orientability , handle decompositions , local flatness , crumpling and the planar and higher-dimensional Schönflies theorem .
In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature /spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.
2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, [ 16 ] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories and with that the definition of general cohomology theories. [ 17 ]
Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis . [ 18 ]
Topological data analysis uses techniques from algebraic topology to determine the large-scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal ). The main method used by topological data analysis is to:
Several branches of programming language semantics , such as domain theory , are formalized using topology. In this context, Steve Vickers , building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties. [ 20 ]
Topology is relevant to physics in areas such as condensed matter physics , [ 21 ] quantum field theory , quantum computing and physical cosmology .
The topological dependence of mechanical properties in solids is of interest in the disciplines of mechanical engineering and materials science . Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. [ 22 ] The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. [ 23 ] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.
A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants . Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory , the theory of four-manifolds in algebraic topology, and the theory of moduli spaces in algebraic geometry. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory.
The topological classification of Calabi–Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. [ 24 ]
In topological quantum computers , the qubits are stored in topological properties , that are by definition invariant with respect to homotopies . [ 25 ]
In cosmology, topology can be used to describe the overall shape of the universe . [ 26 ] This area of research is commonly known as spacetime topology .
In condensed matter, a relevant application to topological physics comes from the possibility of obtaining a one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect , and then generalized in other areas of physics, for instance in photonics [ 27 ] by F.D.M Haldane .
The possible positions of a robot can be described by a manifold called configuration space . [ 28 ] In the area of motion planning , one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. [ 29 ]
Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components. [ 30 ] [ 31 ] [ 32 ]
In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order that surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path . [ 33 ] | https://en.wikipedia.org/wiki/Topology |
In chemistry , topology provides a way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms , topology provides a model for explaining how the atoms ethereal wave functions must fit together. Molecular topology is a part of mathematical chemistry dealing with the algebraic description of chemical compounds so allowing a unique and easy characterization of them.
Topology is insensitive to the details of a scalar field , and can often be determined using simplified calculations. Scalar fields such as electron density , Madelung field, covalent field and the electrostatic potential can be used to model topology. [ 1 ]
Each scalar field has its own distinctive topology and each provides different information about the nature of chemical bonding and structure. The analysis of these topologies, when combined with simple electrostatic theory and a few empirical observations, leads to a quantitative model of localized chemical bonding. In the process, the analysis provides insights into the nature of chemical bonding.
Applied topology explains how large molecules reach their final shapes and how biological molecules achieve their activity.
Circuit topology is a topological property of folded linear polymers. It describes the arrangement of intra-chain contacts. Contacts can be established by intra-chain interactions, the so called hard contacts (h-contacts), or via chain entanglement or soft contacts (s-contacts). This notion has been applied to structural analysis of biomolecules such as proteins, RNAs, and genome.
It is possible to set up equations correlating direct quantitative structure activity relationships with experimental properties, usually referred to as topological indices (TIs). Topological indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. | https://en.wikipedia.org/wiki/Topology_(chemistry) |
Topology broadcast based on reverse-path forwarding (TBRPF) is a link-state routing protocol for wireless mesh networks .
The obvious design for a wireless link-state protocol (such as the optimized link-state routing protocol ) transmits large amounts of routing data, and this limits the utility of a link-state protocol when the network is made of moving nodes. The number and size of the routing transmissions make the network unusable for any but the smallest networks.
The conventional solution is to use a distance-vector routing protocol such as AODV , which usually transmits no data about routing. However, distance-vector routing requires more time to establish a connection [ citation needed ] , and the routes are less optimized than a link-state router [ citation needed ] .
TBRPF transmits only the differences between the previous network state and the current network state. Therefore, routing messages are smaller, and can therefore be sent more frequently. This means that nodes' routing tables are more up-to-date.
TBRPF is controlled under a US patent filed in December 2000 and assigned to SRI International (Patent ID 6845091, issued January 18, 2005). | https://en.wikipedia.org/wiki/Topology_dissemination_based_on_reverse-path_forwarding |
Topology optimization is a mathematical method that optimizes material layout within a given design space, for a given set of loads , boundary conditions and constraints with the goal of maximizing the performance of the system. Topology optimization is different from shape optimization and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
The conventional topology optimization formulation uses a finite element method (FEM) to evaluate the design performance. The design is optimized using either gradient-based mathematical programming techniques such as the optimality criteria algorithm and the method of moving asymptotes or non gradient-based algorithms such as genetic algorithms .
Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering. Currently, engineers mostly use topology optimization at the concept level of a design process . Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from topology optimization is often fine-tuned for manufacturability. Adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from topology optimization can be directly manufactured using additive manufacturing ; topology optimization is thus a key part of design for additive manufacturing .
A topology optimization problem can be written in the general form of an optimization problem as:
The problem statement includes the following:
Evaluating u ( ρ ) {\displaystyle \mathbf {u(\rho )} } often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not have a known analytical solution.
There are various implementation methodologies that have been used to solve topology optimization problems.
Solving topology optimization problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations. [ 1 ] In literature problems with up to 30000 variables have been reported. [ 2 ]
The earlier stated complexities with solving topology optimization problems using binary variables has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the Solid Isotropic Material with Penalisation method (SIMP). [ 3 ] [ 4 ] This interpolation is essentially a power law E = E 0 + ρ p ( E 1 − E 0 ) {\displaystyle E\;=\;E_{0}\,+\,\rho ^{p}(E_{1}-E_{0})} . It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter p {\displaystyle p} is generally taken between [ 1 , 3 ] {\displaystyle [1,\,3]} . This has been shown to confirm the micro-structure of the materials. [ 5 ] In the SIMP method a lower bound on the Young's modulus is added, E 0 {\displaystyle E_{0}} , to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities. [ 6 ]
There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing. [ 7 ]
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance ) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the objective function of the problem is to minimize the strain energy.
On a broad level, one can visualize that the more the material, the less the deflection as there will be more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint. This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
subject to:
But, a straightforward implementation in the finite element framework of such a problem is still infeasible owing to issues such as:
Some techniques such as filtering based on image processing [ 10 ] are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods. [ 11 ]
Fluid-structure-interaction is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references [ 12 ] [ 13 ] [ 14 ] and. [ 15 ] Design solutions solved for different Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.
