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On 2 April 1979, spores of Bacillus anthracis (the causative agent of anthrax ) were accidentally released from a Soviet Armed Forces research facility in the city of Sverdlovsk in the Soviet Union . The ensuing outbreak of the disease resulted in the deaths of at least 68 people, although the exact number of victims remains unknown. [ 1 ] The cause of the outbreak was denied for years by the Soviet authorities , which blamed the deaths on consumption of tainted meat from the area, and subcutaneous exposure due to butchers handling the tainted meat. The accident was the first major indication in the Western world that the Soviet Union had embarked upon an offensive programme aimed at the development and large-scale production of biological weapons .
Sverdlovsk had been a major production center of the Soviet military-industrial complex since World War II . By the 1970s, 87 percent of the city's industrial production was military; only 13 percent for public consumption. [ 2 ] It produced tanks , ballistic missiles , rockets , and other armaments . The city has at times been referred to as Russia's Pittsburgh because of its large steelmaking industry. [ 3 ] During the Cold War, Sverdlovsk became a Soviet " closed city " to which travel was restricted for foreigners. [ 2 ]
The biological warfare (BW) facility in Sverdlovsk was built during the period 1947 to 1949 and was a spin-off of the Soviet Union's main military BW facility in Kirov . It was allocated the former site of the Cherkassk-Sverdlovsk Infantry Academy in Sverdlovsk on Ulitsa Zvezdnaya, 1, and abutted the southern industrialised sector of the city. The new facility, known as the USSR Ministry of Defence's Scientific-Research Institute of Hygiene, became operational on 19 July 1949. [ 4 ] Alibek suggests that the construction of the institute incorporated technical knowledge which had been extracted from captured Japanese scientists who had participated in the Japanese biological warfare programme . [ 5 ] Research was initiated at Sverdlovsk on bacterial pathogens including Bacillus anthracis. In 1951, a programme was launched which focussed on botulinum toxin . [ 4 ] Later in the 1970s, interest in the latter ceased and there was a major shift in focus to B. anthracis . [ 6 ] In 1974, the facility was re-named as the Scientific-Research Institute of Bacterial Vaccine Preparations. [ citation needed ]
The BW facility at Sverdlovsk was located within a military base known as Compound 19 ( 19th gorodok , Russian: 19-й городок ) which itself was created between 1947 and 1949. Compound 19 abutted the southern industrialised sector of the city in the Chkalovskii district. It was located immediately adjacent to the Vtorchermet housing estate. A meat-processing plant was located nearby with a view to supplying components of bacterial nutrient media. There was a high degree of autonomy with regard to the secret base. As well as the military institute, Compound 19 embraced its own 75-bed military hospital, a postal service, a range of shops, a kindergarten, schools, a social club, a sports stadium, parks and walkways, a civil registry office, and its own special prosecutor's office. Sentries and construction workers at the site required special security clearance. [ 7 ] A Russian television crew visited the site shortly after the collapse of the USSR. It is likely that much of what they observed may not have been greatly different from the situation in 1979. They reported that Compound 19 comprised around 200 hectares and was sub-divided into three main zones. The first, residential zone, housed the scientists and their families, around 7,000 inhabitants, along with the ancillary services described above. Nestled within the outer zone there was a check-point which provided restricted access to a so-called industrial zone. At the very heart of Compound 19, guarded by a final check-point and barbed wire, was the most secret special working zone which housed the main administration building together with secret laboratories and production units housed underground. [ 8 ]
During the 1960s, Compound 32, an army base with barracks and apartments for Soviet soldiers serving in armoured and artillery units, and with no connection to BW, was added to the southern edge of Compound 19. [ citation needed ]
In their authoritative account, Leitenberg and Zilinskas with Kuhn report that, at some point during the period 2–3 April 1979, a mass of B. anthracis spores were released from a four-story building located in Compound 19's special zone. The building housed a production unit which produced dry B. anthracis spores for weapons use. The unit was manned by 40 personnel and commanded by Lieutenant Colonel Nikolai Chernyshov. The spores created a plume which the wind carried over parts of Sverdlovsk itself as well as a number of rural villages. Russian sources indicate that the release occurred as a result of a defect in an air-handling system which carried exhaust from a spray dryer. The release, according to one Russian source took place during the evening or night of 2–3 April. [ 6 ] Based on interviews with friends and families of victims, together with a study of wind data, Meselson and his investigative team conclude that the release probably took place during the day of 2 April. [ 1 ]
The precise number of fatalities associated with the military leakage of anthrax spores is not known. Meselson's group report that the incident led directly to the deaths of at least 68 people in Sverdlovsk itself and to cases of animal anthrax in nearby villages (Rudnii, Bol'shoe Sedelnikovo, Maloe Sedelnikovo, Pervomaiskii, Kashino and Abramovo) to the south-east of the city. [ 1 ] Leitenberg and Zilinskas with Kuhn quote a Russian source which indicates that "According to the official data, 95 people were infected, 68 (71.5 per cent) died [but] actually the number of the dead and infected was larger". [ 6 ] The leakage of anthrax hit the ceramics factory, south of Compound 19, the hardest. The factory, which employed 2,180 personnel, was in possession of a ventilation system which sucked air from the outside, directing some to the furnaces with the remainder being directed to the workforce . In the coming weeks at least 18 workers at the factory died. [ 9 ]
In response to the incident, the Soviet authorities acted to mobilise medical teams in the affected district. Tetracycline was administered to affected households, sick rooms were disinfected and potentially contaminated sheets and clothing were collected. Checks for illness in family members were made. Individuals who had fevers were directed to polyclinics and those who were very ill were transferred to the local Hospital 40. A Moscow-controlled Extraordinary Commission was eventually established to manage the response and on the 22 April firemen and factory workers began hosing down buildings with solutions of chlorine. The large-scale vaccination of the population in the affected Chkalovskii district was also undertaken by the authorities. In all, some 80 per cent of around 59,000 eligible individuals were injected with the Soviet STI anthrax vaccine. The latter had been manufactured by the Scientific-Research Institute of Vaccines and Sera based in Tbilisi , Georgia. [ 2 ] The first indication in the West of the accident in Sverdlovsk was a story which appeared in January 1980 in an obscure Frankfurt-based magazine named Possev which was published by a group of Russian emigres. It claimed that there had been an outbreak of anthrax in April 1979 in Sverdlovsk after an explosion at a military settlement south-west of the city. [ 10 ]
In 1986, Matthew Meselson of Harvard University was granted approval by Soviet authorities for a four-day trip to Moscow where he interviewed several senior Soviet health officials about the outbreak. He later issued a report which agreed with the Soviet assessment that the outbreak was caused by a contaminated meat processing plant, concluding the Soviets' official explanation was completely "plausible and consistent with what is known from medical literature and recorded human experiences with anthrax". [ 11 ] [ 12 ] However, the Soviet version of events was fatally undermined when, in October 1991, the Wall Street Journal sent its Moscow Bureau Chief, Peter Gumbel, to Sverdlovsk to investigate the outbreak. After interviewing numerous families, hospital workers and doctors, he is reported to have found the Soviet version of events "riddled with inconsistencies, half-truths and plain falsehoods". [ 13 ]
This was followed by an admission in May 1992 by President Boris Yeltsin , who had been Sverdlovsk's Communist Party chief at the time of the accident, that the KGB had revealed to him that "our military development was the cause". [ 14 ] Based on these reports a team of Western scientists led by Meselson gained access to the region in June 1992. Before they arrived they had been provided by the authorities with a list of 68 known incident victims in Sverdlovsk. By visiting and questioning in their homes surviving relatives of those who had died, the investigating researchers ascertained both where the victims had been living and where they had been during daylight hours at the time during which hospital admission records indicated a possible release into the atmosphere of anthrax dust. When the locations were plotted on maps, the places where the victims lived did not form a clear geographical pattern. However, there was a very precise indication from their reported locations during working hours, that all of the victims had been directly downwind at the time of the release of the spores via aerosol . [ 15 ] [ 16 ] Livestock in the area were also affected. Had the winds been blowing in the direction of the city at that time, it could have resulted in the pathogen being spread to hundreds of thousands of people. Meselson's original contention for many years had been that the outbreak was a natural one and that the Soviet authorities were not lying when they disclaimed having an active offensive bio-warfare program, but the information uncovered in the investigation left no room for doubt. [ 1 ]
In April 1992, President Boris Yeltsin issued a decree On ensuring the implementation of international pledges in the sphere of biological weapons . [ 17 ] Under the reforming president, there was a desire, over time, to shift the Ministry of Defence's BW institutes from military jurisdiction to work for the civil economy. It was against this background, that at some point between 1992 and 1994, a representative from the US investment bank and stock brokerage firm Paine Webber Incorporated, held a meeting with members of Russia's Committee on Convention Problems of Chemical and Biological Weapons which was specifically focused on the potential for cooperation with Compound No. 19 (Ekaterinburg) in the areas of infectious diseases in animals and production of veterinary vaccines. The project eventually floundered because of the Russian military's desire to maintain the "closed" (highly restricted access) status of its biological facilities. [ citation needed ]
Medical anthropologist Jeanne Guillemin , member of the 1992 expedition team and wife of Meselson, published a comprehensive book about the investigation in 1999, titled Anthrax – The Investigation of a Deadly Outbreak. [ 2 ]
In August 2016, the journal Science reported that anthrax scientist Paul Keim of Northern Arizona University (Flagstaff) and colleagues had attempted to sequence the B . anthracis genome from two samples taken from victims of the Sverdlovsk anthrax leak. The samples had been preserved by local Russian pathologists who investigated the outbreak as it was occurring. Later, they shared the material with Professor Meselson during his investigative trip in 1992 (see above). The samples had been fixed in formalin and embedded in paraffin and the DNA was, as a result, badly degraded. Nevertheless, the US researchers were able to isolate the pathogen's DNA and piece together its entire genome, comparing it with hundreds of other anthrax isolates. Keim and his team reported that they had not found any genomic evidence that the Soviet military had attempted to grow an antibiotic- or vaccine-resistant strain or genetically engineered the strain in any way. [ 18 ] Meselson commented that although this was a perfectly ordinary strain, "that doesn't mean it wasn't nasty. It was extracted from people who were killed by it." [ 19 ]
Zilinskas and Mauger in 2018 provided the most up-to-date information regarding the current status of the Sverdlovsk military facility. Under the National System of Chemical and Biological Safety/Security of the Russian Federation, funding has been provided to the Sverdlovsk institute for the renovation of two facilities for the production of antibiotics. Such products could be used in the civil medical sector. Major reconstruction work has also been carried out with regard to a building that used to produce B. anthracis spores. A building used for media and substrate production has also been extensively renovated. Refurbishment has also been underway in recent years of an open-air test site—the so-called Pyshma field test base—for the Sverdlovsk institute. Once complete, it was intended to be used to "assess the effectiveness of means and methods of biological prospecting and the elimination of the consequences of emergency situations". As of late 2015 [update] , this refurbishment project remained incomplete. [ 20 ]
In August 2020, the US Commerce Department's Bureau of Industry and Security (BIS) imposed "blacklist" restrictions on three Russian military biological institutes for their alleged involvement with the Russian biological weapons programme. One of these was the Sverdlovsk institute (now operating under the name 48th Central Scientific Research Institute, Yekaterinburg). [ 21 ] On 2 March 2021, additional US sanctions were imposed on the 48 Central Scientific Research Institute Yekaterinburg (aka 48th TsNII Yekaterinburg) along with its associated military BW institutes in Kirov and Sergiev Posad. [ 22 ] | https://en.wikipedia.org/wiki/Sverdlovsk_anthrax_leak |
Svetozar Lj. Jovanović (1895–1951) was a Serbian chemist [ 1 ] and assistant professor of chemistry from 1925 to 1941. [ 2 ] He specialized in the field of analytical chemistry .
Jovanović developed a new electroanalytical method for the quantitative determination of antimony [ 3 ] and a method for the separation of copper from zinc by rapid electrolysis . He studied the gravrimetric determination of problems, such as the manufacture of drugs and, in particular, the practical utilization of the country's paraffin shales . He co-authored with Momir Jovanović a book on Principles of Qualitative Chemical Analysis [ 4 ] (Serbian: Kvalitativna hemijska aanaliza ). [ 5 ] He also wrote an article for a Serbian medical journal titled "Diagnostic evaluation of Vidal's reaction". | https://en.wikipedia.org/wiki/Svetozar_Lj._Jovanović |
The Swain equation relates the kinetic isotope effect for the protium / tritium combination with that of the protium / deuterium combination according to:
where k H,D,T are the reaction rate constants for the protonated, deuterated and tritiated reactants respectively. | https://en.wikipedia.org/wiki/Swain_equation |
In physical organic chemistry , the Swain–Lupton equation is a linear free energy relationship (LFER) that is used in the study of reaction mechanisms and in the development of quantitative structure activity relationships for organic compounds . It was developed by C. Gardner Swain and Elmer C. Lupton Jr. in 1968 as a refinement of the Hammett equation to include both field effects and resonance effects.
In organic chemistry, the Hammett plot provides a means to assess substituent effects on a reaction equilibrium or rate using the Hammett equation ( 1 ):
Hammett developed this equation from equilibrium constants from the dissociation of benzoic acid and derivatives (Fig. 1):
Hammett defined the equation based on two parameters: the reaction constant (ρ) and the substituent parameter (σ). When other reactions were studied using these parameters, a correlation was not always found due to the specific derivation of these parameters from the dissociation equilibrium of substituted benzoic acids and the original negligence of resonance effects. Therefore, the effects of substituents on an array of compounds must be studied on an individual reaction basis using the equation Hammett derived either for field or resonance effects, but not both.
C. Gardner Swain and Elmer C. Lupton Jr. from the Massachusetts Institute of Technology redefined the substituent parameter, σ, based on the idea that no more than two variables (resonance effects and field effects) are necessary to describe the effects of any given substituent. Field effects, F , are defined to include all effects (inductive and pure field). Likewise, effects due to resonance , R , are due to the average of electron-donating ability and electron-accepting ability. These two effects are assumed to be independent of each other and therefore can be written as a linear combination:
These two parameters are treated as independent terms because of the assumption that Swain and Lupton made; the substituent is kept distant by three or more saturated centers or if the substituent is (CH 3 ) 3 N + . All other terms are then negligible and leads to the Swain–Lupton equation ( 2 ).
The substituent parameter is now defined by field and resonance effects, F and R , which are dependent on the individual substituent. Constants r and f account for the importance of each of the two effects. These constants do not depend on the substituent but instead depend on the set of Hammett substituent parameters (σ m , σ p , σ p+ , σ ' , etc.).
In order to find the weighted constants, r and f , for each set of substituent parameters, one would need to establish the fact that each new substituent parameter σ X could be written as a linear combination of specific reaction substituent parameters, i.e.
where σ 1X and σ 2X are specific substituent parameters (i.e. σ + , σ − , etc.) and c 1 and c 2 are constants independent of the substituent (depend on the reaction conditions, i.e. temperature, solvent, and individual reaction being studied). This can be expressed more generically as:
where i is an intercept to keep from fixing the origin at (0,0). If this was not done, the equation would give exceedingly more weight to the unsubstituted compounds that one is trying to make a comparison to using this equation. [ 1 ] A linear least-squares analysis is used to determine the coefficients/constants a , b , and i (Swain and Lupton used a procedure called DOVE: Dual Obligate Vector Evaluation). [ 2 ] Constants were first based on three previous reactions (σ m , σ p , σ p+ ), which leads to more possible errors since the compiled data is only a minimal combination of a much larger pool. Seeing possible error in this limited pool, the data pool was increased by assigning a scale to begin with. A zero-scale is used for hydrogen, because it is known to neither readily donate or accept electron density when attached to a carbon atom due to similar electronegativities. A value of 1 was assigned to NO 2 , because previous research determined the effect of this substituent was predominantly due to resonance. [ 3 ] Lastly, F was set equal to R for both components so that the field effects could be compared directly to the resonance effects. This then leads to:
Fig. 2 shows some relative F and R values that Swain and Lupton founded. [ 2 ]
Alkyl groups have a low to zero value for F but sensible values for R . This is most commonly explained by hyperconjugation , meaning little to no inductive effects but partial resonance effects.
CF 3 has a much higher R / F ratio than other substituents with high degrees of conjugation. This was studied in greater detail by Swain but is still explained best by fluoride hyperconjugation.
Positively charged substituents (i.e., N(CH 3 ) 3+ and S(CH 3 ) 2+ ) have larger positive F values due to a positive charge that is saturated near the carbon framework in question. Negatively charged substituents (i.e., CO 2− and SO 3− ) have much lower F values because of their ability to resonate electron density amongst the oxygen atoms and stabilize it through hydrogen-bonding with solvents.
Linear free energy relationships are still useful, despite their disadvantages when pushed to the limits. New techniques to solve for Swain–Lupton substituent parameters involve studying chemical shifts through nuclear magnetic resonance spectroscopy . Recently, 15 N NMR chemical shifts and substituent effects of 1,2,3,4,5,6,7,8-octahydroacridine and derivatives were studied. Values for R and F were found for the −N(COCH 3 ) 2 group, which could not be found previously using known methods. [ 4 ]
It is sometime useful to look at the percent resonance (% r ), because r is dependent on the reaction and is the same for all substituents.
One can predict the difference in data comparing two substituents using % r :
The most dominant effect is clear when looking at the ratio of R to F . For example, a tungsten complex was shown to alkylate allyl carbonates A and B . The ratio of products A1 and B1 can be attributed to the para substituent, X ( Fig. 3 ). Using Swain–Lupton parameters (σ = 0.2 F + 0.8 R ) a ρ value of -2.5 was found to be the slope.
This is in agreement with the proposed mechanism (a positive charge forms on the benzylic carbon and is stabilized by resonance; R dominates by a ratio of 0.8/0.2). [ 5 ]
Like any other linear free-energy relationship established, the Swain–Lupton equation will too fail when special circumstances arise, i.e. change in the rate determining step of a mechanism or solvation structure. [ 6 ] | https://en.wikipedia.org/wiki/Swain–Lupton_equation |
A swale is a shady spot, or a sunken or marshy place. [ 1 ] In US usage in particular, it is a shallow channel with gently sloping sides. Such a swale may be either natural or human-made. Artificial swales are often infiltration basins , designed to manage water runoff , filter pollutants , and increase rainwater infiltration . [ 2 ] Bioswales are swales that involve the inclusion of plants or vegetation in their construction, specifically. [ 3 ]
The use of swales has been popularized as a rainwater-harvesting and soil-conservation strategy by Bill Mollison , David Holmgren , and other advocates of permaculture . In this context a swale is usually a water-harvesting ditch on contour, also called a contour bund . [ 4 ] [ 5 ]
Swales as used in permaculture are designed by permaculturalists to slow and capture runoff by spreading it horizontally across the landscape (along an elevation contour line ), facilitating runoff infiltration into the soil. This archetypal form of swale is a dug-out, sloped, often grassed or reeded "ditch" or "lull" in the landform. One option involves piling the soil onto a new bank on the still lower slope, in which case a bund or berm is formed, mitigating the natural (and often hardscape-increased ) risks to slopes below and to any linked watercourse from flash flooding .
In arid and seasonally dry places, vegetation (existing or planted) in the swale benefits heavily from the concentration of runoff. Trees and shrubs along the swale can provide shade and mulch which decrease evaporation.
The term "swale" or "beach swale" is also used to describe long, narrow, usually shallow troughs between ridges or sandbars on a beach, that run parallel to the shoreline. [ 6 ] | https://en.wikipedia.org/wiki/Swale_(landform) |
Swallowing , also called deglutition or inglutition [ 1 ] in scientific and medical contexts, is a physical process of an animal 's digestive tract (e.g. that of a human body ) that allows for an ingested substance (typically food ) to pass from the mouth to the pharynx and then into the esophagus . In colloquial English , the term "swallowing" is also used to describe the action of gulping , i.e. taking in a large mouthful of food without any biting .
Swallowing is performed by an initial push from back part of the tongue (with the tongue tip contacting the hard palate for mechanical anchorage) and subsequent coordinated contractions of the pharyngeal muscles . The portion of food, drink and/or other material (e.g. mucus , secretions and medications ) that moves into the gullet in one swallow is called a bolus , which is then propelled through to the stomach for further digestion by autonomic peristalsis of the esophagus.
Swallowing is an important part of eating and drinking . If the process fails and the bolus to be swallowed mistakenly goes into the trachea , then choking or pulmonary aspiration can occur. In the human body, such incidents are prevented by an automatic trapdoor -like inversion of the epiglottis to temporarily cover the larynx and close off the upper airway , controlled by a complex reflex that facilitates the elevation of the hyoid bone and thyroid cartilage at the same time. The body will also initiate a cough reflex to expel any unwanted material that have accidentally entered the airway. A separate gag reflex , which involves the elevation of the uvula and tightening of the soft palate , prevents food from wrongly entering the nasal cavity above during swallowing.
Swallowing comes so easily to most people that the process rarely prompts much thought. However, from the viewpoints of physiology , of speech–language pathology , and of health care for people with difficulty in swallowing (dysphagia) , it is an interesting topic with extensive scientific literature .
Eating and swallowing are complex neuromuscular activities consisting essentially of three phases, an oral , pharyngeal and esophageal phase. Each phase is controlled by a different neurological mechanism. The oral phase, which is entirely voluntary, is mainly controlled by the medial temporal lobes and limbic system of the cerebral cortex with contributions from the motor cortex and other cortical areas. The pharyngeal swallow is started by the oral phase and subsequently is coordinated by the swallowing center on the medulla oblongata and pons . The reflex is initiated by touch receptors in the pharynx as a bolus of food is pushed to the back of the mouth by the tongue, or by stimulation of the palate (palatal reflex).
Swallowing is a complex mechanism using both skeletal muscle ( tongue ) and smooth muscles of the pharynx and esophagus . The autonomic nervous system (ANS) coordinates this process in the pharyngeal and esophageal phases.
Prior to the following stages of the oral phase, the mandible depresses and the lips abduct to allow food or liquid to enter the oral cavity. Upon entering the oral cavity, the mandible elevates and the lips adduct to assist in oral containment of the food and liquid. The following stages describe the normal and necessary actions to form the bolus, which is defined as the state of the food in which it is ready to be swallowed.
1) Moistening
Food is moistened by saliva from the salivary glands ( parasympathetic ).
2) Mastication
Food is mechanically broken down by the action of the teeth controlled by the muscles of mastication (V 3 ) acting on the temporomandibular joint . This results in a bolus which is moved from one side of the oral cavity to the other by the tongue. Buccinator (VII) helps to contain the food against the occlusal surfaces of the teeth. The bolus is ready for swallowing when it is held together by saliva (largely mucus), sensed by the lingual nerve of the tongue (VII—chorda tympani and IX—lesser petrosal) (V 3 ). Any food that is too dry to form a bolus will not be swallowed.
3) Trough formation
A trough is then formed at the back of the tongue by the intrinsic muscles (XII). The trough obliterates against the hard palate from front to back, forcing the bolus to the back of the tongue.
The intrinsic muscles of the tongue (XII) contract to make a trough (a longitudinal concave fold) at the back of the tongue. The tongue is then elevated to the roof of the mouth (by the mylohyoid (mylohyoid nerve—V 3 ), genioglossus , styloglossus and hyoglossus (the rest XII)) such that the tongue slopes downwards posteriorly. The contraction of the genioglossus and styloglossus (both XII) also contributes to the formation of the central trough.
4) Movement of the bolus posteriorly
At the end of the oral preparatory phase, the food bolus has been formed and is ready to be propelled posteriorly into the pharynx. In order for anterior to posterior transit of the bolus to occur, orbicularis oris contracts and adducts the lips to form a tight seal of the oral cavity. Next, the superior longitudinal muscle elevates the apex of the tongue to make contact with the hard palate and the bolus is propelled to the posterior portion of the oral cavity. Once the bolus reaches the palatoglossal arch of the oropharynx, the pharyngeal phase, which is reflex and involuntary, then begins. Receptors initiating this reflex are proprioceptive (afferent limb of reflex is IX and efferent limb is the pharyngeal plexus- IX and X). They are scattered over the base of the tongue, the palatoglossal and palatopharyngeal arches, the tonsillar fossa, uvula and posterior pharyngeal wall. Stimuli from the receptors of this phase then provoke the pharyngeal phase. In fact, it has been shown that the swallowing reflex can be initiated entirely by peripheral stimulation of the internal branch of the superior laryngeal nerve . This phase is voluntary and involves important cranial nerves : V (trigeminal) , VII (facial) and XII (hypoglossal) .
For the pharyngeal phase to work properly all other egress from the pharynx must be occluded—this includes the nasopharynx and the larynx . When the pharyngeal phase begins, other activities such as chewing, breathing, coughing and vomiting are concomitantly inhibited.
5) Closure of the nasopharynx
The soft palate is tensed by tensor palatini (Vc), and then elevated by levator palatini (pharyngeal plexus—IX, X) to close the nasopharynx. There is also the simultaneous approximation of the walls of the pharynx to the posterior free border of the soft palate, which is carried out by the palatopharyngeus (pharyngeal plexus—IX, X) and the upper part of the superior constrictor (pharyngeal plexus—IX, X).
6) The pharynx prepares to receive the bolus
The pharynx is pulled upwards and forwards by the suprahyoid and longitudinal pharyngeal muscles – stylopharyngeus (IX), salpingopharyngeus (pharyngeal plexus—IX, X) and palatopharyngeus (pharyngeal plexus—IX, X) to receive the bolus. The palatopharyngeal folds on each side of the pharynx are brought close together through the superior constrictor muscles, so that only a small bolus can pass.
7) Opening of the auditory tube
The actions of the levator palatini (pharyngeal plexus—IX, X), tensor palatini (Vc) and salpingopharyngeus (pharyngeal plexus—IX, X) in the closure of the nasopharynx and elevation of the pharynx opens the auditory tube, which equalises the pressure between the nasopharynx and the middle ear. This does not contribute to swallowing, but happens as a consequence of it.
8) Closure of the oropharynx
The oropharynx is kept closed by palatoglossus (pharyngeal plexus—IX, X), the intrinsic muscles of tongue (XII) and styloglossus (XII).
9) Laryngeal closure
The primary laryngopharyngeal protective mechanism to prevent aspiration during swallowing is via the closure of the true vocal folds. The adduction of the vocal cords is affected by the contraction of the lateral cricoarytenoids and the oblique and transverse arytenoids (all recurrent laryngeal nerve of vagus). Since the true vocal folds adduct during the swallow, a finite period of apnea (swallowing apnea) must necessarily take place with each swallow. When relating swallowing to respiration, it has been demonstrated that swallowing occurs most often during expiration, even at full expiration a fine air jet is expired probably to clear the upper larynx from food remnants or liquid. The clinical significance of this finding is that patients with a baseline of compromised lung function will, over a period of time, develop respiratory distress as a meal progresses.
Subsequently, false vocal fold adduction, adduction of the aryepiglottic folds and retroversion of the epiglottis take place. The aryepiglotticus (recurrent laryngeal nerve of vagus) contracts, causing the arytenoids to appose each other (closes the laryngeal aditus by bringing the aryepiglottic folds together), and draws the epiglottis down to bring its lower half into contact with arytenoids, thus closing the aditus. Retroversion of the epiglottis, while not the primary mechanism of protecting the airway from laryngeal penetration and aspiration, acts to anatomically direct the food bolus laterally towards the piriform fossa .
Additionally, the larynx is pulled up with the pharynx under the tongue by stylopharyngeus (IX), salpingopharyngeus (pharyngeal plexus—IX, X), palatopharyngeus (pharyngeal plexus—IX, X) and inferior constrictor (pharyngeal plexus—IX, X). This phase is passively controlled reflexively and involves cranial nerves V, X (vagus) , XI (accessory) and XII (hypoglossal) . The respiratory center of the medulla is directly inhibited by the swallowing center for the very brief time that it takes to swallow. This means that it is briefly impossible to breathe during this phase of swallowing and the moment where breathing is prevented is known as deglutition apnea .
10) Hyoid elevation
The hyoid is elevated by digastric (V & VII) and stylohyoid (VII), lifting the pharynx and larynx up even further.
11) Bolus transits pharynx
The bolus moves down towards the esophagus by pharyngeal peristalsis which takes place by sequential contraction of the superior, middle and inferior pharyngeal constrictor muscles (pharyngeal plexus—IX, X). The lower part of the inferior constrictor ( cricopharyngeus ) is normally closed and only opens for the advancing bolus. Gravity plays only a small part in the upright position—in fact, it is possible to swallow solid food even when standing on one's head. The velocity through the pharynx depends on a number of factors such as viscosity and volume of the bolus. In one study, bolus velocity in healthy adults was measured to be approximately 30–40 cm/s. [ 2 ]
12) Esophageal peristalsis
Like the pharyngeal phase of swallowing, the esophageal phase of swallowing is under involuntary neuromuscular control. However, propagation of the food bolus is significantly slower than in the pharynx. The bolus enters the esophagus and is propelled downwards first by striated muscle (recurrent laryngeal, X) then by the smooth muscle (X) at a rate of 3–5 cm/s. The upper esophageal sphincter relaxes to let food pass, after which various striated constrictor muscles of the pharynx as well as peristalsis and relaxation of the lower esophageal sphincter sequentially push the bolus of food through the esophagus into the stomach.
13) Relaxation phase
Finally the larynx and pharynx move down with the hyoid mostly by elastic recoil. Then the larynx and pharynx move down from the hyoid to their relaxed positions by elastic recoil.
Swallowing therefore depends on coordinated interplay between many various muscles, and although the initial part of swallowing is under voluntary control, once the deglutition process is started, it is quite hard to stop it.
Swallowing becomes a great concern for the elderly since strokes and Alzheimer's disease can interfere with the autonomic nervous system . Speech pathologists commonly diagnose and treat this condition since the speech process uses the same neuromuscular structures as swallowing. Diagnostic procedures commonly performed by a speech pathologist to evaluate dysphagia include Fiberoptic Endoscopic Evaluation of Swallowing and Modified Barium Swallow Study. Occupational Therapists may also offer swallowing rehabilitation services as well as prescribing modified feeding techniques and utensils. Consultation with a dietician is essential, in order to ensure that the individual with dysphagia is able to consume sufficient calories and nutrients to maintain health. In terminally ill patients, a failure of the reflex to swallow leads to a build-up of mucus or saliva in the throat and airways, producing a noise known as a death rattle (not to be confused with agonal respiration , which is an abnormal pattern of breathing due to cerebral ischemia or hypoxia).
Abnormalities of the pharynx and/or oral cavity may lead to oropharyngeal dysphagia . Abnormalities of the esophagus may lead to esophageal dysphagia .
The failure of the lower esophagus sphincter to respond properly to swallowing is called achalasia .
M-Type Swallowing
With practice, people can learn to swallow fluidly without closing the mouth by merely manipulating the tongue and jaw to drive fluids or foods down the esophagus. With a continuous motion, an individual forges breathing and priorities the swallowed matter. This intermediate level of muscle manipulation is similar to the techniques used by sword swallowers.
In many birds, the esophagus is largely a mere gravity chute , and in such events as a seagull swallowing a fish or a stork swallowing a frog , swallowing consists largely of the bird lifting its head with its beak pointing up and guiding the prey with tongue and jaws so that the prey slides inside and down.
In fish , the tongue is largely bony and much less mobile and getting the food to the back of the pharynx is helped by pumping water in its mouth and out of its gills .
In snakes , the work of swallowing is done by raking with the lower jaw until the prey is far enough back to be helped down by body undulations. | https://en.wikipedia.org/wiki/Swallowing |
Swami Sundaranand (April 1926 – 23 December 2020) [ 1 ] was an Indian Yogi , photographer , author and mountaineer who lectured widely in India on threats to the Ganges River and the loss of Himalayan glaciers due to global warming . [ 2 ] [ 3 ] [ 4 ]
Swami Sundarananda was a student of the reclusive yoga master Swami Tapovan Maharaj (1889–1957), who wrote in the late 19th and early 20th centuries about yogic life in the Himalayas in the classic yoga book Wanderings in the Himalayas (Himagiri Vihar). [ 5 ] Sundaranand lived with Swami Tapovan in the then inaccessible area of Gangotri , at the source of the Ganges , which is considered one of India's most sacred places. [ 6 ] [ 7 ]
Since 1948, he has lived by the Ganges in Gangotri, at 10,400 feet, in a modest hut ( kuti ) which his master Swami Tapovan Maharaj later bequeathed to him on his death in 1957. There, Swami Sundaranand has lived in solitude and through the severest of winters without any comforts or conveniences. [ 8 ] He has witnessed up close the gradual shrinking of the Gangotri Glacier from which the Ganges springs forth, and has chronicled his devotion to the natural beauty of the Indian Himalayas as an accomplished photographer. A museum devoted to environmental protection and spiritual guidance, containing Swami Sundaranand's Himalayan photography, is now in the planning stages. It will be located in Gangotri on the property of Sundaranand and his master.
As an ascetic, he took the brahmacharya sadhu vow in 1948 and daily devotes his life to rigorous meditation and other spiritual practices. He continues to be a principal advocate for the ecological preservation of the Himalayas, the Ganges and its source at Gangotri.
He has taken more than 100,000 photos, over a 50-year period, of the shrinking Gangotri glacier in the Indian Himalayas. He travelled through India raising awareness of the Gangotri's rapid decline. [ 9 ]
Nicknamed "the Sadhu Who Clicks" because of his photography, he was also a noted mountain climber, having scaled over 25 Himalayan peaks, and climbing twice with Sir Edmund Hillary and Tenzing Norgay . [ 10 ] Sir Edmund Hillary paid his respects to Swami Sundaranand in the 1980s at his Gangotri hut. [ 11 ] Of the Gangotri glacier, Swami Sundaranand says:
In 1949, when I first saw the glacier, I felt as if all my sins were washed away and I had truly attained rebirth. But now, it is impossible to experience that Ganga of the past. [ 10 ]
Swami Sundaranand is the author of the book Himalaya: Through the Lens of a Sadhu with over 425 photographs spanning 60 years of his work. [ 12 ] The book also contains a letter of endorsement from the former Indian Prime Minister Atal Bihari Vajpayee . [ 13 ] [ 14 ] He sought to capture the Eternal in Nature and to document the region as it once was with a special emphasis on planting the seeds of hope and inspiration to solve the environmental concerns of the area. A lookout point and plaque have been built downriver from Gangotri and dedicated to the Swami's work and efforts.
Swami Sundaranand is the subject of a feature documentary shot at his home in Gangotri titled Personal Time with Swamiji . The film was produced by The Center for Healing Arts and directed by Victor Demko .
Over the past six decades, Swami Sundaranand has used his combined interests to raise awareness about the Ganga. "When I first came to this region, it was one of the most beautiful parts of the Himalayas," he says. "It is difficult to imagine the purity of the Ganga and the abundance of Himalayan vegetation and fauna that was prevalent then. We don't know what we have cruelly destroyed."
Swami Sundaranand has lived in Gangotri since 1948, when he became a renunciate, and arrived there from Andhra Pradesh. In his words: "A lot has changed since then. Although the air is cold here, the sun is harsh. It's becoming hotter every year. People say it is global warming. I say it is a global warning."
The pollution of Ganga in the plains has been an oft-repeated refrain, but, according to Sundaranand, a graver threat is its pollution at the source. He attributed this to the unchecked construction of hotels and ashrams in Gangotri and the dumping of waste from these locations, such as faecal matter and garbage, into the Ganga. According to him, "there are no environment lovers left here, only money lovers". Every year, while the temple town closes during the harsh winter months, unchecked construction and felling of trees is at its peak. According to the sadhu, "many bhoj trees in Bhojbasa, en route Gaumukh have been cut down. Earlier, on my treks to the Gaumukh glacier, I could spot rare animals like the snow leopard and musk deer. They are rarely visible now".
The sadhu was also an avid mountaineer — it was during his treks to the glacier over the last 10–15 years that he saw the glacier retreat more rapidly than ever before. According to him, Gaumukh was barely 1 km away from Bhojbasa, but today, it is 4 km away and that every year, the glacier was retreating by at least 10 metres. He has expressed the view that the pollution of Ganga at its source and melting Himalayan glaciers were the real issues that environmentalists needed to urgently take up, rather than opposing the construction of dams. [ 15 ]
Swami Sundaranand had a strong connection with the Himalayas that few others have. He has climbed dozens of its peaks, several of them over 21,000 feet above sea level, and has lectured at Tensing's Himalayan Institute (a famous mountaineering school). He was also a skilled naturalist who was familiar with thousands of Himalayan plants and he knew the lore and medicinal uses of these species.
He engaged in 3 hours of meditation during the day, and sometimes meditated at night into the early hours of the morning. The most important parts of his life were meditation, japa and pranayama . As a younger man he was an accomplished hatha yogi, mastering 300 postures, and he continued to practice it daily. He was very devoted to the ecosystem in which he has lived for forty years and believed that "God does not reside in temples or mosques - he is scattered everywhere in the courtyard of nature." [ 14 ] [ 16 ]
Sundaranand died on 23 December 2020 at a private hospital in Dehradun . He was aged 96. He had been diagnosed with COVID-19 and had recovered earlier in October. [ 17 ]
Media related to Swami Sundaranand at Wikimedia Commons | https://en.wikipedia.org/wiki/Swami_Sundaranand |
The swamping argument is an objection against Darwinism made by Fleeming Jenkin . He asserted that an accidentally-appearing profitable variety cannot be preserved by natural selection in the population, but should be 'swamped' with ordinary traits. Population genetics helped to overcome this logical difficulty.
Jenkin’s article was published anonymously in the North British Review in June 1867. It took Darwin a year and a half to discover that the author was Fleeming Jenkin, Regius Professor of Engineering at the University of Edinburgh . The critical article was most valuable to Darwin. In 1869 he wrote to Alfred Russel Wallace : "Fleming Jenkyn’s arguments have convinced me". Darwin's son Francis said that Jenkin’s critique was the most valuable ever made on his father's views. Jenkin’s article was a critique intended to be based entirely on science, unlike most other critiques which were based on religion. Jenkin humorously said in his article "we are asked to believe", suggesting [ citation needed ] he opposed the theory because it was too much like a religion.
In his article Jenkin stated that organisms could obtain adaptations through natural selection, but would never gain whole new organs for smell, hearing or sight if they had never possessed them. Jenkin further asserted that once selective pressure was removed, the population would revert to its original condition. He then introduced the 'swamping argument' to deny the possibility that an occasional monstrous individual, a saltation , could supply an escape from this state of affairs and give rise to a permanent adaptation. Jenkin made a mathematical calculation for his argument
…the advantage, whatever it may be, is utterly outbalanced by numerical inferiority. A million creatures are born; ten thousand survive to produce offspring. One of the million has twice as good a chance as any other of surviving; but the chances are fifty to one against the gifted individuals being one of the hundred [ sic ] survivors. No doubt, the chances are twice as great against any one other individual, but this does not prevent their being enormously in favour of some average individual. However slight the advantage may be if it is shared by half the individuals produced, it will probably be present in at least fifty-one of the survivors, and in a larger proportion of their offspring; but the chances are against the preservation of any one ‘sport’ in a numerous tribe…
Jenkin made a mistake in his letter: the hundred survivors should have been the ten thousand survivors . [ 1 ] He continued his essay with a melodramatic story to elaborate on his calculation
... Suppose a white man to have been wrecked on an island inhabited by negroes.... Our shipwrecked hero would probably become king; he would kill a great many blacks in the struggle for existence; he would have a great many wives and children, while many of his subjects would live and die as bachelors.... Our white's qualities would certainly tend very much to preserve him to good old age, and yet he would not suffice in any number of generations to turn his subjects' descendants white.... In the first generation there will be some dozens of intelligent young mulattoes, much superior in average intelligence to the negroes. We might expect the throne for some generations to be occupied by a more or less yellow king; but can any one believe that the whole island will gradually acquire a white, or even a yellow population...? Here is a case in which a variety was introduced, with far greater advantages than any sport every heard of, advantages tending to its preservation, and yet powerless to perpetuate the new variety. [ 2 ]
Darwin agreed that a variation originating in a single individual would not spread across a population, and would invariably be lost. Darwin stated to his colleagues that he was always aware that “swamping” would stamp out saltations. In the fifth edition of On the Origin of Species he responded:
I saw, also, that the preservation in a state of nature [as opposed to under domestication] of any occasional deviation of structure, such as a monstrosity, would be a rare event; and that, if preserved, it would generally be lost by subsequent intercrossing with ordinary individuals. Nevertheless, until reading an able and valuable article in the 'North British Review' (1867), I did not appreciate how rarely single variations, whether slight or strongly-marked could be perpetuated. [ 3 ]
Darwin concluded that natural selection must instead act upon the normal small variations in any given characteristic across all the individuals in the population.
Apart from the swamping argument Jenkin also questioned Darwin’s calculation of the Earth’s age . This calculation was more worrisome for Darwin as he himself agreed that in the timespan given there would not have been enough time for natural selection to take place. Darwin needed a solution to both the swamping argument for non-saltation’s and the Earths age. Darwin theorized that ‘ negative selection ’ by the increased destruction of non-adapted specimens would further speed up the process of natural selection. Darwin added this to the fifth edition of “On the Origin of species”. This solved for him the problem of swamping in large numbers and would shorten the evolutionary process to fit his own calculated age of the Earth.
Because the ‘swamping argument’ was mostly anchored in the ‘ blending inheritance ’ theory. When the ‘blending inheritance’ theory was replaced by mendelian inheritance in the early 1900s the ‘swamping argument’ also became obsolete.
Susan W. Morris. “Fleeming Jenkin and ‘The Origin of Species’: A Reassessment.” The British Journal for the History of Science, vol. 27, no. 3, 1994, pp. 313–343. JSTOR, www.jstor.org/stable/4027601. | https://en.wikipedia.org/wiki/Swamping_argument |
Swan bands are a characteristic of the spectra of carbon stars , comets and of burning hydrocarbon fuels. [ 1 ] [ 2 ] They are named for the Scottish physicist William Swan , who first studied the spectral analysis of radical diatomic carbon (C 2 ) in 1856. [ 3 ]
Swan bands consist of several sequences of vibrational bands scattered throughout the visible spectrum. [ 4 ]
This astrophysics -related article is a stub . You can help Wikipedia by expanding it .
This spectroscopy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Swan_band |
Swanson's law is the observation that the price of solar photovoltaic modules tends to drop 20 percent for every doubling of cumulative shipped volume. At present rates, costs go down 75% about every 10 years. [ 3 ]
It is named after Richard Swanson , the founder of SunPower Corporation , a solar panel manufacturer. [ 4 ] The term Swanson's Law appears to have originated with an article in The Economist published in late 2012. [ 5 ] [ 6 ] Swanson had been presenting such curves at technical conferences for several years. [ 7 ]
Swanson's law has been compared to Moore's law , which predicts the growing computing power of processors. Swanson's Law is a solar industry specific application of the more general Wright's Law which states there will be a fixed cost reduction for each doubling of manufacturing volume.
The method used by Swanson is more commonly referred to as learning curve or more precise experience curve analysis. It was first developed and applied to the aeronautics industry in 1936 by Theodore Paul Wright . [ 8 ] There are reports of it first being applied to the photovoltaics industry in 1975, and saw wider use starting in the early 1990s. [ 9 ]
Crystalline silicon photovoltaic cell prices have fallen from $76.67 per watt in 1977 to $0.36 per watt in 2014. [ 5 ] [ 6 ] [ 10 ] Plotting the module price (in $/Wp) versus time shows a dropping by 10% per year. [ 11 ] | https://en.wikipedia.org/wiki/Swanson's_law |
Swarachakra (Devanagari: स्वरचक्र) is a free text input application developed by the IDIN group at Industrial Design Center (IDC), Indian Institute of Technology Bombay for Indic scripts . [ 1 ] Swarachakra's alphabetical keyboard layout performed better than the Inscript layout (a QWERTY -based design and government standard in India). Currently, it is available for Android devices [ 2 ] [ 3 ] in twelve languages.
Work on other languages is in progress. This is one of the many projects taken up to develop interactive products for developing countries at IDC.
India is a multilingual country with 1,635 languages (including dialects and sub dialects) while Indian currency notes display 15 major languages. Although smartphone penetration is increasing in India, text input in Indic scripts has been a challenge for many years. According to the 2005 India Human Development Survey, the surveyed households reported that 72 per cent of men did not speak any English, 28 per cent spoke some English, and five per cent spoke fluent English. Among women, the corresponding percentages were 83, 17, and 3. While Indian languages are preferred in other means of communication in India (e.g. oral and written communication, radio, TV, newspapers and magazines), English is still the most preferred language for text input.
The focus of this project is to increase the Indian text input on smartphones and tablets. A better text input mechanism will be helpful to people with little or no knowledge of English, and also for those who prefer not to use English.
Text entry in Indian Languages has many challenges. Devanagari, for example, requires 52-65 keys to input just the frequently used characters (25 consonants, 9 semi-vowels, 3 frequent conjuncts, 13 vowels, a halant and the 13 vowel modifiers), whereas 26 keys are sufficient to represent Latin script. The government standard Inscript uses 42 keys but takes away many punctuation marks, including commonly used ones such as question marks and quotation marks. Further, it clubs unrelated characters on same key such as म ण, ह ङ, ऍ १, र ३, ज्ञ ५, त्र ६, क्ष ७, श्र. This is believed to be why Inscript and most other qwerty-based text input mechanisms have not become popular up until now. [ citation needed ]
Researchers at IDC have been working on alternatives to QWERTY-based keyboards for Indic text input since 2001. Earlier projects included the design of hardware devices such as Dynakey, Barakhadi, Keylekh [ 4 ] and e-Lekh. As mobile phones became smarter and more powerful, this group started working on text input alternatives for mobile phones in 2007 with Saral. [ 5 ] Swarachakra and Disha were two designs developed in 2010 for touch-screen devices. [ 6 ] In June 2013, Swarachakra was launched on Android devices. As of August 2017 [update] , more than 20 lakh people have downloaded Swarachakra from Google play store. Swarachakra has received appreciation from local newspapers in India [ 1 ] [ 3 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] and international magazines [ 15 ] [ 16 ] [ 17 ] [ 18 ] | https://en.wikipedia.org/wiki/Swarachakra |
Swarm behaviour , or swarming , is a collective behaviour exhibited by entities, particularly animals, of similar size which aggregate together, perhaps milling about the same spot or perhaps moving en masse or migrating in some direction. It is a highly interdisciplinary topic. [ 1 ]
As a term, swarming is applied particularly to insects, but can also be applied to any other entity or animal that exhibits swarm behaviour. The term flocking or murmuration can refer specifically to swarm behaviour in birds, herding to refer to swarm behaviour in tetrapods , and shoaling or schooling to refer to swarm behaviour in fish. Phytoplankton also gather in huge swarms called blooms , although these organisms are algae and are not self-propelled the way animals are. By extension, the term "swarm" is applied also to inanimate entities which exhibit parallel behaviours, as in a robot swarm , an earthquake swarm , or a swarm of stars.