Thermoelectricity is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The Seebeck effect and the Peltier effect. The Seebeck effect concerns the conversion of thermal energy into electric energy and the Peltier effect concerns the conversion of electric energy into thermal energy. [ 16 ] By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology, [ 17 ] it is possible to exceed performance of the constitutive thermoelectric materials for thermoelectric coolers and thermoelectric generators . [ 18 ]
The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques. [ citation needed ]
Internal contact can be included in topology optimization by applying the third medium contact method . [ 20 ] [ 21 ] The third medium contact (TMC) method is an implicit contact formulation that is continuous and differentiable. This makes TMC suitable for use with gradient-based approaches to topology optimization. Monolithic [ 22 ] as well as staggerede approaches, [ 19 ] [ 23 ] which are more common in topology optimization, have been used to create various design with internal contact. Recently, thermal contact has been included in the TMC topology optmization framework. [ 24 ] | https://en.wikipedia.org/wiki/Topology_optimization |
The toponome is the spatial network code of proteins and other biomolecules in morphologically intact cells and tissues . [ 1 ] It is mapped and decoded by imaging cycler microscopy (ICM) in situ able to co-map many thousand supermolecules in one sample (tissue section or cell sample at high subcellular resolution). The term "toponome" is derived from the ancient Greek nouns "topos" (τόπος: "place, position") and "nomos" (νόμος: "law"), and the term " toponomics " refers to the study of the toponome. It was introduced by Walter Schubert in 2003. [ 2 ] It addresses the fact that the network of biomolecules in cells and tissues follows topological rules enabling coordinated actions. For example, the cell surface toponome provides the spatial protein interaction code for the execution of a cell movement, a "code of conduct". [ 2 ] [ 3 ] [ 4 ] This is intrinsically dependent on the specific spatial arrangement of similar and dissimilar compositions of supermolecules (compositional periodicity) with a specific spatial order along a cell surface membrane. This spatial order is periodically repeated when the cell tries to enter the exploratory state from the spherical state (spatial periodicity). [ 5 ] This spatial toponome code is hierarchically organized with lead biomolecule(s), anti-colocated (absent) biomolecule(s) [ 2 ] [ 3 ] and wildcard molecules which are variably associated with the lead biomolecule(s). It has been shown that inhibition of lead molecule(s) in a surface membrane leads to disassembly of the corresponding biomolecular network and loss of function. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/Toponome |
Toponomics is a discipline in systems biology , molecular cell biology , and histology concerning the study of the toponome of organisms. [ 1 ] [ 2 ] It is the field of study that purposes to decode the complete toponome in health and disease (the human toponome project) [ 3 ] —which is the next big challenge in human biotechnology after having decoded the human genome. [ 3 ] [ 4 ]
A toponome is the spatial network code of proteins and other biomolecules in morphologically intact cells and tissues. [ 2 ] [ 5 ]
The spatial organization of biomolecules in cells is directly revealed by imaging cycler microscopy with parameter- and dimension-unlimited functional resolution. The resulting toponome structures are hierarchically organized and can be described by a three symbol code. [ 1 ] [ 5 ] [ 6 ] [ 7 ]
The terms toponome and toponomics were introduced in 2003 by Walter Schubert based on observations with imaging cycler microscopes (ICM) . [ 1 ]
Toponome derived from the ancient Greek nouns topos (τόπος, 'place, position') and ' nomos ' (νόμος, 'law'). Hence toponomics is a descriptive term addressing the fact that the spatial network of biomolecules in cells follows topological rules enabling coordinated actions. [ 1 ]
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Toponomics |
In mathematics , a topos ( US : / ˈ t ɒ p ɒ s / , UK : / ˈ t oʊ p oʊ s , ˈ t oʊ p ɒ s / ; plural topoi / ˈ t ɒ p ɔɪ / or / ˈ t oʊ p ɔɪ / , or toposes ) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site ). Topoi behave much like the category of sets and possess a notion of localization. [ 1 ] The Grothendieck topoi find applications in algebraic geometry , and more general elementary topoi are used in logic .
The mathematical field that studies topoi is called topos theory .
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme . Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as "bridges" for connecting theories which, albeit written in possibly very different languages, share a common mathematical content. [ 2 ] [ 3 ]
A Grothendieck topos is a category C {\displaystyle C} which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.)
Here Presh ( D ) {\displaystyle \operatorname {Presh} (D)} denotes the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf .
Giraud's axioms for a category C {\displaystyle C} are:
The last axiom needs the most explanation. If X is an object of C , an "equivalence relation" R on X is a map R → X × X in C such that for any object Y in C , the induced map Hom( Y , R ) → Hom( Y , X ) × Hom( Y , X ) gives an ordinary equivalence relation on the set Hom( Y , X ). Since C has colimits we may form the coequalizer of the two maps R → X ; call this X / R . The equivalence relation is "effective" if the canonical map
is an isomorphism.
Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give
rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.
The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point since functors on the singleton category with a single object and only the identity morphism are just specific sets in the category of sets.
Similarly, there is a topos B G {\displaystyle BG} for any group G {\displaystyle G} which is equivalent to the category of G {\displaystyle G} -sets. We construct this as the category of presheaves on the category with one object, but now the set of morphisms is given by the group G {\displaystyle G} . Since any functor must give a G {\displaystyle G} -action on the target, this gives the category of G {\displaystyle G} -sets. Similarly, for a groupoid G {\displaystyle {\mathcal {G}}} the category of presheaves on G {\displaystyle {\mathcal {G}}} gives a collection of sets indexed by the set of objects in G {\displaystyle {\mathcal {G}}} , and the automorphisms of an object in G {\displaystyle {\mathcal {G}}} has an action on the target of the functor.
More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of a scheme . For each scheme X {\displaystyle X} there is a site Open ( X ) {\displaystyle {\text{Open}}(X)} (of objects given by open subsets and morphisms given by inclusions) whose category of presheaves forms the Zariski topos ( X ) Z a r {\displaystyle (X)_{Zar}} . But once distinguished classes of morphisms are considered, there are multiple generalizations of this which leads to non-trivial mathematics. Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras.
To a scheme and even a stack one may associate an étale topos, an fppf topos, or a Nisnevich topos. Another important example of a topos is from the crystalline site . In the case of the étale topos, these form the foundational objects of study in anabelian geometry , which studies objects in algebraic geometry that are determined entirely by the structure of their étale fundamental group .
Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos).
If X {\displaystyle X} and Y {\displaystyle Y} are topoi, a geometric morphism u : X → Y {\displaystyle u:X\to Y} is a pair of adjoint functors ( u ∗ , u ∗ ) (where u ∗ : Y → X is left adjoint to u ∗ : X → Y ) such that u ∗ preserves finite limits. Note that u ∗ automatically preserves colimits by virtue of having a right adjoint.
By Freyd's adjoint functor theorem , to give a geometric morphism X → Y is to give a functor u ∗ : Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales .
If X {\displaystyle X} and Y {\displaystyle Y} are topological spaces and u {\displaystyle u} is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi for the sites Open ( X ) , Open ( Y ) {\displaystyle {\text{Open}}(X),{\text{Open}}(Y)} .
A point of a topos X {\displaystyle X} is defined as a geometric morphism from the topos of sets to X {\displaystyle X} .
If X is an ordinary space and x is a point of X , then the functor that takes a sheaf F to its stalk F x has a right adjoint
(the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x : 1 → X .
For the etale topos ( X ) e t {\displaystyle (X)_{et}} of a space X {\displaystyle X} , a point is a bit more refined of an object. Given a point x : Spec ( κ ( x ) ) → X {\displaystyle x:{\text{Spec}}(\kappa (x))\to X} of the underlying scheme X {\displaystyle X} a point x ′ {\displaystyle x'} of the topos ( X ) e t {\displaystyle (X)_{et}} is then given by a separable field extension k {\displaystyle k} of κ ( x ) {\displaystyle \kappa (x)} such that the associated map x ′ : Spec ( k ) → X {\displaystyle x':{\text{Spec}}(k)\to X} factors through the original point x {\displaystyle x} . Then, the factorization map Spec ( k ) → Spec ( κ ( x ) ) {\displaystyle {\text{Spec}}(k)\to {\text{Spec}}(\kappa (x))} is an etale morphism of schemes.
More precisely, those are the global points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non-trivial topos may fail to have any. Generalized points are geometric morphisms from a topos Y (the stage of definition ) to X . There are enough of these to display the space-like aspect. For example, if X is the classifying topos S [ T ] for a geometric theory T , then the universal property says that its points are the models of T (in any stage of definition Y ).
A geometric morphism ( u ∗ , u ∗ ) is essential if u ∗ has a further left adjoint u ! , or equivalently (by the adjoint functor theorem) if u ∗ preserves not only finite but all small limits.
A ringed topos is a pair ( X , R ), where X is a topos and R is a commutative ring object in X . Most of the constructions of ringed spaces go through for ringed topoi. The category of R -module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent R -modules: these are R -modules that admit a presentation.
Another important class of ringed topoi, besides ringed spaces, are the étale topoi of Deligne–Mumford stacks .
Michael Artin and Barry Mazur associated to the site underlying a topos a pro-simplicial set (up to homotopy ). [ 4 ] (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory. The study of the pro-simplicial set associated to the étale topos of a scheme is called étale homotopy theory . [ 5 ] In good cases (if the scheme is Noetherian and geometrically unibranch ), this pro-simplicial set is pro-finite .
Since the early 20th century, the predominant axiomatic foundation of mathematics has been set theory , in which all mathematical objects are ultimately represented by sets (including functions , which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle . If symmetry under a particular group G is of importance, one can use the topos consisting of all G -sets .
It is also possible to encode an algebraic theory , such as the theory of groups, as a topos, in the form of a classifying topos . The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.
When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise:
A topos is a category that has the following two properties:
Formally, a power object of an object X {\displaystyle X} is a pair ( P X , ∋ X ) {\displaystyle (PX,\ni _{X})} with ∋ X ⊆ P X × X {\displaystyle {\ni _{X}}\subseteq PX\times X} , which classifies relations, in the following sense.
First note that for every object I {\displaystyle I} , a morphism r : I → P X {\displaystyle r\colon I\to PX} ("a family of subsets") induces a subobject { ( i , x ) | x ∈ r ( i ) } ⊆ I × X {\displaystyle \{(i,x)~|~x\in r(i)\}\subseteq I\times X} . Formally, this is defined by pulling back ∋ X {\displaystyle \ni _{X}} along r × X : I × X → P X × X {\displaystyle r\times X:I\times X\to PX\times X} . The universal property of a power object is that every relation arises in this way, giving a bijective correspondence between relations R ⊆ I × X {\displaystyle R\subseteq I\times X} and morphisms r : I → P X {\displaystyle r\colon I\to PX} .
From finite limits and power objects one can derive that
In some applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.
A logical functor is a functor between topoi that preserves finite limits and power objects. Logical functors preserve the structures that topoi have. In particular, they preserve finite colimits, subobject classifiers , and exponential objects . [ 6 ]
A topos as defined above can be understood as a Cartesian closed category for which the notion of subobject of an object has an elementary or first-order definition. This notion, as a natural categorical abstraction of the notions of subset of a set, subgroup of a group, and more generally subalgebra of any algebraic structure , predates the notion of topos. It is definable in any category, not just topoi, in second-order language, i.e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monics m , n from respectively Y and Z to X , we say that m ≤ n when there exists a morphism p : Y → Z for which np = m , inducing a preorder on monics to X . When m ≤ n and n ≤ m we say that m and n are equivalent. The subobjects of X are the resulting equivalence classes of the monics to it.
In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows.
As noted above, a topos is a category C having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the form x : 1 → X as elements x ∈ X . Morphisms f : X → Y thus correspond to functions mapping each element x ∈ X to the element fx ∈ Y , with application realized by composition.
One might then think to define a subobject of X as an equivalence class of monics m : X′ → X having the same image { mx | x ∈ X′ }. The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume that C is concrete in the sense that the functor C (1,-): C → Set is faithful. For example the category Grph of graphs and their associated homomorphisms is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 → G of a graph G correspond only to the self-loops and not the other edges, nor the vertices without self-loops. Whereas the second-order definition makes G and the subgraph of all self-loops of G (with their vertices) distinct subobjects of G (unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via the Yoneda Lemma as described in the Further examples section below, but this then ceases to be first-order. Topoi provide a more abstract, general, and first-order solution.
As noted above, a topos C has a subobject classifier Ω, namely an object of C with an element t ∈ Ω, the generic subobject of C , having the property that every monic m : X′ → X arises as a pullback of the generic subobject along a unique morphism f : X → Ω, as per Figure 1. Now the pullback of a monic is a monic, and all elements including t are monics since there is only one morphism to 1 from any given object, whence the pullback of t along f : X → Ω is a monic. The monics to X are therefore in bijection with the pullbacks of t along morphisms from X to Ω. The latter morphisms partition the monics into equivalence classes each determined by a morphism f : X → Ω, the characteristic morphism of that class, which we take to be the subobject of X characterized or named by f .
All this applies to any topos, whether or not concrete. In the concrete case, namely C (1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monics m : X′ → X are exactly the injections (one-one functions) from X′ to X , and those with a given image { mx | x ∈ X′ } constitute the subobject of X corresponding to the morphism f : X → Ω for which f −1 ( t ) is that image. The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.
To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to X as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject classifier Ω, leaving the notion of subobject of X as an implicit consequence characterized (and hence namable) by its associated morphism f : X → Ω.
Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos).
The categories of finite sets, of finite G -sets ( actions of a group G on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi.
If C is a small category, then the functor category Set C (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos. Such a graph consists of two sets, an edge set and a vertex set, and two functions s,t between those sets, assigning to every edge e its source s ( e ) and target t ( e ). Grph is thus equivalent to the functor category Set C , where C is the category with two objects E and V and two morphisms s,t : E → V giving respectively the source and target of each edge.
The Yoneda lemma asserts that C op embeds in Set C as a full subcategory. In the graph example the embedding represents C op as the subcategory of Set C whose two objects are V' as the one-vertex no-edge graph and E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from V' to E' (both as natural transformations). The natural transformations from V' to an arbitrary graph (functor) G constitute the vertices of G while those from E' to G constitute its edges. Although Set C , which we can identify with Grph , is not made concrete by either V' or E' alone, the functor U : Grph → Set 2 sending object G to the pair of sets ( Grph ( V' , G ), Grph ( E' , G )) and morphism h : G → H to the pair of functions ( Grph ( V' , h ), Grph ( E' , h )) is faithful. That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.
The category of pointed sets with point-preserving functions is not a topos, since it doesn't have power objects: if P X {\displaystyle PX} were the power object of the pointed set X {\displaystyle X} , and 1 {\displaystyle 1} denotes the pointed singleton, then there is only one point-preserving function r : 1 → P X {\displaystyle r\colon 1\to PX} , but the relations in 1 × X {\displaystyle 1\times X} are as numerous as the pointed subsets of X {\displaystyle X} . The category of abelian groups is also not a topos, for a similar reason: every group homomorphism must map 0 to 0.
The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians.
Grothendieck foundational work on topoi:
The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students. Listed in (perceived) order of increasing difficulty. | https://en.wikipedia.org/wiki/Topos |
In chemistry , a topotactic transition involves a structural change to a crystalline solid, which may include loss or gain of material, so that the final lattice is related to that of the original material by one or more crystallographically equivalent, orientational relationships.
An example is a transition in which the relative structure of the anionic array is unaltered but the cations reorganize as in:
β- Li 2 ZnSiO 4 ⇒ γ- Li 2 ZnSiO 4
An alternate example is the oxidation of magnetite to maghemite .
This chemical reaction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Topotactic_transition |
In building construction , topping out (sometimes referred to as topping off ) is a builders' rite traditionally held when the last beam (or its equivalent) is placed at the top of a structure during its construction. Nowadays, the ceremony is often parlayed into a media event for public relations purposes. [ 1 ] It has since come to mean more generally finishing the structure of the building, whether there is a ceremony or not. It is also commonly used to determine the amount of wind on the top of the structure.
A Scandinavian tradition of hoisting a pine tree to the top of framed out buildings had a more functional purpose: when the pine needles fell off, the builders knew the wood frame below had cured/dried out so they could enclose the building. [ 2 ]
The practice of "topping out" a new building can be traced to the ancient Scandinavian religious rite of placing a tree atop a new building to appease the tree-dwelling spirits displaced in its construction. [ 3 ] The tradition also served a functional purpose: a pine tree was used, and after the needles had fallen off the tree, the builders knew the wood frame below had cured/dried out so they could enclose the building. [ 4 ] Long an important component of timber frame building, [ 5 ] it migrated initially to England and Northern Europe, then to the Americas.