From a more abstract point of view, swarm behaviour is the collective motion of a large number of self-propelled entities . [ 2 ] From the perspective of the mathematical modeller, it is an emergent behaviour arising from simple rules that are followed by individuals and does not involve any central coordination. Swarm behaviour is also studied by active matter physicists as a phenomenon which is not in thermodynamic equilibrium , and as such requires the development of tools beyond those available from the statistical physics of systems in thermodynamic equilibrium. In this regard, swarming has been compared to the mathematics of superfluids , specifically in the context of starling flocks (murmuration). [ 3 ]
Swarm behaviour was first simulated on a computer in 1986 with the simulation program boids . [ 4 ] This program simulates simple agents (boids) that are allowed to move according to a set of basic rules. The model was originally designed to mimic the flocking behaviour of birds, but it can be applied also to schooling fish and other swarming entities.
In recent decades, scientists have turned to modeling swarm behaviour to gain a deeper understanding of the behaviour.
Early studies of swarm behaviour employed mathematical models to simulate and understand the behaviour. The simplest mathematical models of animal swarms generally represent individual animals as following three rules:
The boids computer program, created by Craig Reynolds in 1986, simulates swarm behaviour following the above rules. [ 4 ] Many subsequent and current models use variations on these rules, often implementing them by means of concentric "zones" around each animal. In the "zone of repulsion", very close to the animal, the focal animal will seek to distance itself from its neighbours to avoid collision. Slightly further away, in the "zone of alignment", the focal animal will seek to align its direction of motion with its neighbours. In the outermost "zone of attraction", which extends as far away from the focal animal as it is able to sense, the focal animal will seek to move towards a neighbour.
The shape of these zones will necessarily be affected by the sensory capabilities of a given animal. For example, the visual field of a bird does not extend behind its body. Fish rely on both vision and on hydrodynamic perceptions relayed through their lateral lines , while Antarctic krill rely both on vision and hydrodynamic signals relayed through antennae .
However recent studies of starling flocks have shown that each bird modifies its position, relative to the six or seven animals directly surrounding it, no matter how close or how far away those animals are. [ 5 ] Interactions between flocking starlings are thus based on a topological , rather than a metric, rule. It remains to be seen whether this applies to other animals. Another recent study, based on an analysis of high-speed camera footage of flocks above Rome and assuming minimal behavioural rules, has convincingly simulated a number of aspects of flock behaviour. [ 6 ] [ 7 ] [ 8 ] [ 9 ]
In order to gain insight into why animals evolve swarming behaviours, scientists have turned to evolutionary models that simulate populations of evolving animals. Typically these studies use a genetic algorithm to simulate evolution over many generations. These studies have investigated a number of hypotheses attempting to explain why animals evolve swarming behaviours, such as the selfish herd theory [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] the predator confusion effect, [ 15 ] [ 16 ] the dilution effect, [ 17 ] [ 18 ] the many eyes theory, [ 19 ] and the predator-prey survival pressure theory. [ 20 ]
The concept of emergence—that the properties and functions found at a hierarchical level are not present and are irrelevant at the lower levels–is often a basic principle behind self-organizing systems . [ 21 ] An example of self-organization in biology leading to emergence in the natural world occurs in ant colonies. The queen does not give direct orders and does not tell the ants what to do. [ citation needed ] Instead, each ant reacts to stimuli in the form of chemical scents from larvae, other ants, intruders, food and buildup of waste, and leaves behind a chemical trail, which, in turn, provides a stimulus to other ants. Here each ant is an autonomous unit that reacts depending only on its local environment and the genetically encoded rules for its variety. Despite the lack of centralized decision making, ant colonies exhibit complex behaviours and have even been able to demonstrate the ability to solve geometric problems. For example, colonies routinely find the maximum distance from all colony entrances to dispose of dead bodies.
A further key concept in the field of swarm intelligence is stigmergy . [ 22 ] [ 23 ] Stigmergy is a mechanism of indirect coordination between agents or actions. The principle is that the trace left in the environment by an action stimulates the performance of a next action, by the same or a different agent. In that way, subsequent actions tend to reinforce and build on each other, leading to the spontaneous emergence of coherent, apparently systematic activity. Stigmergy is a form of self-organization. It produces complex, seemingly intelligent structures, without need for any planning, control, or even direct communication between the agents. As such it supports efficient collaboration between extremely simple agents, who lack any memory, intelligence or even awareness of each other. [ 23 ]
Swarm intelligence is the collective behaviour of decentralized , self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence . The expression was introduced by Gerardo Beni and Jing Wang in 1989, in the context of cellular robotic systems. [ 24 ]
Swarm intelligence systems are typically made up of a population of simple agents such as boids interacting locally with one another and with their environment. The agents follow very simple rules, and although there is no centralized control structure dictating how individual agents should behave, local, and to a certain degree random, interactions between such agents lead to the emergence of intelligent global behaviour, unknown to the individual agents.
Swarm intelligence research is multidisciplinary. It can be divided into natural swarm research studying biological systems and artificial swarm research studying human artefacts. There is also a scientific stream attempting to model the swarm systems themselves and understand their underlying mechanisms, and an engineering stream focused on applying the insights developed by the scientific stream to solve practical problems in other areas. [ 25 ]
Swarm algorithms follow a Lagrangian approach or an Eulerian approach. [ 26 ] The Eulerian approach views the swarm as a field , working with the density of the swarm and deriving mean field properties. It is a hydrodynamic approach, and can be useful for modelling the overall dynamics of large swarms. [ 27 ] [ 28 ] [ 29 ] However, most models work with the Lagrangian approach, which is an agent-based model following the individual agents (points or particles) that make up the swarm. Individual particle models can follow information on heading and spacing that is lost in the Eulerian approach. [ 26 ] [ 30 ]
Ant colony optimization is a widely used algorithm which was inspired by the behaviours of ants, and has been effective solving discrete optimization problems related to swarming. [ 32 ] The algorithm was initially proposed by Marco Dorigo in 1992, [ 33 ] [ 34 ] and has since been diversified to solve a wider class of numerical problems. Species that have multiple queens may have a queen leaving the nest along with some workers to found a colony at a new site, a process akin to swarming in honeybees . [ 35 ] [ 36 ]
The concept of self-propelled particles (SPP) was introduced in 1995 by Tamás Vicsek et al. [ 38 ] as a special case of the boids model introduced in 1986 by Reynolds. [ 4 ] An SPP swarm is modelled by a collection of particles that move with a constant speed and respond to random perturbations by adopting at each time increment the average direction of motion of the other particles in their local neighbourhood. [ 39 ]
Simulations demonstrate that a suitable "nearest neighbour rule" eventually results in all the particles swarming together, or moving in the same direction. This emerges, even though there is no centralized coordination, and even though the neighbours for each particle constantly change over time. [ 38 ] SPP models predict that swarming animals share certain properties at the group level, regardless of the type of animals in the swarm. [ 40 ] Swarming systems give rise to emergent behaviours which occur at many different scales, some of which are both universal and robust. It has become a challenge in theoretical physics to find minimal statistical models that capture these behaviours. [ 41 ] [ 42 ]
Particle swarm optimization is another algorithm widely used to solve problems related to swarms. It was developed in 1995 by Kennedy and Eberhart and was first aimed at simulating the social behaviour and choreography of bird flocks and fish schools. [ 43 ] [ 44 ] The algorithm was simplified and it was observed to be performing optimization. The system initially seeds a population with random solutions. It then searches in the problem space through successive generations using stochastic optimization to find the best solutions. The solutions it finds are called particles . Each particle stores its position as well as the best solution it has achieved so far. The particle swarm optimizer tracks the best local value obtained so far by any particle in the local neighbourhood. The remaining particles then move through the problem space following the lead of the optimum particles. At each time iteration, the particle swarm optimiser accelerates each particle toward its optimum locations according to simple mathematical rules . In a related approach, Shvalb et al. (2024) introduced a statistical-physics-based framework for controlling large-scale multi-robot systems. By modeling robots as particles within a statistical ensemble, the study leverages macroscopic parameters—such as density and flow fields—to guide collective behavior without the need for individual identification or direct communication between agents. This method enables scalable and robust control of robot swarms, drawing conceptual parallels to particle swarm optimization by utilizing global information to influence local agent dynamics. [ 45 ] Particle swarm optimization has been applied in many areas. It has few parameters to adjust, and a version that works well for a specific applications can also work well with minor modifications across a range of related applications. [ 46 ] A book by Kennedy and Eberhart describes some philosophical aspects of particle swarm optimization applications and swarm intelligence. [ 47 ] An extensive survey of applications is made by Poli. [ 48 ] [ 49 ]
Researchers in Switzerland have developed an algorithm based on Hamilton's rule of kin selection. The algorithm shows how altruism in a swarm of entities can, over time, evolve and result in more effective swarm behaviour. [ 50 ] [ 51 ]
The earliest evidence of swarm behaviour in animals dates back about 480 million years. Fossils of the trilobite Ampyx priscus have been recently described as clustered in lines along the ocean floor. The animals were all mature adults, and were all facing the same direction as though they had formed a conga line or a peloton . It has been suggested they line up in this manner to migrate, much as spiny lobsters migrate in single-file queues; [ 52 ] it has also been suggested that the formation is the precursor for mating, [ 53 ] as with the fly Leptoconops torrens . The findings suggest animal collective behaviour has very early evolutionary origins. [ 54 ]
Examples of biological swarming are found in bird flocks , [ 55 ] fish schools , [ 56 ] [ 57 ] insect swarms , [ 58 ] bacteria swarms , [ 59 ] [ 60 ] molds, [ 61 ] molecular motors , [ 62 ] quadruped herds [ 63 ] and people. [ 64 ] [ 65 ] [ 66 ] [ 67 ]
The behaviour of social insects (insects that live in colonies , such as ants, bees, wasps and termites) has always been a source of fascination for children, naturalists and artists. Individual insects seem to do their own thing without any central control, yet the colony as a whole behaves in a highly coordinated manner. [ 68 ] Researchers have found that cooperation at the colony level is largely self-organized . The group coordination that emerges is often just a consequence of the way individuals in the colony interact. These interactions can be remarkably simple, such as one ant merely following the trail left by another ant. Yet put together, the cumulative effect of such behaviours can solve highly complex problems, such as locating the shortest route in a network of possible paths to a food source. The organised behaviour that emerges in this way is sometimes called swarm intelligence , a form of biological emergence . [ 68 ]
Individual ants do not exhibit complex behaviours, yet a colony of ants collectively achieves complex tasks such as constructing nests, taking care of their young, building bridges and foraging for food. A colony of ants can collectively select (i.e. send most workers towards) the best, or closest, food source from several in the vicinity. [ 69 ] Such collective decisions are achieved using positive feedback mechanisms. Selection of the best food source is achieved by ants following two simple rules. First, ants which find food return to the nest depositing a pheromone chemical. More pheromone is laid for higher quality food sources. [ 70 ] Thus, if two equidistant food sources of different qualities are found simultaneously, the pheromone trail to the better one will be stronger. Ants in the nest follow another simple rule, to favor stronger trails, on average. More ants then follow the stronger trail, so more ants arrive at the high quality food source, and a positive feedback cycle ensures, resulting in a collective decision for the best food source. If there are two paths from the ant nest to a food source, then the colony usually selects the shorter path. This is because the ants that first return to the nest from the food source are more likely to be those that took the shorter path. More ants then retrace the shorter path, reinforcing the pheromone trail. [ 71 ]
Army ants , unlike most ant species, do not construct permanent nests; an army ant colony moves almost incessantly over the time it exists, remaining in an essentially perpetual state of swarming. Several lineages have independently evolved the same basic behavioural and ecological syndrome, often referred to as "legionary behaviour", and may be an example of convergent evolution . [ 72 ]
The successful techniques used by ant colonies have been studied in computer science and robotics to produce distributed and fault-tolerant systems for solving problems. This area of biomimetics has led to studies of ant locomotion, search engines that make use of "foraging trails", fault-tolerant storage and networking algorithms . [ 73 ]
In temperate climates, honey bees usually form swarms in late spring. A swarm typically contains about half the workers together with the old queen, while the new queen stays back with the remaining workers in the original hive. When honey bees emerge from a hive to form a swarm, they may gather on a branch of a tree or on a bush only a few meters from the hive. The bees cluster about the queen and send out 20–50 scouts to find suitable new nest locations. The scouts are the most experienced foragers in the cluster. If a scout finds a suitable location, she returns to the cluster and promotes it by dancing a version of the waggle dance . This dance conveys information about the quality, direction, and distance of the new site. The more excited she is about her findings, the more vigorously she dances. If she can convince others they may take off and check the site she found. If they approve they may promote it as well. In this decision-making process, scouts check several sites, often abandoning their own original site to promote the superior site of another scout. Several different sites may be promoted by different scouts at first. After some hours and sometimes days, a preferred location eventually emerges from this decision-making process. When all scouts agree on the final location, the whole cluster takes off and swarms to it. Sometimes, if no decision is reached, the swarm will separate, some bees going in one direction; others, going in another. This usually results in failure, with both groups dying. A new location is typically a kilometre or more from the original hive, though some species, e.g., Apis dorsata , [ 74 ] may establish new colonies within as little as 500 meters from the natal nest. This collective decision-making process is remarkably successful in identifying the most suitable new nest site and keeping the swarm intact. A good hive site has to be large enough to accommodate the swarm (about 15 litres in volume), has to be well-protected from the elements, receive an optimal amount of sunshine, be some height above the ground, have a small entrance and be capable of resisting ant infestation - that is why tree cavities are often selected. [ 75 ] [ 76 ] [ 77 ] [ 78 ] [ 79 ]
Unlike social insects, swarms of non-social insects that have been studied primarily seem to function in contexts such as mating, feeding, predator avoidance, and migration.
Moths may exhibit synchronized mating, during which pheromones released by females initiate searching and swarming behavior in males. [ 80 ] Males sense pheromones with sensitive antennae and may track females as far as several kilometers away. [ 81 ] Swarm mating involves female choice and male competition. Only one male in the swarm—typically the first—will successfully copulate. [ 82 ] Females maximize fitness benefits and minimize cost by governing the onset and magnitude of pheromone deployed. Too little pheromone will not attract a mate, too much allows less fit males to sense the signal. [ 83 ] After copulation, females lay the eggs on a host plant. Quality of host plant may be a factor influencing the location of swarming and egg-laying. In one case, researchers observed pink-striped oakworm moths ( Anisota virginiensis ) swarming at a carrion site, where decomposition likely increased soil nutrient levels and host plant quality. [ 84 ]
Midges, such as Tokunagayusurika akamusi , form swarms, dancing in the air. Swarming serves multiple purposes, including the facilitation of mating by attracting females to approach the swarm, a phenomenon known as lek mating . Such cloud-like swarms often form in early evening when the sun is getting low, at the tip of a bush, on a hilltop, over a pool of water, or even sometimes above a person. The forming of such swarms is not out of instinct, but an adaptive behavior – a "consensus" – between the individuals within the swarms. It is also suggested that swarming is a ritual , because there is rarely any male midge by itself and not in a swarm. This could have formed due to the benefit of lowering inbreeding by having males of various genes gathering in one spot. [ 85 ] The genus Culicoides , also known as biting midges, have displayed swarming behavior which are believed to cause confusion in predators. [ 86 ]
Cockroaches leave chemical trails in their feces as well as emitting airborne pheromones for mating. Other cockroaches will follow these trails to discover sources of food and water, and also discover where other cockroaches are hiding. Thus, groups of cockroaches can exhibit emergent behaviour , [ 87 ] in which group or swarm behaviour emerges from a simple set of individual interactions.
Cockroaches are mainly nocturnal and will run away when exposed to light. A study tested the hypothesis that cockroaches use just two pieces of information to decide where to go under those conditions: how dark it is and how many other cockroaches there are. The study conducted by José Halloy and colleagues at the Free University of Brussels and other European institutions created a set of tiny robots that appear to the roaches as other roaches and can thus alter the roaches' perception of critical mass . The robots were also specially scented so that they would be accepted by the real roaches. [ 88 ]
Locusts are the swarming phase of the short-horned grasshoppers of the family Acrididae . Some species can breed rapidly under suitable conditions and subsequently become gregarious and migratory. They form bands as nymphs and swarms as adults—both of which can travel great distances, rapidly stripping fields and greatly damaging crops . The largest swarms can cover hundreds of square miles and contain billions of locusts. A locust can eat its own weight (about 2 grams) in plants every day. That means one million locusts can eat more than one tonne of food each day, and the largest swarms can consume over 100,000 tonnes each day. [ 89 ]
Swarming in locusts has been found to be associated with increased levels of serotonin which causes the locust to change colour, eat much more, become mutually attracted, and breed much more easily. Researchers propose that swarming behaviour is a response to overcrowding and studies have shown that increased tactile stimulation of the hind legs or, in some species, simply encountering other individuals causes an increase in levels of serotonin. The transformation of the locust to the swarming variety can be induced by several contacts per minute over a four-hour period. [ 90 ] [ 91 ] [ 92 ] [ 93 ] Notably, an innate predisposition to aggregate has been found in hatchlings of the desert locust, Schistocerca gregaria , independent of their parental phase. [ 94 ]
An individual locust's response to a loss of alignment in the group appears to increase the randomness of its motion, until an aligned state is again achieved. This noise-induced alignment appears to be an intrinsic characteristic of collective coherent motion. [ 95 ]
Insect migration is the seasonal movement of insects , particularly those by species of dragonflies , beetles , butterflies , and moths . The distance can vary from species to species, but in most cases these movements involve large numbers of individuals. In some cases the individuals that migrate in one direction may not return and the next generation may instead migrate in the opposite direction. This is a significant difference from bird migration .
Monarch butterflies are especially noted for their lengthy annual migration. In North America they make massive southward migrations starting in August until the first frost. A northward migration takes place in the spring. The monarch is the only butterfly that migrates both north and south as the birds do on a regular basis. But no single individual makes the entire round trip. Female monarchs deposit eggs for the next generation during these migrations. [ 96 ] The length of these journeys exceeds the normal lifespan of most monarchs, which is less than two months for butterflies born in early summer. The last generation of the summer enters into a non-reproductive phase known as diapause and may live seven months or more. [ 97 ] During diapause, butterflies fly to one of many overwintering sites. The generation that overwinters generally does not reproduce until it leaves the overwintering site sometime in February and March. It is the second, third and fourth generations that return to their northern locations in the United States and Canada in the spring. How the species manages to return to the same overwintering spots over a gap of several generations is still a subject of research; the flight patterns appear to be inherited, based on a combination of the position of the sun in the sky [ 98 ] and a time-compensated Sun compass that depends upon a circadian clock that is based in their antennae. [ 99 ] [ 100 ]
[ 101 ]
Approximately 1800 of the world's 10,000 bird species are long-distance migrants. [ 102 ] The primary motivation for migration appears to be food; for example, some hummingbirds choose not to migrate if fed through the winter. Also, the longer days of the northern summer provide extended time for breeding birds to feed their young. This helps diurnal birds to produce larger clutches than related non-migratory species that remain in the tropics. As the days shorten in autumn, the birds return to warmer regions where the available food supply varies little with the season. These advantages offset the high stress, physical exertion costs, and other risks of the migration such as predation.
Many birds migrate in flocks. For larger birds, it is assumed that flying in flocks reduces energy costs. The V formation is often supposed to boost the efficiency and range of flying birds, particularly over long migratory routes. All the birds except the first fly in the upwash from one of the wingtip vortices of the bird ahead. The upwash assists each bird in supporting its own weight in flight, in the same way a glider can climb or maintain height indefinitely in rising air. Geese flying in a V formation save energy by flying in the updraft of the wingtip vortex generated by the previous animal in the formation. Thus, the birds flying behind do not need to work as hard to achieve lift. Studies show that birds in a V formation place themselves roughly at the optimum distance predicted by simple aerodynamic theory. [ 103 ] Geese in a V-formation may conserve 12–20% of the energy they would need to fly alone. [ 104 ] [ 105 ] Red knots and dunlins were found in radar studies to fly 5 km per hour faster in flocks than when they were flying alone. [ 106 ] The birds flying at the tips and at the front are rotated in a timely cyclical fashion to spread flight fatigue equally among the flock members. The formation also makes communication easier and allows the birds to maintain visual contact with each other.
Other animals may use similar drafting techniques when migrating. Lobsters , for example, migrate in close single-file formation "lobster trains", sometimes for hundreds of miles.
The Mediterranean and other seas present a major obstacle to soaring birds, which must cross at the narrowest points. Massive numbers of large raptors and storks pass through areas such as Gibraltar , Falsterbo , and the Bosphorus at migration times. More common species, such as the European honey buzzard , can be counted in hundreds of thousands in autumn. Other barriers, such as mountain ranges, can also cause funnelling, particularly of large diurnal migrants. This is a notable factor in the Central American migratory bottleneck. This concentration of birds during migration can put species at risk. Some spectacular migrants have already gone extinct, the most notable being the passenger pigeon . During migration the flocks were a mile (1.6 km) wide and 300 miles (500 km) long, taking several days to pass and containing up to a billion birds.
The term "shoal" can be used to describe any group of fish, including mixed-species groups, while "school" is used for more closely knit groups of the same species swimming in a highly synchronised and polarised manner.
Fish derive many benefits from shoaling behaviour including defence against predators (through better predator detection and by diluting the chance of capture), enhanced foraging success, and higher success in finding a mate. [ 108 ] It is also likely that fish benefit from shoal membership through increased hydrodynamic efficiency. [ 109 ]
Fish use many traits to choose shoalmates. Generally they prefer larger shoals, shoalmates of their own species, shoalmates similar in size and appearance to themselves, healthy fish, and kin (when recognised). The "oddity effect" posits that any shoal member that stands out in appearance will be preferentially targeted by predators. This may explain why fish prefer to shoal with individuals that resemble them. The oddity effect would thus tend to homogenise shoals. [ 110 ]
One puzzling aspect of shoal selection is how a fish can choose to join a shoal of animals similar to themselves, given that it cannot know its own appearance. Experiments with zebrafish have shown that shoal preference is a learned ability, not innate. A zebrafish tends to associate with shoals that resemble shoals in which it was reared, a form of imprinting . [ 111 ]
Other open questions of shoaling behaviour include identifying which individuals are responsible for the direction of shoal movement. In the case of migratory movement, most members of a shoal seem to know where they are going. In the case of foraging behaviour, captive shoals of golden shiner (a kind of minnow ) are led by a small number of experienced individuals who knew when and where food was available. [ 112 ]
Radakov estimated herring schools in the North Atlantic can occupy up to 4.8 cubic kilometres (1.2 cu mi) with fish densities between 0.5 and 1.0 fish/cubic metre, totalling several billion fish in one school. [ 113 ]
Between May and July huge numbers of sardines spawn in the cool waters of the Agulhas Bank and then follow a current of cold water northward along the east coast of South Africa. This great migration, called the sardine run , creates spectacular feeding frenzies along the coastline as marine predators, such as dolphins, sharks and gannets attack the schools.
Most krill , small shrimp-like crustaceans , form large swarms, sometimes reaching densities of 10,000–60,000 individual animals per cubic metre. [ 115 ] [ 116 ] [ 117 ] Swarming is a defensive mechanism, confusing smaller predators that would like to pick out single individuals. The largest swarms are visible from space and can be tracked by satellite. [ 118 ] One swarm was observed to cover an area of 450 square kilometres (175 square miles) of ocean, to a depth of 200 meters (650 feet) and was estimated to contain over 2 million tons of krill. [ 119 ] Recent research suggests that krill do not simply drift passively in these currents but actually modify them. [ 119 ] Krill typically follow a diurnal vertical migration . By moving vertically through the ocean on a 12-hour cycle, the swarms play a major part in mixing deeper, nutrient-rich water with nutrient-poor water at the surface. [ 119 ] Until recently it has been assumed that they spend the day at greater depths and rise during the night toward the surface. It has been found that the deeper they go, the more they reduce their activity, [ 120 ] apparently to reduce encounters with predators and to conserve energy.
Later work suggested that swimming activity in krill varied with stomach fullness. Satiated animals that had been feeding at the surface swim less actively and therefore sink below the mixed layer. [ 121 ] As they sink they produce faeces which may mean that they have an important role to play in the Antarctic carbon cycle. Krill with empty stomachs were found to swim more actively and thus head towards the surface. This implies that vertical migration may be a bi- or tri-daily occurrence. Some species form surface swarms during the day for feeding and reproductive purposes even though such behaviour is dangerous because it makes them extremely vulnerable to predators. [ 122 ] Dense swarms may elicit a feeding frenzy among fish, birds and mammal predators, especially near the surface. When disturbed, a swarm scatters, and some individuals have even been observed to moult instantaneously, leaving the exuvia behind as a decoy. [ 123 ] In 2012, Gandomi and Alavi presented what appears to be a successful stochastic algorithm for modelling the behaviour of krill swarms. The algorithm is based on three main factors: " (i) movement induced by the presence of other individuals (ii) foraging activity, and (iii) random diffusion." [ 124 ]
Copepods are a group of tiny crustaceans found in the sea and lakes. Many species are planktonic (drifting in sea waters), and others are benthic (living on the ocean floor). Copepods are typically 1 to 2 millimetres (0.04 to 0.08 in) long, with a teardrop shaped body and large antennae . Although like other crustaceans they have an armoured exoskeleton , they are so small that in most species this thin armour, and the entire body, is almost totally transparent. Copepods have a compound, median single eye, usually bright red, in the centre of the transparent head.
Copepods also swarm. For example, monospecific swarms have been observed regularly around coral reefs and sea grass , and in lakes. Swarms densities were about one million copepods per cubic metre. Typical swarms were one or two metres in diameter, but some exceeded 30 cubic metres. Copepods need visual contact to keep together, and they disperse at night. [ 125 ]
Spring produces blooms of swarming phytoplankton which provide food for copepods. Planktonic copepods are usually the dominant members of the zooplankton , and are in turn major food organisms for many other marine animals. In particular, copepods are prey to forage fish and jellyfish , both of which can assemble in vast, million-strong swarms. Some copepods have extremely fast escape responses when a predator is sensed and can jump with high speed over a few millimetres (see animated image below).
Planktonic copepods are important to the carbon cycle . Some scientists say they form the largest animal biomass on earth. [ 126 ] They compete for this title with Antarctic krill . Because of their smaller size and relatively faster growth rates, however, and because they are more evenly distributed throughout more of the world's oceans, copepods almost certainly contribute far more to the secondary productivity of the world's oceans, and to the global ocean carbon sink than krill , and perhaps more than all other groups of organisms together. The surface layers of the oceans are currently believed to be the world's largest carbon sink, absorbing about 2 billion tonnes of carbon a year, the equivalent to perhaps a third of human carbon emissions , thus reducing their impact. Many planktonic copepods feed near the surface at night, then sink into deeper water during the day to avoid visual predators. Their moulted exoskeletons, faecal pellets and respiration at depth all bring carbon to the deep sea.
Many single-celled organisms called phytoplankton live in oceans and lakes. When certain conditions are present, such as high nutrient or light levels, these organisms reproduce explosively. The resulting dense swarm of phytoplankton is called an algal bloom . Blooms can cover hundreds of square kilometres and are easily seen in satellite images. Individual phytoplankton rarely live more than a few days, but blooms can last weeks. [ 127 ] [ 128 ]
Scientists have attributed swarm behavior to plants for hundreds of years. In his 1800 book, Phytologia: or, The philosophy of agriculture and gardening , Erasmus Darwin wrote that plant growth resembled swarms observed elsewhere in nature. [ 129 ] While he was referring to more broad observations of plant morphology, and was focused on both root and shoot behavior, recent research has supported this claim.
Plant roots , in particular, display observable swarm behavior, growing in patterns that exceed the statistical threshold for random probability, and indicate the presence of communication between individual root apexes . The primary function of plant roots is the uptake of soil nutrients , and it is this purpose which drives swarm behavior. Plants growing in close proximity have adapted their growth to assure optimal nutrient availability. This is accomplished by growing in a direction that optimizes the distance between nearby roots, thereby increasing their chance of exploiting untapped nutrient reserves. The action of this behavior takes two forms: maximization of distance from, and repulsion by, neighboring root apexes. [ 130 ] The transition zone of a root tip is largely responsible for monitoring for the presence of soil-borne hormones, signaling responsive growth patterns as appropriate. Plant responses are often complex, integrating multiple inputs to inform an autonomous response. Additional inputs that inform swarm growth includes light and gravity, both of which are also monitored in the transition zone of a root's apex. [ 131 ] These forces act to inform any number of growing "main" roots, which exhibit their own independent releases of inhibitory chemicals to establish appropriate spacing, thereby contributing to a swarm behavior pattern. Horizontal growth of roots, whether in response to high mineral content in soil or due to stolon growth, produces branched growth that establish to also form their own, independent root swarms. [ 132 ]
Swarming also describes groupings of some kinds of predatory bacteria such as myxobacteria . Myxobacteria swarm together in "wolf packs", actively moving using a process known as bacterial gliding and keeping together with the help of intercellular molecular signals . [ 59 ] [ 133 ]
A collection of people can also exhibit swarm behaviour, such as pedestrians [ 136 ] or soldiers swarming the parapets [ dubious – discuss ] . In Cologne, Germany, two biologists from the University of Leeds demonstrated flock like behaviour in humans. The group of people exhibited similar behavioural pattern to a flock, where if five percent of the flock changed direction the others would follow. If one person was designated as a predator and everyone else was to avoid him, the flock behaved very much like a school of fish. [ 137 ] [ 138 ] Understanding how humans interact in crowds is important if crowd management is to effectively avoid casualties at football grounds, music concerts and subway stations. [ 139 ]
The mathematical modelling of flocking behaviour is a common technology, and has found uses in animation. Flocking simulations have been used in many films [ 140 ] to generate crowds which move realistically. Tim Burton's Batman Returns was the first movie to make use of swarm technology for rendering, realistically depicting the movements of a group of bats using the boids system. The Lord of the Rings film trilogy made use of similar technology, known as Massive , during battle scenes. Swarm technology is particularly attractive because it is cheap, robust, and simple.
An ant-based computer simulation using only six interaction rules has also been used to evaluate aircraft boarding behaviour. [ 141 ] Airlines have also used ant-based routing in assigning aircraft arrivals to airport gates. An airline system developed by Douglas A. Lawson uses swarm theory, or swarm intelligence —the idea that a colony of ants works better than one alone. Each pilot acts like an ant searching for the best airport gate. "The pilot learns from his experience what's the best for him, and it turns out that that's the best solution for the airline," Lawson explains. As a result, the "colony" of pilots always go to gates they can arrive and depart quickly. The program can even alert a pilot of plane back-ups before they happen. "We can anticipate that it's going to happen, so we'll have a gate available," says Lawson. [ 142 ]
Swarm behaviour occurs also in traffic flow dynamics, such as the traffic wave . Bidirectional traffic can be observed in ant trails. [ 143 ] [ 144 ] In recent years this behaviour has been researched for insight into pedestrian and traffic models. [ 145 ] [ 146 ] Simulations based on pedestrian models have also been applied to crowds which stampede because of panic. [ 147 ]
Herd behaviour in marketing has been used to explain the dependencies of customers' mutual behaviour. The Economist reported a recent conference in Rome on the subject of the simulation of adaptive human behaviour. [ 148 ] It shared mechanisms to increase impulse buying and get people "to buy more by playing on the herd instinct." The basic idea is that people will buy more of products that are seen to be popular, and several feedback mechanisms to get product popularity information to consumers are mentioned, including smart card technology and the use of Radio Frequency Identification Tag technology. A "swarm-moves" model was introduced by a Florida Institute of Technology researcher, which is appealing to supermarkets because it can "increase sales without the need to give people discounts."
The application of swarm principles to robots is called swarm robotics , while swarm intelligence refers to the more general set of algorithms.
Partially inspired by colonies of insects such as ants and bees, researchers are modelling the behaviour of swarms of thousands of tiny robots which together perform a useful task, such as finding something hidden, cleaning, or spying. Each robot is quite simple, but the emergent behaviour of the swarm is more complex. [ 150 ] The whole set of robots can be considered as one single distributed system, in the same way an ant colony can be considered a superorganism , exhibiting swarm intelligence . The largest swarms so far created is the 1024 robot Kilobot swarm. [ 151 ] Other large swarms include the iRobot swarm, the SRI International /ActivMedia Robotics Centibots project, [ 152 ] and the Open-source Micro-robotic Project swarm, which are being used to research collective behaviours. [ 153 ] [ 154 ] Swarms are also more resistant to failure. Whereas one large robot may fail and ruin a mission, a swarm can continue even if several robots fail. This could make them attractive for space exploration missions, where failure is normally extremely costly. [ 155 ] In addition to ground vehicles, swarm robotics includes also research of swarms of aerial robots [ 149 ] [ 156 ] and heterogeneous teams of ground and aerial vehicles. [ 157 ] [ 158 ]
In contrast macroscopic robots, colloidal particles at microscale can also be adopted as agents to perform collective behaviors to conduct tasks using mechanical and physical approaches, such as reconfigurable tornado-like microswarm [ 159 ] mimicking schooling fish, [ 160 ] hierarchical particle species [ 161 ] mimicking predating behavior of mammals, micro-object manipulation using a transformable microswarm. [ 162 ] The fabrication of such colloidal particles is usually based on chemical synthesis.
Military swarming is a behaviour where autonomous or partially autonomous units of action attack an enemy from several different directions and then regroup. Pulsing , where the units shift the point of attack, is also a part of military swarming. Military swarming involves the use of a decentralized force against an opponent, in a manner that emphasizes mobility, communication, unit autonomy and coordination or synchronization. [ 163 ] Historically military forces used principles of swarming without really examining them explicitly, but now active research consciously examines military doctrines that draw ideas from swarming.
Merely because multiple units converge on a target, they are not necessarily swarming. Siege operations do not involve swarming, because there is no manoeuvre; there is convergence but on the besieged fortification. Nor do guerrilla ambushes constitute swarms, because they are "hit-and-run". Even though the ambush may have several points of attack on the enemy, the guerillas withdraw when they either have inflicted adequate damage, or when they are endangered.
In 2014 the U. S. Office of Naval Research released a video showing tests of a swarm of small autonomous drone attack boats that can steer and take coordinated offensive action as a group. [ 164 ] | https://en.wikipedia.org/wiki/Swarm_behaviour |
Swarts fluorination is a process whereby the chlorine atoms in a compound – generally an organic compound, but experiments have been performed using silanes – are replaced with fluorine , by treatment with antimony trifluoride in the presence of chlorine or of antimony pentachloride . Some metal fluorides are particularly more useful than others, including silver(I) fluoride, mercurous fluoride, cobalt(II) fluoride and aforementioned antimony.
Heating the mixture of the metal fluoride and the haloalkane (chlorine and bromine are replaced readily) yields the desired fluoroalkane. In some particularly reactive cases, heating is unnecessary; shaking or stirring the reaction mixture is sufficient. This reaction has a good yield.
The process was initially described by Frédéric Jean Edmond Swarts in 1892. [ 1 ]
The active species is antimony trifluorodichloride 2 , which is produced in situ from the reaction between antimony trifluoride 1 and chlorine; this compound can also be produced in bulk, according to a patent of John Weaver. [ 2 ] This then undergoes a halogen exchange with a haloalkane (here trichloroethylsilane ), as in 3 , replacing the halogen atom (here chlorine) with fluorine and affording the fluorinated product 4 . [ 3 ]
This chemical reaction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Swarts_fluorination |
Swatting is a form of criminal harassment that involves deceiving an emergency service (via such means as hoaxing an emergency services dispatcher ) into sending a police or emergency response team to another person's location. This is achieved by false reporting of a serious law enforcement emergency, such as a bomb threat , domestic violence , murder , hostage situation , or a false report of a mental health emergency, such as that a person is suicidal or homicidal and armed, among other things. [ 1 ]
The term is derived from the law enforcement unit SWAT (Special Weapons and Tactics), a specialized type of police unit in the United States . It is not related to the verb " to swat ". [ 2 ] SWAT teams are equipped with tactical gear and weapons that differ from patrol units, and are called to situations that are deemed high-risk. A threat may result in evacuations of schools and businesses. Advocates have called for swatting to be considered terrorism due to its use to intimidate and create the risk of injury or death. [ 3 ] [ 4 ]
Making false reports to emergency services is a criminal offense in many jurisdictions, often punishable by fine or imprisonment. [ 5 ] In March 2019, a California man was sentenced to 20 years in prison for carrying out a fatal 2017 swatting . [ 6 ] Swatting carries a high risk of violence, and causes resources of about US$10,000 per incident to be wasted by a city or county that responds to a false report of a serious law enforcement emergency, as well as police or municipal liability in cases of violence or use of force . [ 7 ] : 1 [ 8 ] [ 9 ] [ 10 ] In California , swatters bear the "full cost" of the response, which can lead to fines of up to $10,000 if great bodily injury or death occur as a result of the swatting. [ 11 ] [ 12 ]
Bomb threats were a concern to police in the 1970s, with public buildings such as airports being evacuated in response to hoax calls designed to cause mass panic and public disruption, [ 13 ] [ 14 ] or to delay exams at educational institutions. [ 15 ] [ 16 ] In recent decades, hoax callers sometimes use techniques to disguise their identity or country of origin. [ 17 ]
Swatting has origins in prank calls to emergency services. Over the years, callers used increasingly sophisticated techniques to direct response units of particular types. In particular, attempts to have SWAT teams be dispatched to particular locations spawned the term swatting . The term was used by the FBI as early as 2008, [ 18 ] and entered Oxford Dictionaries Online in 2015. [ 19 ]
In 2019 the Anti-Defamation League estimated that there were about 1,000 swatting incidents nationwide, each costing about $10,000 of police time. [ 10 ]
Caller ID spoofing , social engineering , prank calls , and phone phreaking techniques may be variously combined by swatting perpetrators, along with TTY systems meant for the use of those with hearing disabilities. 911 systems (including computer telephony systems and human operators) have been tricked by calls placed from cities hundreds of miles away from the location of the purported call, or even from other countries. [ 20 ] The caller typically places a 911 call using a spoofed phone number, hiding the caller's real location.
Swatting is linked to the action of doxing , which is obtaining and broadcasting, often via the Internet, the address and details of an individual with an intent to harass or endanger them. [ 21 ]
In October 2018, the Seattle Police Department instituted a three-part approach to combating swatting: educating 911 dispatchers to identify fraudulent calls; ensuring that responding officers were aware of the potential for a hoax; and creating an opt-in registry for people who feared that they might become victims of swatting, such as journalists, celebrities, and live streamers. Using the registry, these people can provide cautionary information to the police, to inform officers responding to potential swatting attempts that target the victim's address. [ 7 ] [ 22 ]
Security reporter Brian Krebs recommends that police departments take extra care when responding to calls received at their non-emergency numbers, or through speech synthesis systems, since these methods are often employed by out-of-area swatters who cannot connect to regional 911 centers. [ 23 ]
In September 2019, the Seattle Police Department formed the Swatting Mitigation Advisory Committee, composed of expert community and police representatives. Its purpose is to better understand swatting by collecting and analyzing data, formalizing protocols, and advocating broader awareness and prevention. It is currently co-chaired by Naveed Jamali and Sean Whitcomb , creator of the anti-swatting registry. [ 24 ]
In June 2023, the FBI announced that it would create a database to track swattings and improve information-sharing among local police agencies. [ 10 ]
In the United States , swatting can be prosecuted through federal criminal statutes:
In 2011, California State Senator Ted Lieu authored a bill to increase penalties for swatting. His own family became a victim of swatting when the bill was proposed. [ 31 ] A dozen police officers, along with firefighters and paramedics surrounded his family home.
In 2015, New Jersey State Assemblyman Paul D. Moriarty announced a bill [ 32 ] to increase sentences for hoax emergency calls, and was targeted by a hoax. [ 33 ] [ 34 ] The bill proposed prison sentences up to ten years and fines up to $150,000.
A 2015 bipartisan bill in Congress sponsored by Katherine Clark and Patrick Meehan made swatting a federal crime with increased penalties. [ 35 ] [ 36 ] Congresswoman Clark wrote an op-ed in The Hill saying that 2.5 million cases of cyberstalking between 2010 and 2013 had only resulted in 10 cases prosecuted, although a source for this was not provided. [ 37 ] [ 38 ] As revenge for the bill, an anonymous caller fraudulently called police to Rep. Clark's house on January 31, 2016. [ 39 ]
In the United Kingdom , swatting is not recognized as an offence under UK laws unlike the US but may be prosecuted as Perverting the course of justice where false complaints or allegations were made. [ 40 ]
In 2015, 28-year-old Robert Walker-McDaid pleaded guilty at Warwick Crown Court to perverting the course of justice, and was given a 20 month Suspended sentence . Walker-McDaid was also required to complete 200 hours of community service and provide £1000 compensation to Tyran Dobbs, who was the victim of Walker-McDaid's hoax call. [ 41 ]
On January 15, 2015, in Sentinel , Washita County, Oklahoma , dispatchers received 911 calls from someone who identified himself as Dallas Horton and told dispatchers he had placed a bomb in a local preschool . Washita County sheriff's deputies and Sentinel police chief Louis Ross made forced entry into Horton's residence. Ross, who was wearing a bulletproof vest , was shot several times by Horton. Further investigation revealed that the calls did not originate from the residence, and led Oklahoma State Bureau of Investigation agents to believe Horton was unaware that it was law enforcement officers making entry. James Edward Holly confessed to investigators that he made the calls with two "nonfunctioning" phones because he was angry with Horton. [ 42 ] Ross, who was shot multiple times in the chest and arm, was injured, but was treated for his wounds, and released from a local hospital. [ 43 ]
On December 28, 2017, a Wichita police officer shot a civilian named Andrew Finch directly through his chest in Finch's Wichita, Kansas , residence in a swatting incident. Finch later died at a hospital. Based on a series of screenshotted Twitter posts, the Wichita Eagle suggests that Finch was the unintended victim of the swatting after two Call of Duty: WWII players on the same team got into a heated argument about a US$1.50 bet. On December 29, 2017, the Los Angeles Police Department arrested 25-year-old serial swatter Tyler Raj Barriss, known online as "SWAuTistic" and on Xbox Live as "GoredTutor36", in connection with the incident. [ 44 ] [ 45 ] [ 46 ] [ 47 ] In 2018, Barriss was indicted by a federal grand jury along with two others involved in the incident. According to U.S. Attorney Stephen McAllister , the hoax charge carries a maximum punishment of life in federal prison while other charges carry sentences of up to 20 years. [ 48 ] On March 29, 2019, Barriss was sentenced to 20 years' imprisonment. [ 49 ] The gamer who made the bet with Barriss pleaded guilty to felony charges of conspiracy and obstruction of justice, and was sentenced to 15 months in prison and banned from playing video games for two years. [ 50 ]
On April 27, 2020, Mark Herring, a sixty-year-old man from Bethpage, Tennessee , died of a heart attack after police responded to false reports of a woman being killed at his house. The swatting was organized in an attempt to force him to give up his Twitter handle "@tennessee". Shane Sonderman was sentenced to five years in prison for the swatting, and ordered to pay a $250,000 fine. A 16-year-old in the United Kingdom was also involved, but they could not be extradited or identified due to their age as a juvenile. [ 51 ] [ 52 ]
Due to the popularity of streaming services, many broadcasters have been victim of swatting. Two weeks after the Fortnite World Cup Finals, where 16-year-old Kyle "Bugha" Giersdorf won $3 million and the title of best solo Fortnite player, he was swatted while streaming live on Twitch. [ 53 ] Ben "DrLupo" Lupo stated he was swatted three times in one month. [ 54 ] Other popular gaming broadcasters have been victims of swatting, including Tyler "Ninja" Blevins . [ 55 ]
In 2013, a number of U.S. celebrities were victims of swatting, including Sean Combs (P. Diddy) . [ 56 ] There were also swatting incidents at the residences of Ashton Kutcher , Tom Cruise , Chris Brown , Miley Cyrus , Iggy Azalea , Jason Derulo , Snoop Dogg , Justin Bieber and Clint Eastwood . [ 11 ]
In April 2013 California State Senator Ted Lieu , who was arguing at the time for anti-swatting laws in the state, was himself swatted. [ 57 ]
In 2013, a network of fraudsters involved in carding and doxing of public officials using stolen credit reports targeted computer security expert Brian Krebs with malicious police reports. [ 58 ] [ 59 ] Mir Islam, the group's leader, had also used swatting hoaxes against prosecutor Stephen P. Heymann, congressman Mike Rogers , and a woman he was cyberstalking after she declined his romantic proposals. Islam was convicted of doxing and swatting over 50 public figures, including Michelle Obama , Robert Mueller , John Brennan as well as Krebs, and sentenced to two years in prison. [ 60 ] Ukrainian computer hacker Sergey Vovnenko was convicted of trafficking in stolen credit cards, as well as planning to purchase heroin, ship it to Krebs, then swat him. [ 61 ] He was sentenced to 15 months in prison in Italy, and 41 months in prison in New Jersey. [ 62 ]
Hal Finney , a paralyzed computer scientist with amyotrophic lateral sclerosis (ALS), was swatted in 2014 after refusing to pay a $400,000 ransom . He faced cold, unsafe conditions on his lawn for half an hour while police checked his house. He continued receiving threats until his death in August 2014. [ 63 ]
In July 2022, Emmet G. Sullivan , a U.S. federal judge presiding over cases pertaining to the January 6 United States Capitol attack , was the victim of a swatting incident. [ 64 ]
On August 5, 2022, Canadian transgender streamer and political commentator Clara "Keffals" Sorrenti was swatted at her home by unknown individuals who also, posing as Sorrenti, sent a threatening email and a photo of an illegal firearm to London city councillors, presumably part of a harassment campaign carried out by Kiwi Farms that began on March 21, 2022. [ 65 ] Sorrenti claimed she was repeatedly misgendered and deadnamed by London Police officers , and placed into custody for 11 hours before being released without charges. She stated that she considered the incident a hate crime , an example of harassment towards transgender people by anti-LGBTQ groups in the United States. [ 66 ] The London Police Service responded with a statement from Chief of Police Steve Williams, who said that while he could not confirm any language used before Sorrenti's arrest, she was not addressed by her deadname or previous gender while in the agency's holding cells. He also said that any references to Sorrenti's deadname during the investigation seemed to stem from the existence of prior police reports she had accumulated before the event. [ 67 ] Three other streamers, Adin Ross , Nadia Amine, and IShowSpeed were also swatted the same week as Sorrenti. [ 68 ]
In August 2022, U.S. representative Marjorie Taylor Greene was swatted in Georgia by a caller who allegedly opposed her stances on transgender rights. [ 69 ]
In November 2023, Ned Luke , a voice and performance artist for the fictional character Michael De Santa in the video game Grand Theft Auto V , was swatted in his home during a Thanksgiving live-stream of himself playing the game. He took a phone call warning him of the pending police action before he prematurely ended his stream. [ 70 ] [ 71 ]
There have been widespread doxing , swatting and violent threats against American politicians since early December 2023, predominately members of the Republican Party and conservatives . [ 72 ] Beginning in late December 2023, members of the Democratic Party also began to be increasingly targeted. [ 73 ] It is unknown if the hoaxes were perpetrated by one or more individuals, or what their motivations were. [ 72 ]
Maine Sec. of State Shenna Bellows was targeted with a fake emergency call to police that caused officers to respond to her home the day after she removed former President Donald Trump from Maine's Presidential Primary Ballot under the Constitution's insurrection clause. [ 74 ] Bellows and her husband were not home for the holiday weekend. At her request, police conducted an exterior sweep of the house and then checked inside. Nothing suspicious was found, though an investigation was opened to locate the perpetrator(s). No one has yet been charged. | https://en.wikipedia.org/wiki/Swatting |
Sway is a tiling window manager and Wayland compositor , inspired by i3 , and written in C . [ 3 ] Sway is designed as a drop-in replacement for i3 using the more modern Wayland display server protocol and wlroots compositor library. [ 4 ] Sway works with existing i3 configuration files and supports most of i3's features while providing several new features of its own. [ 5 ]
Sway's default controls for manipulating windows are similar to vi . Window focus is controlled by a combination of the Super key and one of the arrow keys or h, j, k, and l . [ 6 ] Window movement is performed by the same combination of keys with the addition of the shift key .