A tree or leafy branch is placed on the topmost wood or iron beam, often with flags and streamers tied to it. A toast is usually drunk and sometimes workers are treated to a meal. In masonry construction the rite celebrates the bedding of the last block or brick. [ citation needed ]
In some cases a topping out event is held at an intermediate point, such as when the roof is dried-in, which means the roof can provide at least semi-permanent protection from the elements. [ 6 ]
The practice remains common in the United Kingdom and assorted Commonwealth countries such as Australia [ 7 ] and Canada, [ 8 ] as well as Germany , Austria , Slovenia , Iceland , Chile , Czech Republic , Slovakia , Poland , Hungary and the Baltic States . In the United States the last beam of a skyscraper is often painted white and signed by all the workers involved. [ 7 ] In New Zealand, completion of the roof to a water-proof state is celebrated through a "roof shout", where workers are treated to cake and beer. [ 9 ]
The tradition of " pannenbier " (literally "(roof) tile beer" in Dutch ) is popular in the Netherlands and Flanders , where a national, regional or city flag is hung once the highest point of a building is reached. It stays in place until the building's owner provides free beer to the workers, after which it is lowered. [ 10 ] Since the workers are treated to free beer as long as the flag is raised, the workers are considered greedy if they fly the flag for more than a few days. [ citation needed ] | https://en.wikipedia.org/wiki/Topping_out |
Topswops (and the variants Topdrops , Bottomswops and Bottomdrops ) are mathematical problems devised and analysed by the British mathematician John Conway in 1973. Contrary to other games and problems introduced by Conway, these problems have not received much attention from the scientific community. Two famous mathematicians who have contributed to the problem are Martin Gardner and Donald Knuth .
In each variant of the problem, Conway uses a deck of playing cards . Since the numerical values of the deck are only relevant, only one suit is used. This is mathematically equivalent to a row of integers from 1 {\displaystyle 1} to N {\displaystyle N} . A shuffled pile of cards is written as A [ 1 ] , . . . , A [ N ] {\displaystyle A[1],...,A[N]} .
For topswops the following algorithm is applied:
The final configuration of the row always starts with 1 {\displaystyle 1} . The topswops problem is occasionally named differently, with naming including deterministic pancake problem , topswops , topswaps , reverse card shuffle and fannkuch . [ 1 ] [ 2 ] [ 3 ]
The problem formulated by Conway is the following:
In literature there are some attempts to find lower and upper bounds for the number of iterations f ( N ) {\displaystyle f(N)} .
Theorem: f ( N ) {\displaystyle f(N)} is bounded by 2 N − 1 {\displaystyle 2^{N-1}} .
Proof by Herbert S. Wilf: [ 2 ] Consider a permutation A [ 1 ] {\displaystyle A[1]} to A [ N ] {\displaystyle A[N]} of the row 1 {\displaystyle 1} to N {\displaystyle N} . As an example, we consider 7 , 2 , 11 , 8 , 5 , 13 , 6 , 1 , 9 , 10 , 3 , 12 , 4 {\displaystyle 7,2,11,8,5,13,6,1,9,10,3,12,4} . We are specifically interested in numbers which are at 'the correct position'. These are: 2, 5, 9, 10, 12. We define the Wilf number as 2 ( 2 − 1 ) + 2 ( 5 − 1 ) + 2 ( 9 − 1 ) + 2 ( 10 − 1 ) + 2 ( 12 − 1 ) = 2833 {\displaystyle 2^{(2-1)}+2^{(5-1)}+2^{(9-1)}+2^{(10-1)}+2^{(12-1)}=2833} .
Claim: after each iteration of the algorithm, the Wilf number increases.
Proof: We perform one iteration of the algorithm. Every number at 'the correct position' and larger than A [ 1 ] {\displaystyle A[1]} , leaves the Wilf number unchanged. The remaining numbers at 'the correct position' will in general not be at 'the correct position' anymore. Nevertheless, the A [ 1 ] {\displaystyle A[1]} 's number is at the correct position. And since the sum of the first A [ 1 ] − 1 {\displaystyle A[1]-1} Wilf numbers is always smaller than the Wilf number of A [ 1 ] {\displaystyle A[1]} , the total Wilf number always increases (with at least 1 per iteration of the algorithm). ◻ {\displaystyle \square }
The maximal Wilf number is found when each number is at the correct position. So the maximal Wilf number is 2 N + 1 − 1 {\displaystyle 2^{N+1}-1} . By refining the proof, the given upper bound can be proven to be a real upper bound for the number of iterations. ◻ {\displaystyle \square }
Theorem: f ( N ) {\displaystyle f(N)} is bounded by the ( N + 1 ) {\displaystyle (N+1)} th Fibonacci number .
Proof by Murray S. Klamkin : [ 4 ] Suppose that during the algorithm, the first number A [ 1 ] {\displaystyle A[1]} takes on in total k {\displaystyle k} distinct values.
Claim: f ( N ) ≤ F k + 1 {\displaystyle f(N)\leq F_{k+1}} .
Proof: We prove the claim by mathematical induction . For k = 1 {\displaystyle k=1} , the algorithm directly terminates, hence, f ( N ) = 0 {\displaystyle f(N)=0} . Thus A [ 1 ] = 1 {\displaystyle A[1]=1} and since F 2 = 1 {\displaystyle F_{2}=1} the claim is proven.
We now take some k ≥ 2 {\displaystyle k\geq 2} . All k {\displaystyle k} values that A [ 1 ] {\displaystyle A[1]} takes on, are ordered and can be written as: d 1 < . . . < d k {\displaystyle d_{1}<...<d_{k}} . Suppose that the largest value of these values, which is d k {\displaystyle d_{k}} , occurs for the first time at position 1 {\displaystyle 1} during iteration r {\displaystyle r} of the algorithm. Denote t = A [ d k ] {\displaystyle t=A[d_{k}]} . During the ( r + 1 ) {\displaystyle (r+1)} 'th iteration, we know A [ 1 ] = t {\displaystyle A[1]=t} and A [ d k ] = d k {\displaystyle A[d_{k}]=d_{k}} . The remaining iterations will always retain A [ d k ] = d k {\displaystyle A[d_{k}]=d_{k}} . Hence A [ 1 ] {\displaystyle A[1]} can now take on at most k − 1 {\displaystyle k-1} values. Using induction for k {\displaystyle k} , it follows that f ( N ) − r ≤ F k {\displaystyle f(N)-r\leq F_{k}} and also that d 1 = 1 {\displaystyle d_{1}=1} . ◻ {\displaystyle \square }
Suppose we would exchange d 1 = 1 {\displaystyle d_{1}=1} and d k {\displaystyle d_{k}} in iteration r {\displaystyle r} Then A [ 1 ] = 1 {\displaystyle A[1]=1} and the algorithm terminates; f ( N ) = r {\displaystyle f(N)=r} . During the algorithm, we are sure that both d k {\displaystyle d_{k}} and t {\displaystyle t} have never been at position A [ 1 ] {\displaystyle A[1]} , unless t = 1 {\displaystyle t=1} .
Suppose t = 1 {\displaystyle t=1} . Then r ≤ F k {\displaystyle r\leq F_{k}} since A [ 1 ] {\displaystyle A[1]} takes on at most k − 1 {\displaystyle k-1} distinct values. So it follows that f ( N ) ≤ r + 1 ≤ F k + 1 {\displaystyle f(N)\leq r+1\leq F_{k+1}} .
Suppose t > 1 {\displaystyle t>1} . Then r ≤ F k − 1 {\displaystyle r\leq F_{k-1}} since A [ 1 ] {\displaystyle A[1]} takes on at most k − 2 {\displaystyle k-2} distinct values. Using the claim, it follows that f ( N ) ≤ F k + r ≤ F k + 1 {\displaystyle f(N)\leq F_{k}+r\leq F_{k+1}} . This proves the theorem. ◻ {\displaystyle \square }
Besides these results, Morales and Sudborough have recently proven that the lower bound for f ( N ) {\displaystyle f(N)} is a quadratic function in N {\displaystyle N} . [ 1 ] The optimal values are, however, still unknown. There have been several attempts to find the optimal values, for example by A. Pepperdine. [ 5 ] For rows with 19 or fewer numbers, the exact solution is known. Larger rows only have lower bounds, which is shown on the right.