Like i3, Sway can be extended and manipulated using its Unix domain socket and JSON -based IPC interface from many programming languages. [ 7 ]
Sway's first stable release was on March 11, 2019, after 3.6 years of development. [ 8 ]
Sway replicates several of i3's features:
Sway also provides several unique features: | https://en.wikipedia.org/wiki/Sway_wm |
A sweat allergy is the exacerbation of atopic dermatitis associated with an elevated body temperature and resulting increases in the production of sweat . It appears as small reddish welts that become visible in response to increased temperature and resulting production of sweat. [ 1 ] It can affect all ages. Sweating can trigger intense itching or cholinergic urticaria . The protein MGL_1304 secreted by mycobiota (fungi) present on the skin such as Malassezia globosa acts as a histamine or antigen . People can be desensitized using their own samples of sweat that have been purified that contains small amounts of the allergen. [ 2 ] [ 3 ] The allergy is not due to the sweat itself but instead to an allergy-producing protein secreted by microorganisms found on the skin. [ 4 ]
Cholinergic urticaria (CU) is one of the physical urticaria (hives) which is provoked during sweating events such as exercise, bathing, staying in a heated environment, or emotional stress. The hives produced are typically smaller than classic hives and are generally shorter-lasting. [ 5 ] [ 6 ]
Multiple subtypes have been elucidated, each of which require distinct treatment. [ 7 ] [ 8 ]
Tannic acid has been found to suppress the allergic response, along with showering. [ 2 ]
This cutaneous condition article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sweat_allergy |
Sweat diagnostics is an emerging non-invasive technique used to provide insights to the health of the human body. Common sweat diagnostic tests include testing for cystic fibrosis [ 1 ] and illicit drugs . [ 2 ] Most testing of human sweat is in reference to the eccrine sweat gland which in contrast to the apocrine sweat gland , has a lower composition of oils. [ 3 ]
Although sweat is mostly water, [ 3 ] there are many solutes which are found in sweat that have at least some relation to biomarkers found in blood. These include: sodium (Na + ), chloride (Cl − ), potassium (K + ), ammonium (NH + 4 ), alcohols , lactate , peptides & proteins . [ 4 ] [ 5 ] Development of devices, sensing techniques and biomarker identification in sweat continues to be an expanding field for medical diagnostics and athletics applications.
The use of smart biosensors for on-skin sweat analysis has been described as internet-enabled Sudorology (iSudorology) by Brasier et al. in 2019. It describes the lab-independent detection of molecular, next-generation digital biomarkers in sweat. [ 6 ]
Some of the earliest, published studies [ 7 ] on sweat composition date back to the 19th century. Further studies [ 8 ] [ 9 ] [ 10 ] in the 20th century began to solidify understanding of the physiology and pharmacology of the eccrine sweat gland. In-vivo and in-vitro studies from this time period, and even those continuing today, have identified numerous structural nuances and new molecules present within sweat. The first commercially adopted use for sweat diagnostics included testing of sodium and chloride levels in children for the diagnosis of cystic fibrosis. Today, one of the most popular devices for this testing is the Macroduct Sweat Collection System from ELITechGroup. [ 11 ]
More recently, numerous studies have identified the plausibility of sweat as an alternative to blood analysis. [ 12 ] [ 13 ] The potential substitution for sweat versus blood analysis has many potential benefits. For example, sweat can be: extracted in a non-invasive manner via iontophoresis ; extracted with little-to-no pain; and monitored continuously. [ 14 ] There are downfalls to the technology, however. For example, demonstration of successful and reliable sweat extraction and analysis on a cohesive device has yet to be demonstrated. Furthermore, although some biomarker partitioning mechanisms are well understood and well studied, partitioning of other useful biomarkers ( cytokines , peptides , etc.) are less understood. [ 4 ]
Patches have been demonstrated to be a promising detection platform for sweat diagnostics. [ 15 ] [ 16 ] [ 17 ] Simple, long-term collection devices which check for drugs of abuse or alcohol are already on the market and operate on the following principle: a user applies the patch which then collects sweat over a period of hours or days, then the patch is analyzed utilizing techniques such as GC-MS which are accurate but have the drawback of lack of continuous measurements and high costs. For example, sweat diagnostic products for illicit drugs and alcohol are manufactured and supplied by PharmChek and AlcoPro, respectively. Recently several efforts [ 18 ] have been made to develop low cost polymer based continuous perspiration monitoring devices and are in early stages of commercialization. [ 19 ]
More recently, startup companies such as Xsensio have begun developing products targeted towards the consumer, healthcare and athletics market for sweat diagnostics. Ultimately, it is the hope that these devices will have the ability to detect changes in human physiology within minutes without the need for repeated sample collection and analysis. [ 20 ]
Temporary tattoo-based sweat diagnostic tools [ 21 ] have been demonstrated by Dr. Joseph Wang's group from University of California, San Diego . Their work includes sweat diagnostics for sodium, lactate, ammonium, pH and biofuel opportunities. [ 22 ] | https://en.wikipedia.org/wiki/Sweat_diagnostics |
The Sweden Solar System is the world's largest permanent scale model of the Solar System . The Sun is represented by the Avicii Arena in Stockholm , the second-largest hemispherical building in the world . The inner planets can also be found in Stockholm but the outer planets are situated northward in other cities along the Baltic Sea . The system was started by Nils Brenning, professor at the Royal Institute of Technology in Stockholm, and Gösta Gahm , professor at the Stockholm University. [ 1 ] [ 2 ] The model represents the Solar System on the scale of 1:20 000 000, i.e. one metre represents 20,000 km. [ 3 ]
The bodies represented in this model include the Sun , the planets (and some of their moons ), dwarf planets and many types of small bodies ( comets , asteroids , trans-Neptunians , etc.), as well as some abstract concepts (like the Termination Shock zone). Because of the existence of many small bodies in the real Solar System , the model can always be further increased.
The Sun is represented by the Avicii Arena (Globen), Stockholm , which is the second-largest hemispherical building in the world, 110 m (360 ft) in diameter. To respect the scale, the globe represents the Sun including its corona . [ citation needed ] | https://en.wikipedia.org/wiki/Sweden_Solar_System |
The Swedish Astronomical Society (Svenska astronomiska sällskapet) a national organization in Sweden aimed at people who want to follow the achievements of astronomical research,. [ 1 ] Founded in 1919 on the initiative of astronomer Nils Nordenmark , [ 2 ] [ 3 ] the society has from the outset aimed to be "an intimate connection between scientists, amateur astronomers and others interested in astronomy" [ 4 ] The society runs a program of lectures and other outreach activities, including since 2012 Sweden's annual Day and Night of Astronomy . [ 5 ] Since 1920 society has published a quarterly Swedish-language magazine, initially Populär Astronomisk Tidskrift , since 2001 published as Populär Astronomi . [ 6 ] [ 7 ] The current of the society is astronomer and writer Peter Linde . [ 8 ]
This astronomy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Swedish_Astronomical_Society |
The Swedish Chemical Society ( Swedish : Svenska kemisamfundet ) was established in 1883 and is a nonprofit organisation to promote the development of chemistry in Sweden. [ 1 ] The society is based on Wallingatan, Stockholm . [ 1 ] Kemivärlden Biotech ( ISSN 1653-5596 ) is the monthly magazine of the Swedish Chemical Society. [ 2 ] The society also awards the annual Arrhenius Plaque for contributions in the field of science or to the society.
This article about a chemistry organization is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Swedish_Chemical_Society |
The Swedish ethyl acetate method ( SweEt ) is a method for chemical analysis of pesticide residues in food using ethyl acetate as an extraction medium followed by analysis with liquid chromatography - tandem mass spectrometry (LC-MS/MS) and gas chromatography -tandem mass spectrometry (GC-MS/MS). It was developed by the Swedish National Food Agency (National Reference Laboratory for pesticide analysis) for quantitative analysis of over 500 pesticides in fruits, vegetables, cereals and products of animal origin. [ 1 ] | https://en.wikipedia.org/wiki/Swedish_ethyl_acetate_method |
In physical simulations , sweep and prune is a broad phase algorithm used during collision detection to limit the number of pairs of solids that need to be checked for collision , i.e. intersection. This is achieved by sorting the starts (lower bound) and ends (upper bound) of the bounding volume of each solid along a number of arbitrary axes. As the solids move, their starts and ends may overlap. When the bounding volumes of two solids overlap in all axes they are flagged to be tested by more precise and time-consuming algorithms.
Sweep and prune exploits temporal coherence as it is likely that solids do not move significantly between two simulation steps. Because of that, at each step, the sorted lists of bounding volume starts and ends can be updated with relatively few computational operations. Sorting algorithms which are fast at sorting almost-sorted lists, such as insertion sort , are particularly good for this purpose.
According with the type of bounding volume used, it is necessary to update the bounding volume dimensions every time a solid is reoriented. To circumvent this, temporal coherence can be used to compute the changes in bounding volume geometry with fewer operations. Another approach is to use bounding spheres or other orientation independent bounding volumes.
Sweep and prune is also known as sort and sweep , [ 1 ] referred to this way in David Baraff's Ph.D. thesis in 1992. [ 2 ] Later works like the 1995 paper about I-COLLIDE by Jonathan D. Cohen et al. [ 3 ] refer to the algorithm as sweep and prune .
This computational physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sweep_and_prune |
Sweep frequency response analysis ( SFRA ) is a method to evaluate the mechanical integrity of core , windings and clamping structures within power transformers by measuring their electrical transfer functions over a wide frequency range .
SFRA is a comparative method , meaning an evaluation of the transformer condition is done by comparing an actual set of SFRA results to reference results. Three methods are commonly used to assess the measured traces:
Transformers generate a unique signature when tested at discrete frequencies and plotted as a curve. The distance between conductors of the transformer forms a capacitance. Any movement of the conductors or windings will change this capacitance. This capacitance being a part of complex L (inductance), R (Resistance) and C (Capacitance) network, any change in this capacitance will be reflected in the curve or signature. [ 1 ]
An initial SFRA test is carried out to obtain the signature of the transformer frequency response by injecting various discreet frequencies. This reference is then used for future comparisons. A change in winding position, degradation in the insulation, etc. will result in change in capacitance or inductance thereby affecting the measured curves. [ 2 ]
Tests are carried out periodically or during major external events like short circuits and results compared against the initial signature to test for any problems. [ citation needed ]
SFRA analysis can detect problems in transformers such as:
SFRA can be used in the following contexts: | https://en.wikipedia.org/wiki/Sweep_frequency_response_analysis |
A swept wing is a wing angled either backward or occasionally forward from its root rather than perpendicular to the fuselage.
Swept wings have been flown since the pioneer days of aviation. Wing sweep at high speeds was first investigated in Germany as early as 1935 by Albert Betz and Adolph Busemann , finding application just before the end of the Second World War . It has the effect of delaying the shock waves and accompanying aerodynamic drag rise caused by fluid compressibility near the speed of sound , improving performance. Swept wings are therefore almost always used on jet aircraft designed to fly at these speeds.
The term "swept wing" is normally used to mean "swept back", but variants include forward sweep , variable sweep wings and oblique wings in which one side sweeps forward and the other back. The delta wing is also aerodynamically a form of swept wing.
There are three main reasons for sweeping a wing: [ 1 ]
1. to arrange the center of gravity of the aircraft and the aerodynamic center of the wing to coincide more closely for longitudinal balance, e.g. Messerschmitt Me 163 Komet and Messerschmitt Me 262 . Although not a swept wing the wing panels on the Douglas DC-1 outboard of the nacelles also had slight sweepback for similar reasons. [ 2 ]
2. to provide longitudinal stability for tailless aircraft, e.g. Messerschmitt Me 163 Komet . [ 2 ]
3. most commonly to increase Mach-number capability by delaying to a higher speed the effects of compressibility (abrupt changes in the density of the airflow), e.g. combat aircraft, airliners and business jets.
Other reasons include:
1. enabling a wing carry-through box position to achieve a desired cabin size, e.g. HFB 320 Hansa Jet .
2. providing static aeroelastic relief which reduces bending moments under high g-loadings and may allow a lighter wing structure. [ 3 ]
For a wing of given span, sweeping it increases the length of the spars running along it from root to tip. This tends to increase weight and reduce stiffness. If the fore-aft chord of the wing also remains the same, the distance between leading and trailing edges reduces, reducing its ability to resist twisting (torsion) forces. A swept wing of given span and chord must therefore be strengthened and will be heavier than the equivalent unswept wing.
A swept wing typically angles backward from its root rather than forwards. Because wings are made as light as possible, they tend to flex under load. This aeroelasticity under aerodynamic load causes the tips to bend upwards in normal flight. Backwards sweep causes the tips to reduce their angle of attack as they bend, reducing their lift and limiting the effect. Forward sweep causes the tips to increase their angle of attack as they bend. This increases their lift causing further bending and hence yet more lift in a cycle which can cause a runaway structural failure. For this reason forward sweep is rare and the wing must be unusually rigid.
There are two sweep angles of importance, one at the leading edge for supersonic aircraft and the other 25% of the way back from the leading edge for subsonic and transonic aircraft. Leading edge sweep is important because the leading edge has to be behind the mach cone to reduce wave drag. [ 4 ] The quarter chord (25%) line is used because subsonic lift due to angle of attack acts there and, up until the introduction of supercritical sections, the crest was usually close to the quarter chord. [ 5 ]
Typical sweep angles vary from 0 for a straight-wing aircraft, to 45 degrees or more for fighters and other high-speed designs.
Shock waves can form on some parts of an aircraft moving at less than the speed of sound. Low-pressure regions around an aircraft cause the flow to accelerate, and at transonic speeds this local acceleration can exceed Mach 1. Localized supersonic flow must return to the freestream conditions around the rest of the aircraft, and as the flow enters an adverse pressure gradient in the aft section of the wing, a discontinuity emerges in the form of a shock wave as the air is forced to rapidly slow and return to ambient pressure.
At the point where the density drops, the local speed of sound correspondingly drops and a shock wave can form. This is why in conventional wings, shock waves form first after the maximum Thickness/Chord and why all airliners designed for cruising in the transonic range (above M0.8) have supercritical wings that are flatter on top, resulting in minimized angular change of flow to upper surface air. The angular change to the air that is normally part of lift generation is decreased and this lift reduction is compensated for by deeper curved lower surfaces accompanied by a reflex curve at the trailing edge. This results in a much weaker shock wave towards the rear of the upper wing surface and a corresponding increase in critical mach number.
Shock waves require energy to form. This energy is taken out of the aircraft, which has to supply extra thrust to make up for this energy loss. Thus the shocks are seen as a form of drag . Since the shocks form when the local air velocity reaches supersonic speeds, there is a certain " critical mach " speed where sonic flow first appears on the wing. There is a following point called the drag divergence mach number where the effect of the drag from the shocks becomes noticeable. This is normally when the shocks start generating over the wing, which on most aircraft is the largest continually curved surface, and therefore the largest contributor to this effect.
Sweeping the wing has the effect of reducing the curvature of the body as seen from the airflow, by the cosine of the angle of sweep. For instance, a wing with a 45 degree sweep will see a reduction in effective curvature to about 70% of its straight-wing value. This has the effect of increasing the critical Mach by 30%. When applied to large areas of the aircraft, like the wings and empennage , this allows the aircraft to reach speeds closer to Mach 1.
One limiting factor in swept wing design is the so-called "middle effect". If a swept wing is continuous - an oblique swept wing - the pressure isobars will be swept at a continuous angle from tip to tip. However, if the left and right halves are swept back equally, as is common practice, the pressure isobars on the left wing in theory will meet the pressure isobars of the right wing on the centerline at a large angle. As the isobars cannot meet in such a fashion, [ why? ] they will tend to curve on each side as they near the centerline, so that the isobars cross the centerline at right angles to the centerline. This causes an "unsweeping" of the isobars in the wing root region. To combat this unsweeping, German aerodynamicist Dietrich Küchemann proposed and had tested a local indentation of the fuselage above and below the wing root. This proved to not be very effective. [ 7 ] During the development of the Douglas DC-8 airliner, uncambered airfoils were used in the wing root area to combat the unsweeping. [ 8 ] [ 9 ]
Swept wings on supersonic aircraft usually lie within the cone-shaped shock wave produced at the nose of the aircraft so they will "see" subsonic airflow and work as subsonic wings. The angle needed to lie behind the cone increases with increasing speed, at Mach 1.3 the angle is about 45 degrees, at Mach 2.0 it is 60 degrees. [ 10 ] The angle of the Mach cone formed off the body of the aircraft will be at about sin μ = 1/M (μ is the sweep angle of the Mach cone). [ 11 ]
When a swept wing travels at high speed, the airflow has little time to react and simply flows over the wing almost straight from front to back. At lower speeds the air does have time to react, and is pushed spanwise by the angled leading edge, towards the wing tip. At the wing root, by the fuselage, this has little noticeable effect, but as one moves towards the wingtip the airflow is pushed spanwise not only by the leading edge, but the spanwise moving air beside it. At the tip the airflow is moving along the wing instead of over it, a problem known as spanwise flow .
The lift from a wing is generated by the airflow over it from front to rear. With increasing span-wise flow the boundary layers on the surface of the wing have longer to travel, and so are thicker and more susceptible to transition to turbulence or flow separation, also the effective aspect ratio of the wing is less and so air "leaks" around the wing tips reducing their effectiveness. The spanwise flow on swept wings produces airflow that moves the stagnation point on the leading edge of any individual wing segment further beneath the leading edge, increasing effective angle of attack of wing segments relative to its neighbouring forward segment. The result is that wing segments farther towards the rear operate at increasingly higher angles of attack promoting early stall of those segments. This promotes tip stall on back-swept wings, as the tips are most rearward, while delaying tip stall for forward-swept wings, where the tips are forward. With both forward and back-swept wings, the rear of the wing will stall first creating a nose-up moment on the aircraft. If not corrected by the pilot the plane will pitch up, leading to more of the wing stalling and more pitch up in a divergent manner. This uncontrollable instability came to be known as the Sabre dance in reference to the number of North American F-100 Super Sabres that crashed on landing as a result. [ 12 ] [ 13 ]
Reducing pitch-up to an acceptable level has been done in different ways such as the addition of a fin known as a wing fence on the upper surface of the wing to redirect the flow to a streamwise direction. The MiG-15 was one example of an aircraft fitted with wing fences. [ 14 ] Another closely related design was the addition of a dogtooth notch to the leading edge, used on the Avro Arrow interceptor. [ 15 ] Other designs took a more radical approach, including the Republic XF-91 Thunderceptor 's wing that grew wider towards the tip to provide more lift at the tip. The Handley Page Victor was equipped with a crescent wing , with three values of sweep, about 48 degrees near the wing root where the wing was thickest, a 38 degree transition length and 27 degrees for the remainder to the tip. [ 16 ] [ 17 ]
Modern solutions to the problem no longer require "custom" designs such as these. The addition of leading-edge slats and large compound flaps to the wings has largely resolved the issue. [ 18 ] [ 19 ] [ 20 ] On fighter designs, the addition of leading-edge extensions , which are typically included to achieve a high level of maneuverability, also serve to add lift during landing and reduce the problem. [ 21 ] [ 22 ]
In addition to pitch-up there are other complications inherent in a swept-wing configuration. For any given length of wing, the actual span from tip-to-tip is shorter than the same wing that is not swept. There is a strong correlation between low-speed drag and aspect ratio , the span compared to chord, so a swept wing always has more drag at lower speeds. In addition, there is extra torque applied by the wing to the fuselage which has to be allowed for when establishing the transfer of wing-box loads to the fuselage. This results from the significant part of the wing lift which lies behind the attachment length where the wing meets the fuselage.
Sweep theory is an aeronautical engineering description of the behavior of airflow over a wing when the wing's leading edge encounters the airflow at an oblique angle. The development of sweep theory resulted in the swept wing design used by most modern jet aircraft, as this design performs more effectively at transonic and supersonic speeds. In its advanced form, sweep theory led to the experimental oblique wing concept.
Adolf Busemann introduced the concept of the swept wing and presented this in 1935 at the Fifth Volta Conference in Rome. [ 23 ] Sweep theory in general was a subject of development and investigation throughout the 1930s and 1940s, but the breakthrough mathematical definition of sweep theory is generally credited to NACA 's Robert T. Jones in 1945. Sweep theory builds on other wing lift theories. Lifting line theory describes lift generated by a straight wing (a wing in which the leading edge is perpendicular to the airflow). Weissinger theory describes the distribution of lift for a swept wing, but does not have the capability to include chordwise pressure distribution. There are other methods that do describe chordwise distributions, but they have other limitations. Jones' sweep theory provides a simple, comprehensive analysis of swept wing performance.
An explanation of how the swept wing works was offered by Robert T. Jones :
"Assume a wing is a cylinder of uniform airfoil cross-section, chord and thickness and is placed in an airstream at an angle of yaw – i.e., it is swept back. Now, even if the local speed of the air on the upper surface of the wing becomes supersonic, a shock wave cannot form there because it would have to be a sweptback shock – swept at the same angle as the wing – i.e., it would be an oblique shock. Such an oblique shock cannot form until the velocity component normal to it becomes supersonic." [ 24 ]
To visualize the basic concept of simple sweep theory, consider a straight, non-swept wing of infinite length, which meets the airflow at a perpendicular angle. The resulting air pressure distribution is equivalent to the length of the wing's chord (the distance from the leading edge to the trailing edge). If we were to begin to slide the wing sideways ( spanwise ), the sideways motion of the wing relative to the air would be added to the previously perpendicular airflow, resulting in an airflow over the wing at an angle to the leading edge. This angle results in airflow traveling a greater distance from leading edge to trailing edge, and thus the air pressure is distributed over a greater distance (and consequently lessened at any particular point on the surface).
This scenario is identical to the airflow experienced by a swept wing as it travels through the air. The airflow over a swept wing encounters the wing at an angle. That angle can be broken down into two vectors, one perpendicular to the wing, and one parallel to the wing. The flow parallel to the wing has no effect on it, and since the perpendicular vector is shorter (meaning slower) than the actual airflow, it consequently exerts less pressure on the wing. In other words, the wing experiences airflow that is slower - and at lower pressures - than the actual speed of the aircraft.
One of the factors that must be taken into account when designing a high-speed wing is compressibility , which is the effect that acts upon a wing as it approaches and passes through the speed of sound . The significant negative effects of compressibility made it a prime issue with aeronautical engineers. Sweep theory helps mitigate the effects of compressibility in transonic and supersonic aircraft because of the reduced pressures. This allows the mach number of an aircraft to be higher than that actually experienced by the wing.
There is also a negative aspect to sweep theory. The lift produced by a wing is directly related to the speed of the air over the wing. Since the airflow speed experienced by a swept wing is lower than what the actual aircraft speed is, this becomes a problem during slow-flight phases, such as takeoff and landing. There have been various ways of addressing the problem, including the variable-incidence wing design on the Vought F-8 Crusader , [ 25 ] and swing wings on aircraft such as the F-14 , F-111 , and the Panavia Tornado . [ 26 ] [ 27 ]
The term "swept wing" is normally used to mean "swept back", but other swept variants include forward sweep , variable sweep wings and oblique wings in which one side sweeps forward and the other back. The delta wing also incorporates the same advantages as part of its layout.
Sweeping a wing forward has approximately the same effect as rearward in terms of drag reduction, but has other advantages in terms of low-speed handling where tip stall problems simply go away. In this case the low-speed air flows towards the fuselage, which acts as a very large wing fence. Additionally, wings are generally larger at the root anyway, which allows them to have better low-speed lift.
However, this arrangement also has serious stability problems. The rearmost section of the wing will stall first causing a pitch-up moment pushing the aircraft further into stall similar to a swept back wing design. Thus swept-forward wings are unstable in a fashion similar to the low-speed problems of a conventional swept wing. However unlike swept back wings, the tips on a forward swept design will stall last, maintaining roll control.
Forward-swept wings can also experience dangerous flexing effects compared to aft-swept wings that can negate the tip stall advantage if the wing is not sufficiently stiff. In aft-swept designs, when the airplane maneuvers at high load factor the wing loading and geometry twists the wing in such a way as to create washout (tip twists leading edge down). This reduces the angle of attack at the tip, thus reducing the bending moment on the wing, as well as somewhat reducing the chance of tip stall. [ 28 ] However, the same effect on forward-swept wings produces a wash-in effect that increases the angle of attack promoting tip stall.
Small amounts of sweep do not cause serious problems, and had been used on a variety of aircraft to move the spar into a convenient location, as on the Junkers Ju 287 or HFB 320 Hansa Jet . [ 29 ] [ 30 ] However, larger sweep suitable for high-speed aircraft, like fighters, was generally impossible until the introduction of fly by wire systems that could react quickly enough to damp out these instabilities. The Grumman X-29 was an experimental technology demonstration project designed to test the forward swept wing for enhanced maneuverability during the 1980s. [ 31 ] [ 32 ] The Sukhoi Su-47 Berkut is another notable demonstrator aircraft implementing this technology to achieve high levels of agility. [ 33 ] To date, no highly swept-forward design has entered production.
The first successful aeroplanes adhered to the basic design of rectangular wings at right angles to the body of the machine. Such a layout is inherently unstable; if the weight distribution of the aircraft changes even slightly, the wing will want to rotate so its front moves up (weight moving rearward) or down (forward) and this rotation will change the development of lift and cause it to move further in that direction. To make an aircraft stable, the normal solution is to place the weight at one end and offset this with an opposite downward force at the other - this leads to the classic layout with the engine in front and the control surfaces at the end of a long boom with the wing in the middle. This layout has long been known to be inefficient. The downward force of the control surfaces needs further lift from the wing to offset. The amount of force can be decreased by increasing the length of the boom, but this leads to more skin friction and weight of the boom itself.
This problem led to many experiments with different layouts that eliminates the need for the downward force. One such wing geometry appeared before World War I , which led to early swept wing designs. In this layout, the wing is swept so that portions lie far in front and in back of the center of gravity (CoG), with the control surfaces behind it. The result is a weight distribution similar to the classic layout, but the offsetting control force is no longer a separate surface but part of the wing, which would have existed anyway. This eliminates the need for separate structure, making the aircraft have less drag and require less total lift for the same level of performance. These layouts inspired several flying wing gliders and some powered aircraft during the interwar years. [ 34 ]
The first to achieve stability was British designer J. W. Dunne who was obsessed with achieving inherent stability in flight. He successfully employed swept wings in his tailless aircraft (which, crucially, used washout ) as a means of creating positive longitudinal static stability . [ 35 ] For a low-speed aircraft, swept wings may be used to resolve problems with the center of gravity , to move the wing spar into a more convenient location, or to improve the sideways view from the pilot's position. [ 34 ] By 1905, Dunne had already built a model glider with swept wings followed by the powered Dunne D.5 , and by 1913 he had constructed successful powered variants that were able to cross the English Channel . The Dunne D.5 was exceptionally aerodynamically stable for the time, [ 36 ] and the D.8 was sold to the Royal Flying Corps ; it was also manufactured under licence by Starling Burgess to the United States Navy amongst other customers. [ 37 ]
Dunne's work ceased with the onset of war in 1914, but afterwards the idea was taken up by G. T. R. Hill in England who designed a series of gliders and aircraft to Dunne's guidelines, notably the Westland-Hill Pterodactyl series. [ 38 ] However, Dunne's theories met with little acceptance amongst the leading aircraft designers and aviation companies at the time. [ 39 ]
The idea of using swept wings to reduce high-speed drag was developed in Germany in the 1930s. At a Volta Conference meeting in 1935 in Italy, Adolf Busemann suggested the use of swept wings for supersonic flight. He noted that the airspeed over the wing was dominated by the normal component of the airflow, not the freestream velocity, so by setting the wing at an angle the forward velocity at which the shock waves would form would be higher (the same had been noted by Max Munk in 1924, although not in the context of high-speed flight). [ 40 ] Albert Betz immediately suggested the same effect would be equally useful in the transonic. [ 41 ] After the presentation the host of the meeting, Arturo Crocco , jokingly sketched "Busemann's airplane of the future" on the back of a menu while they all dined. Crocco's sketch showed a classic 1950s fighter design, with swept wings and tail surfaces, although he also sketched a swept propeller powering it. [ 40 ]
At the time, however, there was no way to power an aircraft to these sorts of speeds, and even the fastest aircraft of the era were only approaching 400 km/h (249 mph).The presentation was largely of academic interest, and soon forgotten. Even notable attendees including Theodore von Kármán and Eastman Jacobs did not recall the presentation 10 years later when it was re-introduced to them. [ 42 ]
Hubert Ludwieg of the High-Speed Aerodynamics Branch at the AVA Göttingen in 1939 conducted the first wind tunnel tests to investigate Busemann's theory. [ 7 ] Two wings, one with no sweep, and one with 45 degrees of sweep were tested at Mach numbers of 0.7 and 0.9 in the 11 x 13 cm wind tunnel. The results of these tests confirmed the drag reduction offered by swept wings at transonic speeds. [ 7 ] The results of the tests were communicated to Albert Betz who then passed them on to Willy Messerschmitt in December 1939. The tests were expanded in 1940 to include wings with 15, 30 and -45 degrees of sweep and Mach numbers as high as 1.21. [ 7 ]
With the introduction of jets in the later half of the Second World War , the swept wing became increasingly applicable to optimally satisfying aerodynamic needs. The German jet-powered Messerschmitt Me 262 and rocket-powered Messerschmitt Me 163 suffered from compressibility effects that made both aircraft very difficult to control at high speeds. In addition, the speeds put them into the wave drag regime, and anything that could reduce this drag would increase the performance of their aircraft, notably the notoriously short flight times measured in minutes. This resulted in a crash program to introduce new swept wing designs, both for fighters as well as bombers . The Blohm & Voss P 215 was designed to take full advantage of the swept wing's aerodynamic properties; however, an order for three prototypes was received only weeks before the war ended and no examples were ever built. [ 43 ] The Focke-Wulf Ta 183 was another swept wing fighter design, but was also not produced before the war's end. [ 44 ] In the post-war era, Kurt Tank developed the Ta 183 into the IAe Pulqui II , but this proved unsuccessful. [ 45 ]
A prototype test aircraft, the Messerschmitt Me P.1101 , was built to research the tradeoffs of the design and develop general rules about what angle of sweep to use. [ 46 ] When it was 80% complete, the P.1101 was captured by US forces and returned to the United States , where two additional copies with US-built engines carried on the research as the Bell X-5 . [ 47 ] Germany's wartime experience with the swept wings and its high value for supersonic flight stood in strong contrast to the prevailing views of Allied experts of the era, who commonly espoused their belief in the impossibility of manned vehicles travelling at such speeds. [ 48 ]
During the immediate post-war era, several nations were conducting research into high speed aircraft. In the United Kingdom, work commenced during 1943 on the Miles M.52 , a high-speed experimental aircraft equipped with a straight wing that was developed in conjunction with Frank Whittle 's Power Jets company, the Royal Aircraft Establishment (RAE) in Farnborough , and the National Physical Laboratory . [ 49 ] The M.52 was envisioned to be capable of achieving 1,000 miles per hour (1,600 km/h) in level flight, thus enabling the aircraft to potentially be the first to exceed the speed of sound in the world. [ 49 ] In February 1946, the programme was abruptly discontinued for unclear reasons. [ 50 ] It has since been widely recognised that the cancellation of the M.52 was a major setback in British progress in the field of supersonic design. [ 34 ]
Another, more successful, programme was the US's Bell X-1 , which also was equipped with a straight wing. According to Miles Chief Aerodynamicist Dennis Bancroft, the Bell Aircraft company was given access to the drawings and research on the M.52. [ 51 ] On 14 October 1947, the Bell X-1 performed the first manned supersonic flight, piloted by Captain Charles "Chuck" Yeager , having been drop launched from the bomb bay of a Boeing B-29 Superfortress and attained a record-breaking speed of Mach 1.06 (700 miles per hour (1,100 km/h; 610 kn)). [ 34 ] The news of a successful straight-wing supersonic aircraft surprised many aeronautical experts on both sides of the Atlantic, as it was increasingly believed that a swept-wing design not only highly beneficial but also necessary to break the sound barrier. [ 48 ]
During the final years of the Second World War, aircraft designer Sir Geoffrey de Havilland commenced development on the de Havilland Comet , which would become the world's first jet airliner. An early design consideration was whether to apply the new swept-wing configuration. [ 52 ] Thus, an experimental aircraft to explore the technology, the de Havilland DH 108 , was developed by the firm in 1944, headed by project engineer John Carver Meadows Frost with a team of 8–10 draughtsmen and engineers. The DH 108 primarily consisted of the pairing of the front fuselage of the de Havilland Vampire to a swept wing and small vertical tail; it was the first British swept wing jet, unofficially known as the "Swallow". [ 53 ] It first flew on 15 May 1946, a mere eight months after the project's go-ahead. Company test pilot and son of the builder, Geoffrey de Havilland Jr ., flew the first of three aircraft and found it extremely fast – fast enough to try for a world speed record. On 12 April 1948, a D.H.108 did set a world's speed record at 973.65 km/h (605 mph), it subsequently became the first jet aircraft to exceed the speed of sound. [ 54 ]
Around this same timeframe, the Air Ministry introduced a program of experimental aircraft to examine the effects of swept wings, as well as the delta wing configuration. [ 55 ] Furthermore, the Royal Air Force (RAF) identified a pair of proposed fighter aircraft equipped with swept wings from Hawker Aircraft and Supermarine , the Hawker Hunter and Supermarine Swift respectively, and successfully pressed for orders to be placed 'off the drawing board' in 1950. [ 56 ] On 7 September 1953, the sole Hunter Mk 3 (the modified first prototype, WB 188 ) flown by Neville Duke broke the world air speed record for jet-powered aircraft, attaining a speed of 727.63 mph (1,171.01 km/h) over Littlehampton , West Sussex . [ 57 ] This world record stood for less than three weeks before being broken on 25 September 1953 by the Hunter's early rival, the Supermarine Swift, being flown by Michael Lithgow. [ 58 ]
In February 1945, NACA engineer Robert T. Jones started looking at highly swept delta wings and V shapes, and discovered the same effects as Busemann. He finished a detailed report on the concept in April, but found his work was heavily criticised by other members of NACA Langley , notably Theodore Theodorsen, who referred to it as "hocus-pocus" and demanded some "real mathematics". [ 40 ] However, Jones had already secured some time for free-flight models under the direction of Robert Gilruth , whose reports were presented at the end of May and showed a fourfold decrease in drag at high speeds. All of this was compiled into a report published on June 21, 1945, which was sent out to the industry three weeks later. [ 59 ] Ironically, by this point Busemann's work had already been passed around.
In May 1945, the American Operation Paperclip reached Braunschweig , where US personnel discovered a number of swept wing models and a mass of technical data from the wind tunnels. One member of the US team was George S. Schairer , who was at that time working at the Boeing company. He immediately forwarded a letter to Ben Cohn at Boeing, communicating the value of the swept wing concept. [ 60 ] [ 61 ] He also told Cohn to distribute the letter to other companies as well, although only Boeing and North American made immediate use of it. [ citation needed ]
Boeing was in the midst of designing the B-47 Stratojet , and the initial Model 424 was a straight-wing design similar to the B-45 , B-46 and B-48 it competed with. Analysis by Boeing engineer Vic Ganzer suggested an optimum sweepback angle of about 35 degrees. [ 62 ] By September 1945, the Braunschweig data had been worked into the design, which re-emerged as the Model 448, a larger six-engine design with more robust wings swept at 35 degrees. [ 40 ] Another re-work moved the engines into strut-mounted pods under the wings due to concerns of the uncontained failure of an internal engine could potentially destroy the aircraft via either fire or vibration. [ 63 ] The resulting B-47 was hailed as the fastest of its class in the world during the late 1940s, [ 64 ] and trounced the straight-winged competition. Boeing's jet-transport formula of swept wings and pylon-mounted engines has since been universally adopted. [ citation needed ]
In fighters, North American Aviation was in the midst of working on a straight-wing jet-powered naval fighter, then known as the FJ-1 ; it was later submitted to the United States Air Force as the XP-86 . [ 65 ] Larry Green, who could read German, studied the Busemann reports and convinced management to allow a redesign starting in August 1945. [ 40 ] [ 66 ] [ 67 ] The performance of the F-86A allowed it to set the first of several official world speed records , attaining 671 miles per hour (1,080 km/h) on 15 September 1948, flown by Major Richard L. Johnson . [ 68 ] With the appearance of the MiG-15, the F-86 was rushed into combat, while straight-wing jets like the Lockheed P-80 Shooting Star and Republic F-84 Thunderjet were quickly relegated to ground attack missions. Some, such as the F-84 and Grumman F-9 Cougar , were later redesigned with swept wings from straight-winged aircraft. [ 69 ] [ 70 ] Later planes, such as the North American F-100 Super Sabre , would be designed with swept wings from the start, though additional innovations such as the afterburner, area-rule and new control surfaces would be necessary to master supersonic flight. [ 71 ] [ 12 ]
The Soviet Union was also quick to investigate the advantages of swept wings on high speed aircraft, when their "captured aviation technology" counterparts to the western Allies spread out across the defeated Third Reich. Artem Mikoyan was asked by the Soviet government's TsAGI aviation research department to develop a test-bed aircraft to research the swept wing idea — the result was the late 1945-flown, unusual MiG-8 Utka pusher canard layout aircraft, with its rearwards-located wings being swept back for this type of research. [ 72 ] The swept wing was applied to the MiG-15 , an early jet-powered fighter, its maximum speed of 1,075 km/h (668 mph) outclassed the straight-winged American jets and piston-engined fighters initially deployed during the Korean War . [ 73 ] The MiG-15 is believed to have been one of the most produced jet aircraft ; in excess of 13,000 would ultimately be manufactured. [ 74 ]
The MiG-15, which could not safely exceed Mach 0.92, served as the basis for the MiG-17 , which was designed to be controllable at higher Mach numbers. [ 75 ] Its wing sweep, 45° near the fuselage ( the same as the F-100 Super Sabre ), changed to 42° for the outboard part of the wing. [ 76 ] A further derivative of the design, designated MiG-19 , featured a relatively thin wing suited to supersonic flight that was designed at TsAGI, the Soviet Central Aerohydrodynamic Institute ; swept back at an angle of 55 degrees, this wing featured a single wing fence on each side. [ 77 ] A specialist high-altitude variant, the Mig-19SV, featured, amongst other changes, an adjustable flap to generate greater lift at higher altitudes, helping to increase the aircraft's ceiling from 17,500 m (57,400 ft) to 18,500 m (60,700 ft). [ 78 ] [ 79 ]
Germany's swept wing research was also obtained by the Swedish aircraft manufacturer SAAB , with the help of ex-Messerschmitt engineers that had fled to Switzerland during late 1945. [ 80 ] [ 81 ] At the time, SAAB saw the need to make aeronautical advances, particularly in the new field of jet propulsion. [ 82 ] The company incorporated both the jet engine and the swept wing to produce the Saab 29 Tunnan fighter; on 1 September 1948, the first prototype conducted its maiden flight, flown by the English test pilot S/L Robert A. 'Bob' Moore, DFC and bar, [ 83 ] Although not well known outside Sweden, the Tunnan was the first Western European fighter to be introduced with such a wing configuration. [ 84 ] [ 85 ] In parallel, SAAB also developed another swept wing aircraft, the Saab 32 Lansen , primarily to serve as Sweden's standard attack aircraft. [ 86 ] Its wing, which had a 10 per cent laminar profile and a 35° sweep, featured triangular fences near the wing roots in order to improve airflow when the aircraft was being flown at a high angle of attack . [ 86 ] [ 87 ] On 25 October 1953, a SAAB 32 Lansen attained a Mach number of at least 1.12 while in a shallow dive, exceeding the sound barrier . [ 87 ]
The successes of aircraft such as the Hawker Hunter, the B-47, and F-86 showed the value of the swept wing research acquired from Germany. Eventually, almost all advanced design efforts for high speed aircraft would incorporate a wing with a swept leading edge, with either a swept wing or delta wing planform . The Boeing B-52, designed in the 1950s, continues in service as a subsonic long-range heavy bomber. [ 88 ] [ 89 ] While the Soviets never matched the performance of the Boeing B-52 Stratofortress with a jet aircraft, the intercontinental range Tupolev Tu-95 turboprop bomber with its near-jet class top speed of 920 km/h, combining swept wings with propeller propulsion, also remains in service today, being the fastest propeller-powered production aircraft. [ 90 ] In Britain, two swept-wing bombers entered service, the Vickers Valiant (1955) [ 91 ] and the Handley Page Victor (1958). [ 92 ]
By the early 1950s, nearly every new fighter had a swept wing. By the 1960s, most civilian jets also adopted swept wings. Most early transonic and supersonic designs such as the MiG-19 and F-100 used long, highly swept wings. Swept wings would reach Mach 2 on the BAC Lightning, and Republic F-105 Thunderchief , built to operate at low level and very high speed primarily for nuclear strike, but with a secondary air-to-air capability. [ 93 ] By the late 1960s, the McDonnell F-4 Phantom II , was used in large numbers by air forces influenced by the United States. Variable geometry wings were employed on the American F-111 , Grumman F-14 Tomcat and Soviet Mikoyan MiG-27 , although the idea would be abandoned for the American SST design. After the 1970s, most newer generation fighters optimized for maneuvering air combat since the USAF F-15 and Soviet Mikoyan MiG-29 have employed relatively short-span fixed wings with relatively large wing area. [ citation needed ] | https://en.wikipedia.org/wiki/Sweep_theory |
Sweet crude oil is a type of petroleum . The New York Mercantile Exchange designates petroleum with less than 0.5% sulfur as sweet . [ 1 ] [ 2 ]
Petroleum containing higher levels of sulfur is called sour crude oil . [ 3 ]
Sweet crude oil contains small amounts of hydrogen sulfide and carbon dioxide . High-quality, low-sulfur crude oil is commonly used for processing into gasoline and is in high demand, particularly in industrialized nations. Light sweet crude oil is the most sought-after version of crude oil as it contains a disproportionately large fraction that is directly processed ( fractionation ) into gasoline ( naphtha ), kerosene , and high-quality diesel ( gas oil ).
The term sweet originates from the fact that a low level of sulfur provides the oil with a relatively sweet taste and pleasant smell, compared to sulfurous oil. Nineteenth-century prospectors would taste and smell small quantities of oil to determine its quality. [ 4 ]
Producers of sweet crude oil include:
The term " price of oil ", as used in the U.S. media, generally means the cost per barrel (42 U.S. gallons) of West Texas Intermediate Crude, to be delivered to Cushing, Oklahoma during the upcoming month. This information is available from NYMEX or the U.S. Energy Information Administration . | https://en.wikipedia.org/wiki/Sweet_crude_oil |
SwellRT was a free and open-source backend-as-a-service and API focused to ease development of apps featuring real-time collaboration . It supported the building of mobile and web apps, and aims to facilitate interoperability and federation .
SwellRT has its origins in the work done within the GRASIA research team at the Universidad Complutense de Madrid , as part of the EU-funded project P2Pvalue (2013–2016), in a team led by Samer Hassan . [ 1 ] In 2014, the developer Pablo Ojanguren took the lead in forking Apache Wave , dropping several components, re-engineering it, and building a "Wave API" to build applications on top. [ 2 ] In 2015, such Wave API became a standalone product named SwellRT. [ 3 ] [ 4 ]
In 2016, several discussions took place within the Apache Wave community, aiming to tackle the stagnation and crisis state of the project. The Apache Software Foundation mentor of Apache Wave, Upayavira, [ 5 ] was concerned on the project stagnation, but framed SwellRT as Wave's potential savior:
Once more Wave is on the brink of retirement. However, this time, an offer has been made of code from SwellRT, which is a fork of Wave itself, and a concall has been scheduled for interested parties to discuss whether it is a go-er. It is my (limited) understanding that many of the complexity issues in the Wave code that have prevented community development have been resolved in SwellRT. [ 6 ]
Eventually, Wave was approved to continue within Apache incubator program, and a copy of SwellRT codebase was placed in the Apache Wave repository in order to grant the Wave community access to it. [ 7 ] In this regard, Intellectual Property of SwellRT was transferred to the Apache Foundation in 2017. [ 8 ]
In both 2016 [ 9 ] [ 10 ] and 2017, [ 11 ] [ 12 ] SwellRT participated in the Google Summer of Code as part of the set of projects from the Berkman Klein Center for Internet and Society at Harvard University . In both years, the contributions were highly relevant. In 2016, SwellRT replaced its XMPP-based federation protocol (inherited from Apache Wave) for the Matrix.org federation protocol. [ 13 ] In 2017, end-to-end encryption was implemented, [ 14 ] [ 15 ] following an innovative approach to encrypt communication in Operational Transformation collaborative documents. [ 16 ]
SwellRT received international recognition within the fields of decentralized technologies [ 17 ] [ 18 ] [ 19 ] [ 20 ] [ 21 ] [ 22 ] [ 23 ] and real-time collaboration. [ 24 ] [ 25 ] [ 26 ] [ 27 ] [ 28 ] [ 29 ] In the Decentralized Web Summit, organized by the Internet Archive in San Francisco, it was selected as one of the current innovative decentralization technologies. [ 30 ] It was also selected by the Redecentralize advocacy group, as one of the redecentralization projects whose founders were interviewed, [ 31 ] [ 32 ] It launched an international contest to develop apps using SwellRT, [ 33 ] which was awarded to free/open source developers in India. [ 34 ] And the project was presented as invited talk in the Center for Research on Computation and Society at Harvard's School of Engineering and Applied Sciences , [ 35 ] [ 36 ] and in several international conferences. [ 37 ] [ 38 ] [ 39 ] [ 40 ] [ 41 ] SwellRT was one of the first adopters of the Contributor Covenant code of conduct. [ 42 ]
The project has not received new commits since 2018. [ 43 ]
SwellRT was a fork from Apache Wave, inheriting some of its architecture and technology stack. However, it grew beyond the limits of Wave, first presenting itself as a web framework and nowadays growing to a backend-as-a-service platform. [ 44 ] [ 45 ] Its technical approach covers the following:
SwellRT provides a programming model based on collaborative objects . A collaborative object is a JSON-like object that can be shared by some users (or groups) that can make changes in real-time. Changes are propagated (and notified) in real-time to any user connected to the object.
A collaborative object can store properties of simple data types (string, integers, etc.) as well as rich-text and references to files or attachments. This approach is suitable to implement any document based collaborative application like text editors or spreadsheets.
Objects and participants are uniquely identified on the Internet, enabling decentralized access from different federated servers.
SwellRT facilitated the development of mobile/web apps, and thus several apps have been built using this technology. Apart from the demos provided by SwellRT, [ 46 ] third-parties developed other demo apps, such as a Q&A site, [ 47 ] an extension to extract keywords, [ 48 ] a collaborative scrollbar, [ 49 ] a political participation Android app, [ 50 ] a Trello-SwellRT connector. [ 51 ] Besides, two fully-fledged apps are currently using SwellRT technology: | https://en.wikipedia.org/wiki/SwellRT |
The swelling capacity of a polymer is the amount of a liquid that can be absorbed by it. This test can done by two methods:
In this method
Some polymers exhibit larger experimentally measured swelling capacity when immersed in pure liquid compared to testing with saturated vapor. This phenomenon is known as the Schroeder's paradox.
This article about polymer science is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Swelling_capacity |
Swelling index may refer to the following material parameters that quantify volume change: | https://en.wikipedia.org/wiki/Swelling_index |
The Swendsen–Wang algorithm is the first non-local or cluster algorithm for Monte Carlo simulation for large systems near criticality . It has been introduced by Robert Swendsen and Jian-Sheng Wang in 1987 at Carnegie Mellon .
The original algorithm was designed for the Ising and Potts models, and it was later generalized to other systems as well, such as the XY model by Wolff algorithm and particles of fluids. The key ingredient was the random cluster model , a representation of the Ising or Potts model through percolation models of connecting bonds, due to Fortuin and Kasteleyn. It has been generalized by Barbu and Zhu [ 1 ] to arbitrary sampling probabilities by viewing it as a Metropolis–Hastings algorithm and computing the acceptance probability of the proposed Monte Carlo move.
The problem of the critical slowing-down affecting local processes is of fundamental importance in the study of second-order phase transitions (like ferromagnetic transition in the Ising model ), as increasing the size of the system in order to reduce finite-size effects has the disadvantage of requiring a far larger number of moves to reach thermal equilibrium. Indeed the correlation time τ {\displaystyle \tau } usually increases as L z {\displaystyle L^{z}} with z ≃ 2 {\displaystyle z\simeq 2} or greater; since, to be accurate, the simulation time must be t ≫ τ {\displaystyle t\gg \tau } , this is a major limitation in the size of the systems that can be studied through local algorithms . SW algorithm was the first to produce unusually small values for the dynamical critical exponents: z = 0.35 {\displaystyle z=0.35} for the 2D Ising model ( z = 2.125 {\displaystyle z=2.125} for standard simulations); z = 0.75 {\displaystyle z=0.75} for the 3D Ising model, as opposed to z = 2.0 {\displaystyle z=2.0} for standard simulations.
The algorithm is non-local in the sense that a single sweep updates a collection of spin variables based on the Fortuin–Kasteleyn representation . The update is done on a "cluster" of spin variables connected by open bond variables that are generated through a percolation process, based on the interaction states of the spins.
Consider a typical ferromagnetic Ising model with only nearest-neighbor interaction.
where J n m > 0 {\displaystyle J_{nm}>0} is the ferromagnetic coupling strength.
This probability distribution has been derived in the following way: the Hamiltonian of the Ising model is
H [ σ ] = ∑ < i , j > − J i , j σ i σ j {\displaystyle H[\sigma ]=\sum \limits _{<i,j>}-J_{i,j}\sigma _{i}\sigma _{j}} ,
and the partition function is
Z = ∑ { σ } e − β H [ σ ] {\displaystyle Z=\sum \limits _{\lbrace \sigma \rbrace }e^{-\beta H[\sigma ]}} .
Consider the interaction between a pair of selected sites n {\displaystyle n} and m {\displaystyle m} and eliminate it from the total Hamiltonian, defining H n m [ σ ] = ∑ < i , j >≠ < n , m > − J i , j σ i σ j . {\displaystyle H_{nm}[\sigma ]=\sum \limits _{<i,j>\neq <n,m>}-J_{i,j}\sigma _{i}\sigma _{j}.}
Define also the restricted sums:
Z n , m s a m e = ∑ { σ } e − β H n m [ σ ] δ σ n , σ m {\displaystyle Z_{n,m}^{same}=\sum \limits _{\lbrace \sigma \rbrace }e^{-\beta H_{nm}[\sigma ]}\delta _{\sigma _{n},\sigma _{m}}} ;
Z n , m d i f f = ∑ { σ } e − β H n m [ σ ] ( 1 − δ σ n , σ m ) . {\displaystyle Z_{n,m}^{diff}=\sum \limits _{\lbrace \sigma \rbrace }e^{-\beta H_{nm}[\sigma ]}\left(1-\delta _{\sigma _{n},\sigma _{m}}\right).}
Z = e β J n m Z n , m s a m e + e − β J n m Z n , m d i f f . {\displaystyle Z=e^{\beta J_{nm}}Z_{n,m}^{same}+e^{-\beta J_{nm}}Z_{n,m}^{diff}.}
Introduce the quantity
Z n m i n d = Z n , m s a m e + Z n , m d i f f {\displaystyle Z_{nm}^{ind}=Z_{n,m}^{same}+Z_{n,m}^{diff}} ;
the partition function can be rewritten as
Z = ( e β J n m − e − β J n m ) Z n , m s a m e + e − β J n m Z n , m i n d . {\displaystyle Z=\left(e^{\beta J_{nm}}-e^{-\beta J_{nm}}\right)Z_{n,m}^{same}+e^{-\beta J_{nm}}Z_{n,m}^{ind}.}
Since the first term contains a restriction on the spin values whereas there is no restriction in the second term, the weighting factors (properly normalized) can be interpreted as probabilities of forming/not forming a link between the sites: P < n , m > l i n k = 1 − e − 2 β J n m . {\displaystyle P_{<n,m>\;link}=1-e^{-2\beta J_{nm}}.} The process can be easily adapted to antiferromagnetic spin systems, as it is sufficient to eliminate Z n , m s a m e {\displaystyle Z_{n,m}^{same}} in favor of Z n , m d i f f {\displaystyle Z_{n,m}^{diff}} (as suggested by the change of sign in the interaction constant).
It can be shown that this algorithm leads to equilibrium configurations. To show this, we interpret the algorithm as a Markov chain , and show that the chain is both ergodic (when used together with other algorithms) and satisfies detailed balance , such that the equilibrium Boltzmann distribution is equal to the stationary distribution of the chain.
Ergodicity means that it is possible to transit from any initial state to any final state with a finite number of updates. It has been shown that the SW algorithm is not ergodic in general (in the thermodynamic limit). [ 2 ] Thus in practice, the SW algorithm is usually used in conjunction with single spin-flip algorithms such as the Metropolis–Hastings algorithm to achieve ergodicity.
The SW algorithm does however satisfy detailed-balance. To show this, we note that every transition between two Ising spin states must pass through some bond configuration in the percolation representation. Let's fix a particular bond configuration: what matters in comparing the probabilities related to it is the number of factors q = e − 2 β J {\displaystyle q=e^{-2\beta J}} for each missing bond between neighboring spins with the same value; the probability of going to a certain Ising configuration compatible with a given bond configuration is uniform (say p {\displaystyle p} ). So the ratio of the transition probabilities of going from one state to another is
P { σ } → { σ ′ } P { σ ′ } → { σ } = P r ( { σ ′ } | B . C . ) P r ( B . C . | { σ } ) P r ( { σ } | B . C . ) P r ( B . C . | { σ ′ } ) = p ⋅ exp [ − 2 β ∑ < l , m > δ σ l , σ m J l m ] p ⋅ exp [ − 2 β ∑ < l , m > δ σ l ′ , σ m ′ J l m ] = e − β Δ E {\displaystyle {\frac {P_{\lbrace \sigma \rbrace \rightarrow \lbrace \sigma '\rbrace }}{P_{\lbrace \sigma '\rbrace \rightarrow \lbrace \sigma \rbrace }}}={\frac {Pr\left(\lbrace \sigma '\rbrace |B.C.\right)Pr\left(B.C.|\lbrace \sigma \rbrace \right)}{Pr\left(\lbrace \sigma \rbrace |B.C.\right)Pr\left(B.C.|\lbrace \sigma '\rbrace \right)}}={\frac {p\cdot \exp \left[-2\beta \sum \limits _{<l,m>}\delta _{\sigma _{l},\sigma _{m}}J_{lm}\right]}{p\cdot \exp \left[-2\beta \sum \limits _{<l,m>}\delta _{\sigma '_{l},\sigma '_{m}}J_{lm}\right]}}=e^{-\beta \Delta E}}
since Δ E = − ∑ < l , m > J l m ( σ l ′ σ m ′ − σ l σ m ) = − ∑ < l , m > J l m [ δ σ l ′ , σ m ′ − ( 1 − δ σ l ′ , σ m ′ ) − δ σ l , σ m + ( 1 − δ σ l , σ m ) ] = − 2 ∑ < l , m > J l m ( δ σ l ′ , σ m ′ − δ σ l , σ m ) {\displaystyle \Delta E=-\sum \limits _{<l,m>}J_{lm}\left(\sigma '_{l}\sigma '_{m}-\sigma _{l}\sigma _{m}\right)=-\sum \limits _{<l,m>}J_{lm}\left[\delta _{\sigma '_{l},\sigma '_{m}}-\left(1-\delta _{\sigma '_{l},\sigma '_{m}}\right)-\delta _{\sigma _{l},\sigma _{m}}+\left(1-\delta _{\sigma _{l},\sigma _{m}}\right)\right]=-2\sum \limits _{<l,m>}J_{lm}\left(\delta _{\sigma '_{l},\sigma '_{m}}-\delta _{\sigma _{l},\sigma _{m}}\right)} .
This is valid for every bond configuration the system can pass through during its evolution, so detailed balance is satisfied for the total transition probability. This proves that the algorithm is correct.
Although not analytically clear from the original paper, the reason why all the values of z obtained with the SW algorithm are much lower than the exact lower bound for single-spin-flip algorithms ( z ≥ γ / ν {\displaystyle z\geq \gamma /\nu } ) is that the correlation length divergence is strictly related to the formation of percolation clusters, which are flipped together. In this way the relaxation time is significantly reduced. Another way to view this is through the correspondence between the spin statistics and cluster statistics in the Edwards-Sokal representation . [ 3 ] Some mathematically rigorous results on the mixing time of this process have been obtained by Guo and Jerrum [1] .
The algorithm is not efficient in simulating frustrated systems , because the correlation length of the clusters is larger than the correlation length of the spin model in the presence of frustrated interactions. [ 4 ] Currently, there are two main approaches to addressing this problem, such that the efficiency of cluster algorithms is extended to frustrated systems.
The first approach is to extend the bond-formation rules to more non-local cells, and the second approach is to generate clusters based on more relevant order parameters. In the first case, we have the KBD algorithm for the fully-frustrated Ising model , where the decision of opening bonds are made on each plaquette, arranged in a checkerboard pattern on the square lattice. [ 5 ] In the second case, we have replica cluster move for low-dimensional spin glasses , where the clusters are generated based on spin overlaps, which is believed to be the relevant order parameter. | https://en.wikipedia.org/wiki/Swendsen–Wang_algorithm |
In organic chemistry , the Swern oxidation , named after Daniel Swern , is a chemical reaction whereby a primary or secondary alcohol ( −OH ) is oxidized to an aldehyde ( −CH=O ) or ketone ( >C=O ) using oxalyl chloride , dimethyl sulfoxide (DMSO) and an organic base , such as triethylamine . [ 1 ] [ 2 ] [ 3 ] It is one of the many oxidation reactions commonly referred to as 'activated DMSO' oxidations. The reaction is known for its mild character and wide tolerance of functional groups . [ 4 ] [ 5 ] [ 6 ] [ 7 ]
The by-products are dimethyl sulfide ((CH 3 ) 2 S), carbon monoxide (CO), carbon dioxide (CO 2 ) and—when triethylamine is used as base— triethylammonium chloride (Et 3 NHCl). Of the volatile by-products, dimethyl sulfide has a strong, pervasive odour and carbon monoxide is acutely toxic, so the reaction and the work-up needs to be performed in a fume hood. Dimethyl sulfide is a volatile liquid (B.P. 37 °C) with an unpleasant odour at even low concentrations. [ 8 ] [ 9 ] [ 10 ]
The first step of the Swern oxidation is the low-temperature reaction of DMSO, 1a , formally as resonance contributor 1b , with oxalyl chloride, 2 . The first intermediate, 3 , quickly decomposes giving off carbon dioxide and carbon monoxide and producing chloro(dimethyl)sulfonium chloride, 4 .
After addition of the alcohol 5 , the chloro(dimethyl)sulfonium chloride 4 reacts with the alcohol to give the key alkoxysulfonium ion intermediate, 6 . The addition of at least 2 equivalents of base — typically triethylamine — will deprotonate the alkoxysulfonium ion to give the sulfur ylide 7 . In a five-membered ring transition state , the sulfur ylide 7 decomposes to give dimethyl sulfide and the desired carbonyl compound 8 .
When using oxalyl chloride as the dehydration agent, the reaction must be kept colder than −60 °C to avoid side reactions. With cyanuric chloride [ 11 ] or trifluoroacetic anhydride [ 12 ] instead of oxalyl chloride, the reaction can be warmed to −30 °C without side reactions. Other methods for the activation of DMSO to initiate the formation of the key intermediate 6 are the use of carbodiimides ( Pfitzner–Moffatt oxidation ), a sulfur trioxide pyridine complex ( Parikh–Doering oxidation ) or acetic anhydride ( Albright-Goldman oxidation ). The intermediate 4 can also be prepared from dimethyl sulfide and N -chlorosuccinimide (the Corey–Kim oxidation ).
In some cases, the use of triethylamine as the base can lead to epimerisation at the carbon alpha to the newly formed carbonyl. Using a bulkier base, such as diisopropylethylamine , can mitigate this side reaction.
Dimethyl sulfide, a byproduct of the Swern oxidation, is one of the most notoriously unpleasant odors known in organic chemistry. Humans can detect this compound in concentrations as low as 0.02 to 0.1 parts per million. [ 13 ] A simple remedy for this problem is to rinse used glassware with bleach or oxone solution, which will oxidize the dimethyl sulfide back to dimethyl sulfoxide or to dimethyl sulfone , both of which are odourless and nontoxic. [ 14 ]
The reaction conditions allow oxidation of acid-sensitive compounds, which might decompose under the acidic oxidation conditions such as Jones oxidation . For example, in Thompson & Heathcock's synthesis of the sesquiterpene isovelleral, [ 15 ] the final step uses the Swern protocol, avoiding rearrangement of the acid-sensitive cyclopropanemethanol moiety. | https://en.wikipedia.org/wiki/Swern_oxidation |
Swift heavy ions are the components of a type of particle beam with high enough energy that electronic stopping dominates over nuclear stopping . [ 1 ] [ 2 ] They are accelerated in particle accelerators to very high energies, typically in the MeV or GeV range and have sufficient energy and mass to penetrate solids on a straight line. In many solids swift heavy ions release sufficient energy to induce permanently modified cylindrical zones, so-called ion tracks . If the irradiation is carried out in an initially crystalline material, ion tracks consist of an amorphous cylinder. [ 1 ] Ion tracks can be produced in many amorphizing materials, but not in pure metals, where the high electronic heat conductivity dissipates away the electronic heating before the ion track has time to form.
Heavy ion beams are generally described in terms of their energy in Mega electron volts (MeV) divided by the mass of the atomic nucleus, written "MeV/u". In order for an ion beam to be considered "swift", the constituent ions should be carbon or heavier, and the energy such that the beam particles have a velocity comparable to the Bohr velocity . [ 3 ]
The mechanisms by which ion tracks are produced are subject to some debate. They can be considered to produce thermal spikes [ 4 ] [ 5 ] in the sense that they lead to strong lattice heating and a transient disordered atom zone. However, at least the initial stage of the damage might be better understood in terms of a Coulomb explosion mechanism. [ 6 ] Regardless of what the heating mechanism is, it is well established that swift heavy ions typically produce a long nearly cylindrical track of damage in insulators, [ 1 ] [ 4 ] which has been shown to be underdense in the middle, at least in SiO 2 . [ 7 ] [ 8 ]
Swift heavy ion tracks have several established and potential practical applications. Ion tracks in polymers can be etched to form a nanometer-thin channel through a polymer foil, so called track etch membranes . These are in industrial use. [ 9 ]
Irradiation of polyimide resists have potential to be used as templates for nanowire growth. [ 10 ] Tracks can also be used to sputter materials. [ 11 ] [ 12 ] They can also be used to elongate nanocrystals embedded in materials. [ 13 ] [ 14 ] [ 15 ] SHI irradiation can also be used for structural modification of nanomaterials. [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/Swift_heavy_ion |
Swiftships is a shipbuilding and marine engineering company headquartered in South Louisiana, USA . [ 1 ] Company operates globally [ 2 ] and specialized in the construction of small to medium sized vessels made of steel , aluminum or fiberglass . [ 1 ] [ 3 ] Swiftships is involved in ship design, construction, repair and maintenance activities.
Founded by Fred Sewart in 1942, Swiftships began as Sewart Machine Works and then as Sewart Seacraft [ 4 ] in 1946. Company became a supplier of “ Swift Boats ” to the US Navy during the Vietnam War (Swiftships delivered 193 Fast Patrol Crafts to the US Navy throughout the conflict). [ 2 ] The mission objective of the Swift Boat was to provide the Navy with a fast boat that could patrol the river shores for enemy soldiers. [ 5 ]
In 1969 the company was renamed as Swiftships. [ 6 ] [ 1 ]
Since 2004 and for the next years, Swiftships built ships for the oil and gas industry of the Gulf of Mexico [ 7 ] [ 8 ] and restored vessels for the Dominican Republic . [ 9 ]
Company has created its first fully unmanned surface vehicle in 2015, called Anaconda (AN-1), and later the Anaconda (AN-2), for which Swiftships teamed with the University of Louisiana at Lafayette [ 10 ] and augmented technology developers. [ 11 ] [ 12 ]
Since 1942 Swiftships has designed and built over 600 naval vessels and commercial platforms. [ 3 ]
In 2008 the company signed a contract with the Egyptian Navy , initiating a co-production program, building vessels in-country. [ 13 ] The partnership includes a yard in Alexandria, where the company produces patrol crafts . [ 14 ]
In 2009, Swiftships was awarded a contract by the U.S. Navy to provide Follow on Technical Support on behalf of the Iraqi Navy that included the establishment of a Ship Repair Facility in Umm Qasr, Iraq . [ 15 ] [ 16 ] [ 17 ] [ 18 ]
In 2020, Swiftships operates 3 yards in USA and 1 co-production yard (JV) with Egyptian Navy in Alexandria, Egypt :
Ship types include:
Preceded by PT boat - Followed by Patrol Craft Fast and US patrol vessels | https://en.wikipedia.org/wiki/Swiftships |
Swimming pool bacteria are the diverse array of bacteria that are present in aquatic environments , primarily swimming pools , which can have effects on human health and water quality . Recreational waters are known to be source of infectious diseases . [ 1 ]
There are different types of bacteria that are found in swimming pools and other types of recreational waters. The most prevalent of them is Staphylococcus aureus ( S. aureus ) . This bacterium is one of the leading causes of skin infections in the world. Such infections could appear as painful boils and rashes . [ 2 ] It is naturally present in humans on skin, in nasal mucous , and inside the intestinal tract . [ 3 ] S. aureus has a strong resistance to chlorine , which is one of the methods by which pools are often cleaned. [ 4 ]
Other bacteria often found in recreational waters are Enterococci , [ 5 ] which is a genus of bacteria found in feces . Fecal contamination is one of the primary public health concerns in swimming pools. [ 4 ] Fecal contamination usually occurs through excretion by bathers, other animals, or contaminated water sources . [ 4 ]
Researchers have studied the quantity of bacteria in recreational waters. In a study of swimming pools in Alexandria , Egypt , [ 4 ] the team studied 10 pools, both indoor and outdoor, over two months during the summer. [ 4 ] The team found that bacteria seemed to be more prevalent in outdoor pools. [ 4 ] Furthermore, they noted that the higher the pH of the pool, the more bacteria were present in the water. 20.2% of the bacteria in the pools were found to be S. aureus . [ 4 ]
Another study reported two experiments involving a large pool and a small pool. [ 5 ] Using 10 volunteers, the team examined how many bacteria could be found in the watershed by the bathers and how many could be found in the water after the bathers were exposed to sand . [ 5 ] The study concluded that bathers shed both S. aureu s and enterococci into the water, and S. aureus was shed the most. [ 5 ] After each cycle, the number of bacteria the bathers shed decreased. [ 5 ]
Enterotoxic Escherichia coli has been found in pools with sub-optimal chlorine levels. [ 1 ]
There are several diseases caused by S. aureus and enterococci . S. aureus has been found to cause sepsis and pneumonia , among other problems, [ 6 ] while Enterococci has been found to cause sepsis and urinary tract infections , [ 7 ] as well as being resistant to antibiotics . [ 8 ] There are various actions taken to prevent swimmers from falling ill. The swimming facilities must ensure that their filtration is working and that their staff are trained and know the appropriate behavior and procedures of the facilities. [ 9 ] Individual swimmers must also take preventative measures. As stated by the Centers for Disease Control and Prevention (CDC), guests must not swim if they have diarrhea , swimmers should not swallow pool water, swimmers should wash themselves before entering the pool, and if the restroom is used, each guest must thoroughly wash themselves. [ 9 ]
In Europe, the cleanliness of pools is monitored by measuring the levels Escherichia coli, enterococci and Pseudomonas aeruginosa. Staphylococcus aureus levels are not monitored despite the detection of the bacteria in recreational waters (and on beaches). [ 10 ] The authors of a 2023 study recommended levels of bacteria of 0 CFU /100 mL for water of excellent quality, less than 20 CFU/100 mL for water of very good quality, less than 50 CFU/100 mL for good quality water, and more than 50 CFU/100 mL for poor quality water. [ 10 ] | https://en.wikipedia.org/wiki/Swimming_pool_bacteria |
Swimming pool sanitation is the process of ensuring healthy conditions in swimming pools . Proper sanitation is needed to maintain the visual clarity of water and to prevent the transmission of infectious waterborne diseases .
Two distinct and separate methods are employed in the sanitation of a swimming pool. The filtration system removes organic waste on a daily basis by using the sieve baskets inside the skimmer and circulation pump and the sand unit with a backwash facility for easy removal of organic waste from the water circulation. Disinfection - normally in the form of hypochlorous acid (HClO) - kills infectious microorganisms . Alongside these two distinct measures within the pool owner's jurisdiction, swimmer hygiene and cleanliness helps reduce organic waste build-up.
The World Health Organization has published international guidelines for the safety of swimming pools and similar recreational-water environments, including standards for minimizing microbial and chemical hazards. [ 1 ] The United States Centers for Disease Control and Prevention also provides information on pool sanitation and water related illnesses for health professionals and the public. [ 2 ] The main organizations providing certifications for pool and spa operators and technicians are the National Swimming Pool Foundation and Association of Pool & Spa Professionals. The certifications are accepted by many state and local health departments. [ 3 ]
Swimming pool contaminants are introduced from environmental sources and swimmers. Affecting primarily outdoor swimming pools, environmental contaminants include windblown dirt and debris, incoming water from unsanitary sources, rain containing microscopic algae spores and droppings from birds possibly harboring disease-causing pathogens. [ 4 ] Indoor pools are less susceptible to environmental contaminants.
Contaminants introduced by swimmers can dramatically influence the operation of indoor and outdoor swimming pools. Contaminants include micro-organisms from infected swimmers and body oils including sweat , cosmetics , suntan lotion , urine , saliva and fecal matter ; for example, it was estimated by researchers that swimming pools contain, on average, 30 to 80 mL of urine for each person that uses the pool. [ 5 ] In addition, the interaction between disinfectants and pool water contaminants can produce a mixture of chloramines and other disinfection by-products . The journal Environmental Science & Technology reported that sweat and urine react with chlorine and produce trichloramine and cyanogen chloride , two chemicals dangerous to human health. An answer to the perennial question: Is it safe to pee in the pool? Nitrosamines are another type of the disinfection by-products that are of concern as a potential health hazard. [ 6 ]
Acesulfame potassium is widely used in the human diet and excreted by the kidneys. It has been used by researchers as a marker to estimate the degree to which swimming pools are contaminated by urine. [ 6 ] It was estimated that a commercial-size swimming pool of 220,000 gallons would contain about 20 gallons of urine, equivalent to about 2 gallons of urine in a typical residential pool. [ 6 ]
Pathogenic contaminants are of greatest concern in swimming pools as they have been associated with numerous recreational water illnesses (RWIs). [ 7 ] Public health pathogens can be present in swimming pools as viruses, bacteria, protozoa and fungi . Diarrhea is the most commonly reported illness associated with pathogenic contaminants, while other diseases associated with untreated pools are Cryptosporidiosis and Giardiasis . [ 8 ] [ 9 ] Other illnesses commonly occurring in poorly maintained swimming pools include otitis externa , commonly called swimmers ear, skin rashes and respiratory infections.
Contamination can be minimized by good swimmer hygiene practices such as showering before and after swimming, and not letting children with intestinal disorders swim. Effective treatments are needed to address contaminants in pool water because preventing the introduction of pool contaminants, pathogenic and non-pathogenic, into swimming pools is, in practice, impossible.
A well-maintained, properly operating pool filtration and re-circulation system is the first barrier, combating the contaminants large enough to be filtered. Rapid removal of these filterable contaminants reduces the impact on the disinfection system thereby limiting the formation of chloramines , restricting the formation of disinfection by-products and optimizing sanitation effectiveness. To kill pathogens and help prevent recreational water illnesses, pool operators must maintain proper levels of chlorine or another sanitizer. [ 10 ] [ 11 ]
Over time, calcium from municipal water tends to accumulate, developing salt deposits in the swimming pool walls and equipment (filters, pumps), reducing their effectiveness. Therefore, it is advised to either completely drain the pool, and refill it with fresh water, or recycle the existing pool water, using reverse osmosis . The advantage of the latter method is that 90% of the water can be reused.
Pool operators must also store and handle cleaning and sanitation chemicals safely.
Disease prevention should be the top priority for every water quality management program for pool and spa operators. Disinfection is critical to protect against pathogens, and is best managed through routine monitoring and maintenance of chemical feed equipment to ensure optimum chemical levels in accordance with state and local regulations. [ 12 ]
Chemical parameters include disinfectant levels according to regulated pesticide label directions. pH should be kept between 7.2 and 7.8. Human tears have a pH of 7.4, making this an ideal point to set a pool. [ 13 ] More often than not, it is improper pH and not the sanitiser that is responsible for irritating swimmers' skin and eyes.
Total alkalinity should be 80–120 ppm and calcium hardness between 200 and 400 ppm. [ 14 ] [ failed verification ]
Good hygienic behavior at swimming pools is also important for reducing health risk factors at swimming pools and spas. Showering before swimming can reduce introduction of contaminants to the pool, and showering again after swimming will help to remove any that may have been picked up by the swimmer.
Those with diarrhea or other gastroenteritis illnesses should not swim within 2 weeks of an outbreak, especially children. Cryptosporidium is chlorine resistant. [ 15 ]
In order to minimize exposure to pathogens, swimmers should avoid getting water into their mouths, and should never swallow pool or spa water. [ 16 ]
Maintaining an effective concentration of disinfectant is critically important in assuring the safety and health of swimming pool and spa users. When any of these pool chemicals are used, it is very important to keep the pH of the pool in the range 7.2 to 7.8 – according to the Langelier Saturation Index, or 7.8 to 8.2 – according to the Hamilton Index; higher pH drastically reduces the sanitizing power of the chlorine due to reduced oxidation-reduction potential (ORP), while lower pH produces more rapid loss of chlorine and causes bather discomfort, especially to the eyes. However, according to the Hamilton Index, a higher pH can reduce unnecessary chlorine consumption while still remaining effective at preventing algae and bacteria growth.
To help ensure the health of bathers and protect pool equipment, it is essential to perform routine monitoring of water quality factors (or "parameters") on a regular basis. This process becomes the essence of an optimum water quality management program.
Conventional halogen-based oxidizers such as chlorine and bromine are convenient and economical primary sanitizers for swimming pools and provide a residual level of sanitizer that remains in the water. Chlorine-releasing compounds are the most popular and frequently used in swimming pools whereas bromine-releasing compounds have found heightened popularity in spas and hot tubs. Both are members of the halogen group with demonstrated ability to destroy and deactivate a wide range of potentially dangerous bacteria and viruses in swimming pools and spas. Both exhibit three essential elements as ideal first-line-of-defense sanitizers for swimming pools and spas: they are fast-acting and enduring, they are effective algaecides, and they oxidize undesired contaminants.
Swimming pools can be disinfected with a variety of chlorine-releasing compounds. The most basic of these compounds is molecular chlorine (Cl 2 ); however, its application is primarily in large commercial public swimming pools. Inorganic forms of chlorine-releasing compounds frequently used in residential and public swimming pools include sodium hypochlorite commonly known as liquid bleach or simply bleach , calcium hypochlorite and lithium hypochlorite . Chlorine residuals from Cl 2 and inorganic chlorine-releasing compounds break down rapidly in sunlight . To extend their disinfectant usefulness and persistence in outdoor settings, swimming pools treated with one or more of the inorganic forms of chlorine-releasing compounds can be supplemented with cyanuric acid – a granular stabilizing agent capable of extending the active chlorine residual half-life (t ½ ) by four to sixfold. [ 17 ]
Chlorinated isocyanurates, a family of organic chlorine-releasing compounds, are stabilized to prevent UV degradation due to the presence of cyanurate as part of their chemical backbone. These are commonly sold for general use in small summer pools, where the water is expected to be used for only a few months and is expected to be regularly topped up with fresh, due to evaporation and splash loss. It is important to change the water frequently, otherwise, levels of cyanuric acid will build up to beyond the point at which the mechanism functions. Excess cyanurates will actually work in reverse and will inhibit the chlorine. A steadily lowering pH value of the water may at first be noticed. Algal growth may become visible, even though chlorine tests show sufficient levels. [ 18 ]
Chlorine reacting with urea in urine and other nitrogen-containing wastes from bathers can produce chloramines . Chloramines typically occur when an insufficient amount of chlorine is used to disinfect a contaminated pool. Chloramines are generally responsible for the noxious, irritating smell prominently occurring in indoor pool settings. A common way to remove chloramines is to "superchlorinate" (commonly called "shocking") the pool with a high dose of inorganic chlorine sufficient to deliver 10 ppm chlorine. Regular superchlorination (every two weeks in summer) helps to eliminate these unpleasant odors in the pool. Levels of chloramines and other volatile compounds in water can be minimized by reducing contaminants that lead to their formation (e.g., urea, creatinine, amino acids and personal care products) as well as by use of non-chlorine "shock oxidizers" such as potassium peroxymonosulfate .
Medium pressure UV technology is used to control the level of chloramines in indoor pools. It is also used as a secondary form of disinfection to address chlorine-tolerant pathogens. A properly sized and maintained UV system should remove the need to shock for chloramines, although shocking would still be used to address a fecal accident in the pool. UV will not replace chlorine but is used to control the level of chloramines, which are responsible for the odor, irritation, and enhanced corrosion at an indoor pool.
Copper ion systems use an electric current across .500 gm bars (solid copper, or a mixture of copper and .100 gm or silver ) to free copper ions into the flow of pool water to kill organisms such as algae in the water and provide a "residual" in the water. Alternative systems also use titanium plates to produce oxygen in the water to help degrade organic compounds .
An electrically operated water pump is the prime motivator in recirculating the water from the pool. Water is forced through a filter and then returned to the pool. Using a water pump by itself is often not sufficient to completely sanitize a pool. Commercial and public pool pumps usually run 24 hours a day for the entire operating season of the pool. Residential pool pumps are typically run for 4 hours per day in winter (when the pool is not in use) and up to 24 hours in summer. To save electricity costs, most pools run water pumps for between 6 hours and 12 hours in summer with the pump being controlled by an electronic timer .
Most pool pumps available today incorporate a small filter basket as the last effort to avoid leaf or hair contamination reaching the close-tolerance impeller section of the pump.
A pressure-fed sand filter is typically placed in line immediately after the water pump. The filter typically contains a medium such as graded sand (called '14/24 Filter Media' in the UK system of grading the size of sand by sifting through a fine brass-wire mesh of 14 to the inch (5.5 per centimeter) to 24 to the inch (9.5 per cm)). A pressure fed sand filter is termed a 'High Rate' sand filter, and will generally filter turbid water of particulates no less than 10 micrometers in size. [ 19 ] The rapid sand filter type are periodically 'back washed' as contaminants reduce water flow and increase back pressure. Indicated by a pressure gauge on the pressure side of the filter reaching into the 'red line' area, the pool owner is alerted to the need to 'backwash' the unit. The sand in the filter will typically last five to seven years before all the "rough edges" are worn off, and the more tightly packed sand no longer works as intended [ citation needed ] . Recommended filtration for public/commercial pools is 1 ton sand per 100,000 liters water (10 ounces avdp. per cubic foot of water) [7.48 US or 6.23 UK gallons].
Introduced in the early 1900s was another type of sand filter – the 'Rapid Sand' filter, whereby water was pumped into the top of a large volume tank (3' 0" or more cube) (1 cubic yard/200US gal/170UK gal/770 liters) containing filter grade sand and returning to the pool through a pipe at the bottom of the tank. As there is no pressure inside this tank, they were also known as "gravity filters". These types of filters are not greatly effective, and are no longer common in home swimming pools, being replaced by the pressure-fed type filter.
Some filters use diatomaceous earth to help filter out contaminants. Commonly referred to as 'D.E.' filters, they exhibit superior filtration capabilities. [ 20 ] Often a D.E. filter will trap waterborne contaminants as small as 1 micrometer in size. D.E. filters are banned in some states, as they must be emptied out periodically and the contaminated media flushed down the sewer, causing a problem in some districts' sewage systems.
As of 2020, several companies now produce regenerative media filters, sometimes called precoat media filters, which use perlite as the filtration media rather than diatomaceous earth. As of 2021, perlite can safely be flushed down the sewer and is approved and NSF listed for use in the United States.
Other filter media that have been introduced to the residential swimming pool market since 1970 include sand particles and paper type cartridge filters of 50 to 150 square feet (4.6 to 13.9 m 2 ) filter area arranged in a tightly packed 12" diameter x 24" long (300 mm x 600 mm) accordion-like circular cartridge. These units can be 'daisy-chained' together to collectively filter almost any size home pool. The cartridges are typically cleaned by removal from the filter body and hosing-off down a sewer connection. They are popular where backwashed water from a sand filter is not allowed to be discharged or go into the aquifer .
Fabric Filters
Traditional pool filters vary in the micron particle sizes that they can capture. Fabric filters can capture particles smaller than that of standard swimming pool filtration systems. This type of filter connects where the water return to the pool after passing through a standard filter. They are usually in the form of a bag. With filtration levels as small as 1 micrometer, users can attain much cleaner water, when using a sand of cartridge filter. These levels are equal or better than that of a diatomaceous earth filter.
Automated pool cleaners more commonly known as "Automatic pool cleaners" and in particular electric, robotic pool cleaners provide an extra measure of filtration, and in fact like the handheld vacuums can microfilter a pool, which a sand filter without flocculation or coagulants is unable to accomplish. [ 21 ]
These cleaners are independent from the pool's main filter and pump system and are powered by a separate electricity source, usually in the form of a set-down transformer that is kept at least 10 feet (3.0 m) from the water in the pool, often on the pool deck. They have two internal motors: one to suck in water through a self-contained filter bag and then return the filtered water at a high speed back into the pool water, and one that is a drive motor connected to tractor-like rubber or synthetic tracks and "brushes" connected by rubber or plastic bands via a metal shaft. The brushes, resembling paint rollers, are located on the front and back of the machine, and help to remove contaminating particles from the pool's floor, walls, and, in some designs, even the pool steps (depending on size and configuration). They also direct the particles into the internal filter bag. [ 22 ] [ 23 ]
Saline chlorination units, electronic oxidation systems, ionization systems, microbe disinfection with ultra-violet lamp systems, and "Tri-Chlor Feeders" are other independent or auxiliary systems for swimming pool sanitation.
A consecutive dilution system is arranged to remove organic waste in stages after it passes through the skimmer. Waste matter is trapped inside one or more sequential skimmer basket sieves, each having a finer mesh to further dilute contaminant size. Dilution here is defined as the action of making something weaker in force, content, or value.
The first basket is placed closely after the skimmer mouth. The second is attached to the circulation pump. Here the 25% of water drawn from the main drain at the bottom of the swimming pool meets the 75% drawn from the surface. The circulation pump sieve basket is easily accessible for service and is to be emptied daily. The third sieve is the sand unit. Here smaller organic waste that has slipped through the previous sieves is trapped by sand.
If not removed regularly, organic waste will continue to rot down and affect water quality. The dilution process allows organic waste to be easily removed. Ultimately the sand sieve can be backwashed to remove smaller trapped organic waste which otherwise leaches ammonia and other compounds into the recirculated water. These additional solutes eventually lead to the formation of disinfection by-products (DBP's). The sieve baskets are easily removed daily for cleaning as is the sand unit, which should be back-washed at least once a week. A perfectly maintained consecutive dilution system drastically reduces the build-up of chloramines and other DBP's. The water returned to the pool should have been cleared of all organic waste above 10 microns in size.
Mineral sanitizers for the swimming pool and spa use minerals , metals , or elements derived from the natural environment to produce water quality benefits that would otherwise be produced by harsh or synthetic chemicals .
Companies are not allowed to sell a mineral sanitizer in the United States unless it has been registered with the United States Environmental Protection Agency (EPA). Currently, two mineral sanitizers are registered with the EPA: one is a silver salt with a controlled release mechanism which is applied to calcium carbonate granules that help neutralize pH; the other uses a colloidal form of silver released into water from ceramic beads. [ 24 ]
Mineral technology takes advantage of the cleansing and filtering qualities of commonly occurring substances. Silver and copper are well-known oligodynamic substances that are effective in destroying pathogens . Silver has been shown to be effective against harmful bacteria , viruses , protozoa and fungi . Copper is widely used as an algicide . [ 25 ] Alumina , derived from aluminates, filters detrimental materials at the molecular level and can be used to control the delivery rate of desirable metals such as copper. Working through the pool or spa filtration system, mineral sanitizers use combinations of these minerals to inhibit algae growth and eliminate contaminants.
Unlike chlorine or bromine , metals and minerals do not evaporate and do not degrade. Minerals can make the water noticeably softer , and by replacing harsh chemicals in the water they lower the potential for red-eye , dry skin and foul odors.
Water is typically drawn from the pool via a rectangular aperture in the wall, connected through to a device fitted into one (or more) wall/s of the pool. The internals of the skimmer are accessed from the pool deck through a circular or rectangle lid, about one foot in diameter. If the pool's water pump is operational water is drawn from the pool over a floating hinged weir (operating from a vertical position to a 90-degree angle away from the pool, in order to stop leaves and debris being back-flooded into the pool by wave action), and down into a removable "skimmer basket", the purpose of which is to entrap leaves, dead insects and other larger floating debris.
The aperture visible from the pool side is typically 1' 0" (300 mm) wide by 6" (150 mm) high, which intersects the water midway through the center of the aperture. Skimmers with apertures wider than this are termed "wide angle" skimmers and may be as much as 2' 0" wide (600 mm). Floating skimmers have the advantage of not being affected by the level of the water as these are adjusted to work with the rate of pump suction and will retain optimum skimming regardless of water level leading to a markedly reduced amount of bio-material in the water. Skimmers should always have a leaf basket or filter between it and the pump to avoid blockages in the pipes leading to the pump and filter.
Prior to the mid 1970s most skimmers were either made of metal like copper or stainless steel either a large round or square shape. Built in concrete pour skimmers were also common on concrete pools before the introduction of PVC Skimmers in the late 1960s
Water returning from the consecutive dilution system is passed through return jets below the surface. These are designed to impact a turbulent flow as the water enters the pool. This flow as a force is far less than the mass of water in the pool and takes the least pressure route upward where eventually surface tension reforms it into a laminar flow on the surface.
As the returned water disturbs the surface, it creates a capillary wave. If the return jets are positioned correctly, this wave creates a circular motion within the surface tension of the water, allowing that on the surface to slowly circulate around the pool walls. Organic waste floating on the surface through this circulation from the capillary wave is slowly drawn past the mouth of the skimmer where it is pulled in due to the laminar flow and surface tension over the skimmer weir. In a well-designed pool, circulation caused by the disturbed returned water aids in removing organic waste from the pool surface, directing it to be trapped inside the consecutive dilution system for easy disposal.
Many return jets are equipped with a swivel nozzle. Used correctly, it induces deeper circulation, further cleaning the water. Turning the jet nozzles at an angle imparts rotation within the entire depth of pool water. Orientation to the left or right would generate clockwise or anti-clockwise rotation respectively. This has the benefit of cleaning the bottom of the pool and slowly moving sunken inorganic debris to the main drain where it is removed by the circulation pump basket sieve.
In a correctly constructed pool, rotation of the water caused by the manner it is returned from the consecutive dilution system will reduce or even waive the need to vacuum the bottom. To gain the maximum rotation force on the main body of water, the consecutive dilution system needs to be as clean and unblocked as possible to allow maximum flow pressure from the pump. As the water rotates, it also disturbs organic waste at lower water layers, forcing it to the top. Rotational force the pool return jets create is the most important part of cleaning the pool water and pushing organic waste across the mouth of the skimmer.
With a correctly designed and operated swimming pool, this circulation is visible and after a period of time, reaches even the deep end, inducing a low-velocity vortex above the main drain due to suction. Correct use of the return jets is the most effective way of removing disinfection by-products caused by deeper decomposing organic waste and drawing it into the consecutive dilution system for immediate disposal.
Another piece of equipment that may be optioned in the recirculation system is a pool water heater. They can be heat pumps , natural gas or propane gas heaters , electric heaters, wood-burning heaters, or Solar hot water panel heaters – increasingly used in the sustainable design of pools.
Diversions to electronic oxidation systems, ionization systems, microbe disinfection with ultra-violet lamp systems, and "Tri-Chlor Feeders" are other auxiliary systems for swimming pool sanitation - as well as solar panels - and are in most cases required to be placed after the filtration equipment, often the last items being placed before the water is returned to the pool.
Features that are part of the water circulation system can extend treatment capacity needs for sizing calculations and can include: artificial streams and waterfalls , in-pool fountains , integrated hot tubs and spas, water slides and sluices, artificial "pebble beaches", submerged seating as bench-ledges or as "stools" at in-pool bars, plunge pools , and shallow children's wading pools. | https://en.wikipedia.org/wiki/Swimming_pool_sanitation |
A power system consists of a number of synchronous machines operating synchronously under all operating conditions. Under normal operating conditions, the relative position of the rotor axis and the resultant magnetic field axis is fixed. The angle between the two is known as the power angle , torque angle , or rotor angle . During any disturbance, the rotor decelerates or accelerates with respect to the synchronously rotating air gap magnetomotive force, creating relative motion. The equation describing the relative motion is known as the swing equation, which is a non-linear second order differential equation that describes the swing of the rotor of synchronous machine. The power exchange between the mechanical rotor and the electrical grid due to the rotor swing (acceleration and deceleration) is called Inertial response .
A synchronous generator is driven by a prime mover. The equation governing the rotor motion is given by: [ 1 ] J d 2 θ m d t 2 = T a = T m − T e , {\displaystyle J{\frac {d^{2}{\theta _{\text{m}}}}{dt^{2}}}=T_{a}=T_{\text{m}}-T_{\text{e}},} where:
Neglecting losses, the difference between the mechanical and electrical torque gives the net accelerating torque T a {\displaystyle T_{a}} . In the steady state, the electrical torque is equal to the mechanical torque and hence the accelerating power is zero. During this period the rotor moves at synchronous speed ω s {\displaystyle \omega _{s}} in rad/s. The electric torque T e {\displaystyle T_{\text{e}}} corresponds to the net air-gap power in the machine and thus accounts for the total output power of the generator plus | I | 2 R {\displaystyle |I|^{2}R} losses in the armature winding. [ 2 ]
The angular position θ {\displaystyle \theta } is measured with a stationary reference frame. Representing it with respect to the synchronously rotating frame gives: θ m = ω s t + δ m , {\displaystyle \theta _{\text{m}}=\omega _{\text{s}}t+\delta _{\text{m}},} where δ m {\displaystyle \delta _{m}} is the angular position with respect to the synchronously rotating reference frame. The derivative of the above equation with respect to time is: d θ m d t = ω s + d δ m d t . {\displaystyle {\frac {d\theta _{\text{m}}}{dt}}=\omega _{\text{s}}+{\frac {d\delta _{\text{m}}}{dt}}.} The above equations show that the rotor angular speed is equal to the synchronous speed only when d δ m / d t {\displaystyle d\delta _{m}/dt} is equal to zero. Therefore, the term d δ m / d t {\displaystyle d\delta _{m}/dt} represents the deviation of the rotor speed from synchronism in rad/s.