It is yet unknown whether this problem is NP-hard .
A similar problem is topdrops , where the same playing cards are used. In this problem, the first card of the pile is shown (and has value A [ 1 ] {\displaystyle A[1]} ). Take the first A [ 1 ] {\displaystyle A[1]} cards of the pile, change the order and place them back on the bottom of the pile (which contrasts topswops , where the cards are placed at the top). This problem allows for infinite loops . As an example, we consider the row 2,1,3,4. By applying the algorithm, the following sequence is obtained:
whereafter the original row is found again.
In this variant, the bottom card of the pile is taken (and again named A [ 1 ] {\displaystyle A[1]} ). Then the first A [ 1 ] {\displaystyle A[1]} cards of the pile are swapped. Unless the bottom card is the highest card in the pile, nothing happens. This makes the problem uninteresting due to the limited behaviour. [ 2 ]
The final variant is botdrops where the bottom card of the pile is taken (again A [ 1 ] {\displaystyle A[1]} ). In this variant, the bottom A [ 1 ] {\displaystyle A[1]} cards are swapped. | https://en.wikipedia.org/wiki/Topswops |
The Torch Triple X (or XXX ) was a UNIX workstation computer produced by the British company Torch Computers , and launched in 1985. It was based on the Motorola 68010 microprocessor and ran a version of UNIX System V .
The Triple X was based on an 8 MHz 68010 CPU , with a Hitachi 6303 "service processor". The CPU was accompanied by a 68451 memory management unit and a 68450 DMA controller. Both VMEbus and a BBC Micro -compatible "1MHz bus" expansion buses were provided, as was a SCSI host adapter, and an optional Ethernet interface. Both RS-423 and X.25 -compatible synchronous serial ports were provided. This latter feature made the Triple X attractive to the UK academic community, where X.25 networks were prevalent at the time.
Standard RAM capacity was 1 MB, expandable to 7 MB via VME cards. A 720 KB, 5.25-in floppy disk drive and ST-506 -compatible 20 MB hard disk were fitted as standard, interfaced to the SCSI bus via an OMTI adapter.
Either a 10 or 13 inch colour monitor was supplied. Two graphics modes were available: 720 × 256 pixels in four colours, or 720 × 512 in two colours.
The Triple X had a novel touch-sensitive "soft" power switch. When switching off, this commanded the operating system to shut down gracefully before powering down.
The Triple X's firmware was called Caretaker . The native operating system was Uniplus+ UNIX System V Release 2. A graphical user interface called OpenTop was also included as standard. [ 1 ]
The Quad X is an enhanced version of the Triple X, with a 68020 processor and three VME expansion slots. This was produced only in small numbers before Torch became insolvent.
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Torch_Triple_X |
The toric code is a topological quantum error correcting code , and an example of a stabilizer code , defined on a two-dimensional spin lattice . [ 1 ] It is the simplest and most well studied of the quantum double models. [ 2 ] It is also the simplest example of topological order — Z 2 topological order
(first studied in the context of Z 2 spin liquid in 1991). [ 3 ] [ 4 ] The toric code can also be considered to be a Z 2 lattice gauge theory in a particular limit. [ 5 ] It was introduced by Alexei Kitaev .
The toric code gets its name from its periodic boundary conditions, giving it the shape of a torus . These conditions give the model translational invariance, which is useful for analytic study. However, some experimental realizations require open boundary conditions, allowing the system to be embedded on a 2D surface. The resulting code is typically known as the planar code. This has identical behaviour to the toric code in most, but not all, cases.
The toric code is defined on a two-dimensional lattice, usually chosen to be the square lattice , with a spin-½ degree of freedom located on each edge. They are chosen to be periodic. Stabilizer operators are defined on the spins around each vertex v {\displaystyle v} and plaquette [ definition needed ] (or face i.e. a vertex of the dual lattice) [ clarification needed ] p {\displaystyle p} of the lattice as follows,
A v = ∏ i ∈ v σ i x , B p = ∏ i ∈ p σ i z . {\displaystyle A_{v}=\prod _{i\in v}\sigma _{i}^{x},\,\,B_{p}=\prod _{i\in p}\sigma _{i}^{z}.}
Where here we use i ∈ v {\displaystyle i\in v} to denote the edges touching the vertex v {\displaystyle v} , and i ∈ p {\displaystyle i\in p} to denote the edges surrounding the plaquette p {\displaystyle p} . The stabilizer space of the code is that for which all stabilizers act trivially, hence for any state | ψ ⟩ {\displaystyle |\psi \rangle } in this space it holds that
A v | ψ ⟩ = | ψ ⟩ , ∀ v , B p | ψ ⟩ = | ψ ⟩ , ∀ p . {\displaystyle A_{v}|\psi \rangle =|\psi \rangle ,\,\,\forall v,\,\,B_{p}|\psi \rangle =|\psi \rangle ,\,\,\forall p.}
For the toric code, this space is four-dimensional, and so can be used to store two qubits of quantum information . This can be proven by considering the number of independent stabilizer operators. The occurrence of errors will move the state out of the stabilizer space, resulting in vertices and plaquettes for which the above condition does not hold. The positions of these violations is the syndrome of the code, which can be used for error correction.
The unique nature of the topological codes, such as the toric code, is that stabilizer violations can be interpreted as quasiparticles . Specifically, if the code is in a state | ϕ ⟩ {\displaystyle |\phi \rangle } such that,
A v | ϕ ⟩ = − | ϕ ⟩ {\displaystyle A_{v}|\phi \rangle =-|\phi \rangle } ,
a quasiparticle known as an e {\displaystyle e} anyon can be said to exist on the vertex v {\displaystyle v} . Similarly violations of the B p {\displaystyle B_{p}} are associated with so called m {\displaystyle m} anyons on the plaquettes. The stabilizer space therefore corresponds to the anyonic vacuum. Single spin errors cause pairs of anyons to be created and transported around the lattice.
When errors create an anyon pair and move the anyons, one can imagine a path connecting the two composed of all links acted upon. If the anyons then meet and are annihilated, this path describes a loop. If the loop is topologically trivial, it has no effect on the stored information. The annihilation of the anyons, in this case, corrects all of the errors involved in their creation and transport. However, if the loop is topologically non-trivial, though re-annihilation of the anyons returns the state to the stabilizer space, it also implements a logical operation on the stored information. The errors, in this case, are therefore not corrected but consolidated.
Consider the noise model for which bit and phase errors occur independently on each spin, both with probability p . When p is low, this will create sparsely distributed pairs of anyons which have not moved far from their point of creation. Correction can be achieved by identifying the pairs that the anyons were created in (up to an equivalence class), and then re-annihilating them to remove the errors. As p increases, however, it becomes more ambiguous as to how the anyons may be paired without risking the formation of topologically non-trivial loops. This gives a threshold probability, under which the error correction will almost certainly succeed. Through a mapping to the random-bond Ising model, this critical probability has been found to be around 11%. [ 6 ]
Other error models may also be considered, and thresholds found. In all cases studied so far, the code has been found to saturate the Hashing bound . For some error models, such as biased errors where bit errors occur more often than phase errors or vice versa, lattices other than the square lattice must be used to achieve the optimal thresholds. [ 7 ] [ 8 ]
These thresholds are upper limits and are useless unless efficient algorithms are found to achieve them. The most well-used algorithm is minimum weight perfect matching . [ 9 ] When applied to the noise model with independent bit and flip errors, a threshold of around 10.5% is achieved. This falls only a little short of the 11% maximum. However, matching does not work so well when there are correlations between the bit and phase errors, such as with depolarizing noise.