By taking the second order derivative of the above equation it becomes: d 2 θ m d t 2 = d 2 δ m d t 2 . {\displaystyle {\frac {d^{2}\theta _{\text{m}}}{dt^{2}}}={\frac {d^{2}\delta _{\text{m}}}{dt^{2}}}.} Substituting the above equation in the equation of rotor motion gives: J d 2 δ m d t 2 = T a = T m − T e . {\displaystyle J{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=T_{a}=T_{\text{m}}-T_{\text{e}}.} Multiplying both sides by the angular velocity of the rotor, given by ω m = d θ m d t , {\displaystyle \omega _{\text{m}}={\frac {d\theta _{\text{m}}}{dt}},} results in J ω m d 2 δ m d t 2 = P a = P m − P e , {\displaystyle J\omega _{\text{m}}{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=P_{a}=P_{\text{m}}-P_{\text{e}},} where P a {\displaystyle P_{a}} , P m {\displaystyle P_{\text{m}}} and P e {\displaystyle P_{\text{e}}} respectively are the accelerating, mechanical and electrical ( active ) power in Watt (W). Intuitivley, the equation can also be derived by taking the time derivative of the rotational energy .
The coefficient J ω m {\displaystyle J\omega _{\text{m}}} is the angular momentum of the rotor at synchronous speed ω s {\displaystyle \omega _{\text{s}}} . In machine data supplied for stability studies this coefficient is often denoted by M {\displaystyle M} and called the inertia constant of the machine. In practice, ω m {\displaystyle \omega _{\text{m}}} does not differ significantly from synchronous speed when the machine is in steady state; allowing for another constant of inertia: H = 1 2 J ω s 2 S rated = 1 2 M ω s S rated = stored kinetic energy in mega joules at synchronous speed machine rating in MVA , {\displaystyle H={\frac {{\frac {1}{2}}J\omega _{\text{s}}^{2}}{S_{\text{rated}}}}={\frac {{\frac {1}{2}}M\omega _{\text{s}}}{S_{\text{rated}}}}={\frac {\text{stored kinetic energy in mega joules at synchronous speed}}{\text{machine rating in MVA}}},} where S rated {\displaystyle S_{\text{rated}}} is the three phase rating of the machine in MVA . Substituting in the above equation 2 H S rated ω s 2 ω m d 2 δ m d t 2 = P m − P e = P a . {\displaystyle 2H{\frac {S_{\text{rated}}}{\omega _{\text{s}}^{2}}}\omega _{\text{m}}{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=P_{\text{m}}-P_{\text{e}}=P_{a}.} Since P m {\displaystyle P_{\text{m}}} , P e {\displaystyle P_{\text{e}}} and P a {\displaystyle P_{a}} in the machine data are given in MW, dividing them by the generator MVA rating gives these quantities in per unit. Dividing the above equation on both sides by S rated {\displaystyle S_{\text{rated}}} gives
2 H ω s d 2 δ d t 2 = P m − P e = P a {\displaystyle {\frac {2H}{\omega _{\text{s}}}}{\frac {d^{2}{\delta }}{dt^{2}}}=P_{\text{m}}-P_{e}=P_{a}} per unit
The above equation describes the behaviour of the rotor dynamics and hence is known as the swing equation . The angle δ {\displaystyle \delta } is the angle of the internal EMF of the generator and it dictates the amount of power that can be transferred. This angle is therefore called the power angle . When considering a network without resistance , the corresponding power angle equation is: [ 3 ] [ 4 ] P e = P m a x sin ( δ ) . {\displaystyle P_{e}=P_{max}\sin(\delta ).} Hence, the swing equation is non-linear and can be solved numerically using, e.g., the fourth-order Runge-Kutta algorithm . When δ {\displaystyle \delta } is small, the equation can be linearized as P e ≈ P m a x δ {\displaystyle P_{e}\approx P_{max}\delta } . [ 5 ] | https://en.wikipedia.org/wiki/Swing_equation |
In algebra , the Swinnerton-Dyer polynomials are a family of polynomials, introduced by Peter Swinnerton-Dyer , that serve as examples where polynomial factorization algorithms have worst-case runtime. They have the property of being reducible modulo every prime, while being irreducible over the rational numbers. They are a standard counterexample in number theory .
Given a finite set P {\displaystyle P} of prime numbers , the Swinnerton-Dyer polynomial associated to P {\displaystyle P} is the polynomial: f P ( x ) = ∏ ( x + ∑ p ∈ P ( ± ) p ) {\displaystyle f_{P}(x)=\prod \left(x+\sum _{p\in P}(\pm ){\sqrt {p}}\right)} where the product extends over all 2 | P | {\displaystyle 2^{|P|}} choices of sign in the enclosed sum. The polynomial f P ( x ) {\displaystyle f_{P}(x)} has degree 2 | P | {\displaystyle 2^{|P|}} and integer coefficients, which alternate in sign. If | P | > 1 {\displaystyle |P|>1} , then f P ( x ) {\displaystyle f_{P}(x)} is reducible modulo p {\displaystyle p} for all primes p {\displaystyle p} , into linear and quadratic factors, but irreducible over Q {\displaystyle \mathbb {Q} } . The Galois group of f P ( x ) {\displaystyle f_{P}(x)} is Z 2 | P | {\displaystyle \mathbb {Z} _{2}^{|P|}} .
The first few Swinnerton-Dyer polynomials are: f { 2 } ( x ) = x 2 − 2 = ( x − 2 ) ( x + 2 ) {\displaystyle f_{\{2\}}(x)=x^{2}-2=(x-{\sqrt {2}})(x+{\sqrt {2}})} f { 2 , 3 } ( x ) = x 4 − 10 x 2 + 1 = ( x − 2 − 3 ) ( x − 2 + 3 ) ( x + 2 − 3 ) ( x + 2 + 3 ) {\displaystyle f_{\{2,3\}}(x)=x^{4}-10x^{2}+1=(x-{\sqrt {2}}-{\sqrt {3}})(x-{\sqrt {2}}+{\sqrt {3}})(x+{\sqrt {2}}-{\sqrt {3}})(x+{\sqrt {2}}+{\sqrt {3}})} f { 2 , 3 , 5 } ( x ) = x 8 − 40 x 6 + 352 x 4 − 960 x 2 + 576. {\displaystyle f_{\{2,3,5\}}(x)=x^{8}-40x^{6}+352x^{4}-960x^{2}+576.} | https://en.wikipedia.org/wiki/Swinnerton-Dyer_polynomial |
Swiss-model (stylized as SWISS-MODEL ) is a structural bioinformatics web-server dedicated to homology modeling of 3D protein structures. [ 1 ] [ 2 ] As of 2024 [update] , homology modeling is the most accurate method to generate reliable three-dimensional protein structure models and is routinely used in many practical applications. Homology (or comparative) modelling methods make use of experimental protein structures (templates) to build models for evolutionary related proteins (targets).
Today, Swiss-model consists of three tightly integrated components: (1) The Swiss-model pipeline – a suite of software tools and databases for automated protein structure modelling, [ 1 ] (2) The Swiss-model Workspace – a web-based graphical user interface workbench, [ 2 ] (3) The Swiss-model Repository – a continuously updated database of homology models for a set of model organism proteomes of high biomedical interest. [ 3 ]
Swiss-model pipeline comprises the four main steps that are involved in building a homology model of a given protein structure:
The Swiss-model Workspace integrates programs and databases required for protein structure prediction and modelling in a web-based workspace. Depending on the complexity of the modelling task, different modes of use can be applied, in which the user has different levels of control over individual modelling steps: automated mode, alignment mode, and project mode. A fully automated mode is used when a sufficiently high sequence identity between target and template (>50%) allows for no human intervention at all. In this case only the sequence or UniProt accession code of the protein is required as input. The alignment mode enables the user to input their own target-template alignments from which the modelling procedure starts (i.e. search for templates step is skipped and rarely only minor changes in the provided alignment are made). The project mode is used in more difficult cases, when manual corrections of target-template alignments are needed to improve the quality of the resulting model. In this mode the input is a project file that can be generated by the DeepView (Swiss Pdb Viewer) visualization and structural analysis tool, [ 4 ] to allow the user to examine and manipulate the target-template alignment in its structural context. In all three cases the output is a pdb file with atom coordinates of the model or a DeepView project file.
The four main steps of homology modelling may be repeated iteratively until a satisfactory model is achieved.
The Swiss-model Workspace is accessible via the ExPASy web server, or it can be used as part of the program DeepView (Swiss Pdb-Viewer). As of September 2015 it has been cited 20000 times in scientific literature, [ 5 ] making it one of the most widely used tools for protein structure modelling. The tool is free for academic use.
The Swiss-model Repository provides access to an up-to-date collection of annotated three-dimensional protein models for a set of model organisms of high general interest. Model organisms include human , [ 6 ] mouse , [ 7 ] C.elegans , [ 8 ] E.coli , [ 9 ] and various pathogens including severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). [ 10 ] Swiss-model Repository is integrated with several external resources, such as UniProt , [ 11 ] InterPro , [ 12 ] STRING , [ 13 ] and Nature Protein Structure Initiative (PSI) SBKB. [ 14 ]
New developments of the Swiss-model expert system feature (1) automated modelling of homo-oligomeric assemblies ; (2) modelling of essential metal ions and biologically relevant ligands in protein structures; (3) local (per-residue) model reliability estimates based on the QMEAN local score function; [ 15 ] (4) mapping of UniProt features to models. (1) and (2) are available when using the automated mode of the Swiss-model Workspace; (3) is always provided when calculating an homology model using the Swiss-model Workspace, and (4) is available in the Swiss-model Repository.
In the past, the accuracy, stability and reliability of the Swiss-model server pipeline was validated by the EVA-CM benchmark project. As of 2024 [update] , the Swiss-model server pipeline is participating in the Continuous Automated Model EvaluatiOn ( CAMEO3D ) project, which continuously evaluates the accuracy and reliability of protein structure prediction services via fully automated means. [ 16 ] | https://en.wikipedia.org/wiki/Swiss-model |
The Swiss cheese model of accident causation is a model used in risk analysis and risk management . It likens human systems to multiple slices of Swiss cheese , which have randomly placed and sized holes in each slice, stacked side by side, in which the risk of a threat becoming a reality is mitigated by the different types of defenses which are "layered" behind each other. Therefore, in theory, lapses and weaknesses in one defense (e.g. a hole in one slice of cheese) do not allow a risk to materialize, since other defenses also exist (e.g. other slices of cheese), to prevent a single point of failure .
The model was originally formally propounded by James T. Reason of the University of Manchester , [ 1 ] and has since gained widespread acceptance. It is sometimes called the "cumulative act effect". Applications include aviation safety , engineering , healthcare , emergency service organizations, and as the principle behind layered security, as used in computer security and defense in depth .
Although the Swiss cheese model is respected and considered a useful method of relating concepts, it has been subject to criticism that it is used too broadly, and without enough other models or support. [ 2 ]
In the Swiss cheese model, an organization's defenses against failure are modeled as a series of imperfect barriers, represented as slices of cheese, specifically Swiss cheese with holes known as " eyes ", such as Emmental cheese . The holes in the slices represent weaknesses in individual parts of the system and are continually varying in size and position across the slices. The system produces failures when holes in each slice momentarily align, permitting (in Reason's words) "a trajectory of accident opportunity", [ 3 ] so that a hazard passes through holes in all of the slices, leading to a failure. [ 4 ] [ 5 ] [ 6 ] [ 7 ]
Frosch [ 8 ] described Reason's model in mathematical terms as a model in percolation theory , which he analyses as a Bethe lattice .
The model includes active and latent failures. Active failures encompass the unsafe acts that can be directly linked to an accident, such as (in the case of aircraft accidents) a navigation error. Latent failures include contributory factors that may lie dormant for days, weeks, or months until they contribute to the accident. Latent failures span the first three domains of failure in Reason's model. [ 9 ]
In the early days of the Swiss cheese model, late 1980 to about 1992, attempts were made to combine two theories: James Reason's multi-layer defence model and Willem Albert Wagenaar's tripod theory of accident causation . This resulted in a period in which the Swiss cheese diagram was represented with the slices of cheese labelled 'active failures', 'preconditions' and 'latent failures'.
These attempts to combine these theories still causes confusion today. A more correct version of the combined theories is shown with the active failures (now called immediate causes ), preconditions and latent failures (now called underlying causes ) shown as the reason each barrier (slice of cheese) has a hole in it, and the slices of cheese as the barriers.
The framework has been applied to a range of areas including aviation safety , various engineering domains, emergency service organizations, and as the principle behind layered security, as used in computer security and defense in depth . [ 11 ]
The model was used in some areas of healthcare . For example, a latent failure could be the similar packaging of two drugs that are then stored close to each other in a pharmacy. This failure would be a contributory factor in the administration of the wrong drug to a patient. Such research led to the realization that medical error can be the result of "system flaws, not character flaws", and that greed, ignorance, malice or laziness are not the only causes of error. [ 12 ]
The Swiss cheese model is nowadays widely used within process safety . Each slice of cheese is usually associated to a safety-critical system , often with the support of bow-tie diagrams . This use has become particularly common when applied to oil and gas drilling and production, both for illustrative purposes and to support other processes, such as asset integrity management and incident investigation . [ 13 ]
Lubnau, Lubnau, and Okray apply the model to the engineering of firefighting systems, aiming to reduce human errors by "inserting additional layers of cheese into the system", namely the techniques of Crew Resource Management . [ 14 ]
Olson and Raz apply the model to improve deception in the methodology of experimental studies, with multiple thin layers of cheese representing subtle components of deception which hide the study hypothesis. [ 15 ] | https://en.wikipedia.org/wiki/Swiss_cheese_model |
Swissmem is the association for Switzerland's mechanical and electrical engineering industries (MEM industries) and related technology-oriented sectors. It represents the interests of the MEM industries in the commercial, political and public spheres, and boosts the competitive capacity of its 1,250 or so member companies with needs-based services. These include training and development courses for employees in the sector, consulting services, networks and a compensation fund. [ 1 ]
Martin Hirzel has been President of Swissmem since 2021. Stefan Brupbacher has been its CEO since 2019.
Swissmem is headquartered in Zurich .
The history of Swissmem began in 1883 with the founding of the Swiss Association of Machinery Manufacturers (VSM). The Association’s goal was: “To safeguard and promote the general interests of the Swiss engineering industry”. Consequently, the engineering industry employers established the Association of Swiss Engineering Employers (ASM) from the ranks of the VSM in 1905. The ASM’s purpose was to safeguard its members’ interests in the area of social policy. Both associations have been operating under the Swissmem name since 1999. [ 2 ] [ 3 ]
In September 2006, the ASM and VSM members voted in favour of continuing integration. In early 2007, the VSM became Swissmem and took on all of the ASM’s activities except those relating to the sector’s collective employment agreement (GAV). [ 4 ] The ASM continues to exist legally as an independent organization and is a contractual partner in the collective employment agreement for the Swiss MEM industries. [ 5 ] The MEM industries’ GAV traditionally blazes a trail for many other Swiss GAVs and has evolved from the “industrial peace agreement” of 1937 to the high-level agreement of today. [ 6 ]
More than 1,250 companies are members of Swissmem. They include ABB , Bucher, Bühler , Geberit , Georg Fischer , Pilatus, Rieter, Schindler , Siemens, Stadler and many more. 85% of all Swissmem members are small and medium-sized enterprises (SMEs).
With around 320,000 employees, including more than 15,000 apprentices, [ 7 ] the MEM industries are among the biggest employers in Switzerland. They generate total annual sales of CHF 79.9 billion (2020). This equates to around 7% (2020) of GDP. The MEM industries thus occupy a key position within the Swiss economy. The sector accounts for almost a third of Switzerland’s total goods exports, with a value of CHF 60.7 billion (2020). [ 8 ]
Swissmem is the voice of the Swiss MEM industries in the commercial, political and public spheres, and campaigns for the sector on relevant matters. The association advocates for good economic policy framework conditions and a liberal labour market, and is committed to a constructive social partnership. [ 9 ]
From basic education, to inspirational seminars, to management training, Swissmem offers practice-oriented training opportunities at a wide variety of levels. [ 10 ] Swissmem’s vocational training school is the centre of excellence for basic professional training for careers as a plant engineer, automation technician, automation fitter, design engineer, electronics technician, multi-skilled mechanic, production mechanic and mechanical practitioner. The centre of excellence supports businesses with a wide range of training offerings for apprentices and professionals. [ 11 ] With the Swissmem Academy, the association maintains its own training centre. Its offering includes courses, seminars and in-house training sessions. In general, it is open to all. Special conditions are available for employees of Swissmem member companies. [ 12 ]
Swissmem members can also access services such as professional advice on employment, commercial, contract and environmental law, energy efficiency, and knowledge and technology transfer. [ 13 ]
Swissmem members are part of a broad industry network. The individual sub-sectors within the MEM industries join together in a total of 28 industry sectors. Each industry sector organizes itself and enjoys a great deal of autonomy within Swissmem. One focus area for the industry sectors is that of sharing experiences and networking. They also collect market-relevant performance indicators and data. Marketing activities are important too, and many industry sectors are also members of European or global umbrella organizations. [ 14 ]
Industry sectors: [ 15 ]
Every year, Swissmem organizes the Swissmem Industry Day. [ 16 ] Over 1,000 decision-makers from industry, business and politics meet to discuss topical issues and make use of networking opportunities. [ 17 ]
In 1937, the ASM and the trade unions concluded the first collective employment agreement, or GAV, for the Swiss MEM industries. Since then, this agreement has formed the core element of the social partnership [ 18 ] and is renegotiated once every five years. The current GAV has been in force since 2018. The contractual partners on the employees’ side are Employees Switzerland, Kaufmännischer Verband Schweiz, SKO, SYNA and Unia. [ 19 ] | https://en.wikipedia.org/wiki/Swissmem |
In electrical engineering , a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit , interrupting the electric current or diverting it from one conductor to another. [ 1 ] [ 2 ] The most common type of switch is an electromechanical device consisting of one or more sets of movable electrical contacts connected to external circuits. When a pair of contacts is touching current can pass between them, while when the contacts are separated no current can flow.
Switches are made in many different configurations; they may have multiple sets of contacts controlled by the same knob or actuator, and the contacts may operate simultaneously, sequentially, or alternately. A switch may be operated manually, for example, a light switch or a keyboard button, or may function as a sensing element to sense the position of a machine part, liquid level, pressure, or temperature, such as a thermostat . Many specialized forms exist, such as the toggle switch , rotary switch , mercury switch , push-button switch, reversing switch , relay , and circuit breaker . A common use is control of lighting, where multiple switches may be wired into one circuit to allow convenient control of light fixtures. Switches in high-powered circuits must have special construction to prevent destructive arcing when they are opened.
The most familiar form of switch is a manually operated electromechanical device with one or more sets of electrical contacts , which are connected to external circuits. Each set of contacts can be in one of two states: either "closed" meaning the contacts are touching and electricity can flow between them, or "open", meaning the contacts are separated and the switch is nonconducting. The mechanism actuating the transition between these two states (open or closed) is usually (there are other types of actions) either an " alternate action " (flip the switch for continuous "on" or "off") or " momentary " (push for "on" and release for "off") type.
A switch may be directly manipulated by a human as a control signal to a system, such as a computer keyboard button, or to control power flow in a circuit, such as a light switch . Automatically operated switches can be used to control the motions of machines, for example, to indicate that a garage door has reached its full open position or that a machine tool is in a position to accept another workpiece. Switches may be operated by process variables such as pressure, temperature, flow, current, voltage, and force, acting as sensors in a process and used to automatically control a system. For example, a thermostat is a temperature-operated switch used to control a heating process. A switch that is operated by another electrical circuit is called a relay . Large switches may be remotely operated by a motor drive mechanism. Some switches are used to isolate electric power from a system, providing a visible point of isolation that can be padlocked if necessary to prevent accidental operation of a machine during maintenance, or to prevent electric shock.
An ideal switch would have no voltage drop when closed, and would have no limits on voltage or current rating. It would have zero rise time and fall time during state changes, and would change state without "bouncing" between on and off positions.
Practical switches fall short of this ideal; as the result of roughness and oxide films, they exhibit contact resistance , limits on the current and voltage they can handle, finite switching time, etc. The ideal switch is often used in circuit analysis as it greatly simplifies the system of equations to be solved, but this can lead to a less accurate solution. Theoretical treatment of the effects of non-ideal properties is required in the design of large networks of switches, as for example used in telephone exchanges.
In the simplest case, a switch has two conductive pieces, often metal , called contacts , connected to an external circuit, that touch to complete (make) the circuit, and separate to open (break) the circuit. The contact material is chosen for its resistance to corrosion, because most metals form insulating oxides that would prevent the switch from working. Contact materials are also chosen on the basis of electrical conductivity , hardness (resistance to abrasive wear), mechanical strength, low cost and low toxicity. The formation of oxide layers at contact surface, as well as surface roughness and contact pressure, determine the contact resistance , and wetting current of a mechanical switch. Sometimes the contacts are plated with noble metals , for their excellent conductivity and resistance to corrosion. They may be designed to wipe against each other to clean off any contamination. Nonmetallic conductors, such as conductive plastic, are sometimes used. To prevent the formation of insulating oxides, a minimum wetting current may be specified for a given switch design.
In electronics, switches are classified according to the arrangement of their contacts. A pair of contacts is said to be " closed " when current can flow from one to the other. When the contacts are separated by an insulating air gap , they are said to be " open ", and no current can flow between them at normal voltages. The terms " make " for closure of contacts and " break " for opening of contacts are also widely used.
The terms pole and throw are also used to describe switch contact variations. The number of " poles " is the number of electrically separate switches which are controlled by a single physical actuator. For example, a " 2-pole " switch has two separate, parallel sets of contacts that open and close in unison via the same mechanism. The number of " throws " is the number of separate wiring path choices other than "open" that the switch can adopt for each pole. A single-throw switch has one pair of contacts that can either be closed or open. A double-throw switch has a contact that can be connected to either of two other contacts, a triple-throw has a contact which can be connected to one of three other contacts, etc. [ 3 ]
In a switch where the contacts remain in one state unless actuated, such as a push-button switch, the contacts can either be normally open (abbreviated " n.o. " or " no ") until closed by operation of the switch, or normally closed (" n.c. " or " nc ") [ nb 1 ] and opened by the switch action. A switch with both types of contact is called a changeover switch or double-throw switch . These may be " make-before-break " (" MBB " or shorting) which momentarily connects both circuits, or may be " break-before-make " (" BBM " or non-shorting) which interrupts one circuit before closing the other.
These terms have given rise to abbreviations for the types of switch which are used in the electronics industry such as " single-pole, single-throw " (SPST) (the simplest type, "on or off") or " single-pole, double-throw " (SPDT), connecting either of two terminals to the common terminal. In electrical power wiring (i.e., house and building wiring by electricians ), names generally involve the suffix "-way" ; however, these terms differ between British English and American English (i.e., the terms two way and three way are used with different meanings).
Form A [ 4 ]
Switches with larger numbers of poles or throws can be described by replacing the "S" or "D" with a number (e.g. 3PST, SP4T, etc.) or in some cases the letter "T" (for "triple") or "Q" (for "quadruple"). In the rest of this article the terms SPST , SPDT and intermediate will be used to avoid the ambiguity.
Contact bounce (also called chatter ) is a common problem with mechanical switches, relays and battery contacts , which arises as the result of electrical contact resistance (ECR) phenomena at interfaces. Switch and relay contacts are usually made of springy metals. When the contacts strike together, their momentum and elasticity act together to cause them to bounce apart one or more times before making steady contact. The result is a rapidly pulsed electric current instead of a clean transition from zero to full current. The effect is usually unimportant in power circuits, but causes problems in some analogue and logic circuits that respond fast enough to misinterpret the on‑off pulses as a data stream. [ 5 ] In the design of micro-contacts, controlling surface structure ( surface roughness ) and minimizing the formation of passivated layers on metallic surfaces are instrumental in inhibiting chatter.
In the Hammond organ , multiple wires are pressed together under the piano keys of the manuals. Their bouncing and non-synchronous closing of the switches is known as Hammond Click and compositions exist that use and emphasize this feature. Some electronic organs have a switchable replica of this sound effect. [ 6 ]
The effects of contact bounce can be eliminated by:
All of these methods are referred to as 'debouncing'.
When the power being switched is sufficiently large, the electron flow across opening switch contacts is sufficient to ionize the air molecules across the tiny gap between the contacts as the switch is opened, forming a gas plasma , also known as an electric arc . The plasma is of low resistance and is able to sustain power flow, even with the separation distance between the switch contacts steadily increasing. The plasma is also very hot and is capable of eroding the metal surfaces of the switch contacts (the same true for vacuum switches). Electric current arcing causes significant degradation of the contacts and also significant electromagnetic interference (EMI), requiring the use of arc suppression methods. [ 7 ]
Where the voltage is sufficiently high, an arc can also form as the switch is closed and the contacts approach. If the voltage potential is sufficient to exceed the breakdown voltage of the air separating the contacts, an arc forms which is sustained until the switch closes completely and the switch surfaces make contact.
In either case, the standard method for minimizing arc formation and preventing contact damage is to use a fast-moving switch mechanism, typically using a spring-operated tipping-point mechanism to assure quick motion of switch contacts, regardless of the speed at which the switch control is operated by the user. Movement of the switch control lever applies tension to a spring until a tipping point is reached, and the contacts suddenly snap open or closed as the spring tension is released.
As the power being switched increases, other methods are used to minimize or prevent arc formation. A plasma is hot and will rise due to convection air currents. The arc can be quenched with a series of non-conductive blades spanning the distance between switch contacts, and as the arc rises, its length increases as it forms ridges rising into the spaces between the blades, until the arc is too long to stay sustained and is extinguished. A puffer may be used to blow a sudden high velocity burst of gas across the switch contacts, which rapidly extends the length of the arc to extinguish it quickly.
Extremely large switches often have switch contacts surrounded by something other than air to more rapidly extinguish the arc. For example, the switch contacts may operate in a vacuum, immersed in mineral oil , or in sulfur hexafluoride .
In AC power service, the current periodically passes through zero; this effect makes it harder to sustain an arc on opening. Manufacturers may rate switches with lower voltage or current rating when used in DC circuits.
When a switch is designed to switch significant power, the transitional state of the switch as well as the ability to withstand continuous operating currents must be considered. When a switch is in the on state, its resistance is near zero and very little power is dropped in the contacts; when a switch is in the off state, its resistance is extremely high and even less power is dropped in the contacts. However, when the switch is flicked, the resistance must pass through a state where a quarter of the load's rated power [ citation needed ] (or worse if the load is not purely resistive) is briefly dropped in the switch.
For this reason, power switches intended to interrupt a load current have spring mechanisms to make sure the transition between on and off is as short as possible regardless of the speed at which the user moves the rocker.
Power switches usually come in two types. A momentary on‑off switch (such as on a laser pointer ) usually takes the form of a button and only closes the circuit when the button is depressed. A regular on‑off switch (such as on a flashlight ) has a constant on-off feature. Dual-action switches incorporate both of these features.
When a strongly inductive load such as an electric motor is switched off, the current cannot drop instantaneously to zero; a spark will jump across the opening contacts. Switches for inductive loads must be rated to handle these cases. The spark will cause electromagnetic interference if not suppressed; a snubber network of a resistor and capacitor in series will quell the spark. [ 8 ]
When turned on, an incandescent lamp draws a large inrush current of about ten times the steady-state current; as the filament heats up, its resistance rises and the current decreases to a steady-state value. A switch designed for an incandescent lamp load can withstand this inrush current. [ 9 ]
Wetting current is the minimum current needing to flow through a mechanical switch while it is operated to break through any film of oxidation that may have been deposited on the switch contacts. [ 10 ] The film of oxidation occurs often in areas with high humidity . Providing a sufficient amount of wetting current is a crucial step in designing systems that use delicate switches with small contact pressure as sensor inputs. Failing to do this might result in switches remaining electrically "open" due to contact oxidation.
The moving part that applies the operating force to the contacts is called the actuator , and may be a toggle or dolly , a rocker , a push-button or any type of mechanical linkage (see photo).
A switch normally maintains its set position once operated. A biased switch contains a mechanism that springs it into another position when released by an operator. The momentary push-button switch is a type of biased switch. The most common type is a "push-to-make" (or normally-open or NO) switch, which makes contact when the button is pressed and breaks when the button is released. Each key of a computer keyboard, for example, is a normally-open "push-to-make" switch. A "push-to-break" (or normally-closed or NC) switch, on the other hand, breaks contact when the button is pressed and makes contact when it is released. An example of a push-to-break switch is a button used to release a door held closed by an electromagnet . The interior lamp of a household refrigerator is controlled by a switch that is held open when the door is closed.
A rotary switch operates with a twisting motion of the operating handle with at least two positions. One or more positions of the switch may be momentary (biased with a spring), requiring the operator to hold the switch in the position. Other positions may have a detent to hold the position when released. A rotary switch may have multiple levels or "decks" in order to allow it to control multiple circuits.
One form of rotary switch consists of a spindle or "rotor" that has a contact arm or "spoke" which projects from its surface like a cam. It has an array of terminals, arranged in a circle around the rotor, each of which serves as a contact for the "spoke" through which any one of a number of different electrical circuits can be connected to the rotor. The switch is layered to allow the use of multiple poles, each layer is equivalent to one pole. Usually such a switch has a detent mechanism so it "clicks" from one active position to another rather than stalls in an intermediate position. Thus a rotary switch provides greater pole and throw capabilities than simpler switches do.
Other types use a cam mechanism to operate multiple independent sets of contacts.
Rotary switches were used as channel selectors on television receivers until the early 1970s, as range selectors on electrical metering equipment, as band selectors on multi-band radios and other similar purposes. In industry, rotary switches are used for control of measuring instruments, switchgear , or in control circuits. For example, a radio controlled overhead crane may have a large multi-circuit rotary switch to transfer hard-wired control signals from the local manual controls in the cab to the outputs of the remote control receiver.
A toggle switch or tumbler switch is a class of electrical switches that are manually actuated by a mechanical lever , handle, or rocking mechanism.
Toggle switches are available in many different styles and sizes, and are used in numerous applications. Many are designed to provide the simultaneous actuation of multiple sets of electrical contacts , or the control of large amounts of electric current or mains voltages.
The word "toggle" is a reference to a kind of mechanism or joint consisting of two arms, which are almost in line with each other, connected with an elbow-like pivot. However, the phrase "toggle switch" is applied to a switch with a short handle and a positive snap-action, whether it actually contains a toggle mechanism or not. Similarly, a switch where a definitive click is heard, is called a "positive on-off switch". [ 11 ] A very common use of this type of switch is to switch lights or other electrical equipment on or off. Multiple toggle switches may be mechanically interlocked to prevent forbidden combinations.
In some contexts, particularly computing , a toggle switch, or the action of toggling, is understood in the different sense of a mechanical or software switch that alternates between two states each time it is activated, regardless of mechanical construction. For example, the caps lock key on a computer causes all letters to be generated in capitals after it is pressed once; pressing it again reverts to lower-case letters.
Switches can be designed to respond to any type of mechanical stimulus: for example, vibration (the trembler switch), tilt, air pressure, fluid level (a float switch ), the turning of a key ( key switch ), linear or rotary movement (a limit switch or microswitch ), or presence of a magnetic field (the reed switch ). Many switches are operated automatically by changes in some environmental condition or by motion of machinery. A limit switch is used, for example, in machine tools to interlock operation with the proper position of tools. In heating or cooling systems a sail switch ensures that air flow is adequate in a duct. Pressure switches respond to fluid pressure.
The mercury switch consists of a drop of mercury inside a glass bulb with two or more contacts. The two contacts pass through the glass, and are connected by the mercury when the bulb is tilted to make the mercury roll on to them.
This type of switch performs much better than the ball tilt switch, as the liquid metal connection is unaffected by dirt, debris and oxidation, it wets the contacts ensuring a very low resistance bounce-free connection, and movement and vibration do not produce a poor contact. These types can be used for precision works.
It can also be used where arcing is dangerous (such as in the presence of explosive vapour) as the entire unit is sealed.
Knife switches consist of a flat metal blade, hinged at one end, with an insulating handle for operation, and a fixed contact. When the switch is closed, current flows through the hinged pivot and blade and through the fixed contact. Such switches are usually not enclosed. The knife and contacts are typically formed of copper , steel , or brass , depending on the application. Fixed contacts may be backed up with a spring. Several parallel blades can be operated at the same time by one handle. The parts may be mounted on an insulating base with terminals for wiring, or may be directly bolted to an insulated switch board in a large assembly. Since the electrical contacts are exposed, the switch is used only where people cannot accidentally come in contact with the switch or where the voltage is so low as to not present a hazard.
Knife switches are made in many sizes from miniature switches to large devices used to carry thousands of amperes. In electrical transmission and distribution, gang-operated switches are used in circuits up to the highest voltages.
The disadvantages of the knife switch are the slow opening speed and the proximity of the operator to exposed live parts. Metal-enclosed safety disconnect switches are used for isolation of circuits in industrial power distribution. Sometimes spring-loaded auxiliary blades are fitted which momentarily carry the full current during opening, then quickly part to rapidly extinguish the arc.
A DPDT switch has six connections, but since polarity reversal is a very common usage of DPDT switches, some variations of the DPDT switch are internally wired specifically for polarity reversal. These crossover switches only have four terminals rather than six. Two of the terminals are inputs and two are outputs. When connected to a battery or other DC source, the 4-way switch selects from either normal or reversed polarity. Such switches can also be used as intermediate switches in a multiway switching system for control of lamps by more than two switches.
In building wiring, light switches are installed at convenient locations to control lighting and occasionally other circuits. By use of multiple-pole switches, multiway switching control of a lamp can be obtained from two or more places, such as the ends of a corridor or stairwell. A wireless light switch allows remote control of lamps for convenience; some lamps include a touch switch which electronically controls the lamp if touched anywhere. In public buildings several types of vandal resistant switches are used to prevent unauthorized use.
Slide switches are mechanical switches using a slider that moves (slides) from the open (off) position to the closed (on) position.
The term switch has since spread to a variety of solid state electronics that perform a switching function, but which are controlled electronically by active devices rather than purely mechanically. These are categorized in the article electronic switch . Electromechanical switches (such as the traditional relay , electromechanical crossbar , and Strowger switch ) bridge the categorization. | https://en.wikipedia.org/wiki/Switch |
A switched mesh is a type of wireless mesh network network that uses multiple dedicated radios to communicate between each neighboring node in the mesh via dedicated mesh backhaul links. Nodes in a switched mesh use separate access and backhaul radios. [ 1 ]
Each dedicated mesh link is on a separate channel, ensuring that forwarded traffic does not use any bandwidth from any other link in the mesh. At each mesh point, traffic is "switched" from one channel to the next, giving rise to the name. [ 1 ] As a result, a switched mesh is capable of much higher capacities and transmission rates than a shared mesh and grows in capacity as nodes are added to the mesh. All of the available bandwidth of each separate radio channel is dedicated to the link to the neighboring node, meaning that total available bandwidth is the sum of the bandwidth of each of the links. [ 1 ]
Switched mesh is one of three distinct types of configuration of wireless mesh networking products in the market today: | https://en.wikipedia.org/wiki/Switched_mesh |
A switching loop or bridge loop occurs in computer networks when there is more than one layer 2 path between two endpoints (e.g. multiple connections between two network switches or two ports on the same switch connected to each other). The loop creates broadcast storms as broadcasts and multicasts are forwarded by switches out every port , the switch or switches will repeatedly rebroadcast the broadcast messages flooding the network. [ 1 ] Since the layer-2 header does not include a time to live (TTL) field, if a frame is sent into a looped topology, it can loop forever.
A physical topology that contains switching or bridge loops is attractive for redundancy reasons, yet a switched network must not have loops. The solution is to allow physical loops, but create a loop-free logical topology using link aggregation , shortest path bridging , spanning tree protocol or TRILL on the network switches.
In the case of broadcast packets over a switching loop, the situation may develop into a broadcast storm .
In a very simple example, a switch with three ports A, B, and C has a normal node connected to port A while ports B and C are connected to each other in a loop. All ports have the same link speed and run in full duplex mode. Now, when a broadcast frame enters the switch through port A, this frame is forwarded to all ports but the source port, i.e. ports B and C. Both frames exiting ports B and C traverse the loop in opposite directions and reenter the switch through their counterpart port. The frame received on port B is then forwarded to ports A and C, the frame received on port C to ports A and B. So, the node on port A receives two copies of its own broadcast frame while the other two copies produced by the loop continue to cycle. Likewise, each broadcast frame entering the system continues to cycle through the loop in both directions, rebroadcasting back to the network in each loop, and broadcasts accumulate. Eventually, the accumulated broadcasts exhaust the egress capacity of the links, the switch begins dropping frames, and communication across the switch becomes unreliable or even impossible.
Switching loops can cause misleading entries in a switch's media access control (MAC) database and can cause endless unicast frames to be broadcast throughout the network. A loop can make a switch receive the same broadcast frames on two different ports, and alternatingly associate the sending MAC address with the one or the other port. It may then incorrectly direct traffic for that MAC address to the wrong port, effectively causing this traffic to be lost, and even causing other switches to incorrectly associate the sender's address with a wrong port as well.
In a redundant switched network it is possible for an end device to receive the same frame multiple times. [ citation needed ]
Routing loops are tempered by a time to live (TTL) field in layer-3 packet header; Packets will circulate the routing loop until their TTL value expires. No TTL concept exists at layer 2 and packets in a switching loop will circulate until dropped, e.g. due to resource exhaustion. | https://en.wikipedia.org/wiki/Switching_loop |
In an electric power transmission grid system , switchyard reactors are large inductors installed at substations to help stabilize the power system.
For transmission lines, the space between the overhead line and the ground forms a capacitor parallel to transmission line, which causes an increase in voltage as the distance increases. To offset the capacitive effect of the transmission line and to regulate the voltage and reactive power of the power system, reactors are connected either at line terminals or at the middle, thereby improving the voltage profile of transmission line.
In large systems with many generators connected in parallel, it may be necessary to use a series reactor to prevent excessively large current flow during a short circuit; this protects transmission line conductors and switching apparatus from damage due to high currents and forces produced during a short circuit.
A shunt reactor is connected in parallel with a transmission line or other load. A series reactor is connected between a load and source.
A bus reactor is an air core inductor , or oil filled inductor, connected between two buses or two sections of the same bus to limit the voltage transients on either bus. It is installed in a bus to maintain system voltage when the load of the bus changes. It adds inductance to the system to offset the capacitance of the line.
A line reactor is placed in line at the point of use or just after a transformer to maintain a stable amperage to the user. When a line is disconnected from the system, the line reactor is also disconnected from the system. Line reactors are often used to compensate line capacitance, mitigate voltage transients due to switching, and to limit fault currents, especially in case of underground transmission lines.
A bus reactor and a line reactor are interchangeable as long as they are rated for the same voltage which is dependent upon substation's physical layout, and bus configuration.
Shunt reactors are used in power systems to counteract the effect of the line parasitic capacitance , thereby stabilizing the system voltage within acceptable limits. [ 1 ] The utility of shunt reactors for voltage control on lightly-loaded transmission lines was examined in a 1926 paper presented at the AIEE by Edith Clarke . [ 2 ] For short lines, we can basically ignore the impact of capacitive current from a voltage regulation point of view, but medium and long lines can have voltages at their receiving end much higher than the sending end, thus creating issues such as over-fluxing of power transformers and over stressing of line insulators. Under light-load conditions, the line produces more VARs , resulting in receiving end voltage being higher than sending end voltage. In order to consume the excess VARs when system is lightly loaded, an inductor is added to the system.
A controlled shunt reactor (CSR) is a variable inductance, smoothly regulated by magnetic biasing of ferromagnetic elements of magnetic circuit. The magnetic system of a CSR single phase consists of two cores. Each core is equipped with control and power windings. In case of regulated DC voltage source connection to the control windings, biasing flow is increasing and directed to different sides in the adjacent cores. This resulted in saturation of CSR cores at relevant half-period of the current. Core saturation is resulted in initiation and increase of the current in the power winding due to non-linear characteristics of the magnetic core. Change in biasing current value leads to the power winding current change, due to which a stepless variation of voltage levels in CSR connection point as well as the value of reactive power consumed by the reactor is ensured.
Series reactors are used as current limiting reactors to increase the impedance of a system. They are also used for neutral earthing. Such reactors are also used to limit the starting currents of synchronous electric motors and to compensate reactive power in order to improve the transmission capacity of power lines. [ 3 ] | https://en.wikipedia.org/wiki/Switchyard_reactor |
A swivel is a connection that allows the connected object, such as a gun , chair , swivel caster , or an anchor rode to rotate horizontally or vertically.
A common design for a swivel is a cylindrical rod that can turn freely within a support structure. The rod is usually prevented from slipping out by a nut, washer or thickening of the rod. The device can be attached to the ends of the rod or the center. Another common design is a sphere that is able to rotate within a support structure. The device is attached to the sphere. A third design is a hollow cylindrical rod that has a rod that is slightly smaller than its inside diameter inside of it. They are prevented from coming apart by flanges. The device may be attached to either end.
A swivel joint for a pipe is often a threaded connection in between which at least one of the pipes is curved, often at an angle of 45 or 90 degrees. The connection is tightened enough to be water- or air-tight and then tightened further so that it is in the correct position.
Swivels are also used in the nautical sector as an element of the anchor rode and in a boat mooring systems. With yachts, the swivel is most commonly used between the anchor and chain. There is a school of thought that anchor swivels should not be connected to the anchor itself, but should be somewhere in the chain rode. [ 4 ]
The anchor swivel is expected to fulfill two purposes:
The biggest concern about anchor swivels is that they might introduce a weak link to the rode. | https://en.wikipedia.org/wiki/Swivel |
A Swivel is a mechanical device used on a drilling rig that hangs directly under the traveling block and directly above the kelly drive , that provides the ability for the kelly (and subsequently the drill string ) to rotate while allowing the traveling block to remain in a stationary rotational position (yet allow vertical movement up and down the derrick) while simultaneously allowing the introduction of drilling fluid into the drill string .
See Drilling rig (petroleum) for a diagram of a drilling rig.
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/Swivel_(drill_rig) |
Swype was a virtual keyboard for touchscreen smartphones and tablets originally developed by Swype Inc., [ 2 ] founded in 2002, where the user enters words by sliding a finger or stylus from the first letter of a word to its last letter, lifting only between words. [ 3 ] It uses error-correction algorithms and a language model to guess the intended word. It also includes a predictive text system, handwriting and speech recognition support. Swype was first commercially available on the Samsung Omnia II running Windows Mobile, [ 4 ] and was originally pre-loaded on specific devices.
In October 2011, Swype Inc. was acquired by Nuance Communications where the company continued its development and implemented its speech recognition algorithm, Dragon Dictation . [ 5 ]
In February 2018, Nuance announced that it had stopped development on the app and that no further updates will be made to it. [ 6 ] The Android app was pulled from the Play Store. The iOS app was also pulled from the App Store . The trial version of Swype is not visible anymore for users in Play Store except users who have installed the app by accessing it in the installed apps part of the Play Store . Cloud features of the paid version such as "Backup&Sync" no longer function, and Nuance Communications has refused to issue refunds to customers who have purchased the app and can no longer reinstall it.
Swype consists of three major components that contribute to its accuracy and speed: an input path analyzer, word search engine with corresponding database, and a manufacturer customizable interface. [ 3 ]
The creators of Swype predict that users will achieve over 50 words per minute , with the chief technical officer (CTO) and founder Cliff Kushler claiming to have reached 55 words per minute. [ 7 ] [ 8 ] On 22 March 2010, a Swype employee by the name of Franklin Page achieved a new Guinness World Record of 35.54 seconds for the fastest text message on a touchscreen mobile phone using Swype on the Samsung i8000 , [ 9 ] [ 10 ] and reportedly improved on 22 August of the same year to 25.94 using a Samsung Galaxy S. [ 11 ] The Guinness world record text message consists of 160 characters in 25 words and was at that time typed in 25.94 seconds, which corresponds to a speed of nearly 58 words per minute, or 370 characters per minute. However, it has since been bettered by the Fleksy app on an Android phone to 18.19 seconds in 2014. [ 12 ]
As of March 2018 [update] , Swype supports the following languages: [ 13 ]
Swype was listed among Time magazine's 50 Best Android Applications for 2013. [ 14 ]
In February 2018, the Android app was pulled from the Play Store . The iOS app was also pulled from the App Store .
Starting from 2018, users need to use a 3rd party service to download the full version of Swype.
In late February 2018, the full version of Swype was discontinued. The trial version of Swype is hidden from the Play Store and App Store . The Swype website was also discontinued and has become a redirect page to XT9 Smart Input.
In a statement emailed to The Verge , Nuance Communications said it would discontinue support of the Swype keyboard app and instead focus on other products. "The core technology behind Swype will continue to be utilized and improved upon across other Nuance offerings—and integrated into our broader AI-powered solutions—most notably in Android-based keyboard solutions for our automotive customers," the company said. | https://en.wikipedia.org/wiki/Swype |
Sycamore is a transmon superconducting quantum processor created by Google's Artificial Intelligence division. [ 1 ] It has 53 qubits . [ 2 ]
In 2019, Sycamore completed a task in 200 seconds that Google claimed, in a Nature paper, would take a state-of-the-art supercomputer 10,000 years to finish. Thus, Google claimed to have achieved quantum supremacy . To estimate the time that would be taken by a classical supercomputer, Google ran portions of the quantum circuit simulation on the Summit , one of the most powerful classical computers in the world. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] Later, IBM made a counter-argument, claiming that the task would take only 2.5 days on a classical system like Summit. [ 9 ] [ 10 ] If Google's claims are upheld, then it would represent an exponential leap in computing power. [ 11 ] [ 12 ] [ 13 ]
In August 2020, quantum engineers working for Google reported the largest chemical simulation on a quantum computer – a Hartree–Fock approximation with Sycamore paired with a classical computer that analyzed results to provide new parameters for the 12-qubit system. [ 14 ] [ 15 ] [ 16 ]
In April 2021, researchers working with Sycamore reported that they were able to realize the ground state of the toric code , a topologically ordered state, with 31 qubits. They showed long-range entanglement properties of the state by measuring non-zero topological entropy , simulating anyon interferometry and their braiding statistics, and preparing a topological quantum error correcting code with one logical qubit. [ 17 ]
In July 2021, a collaboration consisting of Google and multiple universities reported the observation of a discrete time crystal on the Sycamore processor. The chip of 20 qubits was used to obtain a many-body localization configuration of up and down spins. The configuration was stimulated with a laser to achieve a periodically driven " Floquet " system where all up spins are flipped for down and vice versa in periodic cycles which are multiples of the laser's cycles. No energy was absorbed from the laser so the system remained in a protected eigenstate order . [ 18 ] [ 19 ]
In 2022, the Sycamore processor was used to simulate traversable wormhole dynamics. [ 20 ]
The German Forschungszentrum Jülich cooperated with Google in developing the Sycamore quantum computer, and it will be home to the first universal quantum computer developed in Europe as part of the OpenSuperQ project. [ 21 ] [ 22 ] | https://en.wikipedia.org/wiki/Sycamore_processor |
Syclo, LLC (acquired and currently a part of SAP SE ), was a mobile enterprise application platform (MEAP) [ 1 ] [ 2 ] [ 3 ] [ 4 ] and software provider based in the Chicago suburb of Hoffman Estates, Illinois , offering mobile applications to extend enterprise systems , including packaged software for enterprise resource planning (ERP), enterprise asset management (EAM), and customer relationship management (CRM), to handhelds , smartphones , and mobile computers for technicians and staff performing work away from a central office.
The company’s focus in the mobile middleware market is on business environments across industries, including utilities , oil and gas , and life sciences , that utilize field service management processes [ 5 ] and typically have significantly valued hard assets to service and protect.
Syclo offers the Agentry mobile platform, an extensible framework based on fourth-generation programming language for developing, deploying, and managing a wireless business application architecture. [ 6 ] Agentry also includes mobile device management capabilities embedded into its application platform . Syclo's SMART Mobile Suite of workflow -specific products, a software suite built on the Agentry platform, has been implemented by more than 750 customers in 35 countries and 20 different industries. [ 7 ]
On 10 April 2012, SAP announced its plan to acquire 100% of Syclo equity in a deal whose Financial terms were not disclosed. The transaction is expected to be finalized in the second quarter of 2012.
After his consulting company, Competitive Advantage Systems, developed a computer-based wireless order system for Sprint Nextel field workers, Richard Padula started Syclo in 1995 in Barrington, Illinois. Syclo’s flagship Agentry mobile development platform was introduced a year later, and in 1998 Rush University Medical Center , Chicago’s largest healthcare facility, was the first organization to deploy Syclo's SMART Mobile Suite. [ 8 ]
Syclo was acquired by SAP in 2012. [ 9 ] | https://en.wikipedia.org/wiki/Syclo |
Sydney Brenner CH FRS FMedSci MAE (13 January 1927 – 5 April 2019) [ 13 ] [ 14 ] was a South African biologist . In 2002, he shared the Nobel Prize in Physiology or Medicine with H. Robert Horvitz and Sir John E. Sulston . [ 10 ] Brenner made significant contributions to work on the genetic code , and other areas of molecular biology while working in the Medical Research Council (MRC) Laboratory of Molecular Biology in Cambridge , England. He established the roundworm Caenorhabditis elegans as a model organism for the investigation of developmental biology , [ 11 ] [ 15 ] and founded the Molecular Sciences Institute in Berkeley, California , United States. [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ] [ 21 ] [ 22 ] [ 23 ]
Brenner was born in the town of Germiston in the then Transvaal (today in Gauteng ), South Africa, on 13 January 1927. [ 2 ] His parents, Leah [ 24 ] (née Blecher) and Morris Brenner, were Jewish immigrants . His father, a cobbler, came to South Africa from Lithuania in 1910, and his mother from Riga , Latvia, in 1922. He had one sister, Phyllis. [ 25 ] [ 26 ]
He was educated at Germiston High School [ 2 ] and the University of the Witwatersrand . Having joined the university at the age of 15, it was noted during his second year that he would be too young to qualify for the practice of medicine at the conclusion of his six-year medical course, and he was therefore allowed to complete a Bachelor of Science degree in Anatomy and Physiology . During this time he was taught physical chemistry by Joel Mandelstam , microscopy by Alfred Oettle and neurology by Harold Daitz . He also received an introduction to anthropology and paleontology from Raymond Dart and Robert Broom . The histologist Joseph Gillman and director of research in the Anatomy Department persuaded Brenner to continue towards an honours degree and beyond towards an MSc. Brenner accepted though this would mean he would not graduate from medical school and his bursary would be discontinued. He supported himself during this time by working as a laboratory technician. It was during this time, in 1945, that Brenner would publish his first scientific works. His masters thesis was in the field of cytogenetics and publications during this time in the field Brenner would later call Cell Physiology . [ 25 ]
In 1946 Wilfred Le Gros Clark invited Brenner to his Department of Anatomy in Oxford, during a visit to South Africa. Brenner was persuaded to finish his medical education instead. Brenner returned to medical school where he failed Medicine, nearly failed Surgery and achieved a First Class in Obstetrics and Gynecology. Six months later Brenner had finished repeating Medicine and Surgery and in 1951 received the degrees of Bachelor of Medicine, Bachelor of Surgery (MBBCh). [ 25 ]
Brenner received an 1851 Exhibition Scholarship from the Royal Commission for the Exhibition of 1851 which enabled him to complete a Doctor of Philosophy (DPhil) [ 7 ] degree at the University of Oxford as a postgraduate student of Exeter College, Oxford , supervised by Cyril Hinshelwood . [ 27 ]
Following his DPhil, Brenner did postdoctoral research at the University of California, Berkeley . [ 28 ] He spent the next 20 years at the Laboratory of Molecular Biology [ 29 ] in Cambridge. There, during the 1960s, he contributed to molecular biology, then an emerging field. In 1976 he joined the Salk Institute in California . [ 2 ]
Together with Jack Dunitz , Dorothy Hodgkin , Leslie Orgel , and Beryl M. Oughton , he was one of the first people in April 1953 to see the model of the structure of DNA , constructed by Francis Crick and James Watson ; at the time he and the other scientists were working at the University of Oxford 's Chemistry Department. All were impressed by the new DNA model, especially Brenner, who subsequently worked with Crick in the Cavendish Laboratory at the University of Cambridge and the newly opened Medical Research Council (MRC) Laboratory of Molecular Biology (LMB). According to Beryl Oughton, later Rimmer, they all travelled together in two cars once Dorothy Hodgkin announced to them that they were off to Cambridge to see the model of the structure of DNA. [ 30 ]
Brenner made several seminal contributions to the emerging field of molecular biology in the 1960s (see Phage group ). The first was to prove that all overlapping genetic coding sequences were impossible. This insight separated the coding function from structural constraints as proposed in a clever code by George Gamow . This led Francis Crick to propose the concept of a hypothetical molecule (later identified as transfer RNA or tRNA) that transfer the genetic information from RNA to proteins. Brenner gave the name " adaptor hypothesis " in 1955. [ 31 ] The physical separation between the anticodon and the amino acid on a tRNA is the basis for the unidirectional flow of information in coded biological systems. This is commonly known as the central dogma of molecular biology , i.e. information flows from nucleic acid to protein and never from protein to nucleic acid. Following this adaptor insight, Brenner conceived of the concept of messenger RNA during an April 1960 conversation with Crick and François Jacob , and together with Jacob and Matthew Meselson went on to prove its existence later that summer. [ 32 ] Then, with Crick, Leslie Barnett , and Richard J. Watts-Tobin, Brenner genetically demonstrated the triplet nature of the code of protein translation through the Crick, Brenner, Barnett, Watts-Tobin et al. experiment of 1961 , [ 33 ] which discovered frameshift mutations . Brenner collaborating with Sarabhai, Stretton and Bolle in 1964, using amber mutants defective in the bacteriophage T4D major head protein, showed that the nucleotide sequence of the gene is co-linear with the amino acid sequence of the encoded polypeptide chain. [ 34 ]
Together with the decoding work of Marshall Warren Nirenberg and others, the discovery of the triplet nature of the genetic code was critical to deciphering the code. [ 35 ] Barnett helped set up Sydney Brenner's laboratory in Singapore , many years later. [ 36 ] [ 37 ]
Brenner, with George Pieczenik, [ 38 ] created the first computer matrix analysis of nucleic acids using TRAC, which Brenner continued to use. Crick, Brenner, Klug and Pieczenik returned to their early work on deciphering the genetic code with a pioneering paper on the origin of protein synthesis, where constraints on mRNA and tRNA co-evolved allowing for a five-base interaction with a flip of the anticodon loop, and thereby creating a triplet code translating system without requiring a ribosome . This model requires a partially overlapping code. [ 39 ] The published scientific paper is extremely rare in that its collaborators include three authors who independently became Nobel laureates. [ 40 ]
Brenner then focused on establishing a free-living roundworm Caenorhabditis elegans as a model organism for the investigation of animal development including neural development . He chose this 1-millimeter-long soil roundworm mainly because it is simple, is easy to grow in bulk populations, and turned out to be quite convenient for genetic analysis. One of the key methods for identifying important function genes was the screen for roundworms that had some functional defect, such as being uncoordinated , leading to the identification of new sets of proteins, such as the UNC proteins. For this work, he shared the 2002 Nobel Prize in Physiology or Medicine with H. Robert Horvitz and John Sulston . The title of his Nobel lecture in December 2002, "Nature's Gift to Science", is a homage to this nematode ; in it, he considered that having chosen the right organism turned out to be as important as having addressed the right problems to work on. [ 41 ] In fact, the C. elegans community has grown rapidly in recent decades with researchers working on a wide spectrum of problems. [ 42 ]
Brenner founded the Molecular Sciences Institute in Berkeley, California in 1996. [ 4 ] As of 2015 [update] he was associated with the Salk Institute , the Institute of Molecular and Cell Biology , the Singapore Biomedical Research Council , the Janelia Farm Research Campus , and the Howard Hughes Medical Institute . [ 4 ] In August 2005, Brenner was appointed president of the Okinawa Institute of Science and Technology . [ 43 ] He was also on the Board of Scientific Governors at The Scripps Research Institute , [ 44 ] as well as being Professor of Genetics there. [ 3 ] A scientific biography of Brenner was written by Errol Friedberg in the US, for publication by Cold Spring Harbor Laboratory Press in 2010. [ 19 ]
Known for his penetrating scientific insight and acerbic wit, Brenner, for many years, authored a regular column ("Loose Ends") in the journal Current Biology . [ 45 ] [ 46 ] This column was so popular that "Loose ends from Current Biology", a compilation, was published by Current Biology Ltd. [ 47 ] and became a collectors' item. Brenner wrote " A Life in Science ", [ 48 ] a paperback published by BioMed Central . He is also noted for his generosity with ideas and the great number of students and colleagues his ideas have stimulated. [ 49 ] [ 50 ] [ 51 ] [ 52 ]
In 2017, Brenner co-organized a seminal lecture series in Singapore describing ten logarithmic scales of time from the Big Bang to the present, spanning the appearance of multicellular life forms, the evolution of humans, and the emergence of language, culture and technology. [ 53 ] Prominent scientists and thinkers, including W. Brian Arthur , Svante Pääbo , Helga Nowotny and Jack Szostak , spoke during the lecture series. In 2018, the lectures were adapted into a popular science book titled Sydney Brenner's 10-on-10: The Chronicles of Evolution , published by Wildtype Books. [ 54 ]
Brenner also gave four lectures on the history of molecular biology, its impact on neuroscience and the great scientific questions that lie ahead. [ 55 ] The lectures were adapted into the book, In the Spirit of Science: Lectures by Sydney Brenner on DNA, Worms and Brains . [ 56 ]
The "American plan" and "European plan" were proposed by Sydney Brenner as competing models for the way brain cells determine their neural functions. [ 16 ] [ 57 ] [ 58 ] According to the European plan (sometimes referred to as the British plan), the function of cells is determined by their genetic lineage. According to the American plan, a cell's function is determined by the function of its neighbours after cell migration . Further research has shown that most species follow some combination of these methods, albeit in varying degrees, to transfer information to new cells. [ 59 ] [ 60 ]
Brenner received numerous awards and honours, including: [ 61 ] [ 62 ]
Brenner was married to May Brenner ( née Covitz , subsequently Balkind) [ 2 ] from December 1952 until her death in January 2010; [ 2 ] their children include Belinda, Carla, Stefan, and his stepson Jonathan Balkind from his wife's first marriage to Marcus Balkind. He lived in Ely, Cambridgeshire . [ 70 ] [ 71 ] He was an atheist. [ 72 ]
Brenner died on 5 April 2019, in Singapore, at the age of 92. [ 10 ] [ 73 ] [ 74 ] | https://en.wikipedia.org/wiki/Sydney_Brenner |
Syledis ( SY stem LE ger pour mesure la DIS tance) was a terrestrial radio navigation and locating system. The system operated in the UHF segment of 420-450 MHz. It was manufactured in France by Sercel S.A., headquarters Carquefou , and was operational during the 1980s and until about 1995, providing positioning and navigational support for the petroleum sector in the North Sea and to other scientific projects. Syledis has been replaced by GPS .
Determination of the position of mobile vehicles, like f.e. vessels , using Syledis is accomplished by measurement of transit time of radio waves between mobiles and radio stations at known points. There are two modes of operation, active range mode and passive pseudo-range mode .
A vessel is equipped with a transmitter that transmits a coded signal to at least three radio beacons each placed at a known point. The beacons send the code back to the transmitter. The returned coded signal is placed in a timeslot to determine the origin of the returned code. Therefore, in an earlier stage a specified timeslot is connected to a specific beacon.
The elapsed time is proportional to twice the distance between the transmitter and the beacons. After the distance to the beacons is derived the position of the vessel can be calculated. The transmitter computes the distances to the beacons and a computer, connected with the transmitter, computes the position of the vessel.
The Syledis system has a measurement sensitivity that can be expressed in centimeters. Due to weather conditions the wave propagation speed, for electromagnetic waves in air almost the speed of light , can change. In wet conditions, rain or snow, the wave propagation is a little bit slower than in dry conditions. So that gives an inaccuracy in determining the correct position of the vessel.
Another important factor is the length of the cables used to connect the antennas with the radio beacons at the shore, for example placed on lighthouses , or on oil rig platforms and the cable(s) to connect the antenna with the transmitter at the vessel. Normally the antenna will be placed at the top of the mast and the transmitter will be placed in the wheelhouse , where the captain and the mate can see the displays of the transmitter.
Those cables has to be calibrated in order to obtain the smallest possible measurement uncertainty .
This article related to radio communications is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Syledis |
In computing, a syllable is a unit of information that describes the size of data for some digital hardware from the 1960s and 1970s. The size of the unit varies by hardware design in much the same way that word does. The term is not used for modern hardware; standardized terms, such as byte , are used instead.
Examples:
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Syllable_(computing) |
A syllogism ( Ancient Greek : συλλογισμός , syllogismos , 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
In its earliest form (defined by Aristotle in his 350 BC book Prior Analytics ), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. [ 1 ] For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form:
All men are mortal. Socrates is a man. Therefore, Socrates is mortal. [ 2 ]
In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism . [ 3 ] From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. This article is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning , in which facts are predicted by repeated observations.
Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege , in particular his Begriffsschrift ( Concept Script ; 1879). Syllogism, being a method of valid logical reasoning, will always be useful in most circumstances, and for general-audience introductions to logic and clear-thinking. [ 4 ] [ 5 ]
In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. [ 3 ]
Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so." [ 6 ] Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions , including categorical modal syllogisms. [ 7 ]
The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle . Before the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation , works that contributed heavily to the prevailing Old Logic, or logica vetus . The onset of a New Logic, or logica nova , arose alongside the reappearance of Prior Analytics , the work in which Aristotle developed his theory of the syllogism.
Prior Analytics , upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine", leaving very little for thinkers of the day to debate, and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time. This theory of the syllogism would not enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid-14th century by the likes of John Buridan .
Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is, a premise containing the modal words necessarily , possibly , or contingently . Aristotle's terminology in this aspect of his theory was deemed vague, and in many cases unclear, even contradicting some of his statements from On Interpretation . His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, and would be replaced by new distinctions and new theories altogether.
Boethius (c. 475–526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.
Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept, and accompanying theory in the Dialectica —a discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus . With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.
The French philosopher Jean Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica , in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous. [ 8 ]
The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions ( axioms ), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.
In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature. [ 9 ] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation, and leads to discovering and building on axioms to create a more general conclusion. [ 9 ] Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics .
In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift ( Concept Script ). This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.
A noteworthy exception is the logic developed in Bernard Bolzano 's work Wissenschaftslehre ( Theory of Science , 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, because of the intellectual environment at the time in Bohemia , which was then part of the Austrian Empire . In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.
This led to the rapid development of sentential logic and first-order predicate logic , subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. [ original research? ] The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.
One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith , and the Apostolic Tribunal of the Roman Rota , which still requires that any arguments crafted by Advocates be presented in syllogistic format.
George Boole 's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought . [ 10 ] [ 11 ] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought . [ 12 ] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by: [ 12 ]
More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle."
A categorical syllogism consists of three parts:
Each part is a categorical proposition , and each categorical proposition contains two categorical terms. [ 13 ] In Aristotle, each of the premises is in the form "All S are P," "Some S are P", "No S are P" or "Some S are not P", where "S" is the subject-term and "P" is the predicate-term:
More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, this is the minor term (i.e., the subject of the conclusion). For example:
Each of the three distinct terms represents a category. From the example above, humans , mortal , and Greeks : mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term ; in this example, humans . Both of the premises are universal, as is the conclusion.
Here, the major term is die , the minor term is men , and the middle term is mortals . Again, both premises are universal, hence so is the conclusion.
A polysyllogism, or a sorites , is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, one might argue that all lions are big cats, all big cats are predators, and all predators are carnivores. To conclude that therefore all lions are carnivores is to construct a sorites argument.
There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M – Middle, S – subject, P – predicate.):
The premises and conclusion of a syllogism can be any of four types, which are labeled by letters [ 14 ] as follows. The meaning of the letters is given by the table:
In Prior Analytics , Aristotle uses mostly the letters A, B, and C (Greek letters alpha , beta , and gamma ) as term place holders, rather than giving concrete examples. It is traditional to use is rather than are as the copula , hence All A is B rather than All As are Bs . It is traditional and convenient practice to use a, e, i, o as infix operators so the categorical statements can be written succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic:
The convention here is that the letter S is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another classification of syllogisms known as the figure . Given that in each case the conclusion is S-P, the four figures are:
(Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan —reject the fourth figure as a figure distinct from the first.)
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA below is AAA-1, or "A-A-A in the first figure".
The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are sometimes considered to commit the existential fallacy , meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics . All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.
The letters A, E, I, and O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc.
Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is the middle term, the second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P).
The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii ) could also be written as "Some kittens are pets" (MiS in Datisi).
In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate ("logical not") the result of that expression.
It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms. [ 15 ]
Similar: Cesare (EAE-2)
Camestres is essentially like Celarent with S and P exchanged. Similar: Calemes (AEE-4)
Similar: Datisi (AII-3)
Disamis is essentially like Darii with S and P exchanged. Similar: Dimatis (IAI-4)
Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)
Bamalip is exactly like Barbari with S and P exchanged:
Similar: Cesaro (EAO-2)
Similar: Calemos (AEO-4)
Similar: Fesapo (EAO-4)
This table shows all 24 valid syllogisms, represented by Venn diagrams . Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an existential assumption are dashed.
With Aristotle, we may distinguish singular terms , such as Socrates , and general terms, such as Greeks . Aristotle further distinguished types (a) and (b):
Such a predication is known as a distributive , as opposed to non-distributive as in Greeks are numerous . It is clear that Aristotle's syllogism works only for distributive predication, since we cannot reason All Greeks are animals, animals are numerous, therefore all Greeks are numerous . In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, Men can be predicated of Socrates but Socrates cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the terms must be general terms, or categorical terms as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that employ only categorical terms came to be called categorical syllogisms .
It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals . To argue that its validity can be explained by the theory of syllogism would require that we show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men , so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.
If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have existential import with respect to that term. It is ambiguous whether or not a universal statement of the form All A is B is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement All A is B has existential import with respect to A.
It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop a logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence. This is why he leaves no place for fictional entities like goat-stags (or unicorns). [ 16 ]
However, many logic systems developed since do consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not supposit are false.
The following problems arise:
For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:
If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC, AaB->AiC).
These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?
The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus, natural language statements—of the forms All A is B, No A is B , Some A is B , and Some A is not B —can be represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit or not made at all. Consequently, the four forms AaB, AeB, AiB , and AoB can be represented in first order predicate in every combination of existential import—so it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is no .
People often make mistakes when reasoning syllogistically. [ 17 ]
For instance, from the premises some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C. [ 18 ] [ 19 ] However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black things (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions (C). This is because in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the " fallacy of the undistributed middle ". Because of this, it can be hard to follow formal logic, and a closer eye is needed in order to ensure that an argument is, in fact, valid. [ 20 ]
Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.
In simple syllogistic patterns, the fallacies of invalid patterns are: | https://en.wikipedia.org/wiki/Syllogism |
In mathematics, specifically in the field of finite group theory , the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow [ 1 ] that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups .
For a prime number p {\displaystyle p} , a p -group is a group whose cardinality is a power of p ; {\displaystyle p;} or equivalently, the order of each group element is some power of p {\displaystyle p} . A Sylow p -subgroup (sometimes p -Sylow subgroup ) of a finite group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle G} , i.e., a subgroup of G {\displaystyle G} that is a p -group and is not a proper subgroup of any other p {\displaystyle p} -subgroup of G {\displaystyle G} . The set of all Sylow p {\displaystyle p} -subgroups for a given prime p {\displaystyle p} is sometimes written Syl p ( G ) {\displaystyle {\text{Syl}}_{p}(G)} .
The Sylow theorems assert a partial converse to Lagrange's theorem . Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displaystyle p} of the order of a finite group G {\displaystyle G} , there exists a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} of order p n {\displaystyle p^{n}} , the highest power of p {\displaystyle p} that divides the order of G {\displaystyle G} . Moreover, every subgroup of order p n {\displaystyle p^{n}} is a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} , and the Sylow p {\displaystyle p} -subgroups of a group (for a given prime p {\displaystyle p} ) are conjugate to each other. Furthermore, the number of Sylow p {\displaystyle p} -subgroups of a group for a given prime p {\displaystyle p} is congruent to 1 (mod p {\displaystyle p} ).
The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group G {\displaystyle G} to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. | G | = 60 {\displaystyle |G|=60} . [ 2 ]
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of Syl p ( G ) {\displaystyle \operatorname {Syl} _{p}(G)} , all members are actually isomorphic to each other and have the largest possible order: if | G | = p n m {\displaystyle |G|=p^{n}m} with n > 0 {\displaystyle n>0} where p does not divide m , then every Sylow p -subgroup P has order | P | = p n {\displaystyle |P|=p^{n}} . That is, P is a p -group and gcd ( | G : P | , p ) = 1 {\displaystyle {\text{gcd}}(|G:P|,p)=1} . These properties can be exploited to further analyze the structure of G .
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen .
Theorem (1) — For every prime factor p with multiplicity n of the order of a finite group G , there exists a Sylow p -subgroup of G , of order p n {\displaystyle p^{n}} .
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy , and is known as Cauchy's theorem .
Corollary — Given a finite group G and a prime number p dividing the order of G , then there exists an element (and thus a cyclic subgroup generated by this element) of order p in G . [ 3 ]
Theorem (2) — Given a finite group G and a prime number p , all Sylow p -subgroups of G are conjugate to each other. That is, if H and K are Sylow p -subgroups of G , then there exists an element g ∈ G {\displaystyle g\in G} with g − 1 H g = K {\displaystyle g^{-1}Hg=K} .
Theorem (3) — Let p be a prime factor with multiplicity n of the order of a finite group G , so that the order of G can be written as p n m {\displaystyle p^{n}m} , where n > 0 {\displaystyle n>0} and p does not divide m . Let n p {\displaystyle n_{p}} be the number of Sylow p -subgroups of G . Then the following hold:
The Sylow theorems imply that for a prime number p {\displaystyle p} every Sylow p {\displaystyle p} -subgroup is of the same order, p n {\displaystyle p^{n}} . Conversely, if a subgroup has order p n {\displaystyle p^{n}} , then it is a Sylow p {\displaystyle p} -subgroup, and so is conjugate to every other Sylow p {\displaystyle p} -subgroup. Due to the maximality condition, if H {\displaystyle H} is any p {\displaystyle p} -subgroup of G {\displaystyle G} , then H {\displaystyle H} is a subgroup of a p {\displaystyle p} -subgroup of order p n {\displaystyle p^{n}} .
An important consequence of Theorem 2 is that the condition n p = 1 {\displaystyle n_{p}=1} is equivalent to the condition that the Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} is a normal subgroup (Theorem 3 can often show n p = 1 {\displaystyle n_{p}=1} ). However, there are groups that have proper, non-trivial normal subgroups but no normal Sylow subgroups, such as S 4 {\displaystyle S_{4}} . Groups that are of prime-power order have no proper Sylow p {\displaystyle p} -subgroups.
The third bullet point of the third theorem has as an immediate consequence that n p {\displaystyle n_{p}} divides | G | {\displaystyle |G|} .
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow p -subgroup in an infinite group to be a p -subgroup (that is, every element in it has p -power order) that is maximal for inclusion among all p -subgroups in the group. Let Cl ( K ) {\displaystyle \operatorname {Cl} (K)} denote the set of conjugates of a subgroup K ⊂ G {\displaystyle K\subset G} .
Theorem — If K is a Sylow p -subgroup of G , and n p = | Cl ( K ) | {\displaystyle n_{p}=|\operatorname {Cl} (K)|} is finite, then every Sylow p -subgroup is conjugate to K , and n p ≡ 1 ( m o d p ) {\displaystyle n_{p}\equiv 1\ (\mathrm {mod} \ p)} .
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n -gon, D 2 n . For n odd, 2 = 2 1 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n , and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
By contrast, if n is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism , which can be represented by rotation through π/ n , half the minimal rotation in the dihedral group.
Another example are the Sylow p-subgroups of GL 2 ( F q ), where p and q are primes ≥ 3 and p ≡ 1 (mod q ) , which are all abelian . The order of GL 2 ( F q ) is ( q 2 − 1)( q 2 − q ) = ( q )( q + 1)( q − 1) 2 . Since q = p n m + 1 , the order of GL 2 ( F q ) = p 2 n m ′ . Thus by Theorem 1, the order of the Sylow p -subgroups is p 2 n .
One such subgroup P , is the set of diagonal matrices [ x i m 0 0 x j m ] {\displaystyle {\begin{bmatrix}x^{im}&0\\0&x^{jm}\end{bmatrix}}} , x is any primitive root of F q . Since the order of F q is q − 1 , its primitive roots have order q − 1, which implies that x ( q − 1)/ p n or x m and all its powers have an order which is a power of p . So, P is a subgroup where all its elements have orders which are powers of p . There are p n choices for both a and b , making | P | = p 2 n . This means P is a Sylow p -subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow p -subgroups are conjugate to each other, the Sylow p -subgroups of GL 2 ( F q ) are all abelian.
Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple . For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup .
Some non-prime numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 · 5 and n 3 be the number of Sylow 3-subgroups. Then n 3 ∣ {\displaystyle \mid } 5 and n 3 ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, n 5 must divide 3, and n 5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime , the intersection of these two subgroups is trivial, and so G must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 ( up to isomorphism).
A more complex example involves the order of the smallest simple group that is not cyclic . Burnside's p a q b theorem states that if the order of a group is the product of one or two prime powers , then it is solvable , and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5) .
If G is simple, and | G | = 30, then n 3 must divide 10 ( = 2 · 5), and n 3 must equal 1 (mod 3). Therefore, n 3 = 10, since neither 4 nor 7 divides 10, and if n 3 = 1 then, as above, G would have a normal subgroup of order 3, and could not be simple. G then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means G has at least 20 distinct elements of order 3.
As well, n 5 = 6, since n 5 must divide 6 ( = 2 · 3), and n 5 must equal 1 (mod 5). So G also has 24 distinct elements of order 5. But the order of G is only 30, so a simple group of order 30 cannot exist.
Next, suppose | G | = 42 = 2 · 3 · 7. Here n 7 must divide 6 ( = 2 · 3) and n 7 must equal 1 (mod 7), so n 7 = 1. So, as before, G can not be simple.
On the other hand, for | G | = 60 = 2 2 · 3 · 5, then n 3 = 10 and n 5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is A 5 , the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.
Part of Wilson's theorem states that
for every prime p . One may easily prove this theorem by Sylow's third theorem. Indeed,
observe that the number n p of Sylow's p -subgroups
in the symmetric group S p is 1 / p − 1 times the number of p-cycles in S p , ie. ( p − 2)! . On the other hand, n p ≡ 1 (mod p ) . Hence, ( p − 2)! ≡ 1 (mod p ) . So, ( p − 1)! ≡ −1 (mod p ) .
Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if G is a finite group with Sylow p -subgroup P and two subsets A and B normalized by P , then A and B are G -conjugate if and only if they are N G ( P )-conjugate. The proof is a simple application of Sylow's theorem: If B = A g , then the normalizer of B contains not only P but also P g (since P g is contained in the normalizer of A g ). By Sylow's theorem P and P g are conjugate not only in G , but in the normalizer of B . Hence gh −1 normalizes P for some h that normalizes B , and then A gh −1 = B h −1 = B , so that A and B are N G ( P )-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product : if G is a finite group whose Sylow p -subgroup P is contained in the center of its normalizer, then G has a normal subgroup K of order coprime to P , G = PK and P ∩ K = {1}, that is, G is p -nilpotent .
Less trivial applications of the Sylow theorems include the focal subgroup theorem , which studies the control a Sylow p -subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups , and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group . These rely on J. L. Alperin 's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.
The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse, [ 4 ] Scharlau, [ 5 ] Casadio and Zappa, [ 6 ] Gow, [ 7 ] and to some extent Meo. [ 8 ]
One proof of the Sylow theorems exploits the notion of group action in various creative ways. The group G acts on itself or on the set of its p -subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of Wielandt. [ 9 ] In the following, we use a ∣ b {\displaystyle a\mid b} as notation for "a divides b" and a ∤ b {\displaystyle a\nmid b} for the negation of this statement.
Theorem (1) — A finite group G whose order | G | {\displaystyle |G|} is divisible by a prime power p k has a subgroup of order p k .
Let | G | = p k m = p k + r u such that p ∤ u {\displaystyle p\nmid u} , and let Ω denote the set of subsets of G of size p k . G acts on Ω by left multiplication: for g ∈ G and ω ∈ Ω , g ⋅ ω = { g x | x ∈ ω } . For a given set ω ∈ Ω , write G ω for its stabilizer subgroup { g ∈ G | g ⋅ ω = ω } and G ω for its orbit { g ⋅ ω | g ∈ G } in Ω.
The proof will show the existence of some ω ∈ Ω for which G ω has p k elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup G ω , since for any fixed element α ∈ ω ⊆ G , the right coset G ω α is contained in ω ; therefore, | G ω | = | G ω α | ≤ | ω | = p k .
By the orbit-stabilizer theorem we have | G ω | | G ω | = | G | for each ω ∈ Ω , and therefore using the additive p-adic valuation ν p , which counts the number of factors p , one has ν p (| G ω |) + ν p (| G ω |) = ν p (| G |) = k + r . This means that for those ω with | G ω | = p k , the ones we are looking for, one has ν p (| G ω |) = r , while for any other ω one has ν p (| G ω |) > r (as 0 < | G ω | < p k implies ν p (| G ω |) < k ) . Since | Ω | is the sum of | G ω | over all distinct orbits G ω , one can show the existence of ω of the former type by showing that ν p (| Ω |) = r (if none existed, that valuation would exceed r ). This is an instance of Kummer's theorem (since in base p notation the number | G | ends with precisely k + r digits zero, subtracting p k from it involves a carry in r places), and can also be shown by a simple computation:
and no power of p remains in any of the factors inside the product on the right. Hence ν p (| Ω |) = ν p ( m ) = r , completing the proof.
It may be noted that conversely every subgroup H of order p k gives rise to sets ω ∈ Ω for which G ω = H , namely any one of the m distinct cosets Hg .
Lemma — Let H be a finite p -group, let Ω be a finite set acted on by H , and let Ω 0 denote the set of points of Ω that are fixed under the action of H . Then | Ω | ≡ | Ω 0 | (mod p ) .
Any element x ∈ Ω not fixed by H will lie in an orbit of order | H |/| H x | (where H x denotes the stabilizer ), which is a multiple of p by assumption. The result follows immediately by writing | Ω | as the sum of | H x | over all distinct orbits H x and reducing mod p .
Theorem (2) — If H is a p -subgroup of G and P is a Sylow p -subgroup of G , then there exists an element g in G such that g −1 Hg ≤ P . In particular, all Sylow p -subgroups of G are conjugate to each other (and therefore isomorphic ), that is, if H and K are Sylow p -subgroups of G , then there exists an element g in G with g −1 Hg = K .
Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that | Ω 0 | ≡ | Ω | = [ G : P ] (mod p ) . Now p ∤ [ G : P ] {\displaystyle p\nmid [G:P]} by definition so p ∤ | Ω 0 | {\displaystyle p\nmid |\Omega _{0}|} , hence in particular | Ω 0 | ≠ 0 so there exists some gP ∈ Ω 0 . With this gP , we have hgP = gP for all h ∈ H , so g −1 HgP = P and therefore g −1 Hg ≤ P . Furthermore, if H is a Sylow p -subgroup, then | g −1 Hg | = | H | = | P | so that g −1 Hg = P .
Theorem (3) — Let q denote the order of any Sylow p -subgroup P of a finite group G . Let n p denote the number of Sylow p -subgroups of G . Then (a) n p = [ G : N G ( P )] (where N G ( P ) is the normalizer of P ), (b) n p divides | G |/ q , and (c) n p ≡ 1 (mod p ) .
Let Ω be the set of all Sylow p -subgroups of G and let G act on Ω by conjugation. Let P ∈ Ω be a Sylow p -subgroup. By Theorem 2, the orbit of P has size n p , so by the orbit-stabilizer theorem n p = [ G : G P ] . For this group action, the stabilizer G P is given by { g ∈ G | gPg −1 = P } = N G ( P ) , the normalizer of P in G . Thus, n p = [ G : N G ( P )] , and it follows that this number is a divisor of [ G : P ] = | G |/ q .
Now let P act on Ω by conjugation, and again let Ω 0 denote the set of fixed points of this action. Let Q ∈ Ω 0 and observe that then Q = xQx −1 for all x ∈ P so that P ≤ N G ( Q ). By Theorem 2, P and Q are conjugate in N G ( Q ) in particular, and Q is normal in N G ( Q ), so then P = Q . It follows that Ω 0 = { P } so that, by the Lemma, | Ω | ≡ | Ω 0 | = 1 (mod p ) .
The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory .
One proof of the existence of Sylow p -subgroups is constructive: if H is a p -subgroup of G and the index [ G : H ] is divisible by p , then the normalizer N = N G ( H ) of H in G is also such that [ N : H ] is divisible by p . In other words, a polycyclic generating system of a Sylow p -subgroup can be found by starting from any p -subgroup H (including the identity) and taking elements of p -power order contained in the normalizer of H but not in H itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler, [ 10 ] including the algorithm described in Cannon. [ 11 ] These versions are still used in the GAP computer algebra system .
In permutation groups , it has been proven, in Kantor [ 12 ] [ 13 ] [ 14 ] and Kantor and Taylor, [ 15 ] that a Sylow p -subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress, [ 16 ] and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system . | https://en.wikipedia.org/wiki/Sylow_theorems |
Sylvatest is an ultrasonic measuring device that provides an overall diagnosis of a wooden component to test its mechanical strength. This is a specific non-destructive testing tool for wooden beams.
Sylvatest is an acousto - ultrasonic measuring device that provides an overall diagnosis of the wood element, based on the speed and energy absorption of the transmitted ultrasonic wave. [ 1 ] [ 2 ] [ 3 ] The result given reflects the residual health of the wood in the trunk of the tree tested or the beam measured. These are non-destructive tests . [ 4 ] [ 5 ]
The technology was developed by Professor Jean-Luc Sandoz as part of his thesis at EPFL in 1984. [ 4 ] [ 6 ]
Non-destructive ultrasonic testing of wood is a method used to assess its mechanical properties, particularly the modulus of elasticity in different directions. It is based on measuring the propagation velocity of sound waves through the material and the density of the wood.
An ultrasonic wave is emitted through a wood specimen using a transducer (sender) placed at one end. It is received by a second transducer (receiver) at the opposite end. The transit time is used to calculate the propagation velocity V of the wave through the material. This velocity is directly related to the mechanical properties of the wood.
The propagation velocities of ultrasonic waves in a material, such as wood, are directly related to its mechanical properties and density.
Longitudinal (VL) and radial (VR) wave speeds can be expressed as:
V L = c L density = E L density {\displaystyle V_{L}={\sqrt {\frac {c_{L}}{\text{density}}}}={\sqrt {\frac {E_{L}}{\text{density}}}}}
V R = c R density = E R density {\displaystyle V_{R}={\sqrt {\frac {c_{R}}{\text{density}}}}={\sqrt {\frac {E_{R}}{\text{density}}}}}
These relationships are commonly used in non-destructive testing (NDT), particularly in ultrasonic transmission techniques, to assess the internal mechanical condition of wood and other materials.
This method is widely used for:
It was notably applied in the structural assessment of the spire of Notre-Dame Cathedral in Paris, to evaluate the mechanical condition of oak beams without destructive sampling . [ 7 ]
The sylvatest is used in many countries (Brazil, Europe and Asia). [ 4 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ]
It has been used on historic buildings such as the Entrepôts des magasins généraux in Paris, the Forbidden City in Beijing, and the beams of Notre-Dame de Paris [ 7 ] [ 12 ] or even the Château de Valère . [ 13 ]
It is also used to certify the quality of wood for registered designations of origin such as Bois des Alpes [ 14 ] [ 15 ] or Bois de Chatreuse. [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/Sylvatest |
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite .
Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant :
In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M , it can also be shown that the positivity of any nested sequence of n principal minors of M is equivalent to M being positive-definite. [ 1 ]
An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors as illustrated by the Hermitian matrix
A Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative. [ 2 ] [ 3 ]
Suppose M n {\displaystyle M_{n}} is n × n {\displaystyle n\times n} Hermitian matrix M n † = M n {\displaystyle M_{n}^{\dagger }=M_{n}} . Let M k , k = 1 , … n {\displaystyle M_{k},k=1,\ldots n} be the leading principal minor matrices, i.e. the k × k {\displaystyle k\times k} upper left corner matrices. It will be shown that if M n {\displaystyle M_{n}} is positive definite, then the principal minors are positive; that is, det M k > 0 {\displaystyle \det M_{k}>0} for all k {\displaystyle k} .
M k {\displaystyle M_{k}} is positive definite. Indeed, choosing
we can notice that 0 < x † M n x = x → † M k x → . {\displaystyle 0<x^{\dagger }M_{n}x={\vec {x}}^{\dagger }M_{k}{\vec {x}}.} Equivalently, the eigenvalues of M k {\displaystyle M_{k}} are positive, and this implies that det M k > 0 {\displaystyle \det M_{k}>0} since the determinant is the product of the eigenvalues.
To prove the reverse implication, we use induction . The general form of an ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} Hermitian matrix is
where M n {\displaystyle M_{n}} is an n × n {\displaystyle n\times n} Hermitian matrix, v → {\displaystyle {\vec {v}}} is a vector and d {\displaystyle d} is a real constant.
Suppose the criterion holds for M n {\displaystyle M_{n}} . Assuming that all the principal minors of M n + 1 {\displaystyle M_{n+1}} are positive implies that det M n + 1 > 0 {\displaystyle \det M_{n+1}>0} , det M n > 0 {\displaystyle \det M_{n}>0} , and that M n {\displaystyle M_{n}} is positive definite by the inductive hypothesis. Denote
then
By completing the squares, this last expression is equal to
where c → = x n + 1 M n − 1 v → {\displaystyle {\vec {c}}=x_{n+1}M_{n}^{-1}{\vec {v}}} (note that M n − 1 {\displaystyle M_{n}^{-1}} exists because the eigenvalues of M n {\displaystyle M_{n}} are all positive.)
The first term is positive by the inductive hypothesis. We now examine the sign of the second term. By using the block matrix determinant formula
on ( ∗ ) {\displaystyle (*)} we obtain
Consequently, x † M n + 1 x > 0. {\displaystyle x^{\dagger }M_{n+1}x>0.}
Let M n {\displaystyle M_{n}} be an n x n Hermitian matrix. Suppose M n {\displaystyle M_{n}} is semidefinite. Essentially the same proof as for the case that M n {\displaystyle M_{n}} is strictly positive definite shows that all principal minors (not necessarily the leading principal minors) are non-negative.
For the reverse implication, it suffices to show that if M n {\displaystyle M_{n}} has all non-negative principal minors, then for all t>0 , all leading principal minors of the Hermitian matrix M n + t I n {\displaystyle M_{n}+tI_{n}} are strictly positive, where I n {\displaystyle I_{n}} is the n x n identity matrix . Indeed, from the positive definite case, we would know that the matrices M n + t I n {\displaystyle M_{n}+tI_{n}} are strictly positive definite. Since the limit of positive definite matrices is always positive semidefinite, we can take t → 0 {\displaystyle t\to 0} to conclude.
To show this, let M k {\displaystyle M_{k}} be the k th leading principal submatrix of M n . {\displaystyle M_{n}.} We know that q k ( t ) = det ( M k + t I k ) {\displaystyle q_{k}(t)=\det(M_{k}+tI_{k})} is a polynomial in t , related to the characteristic polynomial p M k {\displaystyle p_{M_{k}}} via q k ( t ) = ( − 1 ) k p M k ( − t ) . {\displaystyle q_{k}(t)=(-1)^{k}p_{M_{k}}(-t).} We use the identity in Characteristic polynomial#Properties to write q k ( t ) = ∑ j = 0 k t k − j tr ( ⋀ j M k ) , {\displaystyle q_{k}(t)=\sum _{j=0}^{k}t^{k-j}\operatorname {tr} \left(\textstyle \bigwedge ^{j}M_{k}\right),} where tr ( ⋀ j M k ) {\textstyle \operatorname {tr} \left(\bigwedge ^{j}M_{k}\right)} is the trace of the j th exterior power of M k . {\displaystyle M_{k}.}
From the definition of minors , we know that the entries in the matrix expansion of ⋀ j M k {\displaystyle \bigwedge ^{j}M_{k}} (for j > 0 ) are just the minors of M k . {\displaystyle M_{k}.} In particular, the diagonal entries are the principal minors of M k {\displaystyle M_{k}} , which of course are also principal minors of M n {\displaystyle M_{n}} , and are thus non-negative. Since the trace of a matrix is the sum of the diagonal entries, it follows that tr ( ⋀ j M k ) ≥ 0. {\displaystyle \operatorname {tr} \left(\textstyle \bigwedge ^{j}M_{k}\right)\geq 0.} Thus the coefficient of t k − j {\displaystyle t^{k-j}} in q k ( t ) {\displaystyle q_{k}(t)} is non-negative for all j > 0. For j = 0 , it is clear that the coefficient is 1. In particular, q k ( t ) > 0 {\displaystyle q_{k}(t)>0} for all t > 0 , which is what was required to show. | https://en.wikipedia.org/wiki/Sylvester's_criterion |
In mathematics , a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring . The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant , which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain ).
Sylvester matrices are named after James Joseph Sylvester .
Formally, let p and q be two nonzero polynomials, respectively of degree m and n . Thus:
The Sylvester matrix associated to p and q is then the ( n + m ) × ( n + m ) {\displaystyle (n+m)\times (n+m)} matrix constructed as follows:
Thus, if m = 4 and n = 3, the matrix is:
If one of the degrees is zero (that is, the corresponding polynomial is a nonzero constant polynomial), then there are zero rows consisting of coefficients of the other polynomial, and the Sylvester matrix is a diagonal matrix of dimension the degree of the non-constant polynomial, with the all diagonal coefficients equal to the constant polynomial. If m = n = 0, then the Sylvester matrix is the empty matrix with zero rows and zero columns.
The above defined Sylvester matrix appears in a Sylvester paper of 1840. In a paper of 1853, Sylvester introduced the following matrix, which is, up to a permutation of the rows, the Sylvester matrix of p and q , which are both considered as having degree max( m , n ). [ 1 ] This is thus a 2 max ( n , m ) × 2 max ( n , m ) {\displaystyle 2\max(n,m)\times 2\max(n,m)} -matrix containing max ( n , m ) {\displaystyle \max(n,m)} pairs of rows. Assuming m > n , {\displaystyle m>n,} it is obtained as follows:
Thus, if m = 4 and n = 3, the matrix is:
The determinant of the 1853 matrix is, up to sign, the product of the determinant of the Sylvester matrix (which is called the resultant of p and q ) by p m m − n {\displaystyle p_{m}^{m-n}} (still supposing m ≥ n {\displaystyle m\geq n} ).
These matrices are used in commutative algebra , e.g. to test if two polynomials have a (non-constant) common factor. In such a case, the determinant of the associated Sylvester matrix (which is called the resultant of the two polynomials) equals zero. The converse is also true.
The solutions of the simultaneous linear equations
where x {\displaystyle x} is a vector of size n {\displaystyle n} and y {\displaystyle y} has size m {\displaystyle m} , comprise the coefficient vectors of those and only those pairs x , y {\displaystyle x,y} of polynomials (of degrees n − 1 {\displaystyle n-1} and m − 1 {\displaystyle m-1} , respectively) which fulfill
where polynomial multiplication and addition is used.
This means the kernel of the transposed Sylvester matrix gives all solutions of the Bézout equation where deg x < deg q {\displaystyle \deg x<\deg q} and deg y < deg p {\displaystyle \deg y<\deg p} .
Consequently the rank of the Sylvester matrix determines the degree of the greatest common divisor of p and q :
Moreover, the coefficients of this greatest common divisor may be expressed as determinants of submatrices of the Sylvester matrix (see Subresultant ). | https://en.wikipedia.org/wiki/Sylvester_matrix |
Symbian is a discontinued mobile operating system (OS) and computing platform designed for smartphones . [ 6 ] It was originally developed as a proprietary software OS for personal digital assistants in 1998 by the Symbian Ltd. consortium. [ 7 ] Symbian OS is a descendant of Psion 's EPOC , and was released exclusively on ARM processors , although an unreleased x86 port existed. Symbian was used by many major mobile phone brands, like Samsung , Motorola , Sony Ericsson , and above all by Nokia . It was also prevalent in Japan by brands including Fujitsu , Sharp and Mitsubishi . As a pioneer that established the smartphone industry, it was the most popular smartphone OS on a worldwide average until the end of 2010, at a time when smartphones were in limited use, when it was overtaken by iOS and Android . It was notably less popular in North America .