The means to perform quantum computation on logical information stored within the toric code has been considered, with the properties of the code providing fault-tolerance. It has been shown that extending the stabilizer space using 'holes', vertices or plaquettes on which stabilizers are not enforced, allows many qubits to be encoded into the code. However, a universal set of unitary gates cannot be fault-tolerantly implemented by unitary operations and so additional techniques are required to achieve quantum computing. For example, universal quantum computing can be achieved by preparing magic states via encoded quantum stubs called tidBits used to teleport in the required additional gates when replaced as a qubit. Furthermore, preparation of magic states must be fault tolerant, which can be achieved by magic state distillation on noisy magic states. A measurement based scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two-dimensional architecture. [ 10 ] [ 11 ]
Since the stabilizer operators of the toric code are quasilocal, acting only on spins located near each other on a two-dimensional lattice, it is not unrealistic to define the following Hamiltonian,
H T C = − J ∑ v A v − J ∑ p B p , J > 0. {\displaystyle H_{\rm {TC}}=-J\sum _{v}A_{v}-J\sum _{p}B_{p},\,\,\,J>0.}
The ground state space of this Hamiltonian is the stabilizer space of the code. Excited states correspond to those of anyons, with the energy proportional to their number. Local errors are therefore energetically suppressed by the gap, which has been shown to be stable against local perturbations. [ 12 ] However, the dynamic effects of such perturbations can still cause problems for the code. [ 13 ] [ 14 ]
The gap also gives the code a certain resilience against thermal errors, allowing it to be correctable almost surely for a certain critical time. This time increases with J {\displaystyle J} , but since arbitrary increases of this coupling are unrealistic, the protection given by the Hamiltonian still has its limits.
The means to make the toric code, or the planar code, into a fully self-correcting quantum memory is often considered. Self-correction means that the Hamiltonian will naturally suppress errors indefinitely, leading to a lifetime that diverges in the thermodynamic limit. It has been found that this is possible in the toric code only if long range interactions are present between anyons. [ 15 ] [ 16 ] Proposals have been made for realization of these in the lab [ 17 ] Another approach is the generalization of the model to higher dimensions, with self-correction possible in 4D with only quasi-local interactions. [ 18 ]
As mentioned above, so called e {\displaystyle e} and m {\displaystyle m} quasiparticles are associated with the vertices and plaquettes of the model, respectively. These quasiparticles can be described as anyons , due to the non-trivial effect of their braiding. Specifically, though both species of anyons are bosonic with respect to themselves, the braiding of two e {\displaystyle e} 's or m {\displaystyle m} 's having no effect, a full monodromy of an e {\displaystyle e} and an m {\displaystyle m} will yield a phase of − 1 {\displaystyle -1} . Such a result is not consistent with either bosonic or fermionic statistics , and hence is anyonic.
The anyonic mutual statistics of the quasiparticles demonstrate the logical operations performed by topologically non-trivial loops. Consider the creation of a pair of e {\displaystyle e} anyons followed by the transport of one around a topologically nontrivial loop, such as that shown on the torus in blue on the figure above, before the pair are reannhilated. The state is returned to the stabilizer space, but the loop implements a logical operation on one of the stored qubits. If m {\displaystyle m} anyons are similarly moved through the red loop above a logical operation will also result. The phase of − 1 {\displaystyle -1} resulting when braiding the anyons shows that these operations do not commute, but rather anticommute. They may therefore be interpreted as logical Z {\displaystyle Z} and X {\displaystyle X} Pauli operators on one of the stored qubits. The corresponding logical Pauli's on the other qubit correspond to an m {\displaystyle m} anyon following the blue loop and an e {\displaystyle e} anyon following the red. No braiding occurs when e {\displaystyle e} and m {\displaystyle m} pass through parallel paths, the phase of − 1 {\displaystyle -1} therefore does not arise and the corresponding logical operations commute. This is as should be expected since these form operations acting on different qubits.
Due to the fact that both e {\displaystyle e} and m {\displaystyle m} anyons can be created in pairs, it is clear to see that both these quasiparticles are their own antiparticles. A composite particle composed of two e {\displaystyle e} anyons is therefore equivalent to the vacuum, since the vacuum can yield such a pair and such a pair will annihilate to the vacuum. Accordingly, these composites have bosonic statistics, since their braiding is always completely trivial. A composite of two m {\displaystyle m} anyons is similarly equivalent to the vacuum. The creation of such composites is known as the fusion of anyons, and the results can be written in terms of fusion rules. In this case, these take the form,
e × e = 1 , m × m = 1. {\displaystyle e\times e=1,\,\,\,m\times m=1.}
Where 1 {\displaystyle 1} denotes the vacuum. A composite of an e {\displaystyle e} and an m {\displaystyle m} is not trivial. This therefore constitutes another quasiparticle in the model, sometimes denoted ψ {\displaystyle \psi } , with fusion rule,
e × m = ψ . {\displaystyle e\times m=\psi .}
From the braiding statistics of the anyons we see that, since any single exchange of two ψ {\displaystyle \psi } 's will involve a full monodromy of a constituent e {\displaystyle e} and m {\displaystyle m} , a phase of − 1 {\displaystyle -1} will result. This implies fermionic self-statistics for the ψ {\displaystyle \psi } 's.
The use of a torus is not required to form an error correcting code. Other surfaces may also be used, with their topological properties determining the degeneracy of the stabilizer space. In general, quantum error correcting codes defined on two-dimensional spin lattices according to the principles above are known as surface codes. [ 19 ]
It is also possible to define similar codes using higher-dimensional spins. These are the quantum double models [ 20 ] and string-net models, [ 21 ] which allow a greater richness in the behaviour of anyons, and so may be used for more advanced quantum computation and error correction proposals. [ 22 ] These not only include models with Abelian anyons, but also those with non-Abelian statistics. [ 23 ] [ 24 ] [ 25 ]
The most explicit demonstration of the properties of the toric code has been in state based approaches. Rather than attempting to realize the
Hamiltonian, these simply prepare the code in the stabilizer space. Using this technique, experiments have been able to demonstrate the creation, transport and statistics of the anyons [ 26 ] [ 27 ] [ 28 ] and measurement of the topological entanglement entropy . [ 28 ] More recent experiments have also been able to demonstrate the error correction properties of the code. [ 29 ] [ 28 ]
For realizations of the toric code and its generalizations with a Hamiltonian, much progress has been made using Josephson junctions . The theory of how the Hamiltonians may be implemented has been developed for a wide class of topological codes. [ 30 ] An experiment has also been performed, realizing the toric code Hamiltonian for a small lattice, and demonstrating the quantum memory provided by its degenerate ground state. [ 31 ]
Other theoretical and experimental works towards realizations are based on cold atoms. A toolkit of methods that may be used to realize topological codes with optical lattices has been explored, [ 32 ] as have experiments concerning minimal instances of topological order. [ 33 ] Such minimal instances of the toric code has been realized experimentally within isolated square plaquettes. [ 34 ] Progress is also being made into simulations of the toric model with Rydberg atoms , in which the Hamiltonian and the effects of dissipative noise can be demonstrated. [ 35 ] [ 36 ] Experiments in Rydberg atom arrays have also successfully realized the toric code with periodic boundary conditions in two dimensions by coherently transporting arrays of entangled atoms. [ 37 ] | https://en.wikipedia.org/wiki/Toric_code |
In algebra , a toric ideal is an ideal generated by differences of two monomials . An affine or projective algebraic variety defined by a toric prime ideal or a homogeneous toric ideal is an affine or projective toric variety . [ 1 ] [ 2 ] [ 3 ] [ 4 ]
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Toric_ideal |
In mathematics , a toric manifold is a topological analogue of toric variety in algebraic geometry . It is an even-dimensional manifold with an effective smooth action of an n {\displaystyle n} -dimensional compact torus which is locally standard with the orbit space a simple convex polytope . [ 1 ] [ 2 ]
The aim is to do combinatorics on the quotient polytope and obtain information on the manifold above. For example, the Euler characteristic and the cohomology ring of the manifold can be described in terms of the polytope.