The Symbian OS platform is formed of two components: one being the microkernel -based operating system with its associated libraries , and the other being the user interface (as middleware ), which provides the graphical shell atop the OS. [ 8 ] The most prominent user interface was the S60 (formerly Series 60) platform built by Nokia, first released in 2002 and powering most Nokia Symbian devices. UIQ was a competing user interface mostly used by Motorola and Sony Ericsson that focused on pen -based devices, rather than a traditional keyboard interface from S60. Another interface was the MOAP (S) platform from carrier NTT DoCoMo in the Japanese market. [ 9 ] [ 10 ] Applications for these different interfaces were not compatible with each other, despite each being built atop Symbian OS. Nokia became the largest shareholder of Symbian Ltd. in 2004 and purchased the entire company in 2008. [ 11 ] The non-profit Symbian Foundation was then created to make a royalty-free successor to Symbian OS. Seeking to unify the platform, S60 became the Foundation's favoured interface and UIQ stopped development. The touchscreen -focused Symbian^1 (or S60 5th Edition) was created as a result in 2009. Symbian^2 (based on MOAP) was used by NTT DoCoMo, one of the members of the Foundation, for the Japanese market. Symbian^3 was released in 2010 as the successor to S60 5th Edition, by which time it became fully free software . The transition from a proprietary operating system to a free software project is believed to be one of the largest in history. [ 12 ] Symbian^3 received the Anna and Belle updates in 2011. [ 13 ] [ 14 ]
The Symbian Foundation disintegrated in late 2010 and Nokia took back control of the OS development. [ 15 ] [ 16 ] In February 2011, Nokia, by then the only remaining company still supporting Symbian outside Japan, announced that it would use Microsoft 's Windows Phone 7 as its primary smartphone platform, while Symbian would be gradually wound down. [ 17 ] [ 18 ] Two months later, Nokia moved the OS to proprietary licensing, only collaborating with the Japanese OEMs [ 19 ] and later outsourced Symbian development to Accenture . [ 6 ] [ 20 ] Although support was promised until 2016, including two major planned updates, by 2012 Nokia had mostly abandoned development and most Symbian developers had already left Accenture, [ 21 ] and in January 2014 Nokia stopped accepting new or changed Symbian software from developers. [ 22 ] The Nokia 808 PureView in 2012 was officially the last Symbian smartphone from Nokia. [ 23 ] NTT DoCoMo continued releasing OPP(S) (Operator Pack Symbian, successor of MOAP) devices in Japan, which still act as middleware on top of Symbian. [ 24 ] Phones running this include the F-07F [ ja ] from Fujitsu and SH-07F [ ja ] from Sharp in 2014.
Symbian originated from EPOC32 , an operating system created by Psion in the 1990s. In June 1998, Psion Software became Symbian Ltd. , a major joint venture between Psion and phone manufacturers Ericsson , Motorola , and Nokia .
Afterwards, different software platforms were created for Symbian, backed by different groups of mobile phone manufacturers. They include S60 ( Nokia , Samsung and LG ), UIQ ( Sony Ericsson and Motorola ) and MOAP (S) (Japanese only such as Fujitsu , Sharp etc.).
With no major competition in the smartphone OS market ( Palm OS and Windows Mobile were comparatively small players), Symbian held 67% of the global smartphone market share in 2006. [ 25 ]
Despite its sizable market share, Symbian was at various stages difficult to develop for: First (around early-to-mid-2000s) due to the complexity of the programming languages available, Open Programming Language (OPL) and Symbian C++ , and of the OS; then the stubborn developer bureaucracy, along with high prices of various integrated development environments (IDEs) and software development kits (SDKs), which were prohibitive for independent or very small developers; and then the subsequent fragmentation, which was in part caused by infighting among and within manufacturers, each of which also had their own IDEs and SDKs. All of this discouraged third-party developers, and served to cause the native app ecosystem for Symbian not to evolve to a scale later reached by Apple's App Store or Android's Google Play.
By contrast, iPhone OS (renamed iOS in 2010) and Android had comparatively simpler design, provided easier and much more centralized infrastructure to create and obtain third-party apps, offered certain developer tools and programming languages with a manageable level of complexity, and having abilities such as multitasking and graphics to meet future consumer demands.
Although Symbian was difficult to program for, this issue could be worked around by creating Java Mobile Edition apps, ostensibly under a "write once, run anywhere" slogan. [ 26 ] This wasn't always the case because of fragmentation due to different device screen sizes and differences in levels of Java ME support on various devices.
In June 2008, Nokia announced the acquisition of Symbian Ltd. , and a new independent non-profit organization called the Symbian Foundation was established. Symbian OS and its associated user interfaces S60 , UIQ , and MOAP (S) were contributed by their owners Nokia , NTT DoCoMo , Sony Ericsson , and Symbian Ltd. , to the foundation with the objective of creating the Symbian platform as a royalty-free, Free software , under the Free Software Foundation (FSF) and Open Source Initiative (OSI) approved Eclipse Public License (EPL). The platform was designated as the successor to Symbian OS, following the official launch of the Symbian Foundation in April 2009. The Symbian platform was officially made available as Free software in February 2010. [ 27 ]
Nokia became the major contributor to Symbian's code, since it then possessed the development resources for both the Symbian OS core and the user interface. Since then Nokia maintained its own code repository for the platform development, regularly releasing its development to the public repository. [ 28 ] Symbian was intended to be developed by a community led by the Symbian Foundation , which was first announced in June 2008 and which officially launched in April 2009. Its objective was to publish the source code for the entire Symbian platform under the EPL. This was accomplished on 4 February 2010; the Symbian Foundation reported this event to be the largest codebase moved to Free software in history. [ 27 ] [ 29 ]
However, some important components within Symbian OS were licensed from third parties, which prevented the foundation from publishing the full source under EPL immediately; instead much of the source was published under a more restrictive Symbian Foundation License (SFL) and access to the full source code was limited to member companies only, although membership was open to any organisation. [ 30 ] Also, the Free software Qt framework was introduced to Symbian in 2010, as the primary upgrade path to MeeGo , which was to be the next mobile operating system to replace and supplant Symbian on high-end devices; Qt was by its nature free and very convenient to develop with. Several other frameworks were deployed to the platform, among them Standard C and C++, Python , Ruby , and Adobe Flash Lite . IDEs and SDKs were developed and then released for free, and application software (app) development for Symbian picked up.
In November 2010, the Symbian Foundation announced that due to changes in global economic and market conditions (and also a lack of support from members such as Samsung [ 31 ] and Sony Ericsson ), it would transition to a licensing-only organisation; [ 30 ] Nokia announced it would take over the stewardship of the Symbian platform. Symbian Foundation would remain the trademark holder and licensing entity and would only have non-executive directors involved.
With market share sliding from 39% in Q32010 to 31% in Q42010, [ 32 ] Symbian was losing ground to iOS and Android quickly, eventually falling behind Android in Q42010. [ 33 ] Stephen Elop was appointed the CEO of Nokia in September 2010, and on 11 February 2011, he announced a partnership with Microsoft that would see Nokia adopt Windows Phone as its primary smartphone platform, [ 34 ] and Symbian would be gradually phased out, together with MeeGo. [ 18 ] As a consequence, Symbian's market share fell, and application developers for Symbian dropped out rapidly. Research in June 2011 indicated that over 39% of mobile developers using Symbian at the time of publication were planning to abandon the platform. [ 35 ]
By 5 April 2011, Nokia ceased to make free any portion of the Symbian software and reduced its collaboration to a small group of preselected partners in Japan. [ 5 ] Source code released under the original EPL remains available in third party repositories, [ 36 ] including a full set of all public code from the project as of 7 December 2010. [ 37 ]
On 22 June 2011, Nokia had made an agreement with Accenture for an outsourcing program. Accenture will provide Symbian-based software development and support services to Nokia through 2016. [ 20 ] The transfer of Nokia employees to Accenture was completed on 30 September 2011 and 2,800 Nokia employees became Accenture employees as of October 2011. [ 6 ]
Nokia had terminated its support of software development and maintenance for Symbian with effect from 1 January 2014, thereafter refusing to publish new or changed Symbian applications or content in the Nokia Store and terminating its 'Symbian Signed' program for software certification. [ 38 ]
Symbian has had a native graphics toolkit since its inception, known as AVKON (formerly known as Series 60 ). S60 was designed to be manipulated by a keyboard-like interface metaphor, such as the ~15-key augmented telephone keypad, or the mini-QWERTY keyboards. AVKON-based software is binary-compatible with Symbian versions up to and including Symbian^3.
Symbian^3 includes the Qt framework , which became the recommended user interface toolkit for new applications. Qt can also be installed on older Symbian devices.
Symbian^4 was planned to introduce a new GUI library framework specifically designed for a touch-based interface, known as "UI Extensions for Mobile" or UIEMO (internal project name "Orbit"), which was built on top of Qt Widget; a preview was released in January 2010, however in October 2010 Nokia announced that Orbit/UIEMO had been cancelled.
Nokia later recommended that developers use Qt Quick with QML , the new high-level declarative UI and scripting framework for creating visually rich touchscreen interfaces that allowed development for both Symbian and MeeGo ; it would be delivered to existing Symbian^3 devices as a Qt update. When more applications gradually feature a user interface reworked in Qt, the legacy S60 framework (AVKON) would be deprecated and no longer included with new devices at some point, thus breaking binary compatibility with older S60 applications. [ 39 ] [ 40 ]
Symbian^3 and earlier have a built-in WebKit based browser . Symbian was the first mobile platform to make use of WebKit (in June 2005). [ 41 ] Some older Symbian models have Opera Mobile as their default browser.
Nokia released a new browser with the release of Symbian Anna with improved speed and an improved user interface. [ 42 ]
Symbian had strong localization support enabling manufacturers and 3rd party application developers to localize Symbian based products to support global distribution. Nokia made languages available in the device, in language packs : a set of languages which cover those commonly spoken in the area where a device variant is to be sold. All language packs have in common English, or a locally relevant dialect of it. The last release, Symbian Belle, supports these 48 languages, with [dialects], and (scripts):
Symbian Belle marks the introduction of Kazakh, while Korean is no longer supported.
From 2010, Symbian switched to using standard C++ with Qt as the main SDK, which can be used with either Qt Creator or Carbide.c++ . Qt supports the older Symbian/S60 3rd (starting with Feature Pack 1, a.k.a. S60 3.1) and Symbian/S60 5th Edition (a.k.a. S60 5.01b) releases, as well as the new Symbian platform. It also supports Maemo and MeeGo , Windows, Linux and Mac OS X. [ 43 ] [ 44 ]
Alternative application development can be done using Python (see Python for S60 ), Adobe Flash Lite or Java ME .
Symbian OS previously used a Symbian specific C++ version, along with CodeWarrior and later Carbide.c++ integrated development environment (IDE), as the native application development environment.
Web Runtime (WRT) is a portable application framework that allows creating widgets on the S60 Platform ; it is an extension to the S60 WebKit based browser that allows launching multiple browser instances as separate JavaScript applications. [ 45 ] [ 46 ]
As of 2010, the SDK for Symbian is standard C++, using Qt . It can be used with either Qt Creator , or Carbide (the older IDE previously used for Symbian development). [ 43 ] [ 47 ] A phone simulator allows testing of Qt apps. Apps compiled for the simulator are compiled to native code for the development platform, rather than having to be emulated. [ 48 ] Application development can either use C++ or QML .
As Symbian OS is written in C++ using Symbian Software's coding standards, it is possible to develop using Symbian C++, although it is not a standard implementation. Before the release of the Qt SDK, this was the standard development environment. There were multiple platforms based on Symbian OS that provided software development kits (SDKs) for application developers wishing to target Symbian OS devices, the main ones being UIQ and S60. Individual phone products, or families, often had SDKs or SDK extensions downloadable from the maker's website too.
The SDKs contain documentation, the header files and library files needed to build Symbian OS software, and a Windows-based emulator ("WINS"). Up until Symbian OS version 8, the SDKs also included a version of the GNU Compiler Collection (GCC) compiler (a cross-compiler ) needed to build software to work on the device.
Symbian OS 9 and the Symbian platform use a new application binary interface (ABI) and needed a different compiler. A choice of compilers is available including a newer version of GCC (see external links below).
Symbian C++ programming has a steep learning curve , as Symbian C++ requires the use of special techniques such as descriptors, active objects and the cleanup stack. This can make even relatively simple programs initially harder to implement than in other environments. It is possible that the techniques, developed for the much more restricted mobile hardware and compilers of the 1990s, caused extra complexity in source code because programmers are required to concentrate on low-level details instead of more application-specific features. As of 2010, these issues are no longer the case when using standard C++, with the Qt SDK.
Symbian C++ programming is commonly done with an integrated development environment (IDE). For earlier versions of Symbian OS, the commercial IDE CodeWarrior for Symbian OS was favoured. The CodeWarrior tools were replaced during 2006 by Carbide.c++ , an Eclipse -based IDE developed by Nokia. Carbide.c++ is offered in four different versions: Express, Developer, Professional, and OEM, with increasing levels of capability. Fully featured software can be created and released with the Express edition, which is free. Features such as UI design, crash debugging etc. are available in the other, charged-for, editions. Microsoft Visual Studio 2003 and 2005 are also supported via the Carbide.vs plugin.
Symbian devices can also be programmed using Python , Java ME , Flash Lite , Ruby , .NET , Web Runtime (WRT) Widgets and Standard C / C++ . [ 49 ]
Visual Basic programmers can use NS Basic to develop apps for S60 3rd Edition and UIQ 3 devices.
In the past, Visual Basic , Visual Basic .NET , and C# development for Symbian were possible through AppForge Crossfire, a plug-in for Microsoft Visual Studio. On 13 March 2007 AppForge ceased operations; Oracle purchased the intellectual property, but announced that they did not plan to sell or provide support for former AppForge products. Net60, a .NET compact framework for Symbian, which is developed by redFIVElabs, is sold as a commercial product. With Net60, VB.NET, and C# (and other) source code is compiled into an intermediate language (IL) which is executed within the Symbian OS using a just-in-time compiler. (As of 18 January 2010, RedFiveLabs has ceased development of Net60 with this announcement on their landing page: "At this stage we are pursuing some options to sell the IP so that Net60 may continue to have a future.")
There is also a version of a Borland IDE for Symbian OS. Symbian development is also possible on Linux and macOS using tools and methods developed by the community, partly enabled by Symbian releasing the source code for key tools. A plug-in that allows development of Symbian OS applications in Apple's Xcode IDE for Mac OS X was available. [ 50 ]
Java ME applications for Symbian OS are developed using standard techniques and tools such as the Sun Java Wireless Toolkit (formerly the J2ME Wireless Toolkit). They are packaged as JAR (and possibly JAD) files. Both CLDC and CDC applications can be created with NetBeans . Other tools include SuperWaba , which can be used to build Symbian 7.0 and 7.0s programs using Java.
Nokia S60 phones can also run Python scripts when the interpreter Python for S60 is installed, with a custom made API that allows for Bluetooth support and such. There is also an interactive console to allow the user to write Python scripts directly from the phone.
Once developed, Symbian applications need to find a route to customers' mobile phones. They are packaged in SIS files which may be installed over-the-air, via PC connect, Bluetooth or on a memory card. An alternative is to partner with a phone manufacturer and have the software included on the phone itself. Applications must be Symbian Signed for Symbian OS 9.x to make use of certain capabilities (system capabilities, restricted capabilities and device manufacturer capabilities). [ 51 ] Applications could be signed for free in 2010. [ 52 ]
Symbian's design is subdivided into technology domains , [ 53 ] each of which comprises a set of software packages . [ 54 ] Each technology domain has its own roadmap, and the Symbian Foundation has a team of technology managers who manage these technology domain roadmaps.
Every package is allocated to exactly one technology domain, based on the general functional area to which the package contributes and by which it may be influenced. By grouping related packages by themes, the Symbian Foundation hopes to encourage a strong community to form around them and to generate discussion and review.
The Symbian System Model [ 55 ] illustrates the scope of each of the technology domains across the platform packages.
Packages are owned and maintained by a package owner, a named individual from an organization member of the Symbian Foundation, who accepts code contributions from the wider Symbian community and is responsible for package.
The Symbian kernel ( EKA2 ) supports sufficiently fast real-time response to build a single-core phone around it – that is, a phone in which a single processor core executes both the user applications and the signalling stack . [ 56 ] The real-time kernel has a microkernel architecture containing only the minimum, most basic primitives and functionality, for maximum robustness, availability and responsiveness. It has been termed a nanokernel , because it needs an extended kernel to implement any other abstractions. It contains a scheduler , memory management and device drivers , with networking , telephony, and file system support services in the OS Services Layer or the Base Services Layer. The inclusion of device drivers means the kernel is not a true microkernel.
Symbian features pre-emptive multitasking and memory protection , like other operating systems (especially those created for use on desktop computers). EPOC's approach to multitasking was inspired by VMS and is based on asynchronous server-based events.
Symbian OS was created with three systems design principles in mind:
To best follow these principles, Symbian uses a microkernel , has a request-and-callback approach to services, and maintains separation between user interface and engine. The OS is optimised for low-power battery-based devices and for read-only memory (ROM)-based systems (e.g. features like XIP and re-entrancy in shared libraries). The OS, and application software , follows an object-oriented programming design named model–view–controller (MVC).
Later OS iterations diluted this approach in response to market demands, notably with the introduction of a real-time kernel and a platform security model in versions 8 and 9.
There is a strong emphasis on conserving resources which is exemplified by Symbian-specific programming idioms like descriptors and a cleanup stack. Similar methods exist to conserve storage space. Further, all Symbian programming is event-based, and the central processing unit (CPU) is switched into a low power mode when applications are not directly dealing with an event. This is done via a programming idiom called active objects . Similarly the Symbian approach to threads and processes is driven by reducing overheads.
The All over Model contains the following layers, from top to bottom:
The Base Services Layer is the lowest level reachable by user-side operations; it includes the File Server and User Library, a Plug-In Framework which manages all plug-ins, Store, Central Repository, DBMS and cryptographic services. It also includes the Text Window Server and the Text Shell: the two basic services from which a completely functional port can be created without the need for any higher layer services.
Symbian has a microkernel architecture, which means that the minimum necessary is within the kernel to maximise robustness, availability and responsiveness. It contains a scheduler , memory management and device drivers, but other services like networking, telephony and file system support are placed in the OS Services Layer or the Base Services Layer. The inclusion of device drivers means the kernel is not a true microkernel. The EKA2 real-time kernel, which has been termed a nanokernel , contains only the most basic primitives and requires an extended kernel to implement any other abstractions.
Symbian is designed to emphasise compatibility with other devices, especially removable media file systems. Early development of EPOC led to adopting File Allocation Table (FAT) as the internal file system, and this remains, but an object-oriented persistence model was placed over the underlying FAT to provide a POSIX -style interface and a streaming model. The internal data formats rely on using the same APIs that create the data to run all file manipulations. This has resulted in data-dependence and associated difficulties with changes and data migration .
There is a large networking and communication subsystem, which has three main servers called: ETEL (EPOC telephony), ESOCK (EPOC sockets) and C32 (responsible for serial communication). Each of these has a plug-in scheme. For example, ESOCK allows different ".PRT" protocol modules to implement various networking protocol schemes. The subsystem also contains code that supports short-range communication links, such as Bluetooth , IrDA and USB .
There is also a large volume of user interface (UI) Code. Only the base classes and substructure were contained in Symbian OS, while most of the actual user interfaces were maintained by third parties. This is no longer the case. The three major UIs – S60, UIQ and MOAP – were contributed to Symbian in 2009. Symbian also contains graphics, text layout and font rendering libraries.
All native Symbian C++ applications are built up from three framework classes defined by the application architecture: an application class, a document class and an application user interface class. These classes create the fundamental application behaviour. The remaining needed functions, the application view, data model and data interface, are created independently and interact solely through their APIs with the other classes.
Many other things do not yet fit into this model – for example, SyncML , Java ME providing another set of APIs on top of most of the OS and multimedia . Many of these are frameworks, and vendors are expected to supply plug-ins to these frameworks from third parties (for example, Helix Player for multimedia codecs ). This has the advantage that the APIs to such areas of functionality are the same on many phone models, and that vendors get a lot of flexibility. But it means that phone vendors needed to do a great deal of integration work to make a Symbian OS phone.
Symbian includes a reference user-interface called "TechView". It provides a basis for starting customisation and is the environment in which much Symbian test and example code runs. It is very similar to the user interface from the Psion Series 5 personal organiser and is not used for any production phone user interface.
The boot process of Symbian is started from the ROM bootloader, later the ROM bootloader load Symbian from flash . [ 57 ]
Symbian, as it advanced to OS version 7.0, spun off into several different graphical user interfaces , each backed by a certain company or group of companies. Unlike Android OS 's cosmetic GUIs, Symbian GUIs are referred to as "platforms" due to more significant modifications and integrations. Things became more complicated when applications developed for different Symbian GUI platforms were not compatible with each other, and this led to OS fragmentation. [ 58 ]
User Interfaces platforms that run on or are based on Symbian OS include:
Samsung: i8910 Omnia HD , [ 77 ]
Sony Ericsson: Satio , Vivaz , Vivaz Pro
* Manufactured by Fujitsu † Manufactured by Sharp ▲ Software update service for Nokia Belle and Symbian (S60) phones is discontinued at the end of December 2015
In Q1 2004 2.4 million Symbian phones were shipped, double the number as in Q1 2003. Symbian Ltd. was particularly impressed by progress made in Japan. [ 78 ]
3.7 million devices were shipped in Q3 2004, a growth of 201% compared to Q3 2003 and market share growing from 30.5% to 50.2%. However, in the United States it was much less popular, with a 6% market share in Q3 2004, well behind Palm OS (43%) and Windows Mobile (25%). This has been attributed to North American customers preferring wireless PDAs over smartphones, as well as Nokia's low popularity there. [ 79 ]
On 16 November 2006, the 100 millionth smartphone running the OS was shipped. [ 80 ] As of 21 July 2009, more than 250 million devices running Symbian OS had been produced. [ 81 ]
In 2006, Symbian had 73% of the smartphone market, [ 82 ] compared with 22.1% of the market in the second quarter of 2011. [ 83 ]
By the end of May 2006, 10 million Symbian-powered phones were sold in Japan, representing 11% of Symbian's total worldwide shipments of 89 million. [ 84 ] By November 2007 the figure was 30 million, achieving a market share of 65% by June 2007 in the Japanese market. [ 85 ]
Symbian has lost market share over the years as the market has dramatically grown, with new competing platforms entering the market, though its sales have increased during the same timeframe. E.g., although Symbian's share of the global smartphone market dropped from 52.4% in 2008 to 47.2% in 2009, shipments of Symbian devices grew 4.8%, from 74.9 million units to 78.5 million units. [ 86 ] From Q2 2009 to Q2 2010, shipments of Symbian devices grew 41.5%, by 8.0 million units, from 19,178,910 units to 27,129,340; compared to an increase of 9.6 million units for Android, 3.3 million units for RIM, and 3.2 million units for Apple. [ 87 ]
Prior reports on device shipments as published in February 2010 showed that the Symbian devices formed a 47.2% share of the smart mobile devices shipped in 2009, with RIM having 20.8%, Apple having 15.1% (via iOS ), Microsoft having 8.8% (via Windows CE and Windows Mobile ) and Android having 4.7%. [ 86 ]
In the number of "smart mobile device" sales, Symbian devices were the market leaders for 2010. Statistics showed that Symbian devices formed a 37.6% share of smart mobile devices sold, with Android having 22.7%, RIM having 16%, and Apple having 15.7% (via iOS ). [ 88 ] Some estimates indicate that the number of mobile devices shipped with the Symbian OS up to the end of Q2 2010 is 385 million. [ 89 ]
Over the course of 2009–10, Motorola , Samsung , LG , and Sony Ericsson announced their withdrawal from Symbian in favour of alternative platforms including Google's Android , Microsoft's Windows Phone . [ 90 ] [ 91 ] [ 92 ] [ 93 ]
In Q2 2012, according to IDC worldwide market share had dropped to an all-time low of 4.4%. [ 94 ]
The users of Symbian in the countries with non-Latin alphabets (such as Russia, Ukraine and others) have been criticizing the complicated method of language switching for many years. [ 95 ] For example, if a user wants to type a Latin letter, they must call the menu, click the languages item, use arrow keys to choose, for example, the English language from among many other languages, and then press the 'OK' button. After typing the Latin letter, the user must repeat the procedure to return to their native keyboard. This method slows down typing significantly. In touch-phones and QWERTY phones the procedure is slightly different but remains time-consuming. All other mobile operating systems, as well as Nokia's S40 phones, enable switching between two initially selected languages by one click or a single gesture.
Early versions of the firmware for the original Nokia N97 , running on Symbian^1/Series 60 5th Edition have been heavily criticized as buggy (also contributed by the low amount of RAM installed in the phone). [ 96 ]
In November 2010, Smartphone blog All About Symbian criticized the performance of Symbian's default web browser and recommended the alternative browser Opera Mobile . [ 97 ] Nokia's Senior Vice President Jo Harlow promised an updated browser in the first quarter of 2011. [ 98 ]
There were many different versions and editions of Symbian, which led to fragmentation. Apps and software may be incompatible when installed across different versions of Symbian. [ 99 ]
Symbian OS is subject to a variety of viruses, the best known of which is Cabir . Usually these send themselves from phone to phone by Bluetooth. So far, none have exploited any flaws in Symbian OS. Instead, they have all asked the user whether they want to install the software, with somewhat prominent warnings that it can't be trusted, although some rely on social engineering , often in the form of messages that come with the malware: rogue software purporting to be a utility, game, or some other application for Symbian.
However, with a view that the average mobile phone user shouldn't have to worry about security, Symbian OS 9.x adopted a Unix -style capability model (permissions per process, not per object). Installed software is theoretically unable to do damaging things (such as costing the user money by sending network data) without being digitally signed – thus making it traceable. Commercial developers who can afford the cost can apply to have their software signed via the Symbian Signed program. Developers also have the option of self-signing their programs. However, the set of available features does not include access to Bluetooth, IrDA, GSM CellID, voice calls, GPS and few others. Some operators opted to disable all certificates other than the Symbian Signed certificates.
Some other hostile programs are listed below, but all of them still require the input of the user to run.
A new form of malware threat to Symbian OS in the form of 'cooked firmware' was demonstrated at the International Malware Conference, Malcon , December 2010, by Indian hacker Atul Alex. [ 100 ] [ 101 ]
Symbian OS 9.x devices can be hacked to remove the platform security introduced in OS 9.1 onwards, allowing users to execute unsigned code. [ 102 ] This allows altering system files, and access to previously locked areas of the OS. The hack was criticised by Nokia for potentially increasing the threat posed by mobile viruses as unsigned code can be executed. [ 103 ]
EPOC16 featured a primarily monochrome, keyboard-operated graphical interface [ 104 ] – the hardware for which it was designed originally had pointer input in the form of a digitiser panel.
In the late 1990s, the operating system was referred to as EPOC16 to distinguish it from Psion's then-new EPOC32 OS.
The EPOC32 operating system, at the time simply referred to as EPOC, was later renamed Symbian OS. Adding to the confusion with names, before the change to Symbian, EPOC16 was often referred to as SIBO to distinguish it from the "new" EPOC. Despite the similarity of the names, EPOC32 and EPOC16 were completely different operating systems, EPOC32 being written in C++ from a new codebase with development beginning during the mid-1990s.
EPOC32 was a pre-emptive multitasking , single user operating system with memory protection, which encourages the application developer to separate their program into an engine and an interface . The Psion line of PDAs come with a graphical user interface called EIKON which is specifically tailored for handheld machines with a keyboard (thus looking perhaps more similar to desktop GUIs than palmtop GUIs [ 105 ] ). However, one of EPOC's characteristics is the ease with which new GUIs can be developed based on a core set of GUI classes, a feature which has been widely explored from Ericsson R380 and onwards.
EPOC32 was originally developed for the ARM family of processors, including the ARM7 , ARM9 , StrongARM and Intel's XScale , but can be compiled towards target devices using several other processor types.
During the development of EPOC32, Psion planned to license EPOC to third-party device manufacturers, and spin off its software division as Psion Software. One of the first licensees was the short-lived Geofox , which halted production with less than 1,000 units sold. Ericsson marketed a rebranded Psion Series 5mx called the MC218 , and later created the EPOC Release 5.1 based smartphone , the R380 . Oregon Scientific also released a budget EPOC device, the Osaris (notable as the only EPOC device to ship with Release 4).
Work started on the 32-bit version in late 1994.
The Series 5 device, released in June 1997, used the first iterations of the EPOC32 OS, codenamed "Protea", and the "Eikon" graphical user interface.
The Oregon Scientific Osaris was the only PDA to use the ER4.
The Psion Series 5mx , Psion Series 7 , Psion Revo , Diamond Mako , Psion netBook and Ericsson MC218 were released in 1999 using ER5. A phone project was announced at CeBIT , the Phillips Illium/Accent, but did not achieve a commercial release. This release has been retrospectively dubbed Symbian OS 5.
The first phone using ER5u, the Ericsson R380 was released in November 2000. It was not an open device: software could not be installed. Notably, several never-released Psion prototypes for next generation PDAs, including a Bluetooth Revo successor codenamed Conan , were using ER5u. The 'u' in the name refers to it supporting Unicode .
In June 1998, Psion Software became Symbian Ltd. , a major joint venture between Psion and phone manufacturers Ericsson , Motorola , and Nokia . As of Release 6, EPOC was renamed Symbian OS.
The first 'open' Symbian OS phone, the Nokia 9210 Communicator, was released in June 2001. Bluetooth support was added. Almost 500,000 Symbian phones were shipped in 2001, rising to 2.1 million the following year.
Development of different UIs was made generic with a "reference design strategy" for either 'smartphone' or 'communicator' devices, subdivided further into keyboard- or tablet-based designs. Two reference UIs (DFRDs or Device Family Reference Designs) were shipped: Quartz and Crystal. The former was merged with Ericsson's Ronneby design and became the basis for the UIQ interface; the latter reached the market as the Nokia Series 80 UI.
Later DFRDs were Sapphire, Ruby, and Emerald. Only Sapphire came to market, evolving into the Pearl DFRD and finally the Nokia Series 60 UI, a keypad-based 'square' UI for the first true smartphones. The first one of them was the Nokia 7650 smartphone (featuring Symbian OS 6.1), which was also the first with a built-in camera, with VGA (0.3 Mpx = 640×480) resolution. Other notable S60 Symbian 6.1 devices are the Nokia 3650 , the short lived Sendo X and Siemens SX1 , the first and the last Symbian phone from Siemens.
Despite these efforts to be generic, the UI was clearly split between competing companies: Crystal or Sapphire was Nokia, Quartz was Ericsson. DFRD was abandoned by Symbian in late 2002, as part of an active retreat from UI development in favour of headless delivery. Pearl was given to Nokia, Quartz development was spun off as UIQ Technology AB, and work with Japanese firms was quickly folded into the MOAP standard.
One million Symbian phones were shipped in Q1 2003, with the rate increasing to one million a month by the end of 2003.
Symbian OS 7.0s was a version of 7.0 special adapted to have greater backward compatibility with Symbian OS 6.x, partly for compatibility between the Communicator 9500 and its predecessor the Communicator 9210.
In 2004, Psion sold its stake in Symbian. The same year, the first worm for mobile phones using Symbian OS, Cabir , was developed, which used Bluetooth to spread itself to nearby phones. See Cabir and Symbian OS threats .
Also included were new APIs to support CDMA , 3G , two-way data streaming, DVB-H , and OpenGL ES with vector graphics and direct screen access.
The first and maybe the most famous smartphone featuring Symbian OS 8.1a was Nokia N90 in 2005, Nokia 's first in Nseries .
Symbian OS has generally maintained reasonable binary code compatibility . In theory the OS was BC from ER1-ER5, then from 6.0 to 8.1b. Substantial changes were needed for 9.0, related to tools and security, but this should be a one-off event. The move from requiring ARMv4 to requiring ARMv5 did not break backwards compatibility.
Symbian 9.1 introduced capabilities and a Platform Security framework. To access certain APIs, developers have to sign their application with a digital signature . Basic capabilities are user-grantable and developers can self-sign them, while more advanced capabilities require certification and signing via the Symbian Signed program, which uses independent 'test houses' and phone manufacturers for approval. For example, file writing is a user-grantable capability while access to Multimedia Device Drivers require phone manufacturer approval. A TC TrustCenter ACS Publisher ID certificate is required by the developer for signing applications.
Nokia phones with Symbian OS 9.2 OS include the Nokia E71 , Nokia E90 , Nokia N95 , Nokia N82 , Nokia N81 and Nokia 5700 .
Used as the basis for Symbian^1, the first Symbian platform release.
The release is also better known as S60 5th edition , as it is the bundled interface for the OS.
On 24 August 2011, Nokia announced it officially for three new smartphones, the Nokia 600 (later replaced by Nokia 603 ), Nokia 700 , and Nokia 701 . [ 110 ]
Nokia officially renamed Symbian Belle to Nokia Belle in a company blog post. [ 111 ] [ 112 ]
Nokia Belle adds to the Anna improvements with a pull-down status/notification bar, deeper near field communication integration, free-form re-sizable homescreen widgets, and six homescreens instead of the previous three. As of 7 February 2012, Nokia Belle update is available for most phone models through Nokia Suite, coming later to Australia. Users can check the availability at the Nokia homepage. [ 113 ]
On 1 March 2012, Nokia announced a Feature Pack 1 update for Nokia Belle which will be available as an update to Nokia 603, 700, 701 (excluding others), and for Nokia 808 PureView natively.
Symbian Carla and Donna were the planned follow-up releases to Belle, to be released in late 2012 and late 2013 respectively. However it was acknowledged in May 2012 that these had been cancelled and that the upcoming Belle Feature Pack 2 would be the last version of the operating system. [ 114 ]
The latest software release for Nokia 1st generation Symbian Belle smartphones (Nokia N8, C7 , C6-01 , Oro, 500 , X7 , E7 , E6 ) is Nokia Belle Refresh (111.040.1511). [ 115 ]
In October 2012, the Nokia Belle Feature Pack 2, widely considered the last major update for Symbian, was released for Nokia 603, 700, 701, and 808 PureView. [ 116 ] | https://en.wikipedia.org/wiki/Symbian |
Symbiobacterium thermophilum is a symbiotic thermophile that depends on co-culture with a Bacillus strain for growth. It is Gram-negative and tryptophanase -positive, with type strain T(T) (= IAM 14863 T ). It is the type species of its genus. [ 1 ] Symbiobacterium is related to the Gram-positive Bacillota and Actinomycetota , but belongs to a lineage that is distinct from both. [ 2 ] S. thermophilum has a bacillus shaped cell structure with no flagella. [ 3 ] This bacterium is located throughout the environment in soils and fertilizers. [ 4 ]
Although Gram staining S. thermophilum shows a negative lab result, there are key Gram-negative membrane biosynthesis proteins that it lacks, such as LPS:glycosyltransferase and polysaccharide transporters. [ 3 ] Instead, the cell structure of S. thermophilum includes proteins STH61, 969, 1321, 2197, 2492, and 3168 which are associated with the enveloped S-layer bacteria. [ 3 ] The bacillus shape of S. thermophilum cells may be caused by the mreBCD (STH372-4) gene, located adjacent to the min locus. [ 3 ] Although it has no flagella, the genome of S. thermophilum does include a flagella biosynthesis gene cluster. S. thermophilum is found to produce endospores in specific conditions. [ 3 ] There is less research on the spore-like structure of S. thermophilum as it is the rarer form.
Its genome has been sequenced, and has a size of 3.57 Mbp , with 3338 protein-coding genes. [ 3 ] Characteristics of S. thermophilum such as the production of tryptophanase and β-tyrosinase, the cell surface structure, and a negative gram stain results indicate that the bacteria is Gram-negative. However, the sequence of 16S rRNA gene led to the complete phylogenic analysis of S. thermophilum , concluding it was in fact Gram-positive. [ 5 ] High-G+C content (68.7%) along with its Gram stain results indicates that S. thermophilum belongs to the Actinomyces phylum, but the genome and proteins are more closely related to the Firmicutes, a Gram-positive phylum with low-G+C content. S. thermophilum further defies the knowledge that endospore forming genes are unique to the Bacillus-Clostridium group, showing genes involved in the formation of endospores. [ 5 ] Sequencing of proteins proved biological roles in 2,082 of the 3,338 CDSs. [ 3 ] The genome of S. thermophilum is not even partially alike other prokaryotic genomes sequenced at this point in time, as indicated by a CDS similarity matrix search. [ 3 ]
S. thermophilum depends on other strains of Bacillus to grow, in a co-culture mechanism. [ 1 ] This is known as microbial commensalism and often occurs in composts. [ 1 ] S. thermophilum is one of many cultures that arise from compost derivatives. Under optimal conditions, the growth rate maximizes at 5x10^8 cells/mL. [ 1 ]
S. thermophilum uses the non-oxidative branch of the pentose-phosphate glycolytic pathway for metabolism. [ 1 ] Despite not using the Entner-Doudoroff pathway and lacking both cellulose-degrading and amylose-degrading enzymes, it has the genes and ability to metabolize glycerol, gluconate, cellobiose, N-acetylgalactosamine, tyrosine, and tryptophan. [ 1 ] S. thermophilum contains genes for ferredoxin oxidoreductases, pyruvate, and 2-oxoacid. [ 1 ] S. thermophilum lacks the genes for methionine and lysine biosynthesis but has the enzymes that are utilized to biosynthesize amino acids. [ 1 ]
The variety of respiratory enzymes possessed by S. thermophilum enables the bacterium to grow in both aerobic and anaerobic conditions. [ 1 ] The ability to grow in both aerobic and anaerobic conditions is indicated by the presence of both aerobic glycerol-3-phosphate dehydrogenase and anaerobic glycerol-3-phosphate dehydrogenase. [ 1 ] The presence of the Nap nitrate reductase gene cluster and Nar nitrate reductase suggest that S. thermophilum utilizes nitrate respiration. [ 1 ]
Due to the thermophilic nature of S. thermophilum , areas that are ideal for the survival of the bacteria would be ones that have increased temperatures and are nutrient dense. [ 4 ] The habitats that are most suited for S. thermophilum would be in the intestinal tract of animals and also in composts. [ 4 ] This is because both of those areas contain the essentials for the bacteria to survive. [ 4 ]
S. thermophilum is a bacterium that is widely distributed throughout the environment. It can be found in many different types of soil and fertilizers that contain animal feces, as well as inside animal intestines, and in the feed that is given to the animals. [ 4 ] To determine the distribution of S. thermophilum , tests were done to check for growth of the bacterium and whether or not the item being tested contained tryptophanase. [ 4 ]
In a study done at the Department of Applied Biological Sciences in Nihon University, Fujisawa, Japan, there was a random sample of Symbiobacterium that was cloned and it determined that out of the 31 samples taken, 16 of the cases showed that the sample had a more diverse genetic structure, where as the other 15 samples had less diverse genetics due to the results showing that the genetics were almost identical to S. thermophilum . [ 4 ] | https://en.wikipedia.org/wiki/Symbiobacterium_thermophilum |
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Symbiodinium is a genus of dinoflagellates that encompasses the largest and most prevalent group of endosymbiotic dinoflagellates known and have photosymbiotic relationships with many species. These unicellular microalgae commonly reside in the endoderm of tropical cnidarians such as corals , sea anemones , and jellyfish , where the products of their photosynthetic processing are exchanged in the host for inorganic molecules. They are also harbored by various species of demosponges , flatworms , mollusks such as the giant clams , foraminifera ( soritids ), and some ciliates . Generally, these dinoflagellates enter the host cell through phagocytosis , persist as intracellular symbionts , reproduce, and disperse to the environment. The exception is in most mollusks, where these symbionts are intercellular (between the cells). Cnidarians that are associated with Symbiodinium occur mostly in warm oligotrophic (nutrient-poor), marine environments where they are often the dominant constituents of benthic communities. These dinoflagellates are therefore among the most abundant eukaryotic microbes found in coral reef ecosystems.
Symbiodinium are colloquially called zooxanthellae , and animals symbiotic with algae in this genus are said to be "zooxanthellate". The term was loosely used to refer to any golden-brown endosymbionts , including diatoms and other dinoflagellates. Continued use of the term in the scientific literature is discouraged because of the confusion caused by overly generalizing taxonomically diverse symbiotic relationships. [ 2 ]
In 2018, the systematics of Symbiodiniaceae was revised, and the distinct clades have been reassigned into seven genera. [ 3 ] Following this revision, the name Symbiodinium is now sensu stricto a genus name for only species that were previously classified as Clade A. [ 3 ] The other clades were reclassified as distinct genera (see Molecular Systematics below).
Many Symbiodinium species are known primarily for their role as mutualistic endosymbionts . In hosts , they usually occur in high densities, ranging from hundreds of thousands to millions per square centimeter. [ 4 ] The successful culturing of swimming gymnodinioid cells from coral led to the discovery that "zooxanthellae" were actually dinoflagellates. [ 5 ] [ 6 ] Each Symbiodinium cell is coccoid in hospite (living in a host cell) and surrounded by a membrane that originates from the host cell plasmalemma during phagocytosis . This membrane probably undergoes some modification to its protein content, which functions to limit or prevent phago-lysosome fusion. [ 7 ] [ 8 ] [ 9 ] The vacuole structure containing the symbiont is therefore termed the symbiosome . A single symbiont cell occupies each symbiosome. It is unclear how this membrane expands to accommodate a dividing symbiont cell. Under normal conditions, symbiont and host cells exchange organic and inorganic molecules that enable the growth and proliferation of both partners.
Symbiodinium is one of the most studied symbionts. Their mutualistic relationships with reef-building corals form the basis of a highly diverse and productive ecosystem . Coral reefs have economic benefits – valued at hundreds of billions of dollars each year – in the form of ornamental , subsistence and commercial fisheries, tourism and recreation, coastal protection from storms, a source of bioactive compounds for pharmaceutical development, and more. [ 10 ]
The study of Symbiodinium biology is driven largely by a desire to understand global coral reef decline. A chief mechanism for widespread reef degradation has been stress-induced coral bleaching caused by unusually high seawater temperature. Bleaching is the disassociation of the coral and the symbiont and/or loss of chlorophyll within the alga, resulting in a precipitous loss in the animal's pigmentation . Many Symbiodinium -cnidarian associations are affected by sustained elevation of sea surface temperatures, [ 11 ] but may also result from exposure to high irradiance levels (including UVR ), [ 12 ] [ 13 ] extreme low temperatures, [ 14 ] low salinity [ 15 ] and other factors. [ 16 ] The bleached state is associated with decreased host calcification , [ 17 ] increased disease susceptibility [ 18 ] and, if prolonged, partial or total mortality. [ 19 ] The magnitude of mortality from a single bleaching event can be global in scale as it was in 2015. These episodes are predicted to become more common and severe as temperatures worldwide continue to rise. [ 20 ] The physiology of a resident Symbiodinium species often regulates the bleaching susceptibility of a coral. [ 21 ] [ 22 ] Therefore, a significant amount of research has focused on characterizing the physiological basis of thermal tolerance [ 23 ] [ 24 ] [ 25 ] [ 26 ] and in identifying the ecology and distribution of thermally tolerant symbiont species. [ 27 ] [ 28 ] [ 29 ]
The symbiosis Symbodinium -coral could provide higher resistance to multiple stress ( desiccation , high UVR ) to the coral holobiont through its mycosporine-like amino acids (MAAs). The concentration of MAAs increases with stress and ROS in Symbodinium . [ 30 ] These UV-absorbing MAAs may also support light-harvesting pigments during photosynthesis, be source of nitrogen storage and for reproduction. More than half of the Symbodinium taxa contain MAAs. [ 31 ] [ 32 ] [ 33 ]
Symbiodinium trenchi is a stress-tolerant species and is able to form mutualistic relationships with many species of coral. It is present in small numbers in coral globally and is common in the Andaman Sea , where the water is about 4 °C (7 °F) warmer than in other parts of the Indian Ocean . [ 34 ] [ self-published source? ] In the Caribbean Sea in late 2005, water temperature was elevated for several months and it was found that S. trenchi , a symbiont not normally abundant, took up residence in many corals in which it had not previously been observed. Those corals did not bleach. Two years later, it had largely been replaced as a symbiont by the species normally found in the Caribbean. [ 28 ]
S. thermophilum was recently found to make up the bulk of the algal population inside the corals of the Persian Gulf . It is also present in the Gulf of Oman and the Red Sea , at a much lower concentration. Coral that hosted this species was able to tolerate the 35 °C (95 °F) waters of the Persian Gulf, much hotter than the 31 °C (88 °F) of coral reefs globally. [ 35 ]
The advent of DNA sequence comparison initiated a rebirth in the ordering and naming of all organisms. The application of this methodology helped overturn the long-held belief that (traditional understood) Symbiodinium comprised a single genus, a process which began in earnest with the morphological , physiological , and biochemical comparisons of cultured isolates . Currently, genetic markers are exclusively used to describe ecological patterns and deduce evolutionary relationships among morphologically cryptic members of this group. Foremost in the molecular systematics of Symbiodinium is to resolve ecologically relevant units of diversity (i.e. species).
The earliest ribosomal gene sequence data indicated that Symbiodinium had lineages whose genetic divergence was similar to that seen in other dinoflagellates from different genera , families , and even orders . [ 36 ] This large phylogenetic disparity among clades A, B, C, etc. was confirmed by analyses of the sequences of the mitochondrial gene coding for cytochrome c oxidase subunit I (CO1) among Dinophyceae . [ 37 ] Most of these clade groupings comprise numerous reproductively isolated, genetically distinct lineages (see ‘Species diversity’ ), exhibiting different ecological and biogeographic distributions (see ‘Geographic distributions and patterns of ‘diversity’ ).