This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polytope.
This algebraic geometry –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Toric_manifold |
A toric section is an intersection of a plane with a torus , just as a conic section is the intersection of a plane with a cone . Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux . [ 1 ]
In general, toric sections are fourth-order ( quartic ) plane curves [ 1 ] of the form
A special case of a toric section is the spiric section , in which the intersecting plane is parallel to the rotational symmetry axis of the torus . They were discovered by the ancient Greek geometer Perseus in roughly 150 BC. [ 2 ] Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli .
Another special case is the Villarceau circles , in which the intersection is a circle despite the lack of any of the obvious sorts of symmetry that would entail a circular cross-section. [ 3 ]
More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or oblique to the rotational symmetry axis.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Toric_section |
The Torino scale is a method for categorizing the impact hazard associated with near-Earth objects (NEOs) such as asteroids and comets . It is intended as a communication tool for astronomers and the public to assess the seriousness of collision predictions, by combining probability statistics and known kinetic damage potentials into a single threat value. The Palermo scale is a similar, but more complex scale.
Near-Earth objects with a Torino scale of 1 are discovered several times a year, and may last a few weeks until they have a longer observation arc that eliminates any possibility of a collision. The only objects on the Torino scale that have ever ranked higher are asteroids 99942 Apophis , which had a rating of 4 for four days in late 2004, the highest recorded rating; (144898) 2004 VD 17 , with a historical rating of 2 from February to May 2006; and 2024 YR 4 , with a rating of 3 from January 27, 2025 to February 20, 2025. [ 1 ]
The Torino scale uses an integer scale from 0 to 10. A 0 indicates an object has a negligibly small chance of collision with the Earth , compared with the usual "background noise" of collision events, or is too small to penetrate Earth's atmosphere intact. A 10 indicates that a collision is certain, and the impacting object is large enough to precipitate a global disaster.
An object is assigned a 0 to 10 value based on its collision probability and the kinetic energy of the possible collision.
The Torino scale is defined only for potential impacts less than 100 years in the future.
"For an object with multiple potential collisions on a set of dates, a Torino scale value should be determined for each date. It may be convenient to summarize such an object by the greatest Torino scale value within the set." [ 2 ]
The Torino scale was created by Professor Richard P. Binzel in the Department of Earth, Atmospheric, and Planetary Sciences, at the Massachusetts Institute of Technology (MIT). The first version, called "A Near-Earth Object Hazard Index", was presented at a United Nations conference in 1995 and was published by Binzel in the subsequent conference proceedings ( Annals of the New York Academy of Sciences, volume 822, 1997.)
A revised version of the "Hazard Index" was presented at a June 1999 international conference on NEOs held in Torino ( Turin ), Italy. The conference participants voted to adopt the revised version, where the bestowed name "Torino scale" recognizes the spirit of international cooperation displayed at that conference toward research efforts to understand the hazards posed by NEOs. ("Torino scale" is the proper usage, not "Turin scale.") [ 2 ] This version was published in a subsequent peer-reviewed article. [ 3 ]
Due to exaggerated press coverage of Level 1 asteroids, a rewording of the Torino scale was published in 2005, adding more details and renaming the categories: in particular, Level 1 was changed from "Events meriting careful monitoring" to "Normal".
The Torino scale has served as the model for the Rio scale , which quantifies the validity and societal impact of SETI data. [ 4 ]
Ratings are assigned based on the parameters of impact calculations. Each rating has a defined meaning which is to inform the public.
Ratings are determined on the basis of the impact probability ( p ), expressed as a real number between 0 for no chance of impact and 1 for a certain impact; and the estimated impact energy ( E ), expressed in megatons of TNT .
The Torino scale also uses a color code scale: white, green, yellow, orange, red. Each color code has an overall meaning: [ 5 ]
No incoming object has ever been rated above level 4, though over Earth's history impacts have spanned the full range of damage described by the scale .
For NASA , a unit of the Jet Propulsion Laboratory (JPL), the Center for Near-Earth Object Studies (CNEOS) calculates impact risks and assigns ratings in its Sentry Risk Table , [ 6 ] while another unit of JPL, Solar System Dynamics (SSD) provides orbital and close approach data. [ 7 ] For ESA , similar services are provided by its Near-Earth Object Coordination Centre (NEOCC), which maintains its own Risk List [ 8 ] and Close Approaches List. [ 9 ]
The basis for the risk evaluation is the most recent orbit calculation based on all known reliable observations. Along the calculated orbit, close approaches with Earth are determined. Due to measurement and model imprecision, the orbit calculation has an uncertainty, which can be quantified for the close approach distance. Assuming a two-dimensional Gaussian probability distribution in the plane perpendicular to the asteroid's orbit (the B-plane), the uncertainty can be characterized by the standard deviation (sigma) the close approach point in the directions along the asteroid's orbit and perpendicular to it, where the former is usually much larger. The one-sigma margin, which is used by ESA NEOCC one-sigma, [ 10 ] means that the close approach point is within those bounds with a 68.3% probability, while the 3-sigma margin, used by NASA JPL SSD, corresponds to 99.7% probability. The probability of an impact is the integral of the probability distribution over the cross section of Earth in the B-plane.
When the close approach of a newly discovered asteroid is first put on a risk list with a significant risk, it is normal for the risk to first increase, regardless whether the potential impact will eventually be ruled out or confirmed with the help of additional observations. [ 11 ] After discovery, Earth will be close to the center of the probability distribution, that is, the 3-sigma uncertainty margin will be much bigger than the nominal close encounter distance. With additional observations, the uncertainty will decrease, thus the 3-sigma uncertainty region will shrink, thus Earth will initially cover an increasing part of the probability distribution, resulting in increased risk, and potentially an increasing rating. If the real orbit bypasses Earth, with further observations, Earth will only intersect the tail of the probability distribution (the 3-sigma region will shrink to exclude the Earth) and the impact risk will fall towards zero; while in case the asteroid will hit the Earth, the probability distribution will contract towards its intersection (the 3-sigma region will shrink into Earth's intersection in the B-plane) and the risk will rise towards 100%. [ 12 ]
The Chicxulub impact , believed by most scientists to have been a significant factor in the extinction of the non-avian dinosaurs , has been estimated at 100 million (10 8 ) megatons. Were an equivalent impact predicted with a probability of 99% or more, it would rate 10 on the Torino scale. The impacts that created the Barringer Crater and the 1908 Tunguska event are both estimated to have been in the 3–10 megaton range, [ 13 ] thus, if a similar impact had been predicted with near certainty, it would correspond to Torino scale 8. The 2013 Chelyabinsk meteor had a total kinetic energy prior to impact of about 0.5 megatons, thus, regardless of impact probability, it would only rate 0 on the Torino scale - despite breaking over 3600 windows and injuring around 1500 people. [ 14 ] Between 2000 and 2013, 26 atmospheric asteroid impacts with an energy of 1–600 kilotons were detected by the network of infrasound sensors operated by the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization . [ 15 ]
The biggest hydrogen bomb ever exploded, the Tsar Bomba , was around 50 megatons. The 1883 eruption of Krakatoa was the equivalent of roughly 200 megatons.
The comet C/2013 A1 , which passed close to Mars in 2014, was originally estimated to have a potential impact energy of 5 million to 24 billion megatons, and in March 2013 was estimated to have a Mars impact probability of ~1:1250, corresponding to the Martian equivalent of Torino scale 6. [ 16 ] The impact probability was reduced to ~1:120000 in April 2013, corresponding to Torino scale 1 or 2. [ 17 ]
This article incorporates public domain material from websites or documents of the National Aeronautics and Space Administration . | https://en.wikipedia.org/wiki/Torino_scale |
In coding theory , Tornado codes are a class of erasure codes that support error correction . Tornado codes require a constant C more redundant blocks than the more data-efficient Reed–Solomon erasure codes , but are much faster to generate and can fix erasures faster. Software-based implementations of tornado codes are about 100 times faster on small lengths and about 10,000 times faster on larger lengths than Reed–Solomon erasure codes. [ 1 ] Since the introduction of Tornado codes, many other similar erasure codes have emerged, most notably Online codes , LT codes and Raptor codes .
Tornado codes use a layered approach. All layers except the last use an LDPC error correction code, which is fast but has a chance of failure. The final layer uses a Reed–Solomon correction code, which is slower but is optimal in terms of failure recovery. Tornado codes dictates how many levels, how many recovery blocks in each level, and the distribution used to generate blocks for the non-final layers.
The input data is divided into blocks. Blocks are sequences of bits that are all the same size. Recovery data uses the same block size as the input data. The erasure of a block (input or recovery) is detected by some other means. (For example, a block from disk does not pass a CRC check or a network packet with a given sequence number never arrived.)
The number of recovery blocks is given by the user. Then the number of levels is determined along with the number of blocks in each level. The number in each level is determined by a factor B which is less than one. If there are N input blocks, the first recovery level has B*N blocks, the second has B×B×N, the third has B×B×B×N, and so on.
All levels of recovery except the final one use an LDPC, which works by xor (exclusive-or). Xor operates on binary values, 1s and 0s. A xor B is 1 if A and B have different values and 0 if A and B have the same values. If you are given result of (A xor B) and A, you can determine the value for B. (A xor B xor A = B) Similarly, if you are given result of (A xor B) and B, you can determine the value for A. This extends to multiple values, so given result of (A xor B xor C xor D) and any 3 of the values, the missing value can be recovered.
So the recovery blocks in level one are just the xor of some set of input blocks. Similarly, the recovery blocks in level two are each the xor of some set of blocks in level one. The blocks used in the xor are chosen randomly, without repetition. However, the number of blocks xor'ed to make a recovery block is chosen from a very specific distribution for each level.
Since xor is a fast operation and the recovery blocks are an xor of only a subset of the blocks in the input (or at a lower recovery level), the recovery blocks can be generated quickly.
The final level is a Reed–Solomon code. Reed–Solomon codes are optimal in terms of recovering from failures, but slow to generate and recover. Since each level has fewer blocks than the one before, the Reed–Solomon code has a small number of recovery blocks to generate and to use in recovery. So, even though Reed–Solomon is slow, it only has a small amount of data to handle.
During recovery, the Reed–Solomon code is recovered first. This is guaranteed to work if the number of missing blocks in the next-to-final level is less than the present blocks in the final level.
Going lower, the LDPC (xor) recovery level can be used to recover the level beneath it with high probability if all the recovery blocks are present and the level beneath is missing at most C' fewer blocks than the recovery level. The algorithm for recovery is to find some recovery block that has only one of its generating set missing from the lower level. Then the xor of the recovery block with all of the blocks that are present is equal to the missing block.
Tornado codes were formerly patented inside the United States of America. [ 2 ] Patents US6163870 A (filed Nov 6, 1997) and US 6081909 A (filed Nov 6, 1997) describe Tornado codes, and have expired as of November 6, 2017. Patents US6307487 B1 (filed Feb 5, 1999) and US6320520 B1 (filed Sep 17, 1999) also mention Tornado codes, and have expired as of February 5, 2019, and September 17, 2019, respectively.
Michael Luby created the Tornado codes. [ 3 ] [ 4 ]
A readable description from CMU (PostScript) [1] and another from Luby at the International Computer Science Institute (PostScript) [2] . | https://en.wikipedia.org/wiki/Tornado_code |
In mathematics , a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. [ 1 ] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle , then the object is called a torus .
The term toroid is also used to describe a toroidal polyhedron . In this context a toroid need not be circular and may have any number of holes. A g -holed toroid can be seen as approximating the surface of a torus having a topological genus , g , of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1− g ). [ 2 ]
The torus is an example of a toroid, which is the surface of a doughnut . Doughnuts are an example of a solid torus created by rotating a disk, and are not toroids.
Toroidal structures occur in both natural and synthetic materials. [ 3 ]
A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
Pappus's centroid theorem generalizes the formulas here to arbitrary surfaces of revolution. | https://en.wikipedia.org/wiki/Toroid |
The terms toroidal and poloidal refer to directions relative to a torus of reference. They describe a three-dimensional coordinate system in which the poloidal direction follows a small circular ring around the surface, while the toroidal direction follows a large circular ring around the torus, encircling the central void.
The earliest use of these terms cited by the Oxford English Dictionary is by Walter M. Elsasser (1946) in the context of the generation of the Earth's magnetic field by currents in the core, with "toroidal" being parallel to lines of constant latitude and "poloidal" being in the direction of the magnetic field (i.e. towards the poles ).
The OED also records the later usage of these terms in the context of toroidally confined plasmas, as encountered in magnetic confinement fusion . In the plasma context, the toroidal direction is the long way around the torus, the corresponding coordinate being denoted by z in the slab approximation or ζ or φ in magnetic coordinates; the poloidal direction is the short way around the torus, the corresponding coordinate being denoted by y in the slab approximation or θ in magnetic coordinates. (The third direction, normal to the magnetic surfaces, is often called the "radial direction", denoted by x in the slab approximation and variously ψ , χ , r , ρ , or s in magnetic coordinates.)
As a simple example from the physics of magnetically confined plasmas, consider an axisymmetric system with circular, concentric magnetic flux surfaces of radius r {\displaystyle r} (a crude approximation to the magnetic field geometry in an early tokamak but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by ζ {\displaystyle \zeta } and the poloidal angle by θ {\displaystyle \theta } .
Then the toroidal/poloidal coordinate system relates to standard Cartesian coordinates by these transformation rules:
where s θ = ± 1 , s ζ = ± 1 {\displaystyle s_{\theta }=\pm 1,s_{\zeta }=\pm 1} .
The natural choice geometrically is to take s θ = s ζ = + 1 {\displaystyle s_{\theta }=s_{\zeta }=+1} , giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes r , θ , ζ {\displaystyle r,\theta ,\zeta } a left-handed curvilinear coordinate system. As it is usually assumed in setting up flux coordinates for describing magnetically confined plasmas that the set r , θ , ζ {\displaystyle r,\theta ,\zeta } forms a right -handed coordinate system, ∇ r ⋅ ∇ θ × ∇ ζ > 0 {\displaystyle \nabla r\cdot \nabla \theta \times \nabla \zeta >0} , we must either reverse the poloidal direction by taking s θ = − 1 , s ζ = + 1 {\displaystyle s_{\theta }=-1,s_{\zeta }=+1} , or reverse the toroidal direction by taking s θ = + 1 , s ζ = − 1 {\displaystyle s_{\theta }=+1,s_{\zeta }=-1} . Both choices are used in the literature.
To study single particle motion in toroidally confined plasma devices, velocity and acceleration vectors must be known. Considering the natural choice s θ = s ζ = + 1 {\displaystyle s_{\theta }=s_{\zeta }=+1} , the unit vectors of toroidal and poloidal coordinates system ( r , θ , ζ ) {\displaystyle \left(r,\theta ,\zeta \right)} can be expressed as:
according to Cartesian coordinates. The position vector is expressed as:
The velocity vector is then given by:
and the acceleration vector is: | https://en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates |
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