Recently (2018), these distinct clades within the family of Symbiodiniaceae have been reassigned, although not exclusively, into seven genera: Symbiodinium (understood sensu stricto , i. e. clade A), Breviolum (clade B), Cladocopium (clade C), Durusdinium (clade D), Effrenium (clade E), Fugacium (clade F), and Gerakladium (clade G). [ 3 ]
The recognition of species diversity in this genus remained problematic for many decades due to the challenges of identifying morphological and biochemical traits useful for diagnosing species. [ 38 ] Presently, phylogenetic, ecological, and population genetic data can be more rapidly acquired to resolve Symbiodinium into separate entities that are consistent with Biological, Evolutionary, and Ecological Species Concepts. [ 39 ] [ 40 ] Most genetics-based measures of diversity have been estimated from the analysis of one genetic marker (e. g. LSU , ITS2 , or cp23S [ 41 ] ), yet in recent studies these and other markers were analyzed in combination. The high concordance found among nuclear, mitochondrial and chloroplast DNA argues that a hierarchical phylogenetic scheme, combined with ecological and population genetic data, can unambiguously recognize and assign nomenclature to reproductively isolated lineages, i.e. species. [ 3 ] [ 39 ] [ 40 ]
The analysis of additional phylogenetic markers show that some Symbiodinium that were initially identified by slight differences in ITS sequences may comprise members of the same species [ 40 ] whereas, in other cases, two or more genetically divergent lineages can possess the same ancestral ITS sequence. [ 42 ] [ 43 ] When analysed in the context of the major species concepts, [ 44 ] the majority of ITS2 sequence data provide a reasonable proxy for species diversity. [ 39 ] [ 40 ] [ 45 ] Currently, ITS2 types number in the hundreds, but most communities of symbiotic cnidaria around the world still require comprehensive sampling. Furthermore, there appears to be a large number of unique species found in association with equally diverse species assemblages of soritid foraminifera, [ 46 ] as well as many other Symbiodinium that are exclusively free-living and found in varied, often benthic , habitats . [ 47 ] Given the potential species diversity of these ecologically cryptic Symbiodinium , the total species number may never be accurately assessed. [ 46 ]
Through the use of microsatellite markers, multilocus genotypes identifying a single clonal line of Symbiodinium can be resolved from samples of host tissue. It appears that most individual colonies harbor a single multilocus genotype (i.e. clone). [ 48 ] [ 49 ] Extensive sampling within colonies confirms that many colonies harbor a homogeneous (clonal) Symbiodinium population. Additional genotypes do occur in some colonies, yet rarely more than two or three are found. When present in the same colony, multiple clones often exhibit narrow zones of overlap. [ 49 ] Colonies adjacent to each other on a reef may harbor identical clones, but across the host population the clone diversity of a particular Symbiodinium species is potentially large and comprises recombinant genotypes that are the product of sexual recombination . A clone tends to remain dominant in a colony over many months and years, but may be occasionally displaced or replaced. The few studies examining clone dispersal find that most genotypes have limited geographic distributions, but that dispersal and gene flow is likely influenced by host life history and mode of symbiont acquisition (e. g. horizontal vs. vertical ). [ 49 ] [ 48 ]
Symbiodinium are perhaps the best group for studying micro-eukaryote physiology and ecology for several reasons. Firstly, available phylogenetic and population genetic markers allow for detailed examination of their genetic diversity over broad spatial and temporal scales. Also, large quantities of Symbiodinium cells are readily obtained through the collection of hosts that harbor them. Lastly, their association with animals provides an additional axis by which to compare and contrast ecological distributions. [ citation needed ]
The earliest genetic methods for assessing Symbiodinium diversity relied on low-resolution molecular markers that separated the genus into a few evolutionarily divergent lineages, referred to as " clades ". Previous characterizations of geographic distribution and dominance have focused on the clade-level of genetic resolution, but more detailed assessments of diversity at the species level are needed. While members of a given clade may be ubiquitous, the species diversity within each clade is potentially large, with each species often having different ecological and geographic distributions related to their dispersal ability, host biogeography, and external environmental conditions. A small number of species occur in temperate environments where few symbiotic animals occur. As a result, these high latitude associations tend to be highly species specific. [ citation needed ]
The large diversity of Symbiodinium revealed by genetic analyses is distributed non-randomly and appears to comprise several guilds with distinct ecological habits. Of the many Symbiodinium characterized genetically, most are host-specific , mutualistic , and dominate their host. [ 50 ] Others may represent compatible symbionts that remain as low-abundance background populations because of competitive inferiority under the prevailing external environmental conditions (e.g. high light vs. low light). [ 51 ] Some may also comprise opportunistic species that may proliferate during periods of physiological stress and displace the normal resident symbiont and remain abundant in the host's tissues for months to years before being replaced by the original symbiont. [ 28 ] [ 52 ] [ 53 ] There are also those that rapidly infect and establish populations in host juveniles until being replaced by symbionts that normally associate with host adult colonies. [ 54 ] Finally, there appears to be another group of Symbiodinium that are incapable of establishing endosymbiosis yet exist in environments around the animal or associate closely with other substrates (i.e. macro-algal surfaces, sediment surface) [ 47 ] [ 55 ] Symbiodinium from functional groups 2, 3, and 4 are known to exist because they culture easily, however species with these life histories are difficult to study because of their low abundance in the environment.
There are few examples of documented populations of free-living Symbiodinium . [ 47 ] Given that most host larvae must initially acquire their symbionts from the environment, viable Symbiodinium cells occur outside the host. The motile phase is probably important in the external environment and facilitates the rapid infection of host larvae. The use of aposymbiotic host polyps deployed as "capture vessels" and the application of molecular techniques has allowed for the detection of environmental sources of Symbiodinium . [ 53 ] [ 56 ] With these methods employed, investigators may resolve the distribution of different species on various benthic surfaces [ 55 ] and cell densities suspended in the water column . [ 57 ] The genetic identities of cells cultured from the environment are often dissimilar to those found in hosts. These likely do not form endosymbioses and are entirely free-living; they are different from "dispersing" symbiotic species. [ 50 ] Learning more about the "private lives" of these environmental populations and their ecological function will further our knowledge about the diversity, dispersal success, and evolution among members within this large genus.
Certain Symbiodinium strains and/or species are more easily cultured and can persist in artificial or supplemented seawater media (e.g. ASP–8A, F/2) [ 58 ] for decades. The comparison of cultured isolates under identical conditions show clear differences in morphology, size, biochemistry, gene expression, swimming behavior, growth rates, etc. [ 59 ] [ 60 ] [ 61 ] This pioneering comparative approach initiated a slow paradigm shift in recognizing that the traditional genus sensu lato comprised more than a single real genus.
Culturing is a selective process , and many Symbiodinium isolates growing on artificial media are not typical of the species normally associated with a particular host. Indeed, most host–specific species have yet to be cultured. Samples for genetic analysis should be acquired from the source colony in order to match the resulting culture with the identity of the dominant and ecologically relevant symbiont originally harbored by the animal. [ 50 ] [ 62 ] [ 63 ]
The life cycle of Symbiodinium was first described from cells growing in culture media. For isolates that are in log phase growth, division rates occur every 1–3 days, with Symbiodinium cells alternating between a spherical, or coccoid, morphology and a smaller flagellated motile mastigote stage. While several similar schemes are published that describe how each morphological state transitions to other, the most compelling life history reconstruction was deduced from light and electron microscopy and nuclear staining evidence. [ 64 ] During asexual propagation (sometimes referred to as mitotic or vegetative growth ), cells undergo a diel cycle of karyokinesis ( chromosome /nuclear division) in darkness. The mother cell then divides ( cytokinesis ) soon after exposure to light and releases two motile cells. The initiation and duration of motility varies among species. [ 64 ] Approaching or at the end of the photoperiod the mastigotes cease swimming, release their flagella , and undergo a rapid metamorphosis into the coccoid form. As cultures reach stationary growth phase, fewer and fewer motile cells are observed, indicating slower division rates.
Large tetrads are occasionally observed, particularly when cells in stationary growth phase are transferred to fresh media. However, it is unknown whether this stage is the product of two consecutive mitotic divisions, or perhaps a process that generates sexually competent motile cells (i.e. gametes ), or is the end result of meiosis (i. e. as meiotic tetrads ) following gamete fusion . There is no cytological evidence for sexual recombination, and meiosis has never been observed, but population genetic evidence supports the view that Symbiodinium periodically undergo events of sexual recombination. How, when, and where the sexual phase in their life history occurs remains unknown. [ 42 ] [ 65 ] [ 66 ]
The morphological description of the genus Symbiodinium is originally based on the type species ( holotype ) S. microadriaticum . [ 38 ] [ 67 ] Because these dinoflagellates possess two major stages in their life history (see above), namely the mastigote (motile) and coccoid (non-motile) stages, the morphology of both is described in order to provide a complete diagnosis of the organism.
The motile flagellated form is gymnodinioid and athecate ("nude"). [ 68 ] The relative dimensions of the episoma (cell portion above the groove) and hyposoma (cell portion below the groove) differ among species. [ 38 ] The alveoli are most visible in the motile phase but lack fibrous cellulosic structures found in thecate ("armored") dinoflagellates. Between the points of origin of the two flagella is an extensible structure of unknown function called the peduncle. In other dinoflagellates, an analogous structure has been implicated in heterotrophic feeding and sexual recombination . In Symbiodinium , it has been suggested that the peduncle may be involved in substrate attachment, explaining why certain cells seem to spin in place. [ 67 ] Compared to other Gymnodiniales genera, there is little or no displacement at the sulcus where the ends of the cingulum (or cigulum) groove converge.
The internal organelles of the mastigote are essentially the same as described in the coccoid cell (see below). The transition from mastigote to coccoid stage in Symbiodinium occurs rapidly, but details about cellular changes are unknown. Mucocysts (an ejectile organelle [ 69 ] ) located beneath the plasmalemma are found in S. pilosum and their function is unknown, but may be involved in heterotrophic feeding.
The coccoid cell of Symbiodinium is spherical and ranges in average diameter from 6 to 13 μm, depending on the species (Blank et al. 1989). This stage is often wrongly interpreted as a dinocyst ; hence, in published literature, the alga in hospite is often referred to as a vegetative cyst. [ 67 ] The term cyst usually refers to a dormant, metabolically quiescent stage in the life history of other dinoflagellates, initiated by several factors, including nutrient availability, temperature, and day length. [ 70 ] Such cysts permit extended resistance to unfavourable environmental conditions. Instead, coccoid Symbiodinium cells are metabolically active, as they photosynthesize , undergo mitosis, and actively synthesize proteins and nucleic acids. While most dinoflagellates undergo mitosis as a mastigote, in Symbiodinium , mitosis occurs exclusively in the coccoid cell. [ 64 ]
The coccoid cell is surrounded by a cellulosic , usually smooth cell wall that contains large-molecular-weight proteins and glycoproteins . [ 38 ] [ 71 ] Cell walls grow thicker in culture than in hospite . [ 7 ] The cell membrane ( plasmalemma ) is located beneath the cell wall, yet little is known about its composition and function in terms of the regulation of trans-membrane transport of metabolites . During karyokinesis and cytokinesis , the cell wall remains intact until the mastigotes escape the mother cell. In culture, the discarded walls accumulate at the bottom of the culture vessel. It is not known what becomes of the walls from divided cells in hospite . [ 72 ] One species, S. pilosum , possesses tufts of hair-like projections from the cell wall; this is the only known surface characteristic used to diagnose a species in the genus.
Most described species possess a single, peripheral, reticulated chloroplast bounded by three membranes. The volume of the cell occupied by the chloroplast varies among species. [ 38 ] The lamellae comprise three closely appressed (stacked) thylakoids , and are attached by two stalks to the pyrenoid [ 38 ] surrounded by a starch sheath. In three of the described species, the thylakoids are in parallel arrays, but in S. pilosum , there are also peripheral lamellae. There are no thylakoid membranes invading the pyrenoid, which is unlike other symbiotic dinoflagellates. [ 73 ] [ 74 ] The lipid components of thylakoids include the galactolipids as
monogalactosyl-diglycerides (MGDG) [ 75 ] and
digalactosyl-diglycerides( DGDG),; [ 76 ] the sulpholipid
sulphoquinovosyl-diglyceride (SQDG), [ 77 ] phosphatidyl glycerol , [ 78 ] and phosphatidyl choline .
Associated with these are various fatty acids . [ 79 ] The light-harvesting and reaction centre components in the thylakoid membrane include a water-soluble peridinin-chlorophyll a-protein complex (PCP or PerCP), [ 80 ] and a membrane-bound
chlorophyll a-chlorophyll c 2 -peridinin-protein-complex (acpPC), [ 81 ] along with typical photosynthetic electron transport systems such as the photosystem II reaction centre and the chlorophyll-a-P700 reaction centre complex of photosystem I . [ 82 ] [ 83 ] Also associated with the thylakoids are the xanthophylls dinoxanthin , diadinoxanthin , diatoxanthin and the carotene , β-Carotene .
The pyrenoid contains the nuclear - encoded enzyme type II Ribulose-1,5-bis-phosphate-carboxylase-oxygenase ( RuBisCO ), [ 84 ] which is responsible for the catalysis of inorganic carbon dioxide (CO 2 ) into organic compounds .
All cultured isolates (i.e. strains ) are capable of phenotypic adjustment in their capacity for light harvesting (i.e. photoacclimation), such as by altering the cellular Chl. a and peridinin quota, as well as the size and number of photosynthetic units. [ 85 ] However, the ability to acclimate is a reflection of genetic differences among species that are differently adapted (evolved) to a particular photic environment. [ 86 ] [ 87 ] For example, S. pilosum is characterized as a high light-adapted species, while others are low light-adapted ( S. kawagutii ) or adapted to a larger range in varying light fields ( S. microadriaticum ).
In general, the nucleus is centrally located and the nucleolus is often associated with the inner nuclear membrane . The chromosomes , as in other dinoflagellates, are seen as ‘permanently super-coiled’ DNA in transmission electron micrographs (TEM). [ 88 ] The described species of Symbiodinium possess distinct chromosome numbers (ranging from 26 to 97 [ 38 ] ), which remain constant throughout all phases of the nuclear cycle. However, during M-phase , the volume of each chromosome is halved, as is the volume of each of the two resulting nuclei. Thus, the ratio of chromosome volume to nuclear volume remains constant. These observations are consistent with the interpretation that the algae are haploid , a conclusion supported by molecular genetic data. [ 89 ] During S-phase of the nuclear cycle the chromosomes do uncoil to facilitate DNA synthesis , and the volumes of both the chromosomes and the nucleus return to those seen in the G 2 2 phase . [ 88 ]
There are several additional organelles found in the cytoplasm of Symbiodinium . The most obvious of these is the structure referred to as the "accumulation body". This is a membrane-bound vesicle ( vacuole ) with contents that are unrecognizable, but appear red or yellow under the light microscope. It may serve to accumulate cellular debris or act as an autophagic vacuole in which non-functional organelles are digested and their components recycled. During mitosis, only one daughter cell appears to acquire this structure. There are other vacuoles that may contain membranous inclusions, [ 90 ] while still others contain crystalline material variously interpreted as oxalate crystals or crystalline uric acid .
The following species are recognised by the World Register of Marine Species : [ 1 ] | https://en.wikipedia.org/wiki/Symbiodinium |
Symbiosis ( Ancient Greek συμβίωσις symbíōsis : living with, companionship < σύν sýn : together; and βίωσις bíōsis : living) [ 2 ] is any type of a close and long-term biological interaction , between two organisms of different species . The two organisms, termed symbionts , can for example be in mutualistic , commensalistic , or parasitic relationships. [ 3 ] In 1879, Heinrich Anton de Bary defined symbiosis as "the living together of unlike organisms".
The term is sometimes more exclusively used in a restricted, mutualistic sense, where both symbionts contribute to each other's subsistence. This means that they benefit each other in some way. [ 3 ]
Symbiosis can be obligate (or obligative ), which means that one, or both of the organisms depend on each other for survival, or facultative (optional), when they can also subsist independently.
Symbiosis is also classified by physical attachment. Symbionts forming a single body live in conjunctive symbiosis, while all other arrangements are called disjunctive symbiosis. [ 4 ] When one organism lives on the surface of another, such as head lice on humans, it is called ectosymbiosis ; when one partner lives inside the tissues of another, such as Symbiodinium within coral , it is termed endosymbiosis . [ 5 ] [ 6 ]
The definition of symbiosis was a matter of debate for 130 years. [ 7 ] In 1877, Albert Bernhard Frank used the term symbiosis to describe the mutualistic relationship found in lichens . [ 8 ] [ 9 ] In 1878, the German mycologist Heinrich Anton de Bary defined it more broadly as "the living together of unlike organisms". [ 10 ] [ 11 ] [ 12 ] Over time, the definition has varied among scientists. Some have argued that it should refer only to persistent mutualisms , while others have proposed that it should include all long-term biological interactions (i.e., mutualism, commensalism , and parasitism ), but exclude brief interactions such as predation . [ 13 ]
In 1949, Edward Haskell proposed an integrative approach with a classification of "co-actions", [ 14 ] which was later adopted by biologists as "interactions". [ 15 ] [ 16 ] [ 17 ] [ 18 ]
Relationships can be obligate, meaning that one or both of the symbionts entirely depend on each other for survival. For example, in lichens , which consist of fungal and photosynthetic symbionts, the fungal partners cannot live on their own. [ 11 ] [ 19 ] [ 20 ] [ 21 ] The algal or cyanobacterial symbionts in lichens, such as Trentepohlia , can generally live independently, and their part of the relationship is therefore described as facultative (optional), or non-obligate. [ 22 ] When one of the participants in a symbiotic relationship is capable of photosynthesis, as with lichens, it is called photosymbiosis. [ 23 ] [ 24 ]
Ectosymbiosis is any symbiotic relationship in which the symbiont lives on the body surface of the host , including the inner surface of the digestive tract or the ducts of exocrine glands . [ 6 ] [ 25 ] Examples of this include ectoparasites such as lice ; commensal ectosymbionts such as the barnacles , which attach themselves to the jaw of baleen whales ; and mutualist ectosymbionts such as cleaner fish .
Contrastingly, endosymbiosis is any symbiotic relationship in which one symbiont lives within the tissues of the other, either within the cells or extracellularly. [ 6 ] [ 26 ] Examples include diverse microbiomes : rhizobia , nitrogen-fixing bacteria that live in root nodules on legume roots; actinomycetes , nitrogen-fixing bacteria such as Frankia , which live in alder root nodules; single-celled algae inside reef-building corals ; and bacterial endosymbionts that provide essential nutrients to about 10%–15% of insects. [ 27 ]
In endosymbiosis, the host cell lacks some of the nutrients which the endosymbiont provides. As a result, the host favors endosymbiont's growth processes within itself by producing some specialized cells. These cells affect the genetic composition of the host in order to regulate the increasing population of the endosymbionts and ensure that these genetic changes are passed onto the offspring via vertical transmission ( heredity ). [ 28 ]
As the endosymbiont adapts to the host's lifestyle, the endosymbiont changes dramatically. There is a drastic reduction in its genome size, as many genes are lost during the process of metabolism , and DNA repair and recombination, while important genes participating in the DNA-to-RNA transcription , protein translation and DNA/RNA replication are retained. The decrease in genome size is due to loss of protein coding genes and not due to lessening of inter-genic regions or open reading frame (ORF) size. Species that are naturally evolving and contain reduced sizes of genes can be accounted for an increased number of noticeable differences between them, thereby leading to changes in their evolutionary rates. When endosymbiotic bacteria related with insects are passed on to the offspring strictly via vertical genetic transmission, intracellular bacteria go across many hurdles during the process, resulting in the decrease in effective population sizes, as compared to the free-living bacteria. The incapability of the endosymbiotic bacteria to reinstate their wild type phenotype via a recombination process is called Muller's ratchet phenomenon. Muller's ratchet phenomenon, together with less effective population sizes, leads to an accretion of deleterious mutations in the non-essential genes of the intracellular bacteria. [ 29 ] This can be due to lack of selection mechanisms prevailing in the relatively "rich" host environment. [ 30 ] [ 31 ]
Competition can be defined as an interaction between organisms or species, in which the fitness of one is lowered by the presence of another. [ 32 ] Competition can also occur between cells within the same organism which is why older cells are usually eliminated from tissues. This allows the organism to stay as healthy as possible by constantly eliminating old cells and making new ones. [ 33 ] Limited supply of at least one resource (such as food , water , and territory ) used by both usually facilitates this type of interaction, although the competition can also be for other resources. [ 34 ]
Amensalism is a non-symbiotic, asymmetric interaction where one species is harmed or killed by the other, and one is unaffected by the other. [ 35 ] [ 36 ] There are two types of amensalism, competition and antagonism (or antibiosis). Competition is where a larger or stronger organism deprives a smaller or weaker one of a resource. Antagonism occurs when one organism is damaged or killed by another through a chemical secretion. An example of competition is a sapling growing under the shadow of a mature tree. The mature tree can rob the sapling of necessary sunlight and, if the mature tree is very large, it can take up rainwater and deplete soil nutrients. Throughout the process, the mature tree is unaffected by the sapling. Indeed, if the sapling dies, the mature tree gains nutrients from the decaying sapling. An example of antagonism is Juglans nigra (black walnut), secreting juglone , a substance which destroys many herbaceous plants within its root zone. [ 37 ]
The term amensalism is often used to describe strongly asymmetrical competitive interactions, such as between the Spanish ibex and weevils of the genus Timarcha which feed upon the same type of shrub. Whilst the presence of the weevil has almost no influence on food availability, the presence of ibex has an enormous detrimental effect on weevil numbers, as they consume significant quantities of plant matter and incidentally ingest the weevils upon it. [ 38 ]
Commensalism describes a relationship between two living organisms where one benefits and the other is not significantly harmed or helped. It is derived from the English word commensal , used of human social interaction . It derives from a medieval Latin word meaning sharing food, formed from com- (with) and mensa (table). [ 39 ] [ 40 ]
Commensal relationships may involve one organism using another for transportation ( phoresy ) or for housing ( inquilinism ), or it may also involve one organism using something another created, after its death ( metabiosis ). Examples of metabiosis are hermit crabs using gastropod shells to protect their bodies, and spiders building their webs on plants .
Mutualism or interspecies reciprocal altruism is a long-term relationship between individuals of different species where both individuals benefit. [ 39 ] Mutualistic relationships may be either obligate for both species, obligate for one but facultative for the other, or facultative for both.
Many herbivores have mutualistic gut flora to help them digest plant matter, which is more difficult to digest than animal prey. [ 5 ] This gut flora comprises cellulose-digesting protozoans or bacteria living in the herbivores' intestines. [ 41 ] Coral reefs result from mutualism between coral organisms and various algae living inside them. [ 42 ] Most land plants and land ecosystems rely on mutualism between the plants, which fix carbon from the air, and mycorrhyzal fungi, which help in extracting water and minerals from the ground. [ 43 ]
An example of mutualism is the relationship between the ocellaris clownfish that dwell among the tentacles of Ritteri sea anemones . The territorial fish protects the anemone from anemone-eating fish, and in turn, the anemone stinging tentacles protect the clownfish from its predators . A special mucus on the clownfish protects it from the stinging tentacles. [ 44 ]
A further example is the goby , a fish which sometimes lives together with a shrimp . The shrimp digs and cleans up a burrow in the sand in which both the shrimp and the goby fish live. The shrimp is almost blind, leaving it vulnerable to predators when outside its burrow. In case of danger, the goby touches the shrimp with its tail to warn it, and both quickly retreat into the burrow. [ 45 ] Different species of gobies ( Elacatinus spp. ) also clean up ectoparasites in other fish, possibly another kind of mutualism. [ 46 ]
A spectacular example of obligate mutualism is the relationship between the siboglinid tube worms and symbiotic bacteria that live at hydrothermal vents and cold seeps . The worm has no digestive tract and is wholly reliant on its internal symbionts for nutrition. The bacteria oxidize either hydrogen sulfide or methane, which the host supplies to them. These worms were discovered in the late 1980s at the hydrothermal vents near the Galapagos Islands and have since been found at deep-sea hydrothermal vents and cold seeps in all of the world's oceans. [ 47 ]
Mutualism improves both organism's competitive ability and will outcompete organisms of the same species that lack the symbiont. [ 48 ]
A facultative symbiosis is seen in encrusting bryozoans and hermit crabs . The bryozoan colony ( Acanthodesia commensale ) develops a cirumrotatory growth and offers the crab ( Pseudopagurus granulimanus ) a helicospiral-tubular extension of its living chamber that initially was situated within a gastropod shell. [ 49 ]
In a parasitic relationship, the parasite benefits while the host is harmed. [ 50 ] Parasitism takes many forms, from endoparasites that live within the host's body to ectoparasites and parasitic castrators that live on its surface and micropredators like mosquitoes that visit intermittently. Parasitism is an extremely successful mode of life; about 40% of all animal species are parasites, and the average mammal species is host to 4 nematodes, 2 cestodes, and 2 trematodes. [ 51 ]
Mimicry is a form of symbiosis in which a species adopts distinct characteristics of another species to alter its relationship dynamic with the species being mimicked, to its own advantage. Among the many types of mimicry are Batesian and Müllerian, the first involving one-sided exploitation, the second providing mutual benefit. Batesian mimicry is an exploitative three-party interaction where one species, the mimic, has evolved to mimic another, the model, to deceive a third, the dupe. In terms of signalling theory , the mimic and model have evolved to send a signal; the dupe has evolved to receive it from the model. This is to the advantage of the mimic but to the detriment of both the model, whose protective signals are effectively weakened, and of the dupe, which is deprived of an edible prey. For example, a wasp is a strongly defended model, which signals with its conspicuous black and yellow coloration that it is an unprofitable prey to predators such as birds which hunt by sight; many hoverflies are Batesian mimics of wasps, and any bird that avoids these hoverflies is a dupe. [ 52 ] [ 53 ] In contrast, Müllerian mimicry is mutually beneficial as all participants are both models and mimics. [ 54 ] [ 55 ] For example, different species of bumblebee mimic each other, with similar warning coloration in combinations of black, white, red, and yellow, and all of them benefit from the relationship. [ 56 ]
Cleaning symbiosis is an association between individuals of two species, where one (the cleaner) removes and eats parasites and other materials from the surface of the other (the client). [ 57 ] It is putatively mutually beneficial, but biologists have long debated whether it is mutual selfishness, or simply exploitative. Cleaning symbiosis is well known among marine fish, where some small species of cleaner fish – notably wrasses , but also species in other genera – are specialized to feed almost exclusively by cleaning larger fish and other marine animals. [ 58 ] In a supreme situation, the host species (fish or marine life) will display itself at a designated station deemed the "cleaning station". [ 59 ]
Cleaner fish play an essential role in the reduction of parasitism on marine animals. Some shark species participate in cleaning symbiosis, where cleaner fish remove ectoparasites from the body of the shark. [ 60 ] A study by Raymond Keyes addresses the atypical behavior of a few shark species when exposed to cleaner fish. In this experiment, cleaner wrasse (Labroides dimidiatus) and various shark species were placed in a tank together and observed. The different shark species exhibited different responses and behaviors around the wrasse. For example, Atlantic and Pacific lemon sharks consistently react to the wrasse fish in a fascinating way. During the interaction, the shark remains passive and the wrasse swims to it. It begins to scan the shark's body, sometimes stopping to inspect specific areas. Commonly, the wrasse would inspect the gills, labial regions, and skin. When the wrasse makes its way to the mouth of the shark, the shark often ceases breathing for up to two and a half minutes so that the fish is able to scan the mouth. Then, the fish passes further into the mouth to examine the gills, specifically the buccopharyngeal area, which typically holds the most parasites. When the shark begins to close its mouth, the wrasse finishes its examination and goes elsewhere. Male bull sharks exhibit slightly different behavior at cleaning stations: as the shark swims into a colony of wrasse fish, it drastically slows its speed to allow the cleaners to do their job. After approximately one minute, the shark returns to normal swimming speed. [ 60 ]
Symbiosis is increasingly recognized as an important selective force behind evolution; [ 5 ] [ 61 ] many species have a long history of interdependent co-evolution . [ 62 ]
Although symbiosis was once discounted as an anecdotal evolutionary phenomenon, evidence is now overwhelming that obligate or facultative associations among microorganisms and between microorganisms and multicellular hosts had crucial consequences in many landmark events in evolution and in the generation of phenotypic diversity and complex phenotypes able to colonise new environments. [ 63 ]
Mutualistic symbiosis can sometimes evolve from parasitism or commensalism , Fungi's relationship to plants in the form of mycelium evolved from parasitism and commensalism . Under certain conditions species of fungi previously in a state of mutualism can turn parasitic on weak or dying plants. [ 64 ] Likewise the symbiotic relationship of clown fish and sea anemones emerged from a commensalist relationship. [ 65 ] [ 66 ] [ 67 ]
Evolution originated from changes in development where variations within species are selected for or against because of the symbionts involved. [ 68 ] The hologenome theory relates to the holobiont and symbionts genome together as a whole. [ 69 ] Microbes live everywhere in and on every multicellular organism. [ 70 ] Many organisms rely on their symbionts in order to develop properly, this is known as co-development. In cases of co-development the symbionts send signals to their host which determine developmental processes. Co-development is commonly seen in both arthropods and vertebrates. [ 68 ]
One hypothesis for the origin of the nucleus in eukaryotes (plants, animals, fungi, and protists ) is that it developed from a symbiogenesis between bacteria and archaea. [ 5 ] [ 71 ] [ 72 ] It is hypothesized that the symbiosis originated when ancient archaea, similar to modern methanogenic archaea, invaded and lived within bacteria similar to modern myxobacteria, eventually forming the early nucleus. This theory is analogous to the accepted theory for the origin of eukaryotic mitochondria and chloroplasts, which are thought to have developed from a similar endosymbiotic relationship between proto-eukaryotes and aerobic bacteria. [ 73 ] Evidence for this includes the fact that mitochondria and chloroplasts divide independently of the cell, and that these organelles have their own genome. [ 74 ]
The biologist Lynn Margulis , famous for her work on endosymbiosis , contended that symbiosis is a major driving force behind evolution . She considered Darwin 's notion of evolution, driven by competition, to be incomplete and claimed that evolution is strongly based on co-operation , interaction , and mutual dependence among organisms. According to Margulis and her son Dorion Sagan , " Life did not take over the globe by combat , but by networking ." [ 75 ]
About 80% of vascular plants worldwide form symbiotic relationships with fungi, in particular in arbuscular mycorrhizas . [ 76 ] The mutualism evolved due to the limitation of plant root capacity for absorbing soil nutrients, especially phosphate and nitrogen, which are crucial for plant growth. [ 77 ]
Flowering plants and the animals that pollinate them have co-evolved. Many plants that are pollinated by insects (in entomophily ), bats , or birds (in ornithophily ) have highly specialized flowers modified to promote pollination by a specific pollinator that is correspondingly adapted. The first flowering plants in the fossil record had relatively simple flowers. Adaptive speciation quickly gave rise to many diverse groups of plants, and, at the same time, corresponding speciation occurred in certain insect groups . Some groups of plants developed nectar and large sticky pollen, while insects evolved more specialized morphologies to access and collect these rich food sources. In some taxa of plants and insects, the relationship has become dependent, [ 78 ] where the plant species can only be pollinated by one species of insect. [ 79 ]
The acacia ant ( Pseudomyrmex ferruginea ) is an obligate plant ant that protects at least five species of "Acacia" ( Vachellia ) [ a ] from preying insects and from other plants competing for sunlight, and the tree provides nourishment and shelter for the ant and its larvae. [ 80 ] [ 81 ]
Seed dispersal is the movement, spread or transport of seeds away from the parent plant. Plants have limited mobility and rely upon a variety of dispersal vectors to transport their propagules, including both abiotic vectors such as the wind and living ( biotic ) vectors like birds. In order to attract animals, these plants evolved a set of morphological characters such as fruit colour, mass, and persistence correlated to particular seed dispersal agents. [ 82 ] For example, plants may evolve conspicuous fruit colours to attract avian frugivores, and birds may learn to associate such colours with a food resource. [ 83 ]
Rhizobia are bacteria that fix nitrogen inside the root nodules of legumes such as beans and clover . To express genes for nitrogen fixation, rhizobia require a plant host ; they cannot independently fix nitrogen. [ 84 ] Rhizobia infect the legume's roots and produce nodules where they fix nitrogen gas (N 2 ) from the atmosphere, turning it into a more readily useful form of nitrogen. From here, the nitrogen is exported and used for growth in the legume. [ 85 ]
A lichen is a hybrid colony of algae or cyanobacteria living symbiotically among the hyphal filaments of multiple species of fungi and other micro-organisms embedded in the outer layer or cortex, in a mutualistic relationship. Lichens are able to flourish in harsh environments such as bare rocks. [ 86 ] [ 87 ] [ 88 ] | https://en.wikipedia.org/wiki/Symbiosis |
The biological term symbiosis was first used in chemistry by C. K. Jørgensen in 1964, [ 1 ] to refer to the process by which a hard ligand on a metal predisposes the metal to receive another hard ligand rather than a soft one. Two superficially antithetical phenomena occur: symbiosis and antisymbiosis .
This is found principally with soft metals. Two soft ligands in mutual trans position will have a destabilizing effect on each other. [ 2 ] [ 3 ] The effect is also found with borderline metals in the presence of high trans effect ligands. For example the selenocyanate ion trans to the soft carbon dioxide in trans-Rh(PPh 3 ) 2 (CO)(NCSe) bonds via the nitrogen, the harder of its two donors. [ 4 ] The phenomenon may be explained in terms of a trans influence:
“With two π-acid ligands in mutual trans positions at a class-b metal, there would be a destabilizing competition for the dπ electrons on the metal. A π-acid bonded to a soft metal thus makes a metal a harder Lewis acid . Similarly a soft σ-donor will tend to polarize the electron density on a soft metal, causing it to favour an electrovalently bonded ligand in the trans position.” [ 5 ]
This effect occurs with class-a metals such as iron (II). The Cyclopentadienyl complex (C 5 H 5 )Fe(CO) 2 (SCN) is an example of chemical symbiosis. The cyclopentadienyl directs the thiocyanate to bond through its softer Sulphur donor. [ 6 ] A more definitive example are the halopentamminocobalt(III) ions, Co(NH 3 ) 5 X 2+ , which are more stable when the halogen , X, is fluoride than with iodide, and the halopentcyanocobalt(III) ions, Co(CN) 5 X 3− , which are most stable when the halogen is iodine. [ 7 ]
“ Hard bases (electronegative donor atoms) retain their valence (outer shell) electrons when attached to a given central metal ion, thus enabling the metal ion to retain more of its positive charge, making it a hard Lewis acid . With soft bases the central metal atom is made a softer Lewis acid , because the metal’s positive charge is reduced by delocalization of electron density from the ligand into the ligand-metal bond. But we have the distinction that with a class-a metal there is little concomitant polarization of the electron density away from the trans position of the metal. In addition, symbiosis , unlike antisymbiosis , is probably not specifically trans directional, and is just as effective in, say, tetrahedral complexes.” [ 8 ] | https://en.wikipedia.org/wiki/Symbiosis_(chemical) |
Amoebozoa of the free living genus Acanthamoeba and the social amoeba genus Dictyostelium are single celled eukaryotic organisms that feed on bacteria, fungi, and algae through phagocytosis, with digestion occurring in phagolysosomes. Amoebozoa are present in most terrestrial ecosystems including soil and freshwater. [ 1 ] Amoebozoa contain a vast array of symbionts that range from transient to permanent infections, confer a range of effects from mutualistic to pathogenic, and can act as environmental reservoirs for animal pathogenic bacteria. [ 2 ] As single celled phagocytic organisms, amoebas simulate the function and environment of immune cells like macrophages , and as such their interactions with bacteria and other microbes are of great importance in understanding functions of the human immune system, as well as understanding how microbiomes can originate in eukaryotic organisms. [ 3 ] [ 4 ]
Some microorganisms have evolved to become resistant to Amoebozoa, and are able to survive, grow, and exit free-living amoebae after phagocytosis. In order for an organism to survive in an Amoebozoa, they have developed a way to avoid or survive digestion by their host's acidic and oxidative phagolysosomes. [ 5 ] Many of these amoeba-resistant microorganisms (ARMs) survive either in the amoeba cytoplasm or in host derived vacuoles surrounded by plasma membrane, [ 6 ] allowing them to not only avoid digestion, but actively reproduce inside their host with some are capable of lysing the amoeba host cell. [ 7 ] Known symbionts of Amoebozoa include bacteria from Alphaproteobacteria , Betaproteobacteria , Bacteroidetes , Firmicutes , Proteobacteria , Chlamydiae , and Paraburkholderia , all with different effects on their host, even within the same phylum. [ 5 ] For example, some Chlamydiae bacteria are able to increase the growth rates of their hosts or increase motility, other Chlamydiae strains are able to fight off other pathogenic symbionts like legionella , and some Chlamydiae are parasitic and decrease host fitness. [ 8 ]
Many free living amoeba species inhabit aquatic environments, including manufactured water systems. While in their encysted state , amoebas have a high resistance to extreme temperatures, UV radiation , osmolarity , and pH. [ 2 ] Some species of pathogenic bacteria are able to take advantage of this resistance and survive in environments that would usually destroy them, and are able to use the amoebas as a " Trojan horse " to travel to new environments and animal hosts. [ 2 ] Legionella pneumophila , a known human pathogen, has been observed in at least 13 different species of amoeba. [ 2 ] Legionella has been shown to survive inside of an encysted amoeba host in chlorine treated water , and can release from the host in respirable vesicles when treated with biocides , with each vesicle possibly containing hundreds of legionella bacteria spread by aerosolized water. [ 9 ] Recent human outbreaks of Legionella are likely due to aerosolized water containing amoeba derived Legionella vesicles produced by modern devices such as air-conditioning systems, water cooling towers , showers, clinical respiration devices, and whirlpool baths that have been contaminated with host amoebae. [ 2 ]
Another unique example of symbiosis occurs in the social amoeba Dictyostelium discoideum . D. discoideum and other social amoeba differ from free living Acanthamoeba in that instead of encysting, they undergo a social cycle where individual D. discoideum cells aggregate together in a food scarce environment. This social cycle results in a differentiation between cells: ~20% are sacrificed to form a structural stalk, [ 10 ] some transform into sentinel cells with immune like and detoxifying functions, [ 4 ] and the rest of the aggregated amoeba form a ball of spores located in protective fruiting body. [ 8 ] This fruiting body gives some amoeba from that population a chance to be transported to a food rich environment and survive. If they are not transported to a food rich environment, then the amoebas of that fruiting body will starve. Some D. discoideum amoebas contain Burkholderia bacteria that have been found to form a type of farming symbiosis with their discoideum hosts, who have reduced sentinel cell numbers. [ 11 ] Burkholderia are able to persist in the fruiting bodies of their hosts that are carried by an animal or environmental path to a new environment. If there are few food bacteria in that new environment, then the social amoeba are able to seed the area with the contained Burkholderia and thus develop a food source. Farmer amoebas do produce fewer spores in a food rich environment than non-farmer amoebas, but this cost is countered by farmers’ ability to replenish their food supply when dispersing to food-poor environments. [ 11 ] Additionally, some farmed Burkholderia produce compounds that are not detrimental to the amoeba host, but are detrimental to nonfarmer amoebas, giving the farmer amoebas a competitive advantage in mixed populations. [ 11 ]
Giant viruses , or nucleocytoplasmic large DNA viruses , frequently infect Amoebozoa and other protists causing amoeba lysis and cell rounding in 12 hours and amoeba population collapse in 55 hours. [ 3 ] As such, there is a strong selective pressure on both Amoebozoa and their symbionts to resist these viruses. Acanthamoeba hattchettii is one species affected by giant viruses, and some use a bacterial symbiont ( Parachlamydia acanthamoebae) to counter giant viruses from Marseilleviridae and Mimiviridae . [ 3 ] Acanthamoeba that are infected with the chlamydiae symbiont and giant viruses are able to avoid lysis and cell rounding that normally occur with infection by repressing viral replication after the virus enters the cell, likely due to secondary metabolites produced by the symbiont. [ 3 ] Acanthamoeba that are not infected by the symbiont or virus have the highest fitness with a doubling time that is twice as fast, Acanthamoeba that are infected with the chlamydia symbiont have the same fitness when infected by the virus and when not, and Acanthamoeba that are infected with just the virus have the lowest fitness with a total population collapse. Therefore, the chlamydiae symbiont acts as a mutualist with a significantly positive fitness effect during viral predation, but is also consistent with the parasitic lifestyle of many chlamydiae when acanthamoeba is not a victim of viral predation. [ 3 ] As an obligate intracellular bacterium, the chlamydiae symbiont is effectively competing in the same niche as other giant viruses, and so has evolved to protect its host from its natural competitor. [ 3 ] | https://en.wikipedia.org/wiki/Symbiosis_in_Amoebozoa |
Symbiosis in lichens is the mutually beneficial symbiotic relationship of green algae and/or blue-green algae ( cyanobacteria ) living among filaments of a fungus , forming lichen . [ 1 ] [ 2 ] [ 3 ]
Living as a symbiont in a lichen appears to be a successful way for a fungus to derive essential nutrients, as about 20% of all fungal species have adopted this mode of life. [ 4 ] The autotrophic symbionts occurring in lichens are a wide variety of simple, photosynthetic organisms commonly and traditionally known as “algae”. These symbionts include both prokaryotic and eukaryotic organisms. [ 5 ] [ 4 ] [ 3 ] [ 2 ]
A lichen is a combination of fungus and/or algae and/or cyanobacteria that has a very different form ( morphology ), physiology , and biochemistry than any of the constituent species growing separately. The algae or cyanobacteria benefit their fungal partner by producing organic carbon compounds through photosynthesis . In return, the fungal partner benefits the algae or cyanobacteria by protecting them from the environment by its filaments, which also gather moisture and nutrients from the environment, and (usually) provide an anchor to it. [ 5 ] [ 4 ] [ 3 ] [ 2 ]
The majority of the lichens contain eukaryotic autotrophs belonging to the Chlorophyta (green algae) or to the Xanthophyta (yellow-green algae). About 90% of all known lichens have a green alga as a symbiont. Among these, Trebouxia is the most common genus, occurring in about 20% of all lichens. [ 6 ] The second most commonly represented green alga genus is Trentepohlia . Overall, about 100 species are known to occur as autotrophs in lichens. All the algae and cyanobacteria are believed to be able to survive separately, as well as within the lichen; that is, at present no algae or cyanobacteria are known which can only survive naturally as part of a lichen. [ 7 ] Common algal partners are Trebouxia , Pseudotrebouxia , or Myrmecia . [ 5 ]
The prokaryotes belong to the Cyanobacteria , which are often called by their old name “ bluegreen algae ”. Cyanobacteria occur as symbionts only in about 8% of known lichens. The most commonly occurring genera of symbiotic cyanobacteria are Nostoc [ 7 ] and Scytonema . [ 4 ]
Both the lichen and the fungus partner bear the same scientific name, and the lichens are being integrated into the classification schemes for fungi. Depending on context, the taxonomic name can be meant to refer to the entire lichen, or just the fungus that is part of the lichen.
The alga or cyanobacterium bears its own scientific name, which has no relationship to either the name of the lichen or the fungus. [ 8 ]
About 20% of all fungal species are able to form lichens. The fungal partner may be an Ascomycete or Basidiomycete . [ 4 ] Overall, about 98% of lichens have an ascomycetous mycobiont. Next to the Ascomycota, the largest number of lichenized fungi occur in the unassigned fungi imperfecti . Comparatively few basidiomycetes are lichenized, but these include agarics , such as species of Lichenomphalia , clavarioid fungi , such as species of Multiclavula , and corticioid fungi , such as species of Dictyonema .
The largest number of lichenized fungi occur in the Ascomycota , with about 40% of species forming such an association. [ 8 ] Some of these lichenized fungi occur in orders with nonlichenized fungi that live as saprotrophs or plant parasites (for example, the Leotiales , Dothideales , and Pezizales ).
Other lichen fungi occur in only five orders in which all members are engaged in this habit (Orders Graphidales , Gyalectales , Peltigerales , Pertusariales , and Teloschistales ). Lichenized and nonlichenized fungi can even be found in the same genus or species. [ citation needed ]
The photosynthetic component of a lichen is called the photobiont or phycobiont . [ 9 ] The layer of tissue containing the cells of the photobiont is called the “photobiontic layer”. [ 9 ]
Approximately 100 species of photosynthetic partners from 40 genera and 5 distinct classes (prokaryotic: Cyanophyceae ; eukaryotic: Trebouxiophyceae , Phaeophyceae , Chlorophyceae ) have been found to associate with the lichen-forming fungi. [ 10 ]
A particular fungus species and algal species are not necessarily always associated together in a lichen. One fungus, for example, can form lichens with a variety of different algae. The thalli produced by a given fungal symbiont with its differing partners will be similar, and the secondary metabolites identical, indicating that the fungus has the dominant role in determining the morphology of the lichen. Further, the same algal species can occur in association with different fungal partners. Lichens are known in which there is one fungus associated with two or even three algal species. Rarely, the reverse can occur, and two or more fungal species can interact to form the same lichen. [ 7 ]
About 90% of all known lichens have a green alga as a symbiont. [ 11 ]
Although the photobionts are almost always green algae (Chlorophyta), sometimes the lichen contains Cyanobacteria, taxonomically bacteria, and sometimes both types of photobionts are found in the same lichen.
A cyanolichen is a lichen with a cyanobacterium as its main photosynthetic component ( photobiont ). [ 12 ] Many cyanolichens are small and black, and have limestone as the substrate.
Another cyanolichen group, the jelly lichens (e.g., from the genera Collema or Leptogium ) are large and foliose (e.g., species of Peltigera , Lobaria , and Degelia . These lichen species are grey-blue, especially when dampened or wet. Many of these characterize the Lobarion communities of higher rainfall areas in western Britain, e.g., in the Celtic Rainforest .
The process by which the fungus and the photobioant comes together is called "lichenization". There are five steps to this process: [ 14 ]
Scientists have successfully replicated lichenization in the laboratory; it takes anywhere from a few months to a few years for the isolated fungus and photobiont of a lichen to grow back into a complete thallus. It is unclear how long the process takes in the wild. [ 14 ] Throughout lichenization (and after its completion), the fungus and the alga continue to exchange different chemical signals. [ 14 ]
Some fungi can only be found living on lichens as obligate parasites ; They are not considered part of the lichen. These are referred to as “ lichenolous fungi ”.
Some of these parasitic lichenolous fungi form their own thalli and become lichen themselves; they are called "lichenicolous lichens". They steal the symbioant of another lichen (kleptosymbiosis) into their own structure (trans-lochenization). [ 14 ] | https://en.wikipedia.org/wiki/Symbiosis_in_lichens |
A symbiosome is a specialised compartment in a host cell that houses an endosymbiont in a symbiotic relationship. [ 1 ]
The term was first used in 1983 to describe the vacuole structure in the symbiosis between the animal host the Hydra , and the endosymbiont Chlorella . Symbiosomes are also seen in other cnidaria - dinoflagellate symbioses, including those found in coral - algal symbioses. In 1989 the concept was applied to the similar structure found in the nitrogen-fixing root nodules of certain plants. [ 1 ]
The symbiosome in the root nodules has been much more successfully researched due in part to the complexity of isolating the symbiosome membrane in animal hosts. [ 1 ] The symbiosome in a root nodule cell in a plant is an organelle-like structure that has formed in a symbiotic relationship with nitrogen-fixing bacteria . The plant symbiosome is unique to those plants that produce root nodules. [ 2 ] The majority of such symbioses are made between legumes and diazotrophic Rhizobia bacteria . The rhizobia-legume symbioses are the most studied due to the importance in agriculture. [ 3 ] [ 4 ]
Each symbiosome in a root nodule cell encloses a single rhizobium that differentiates into a bacteroid. However, in some cases a symbiosome may house several bacteroids. [ 5 ] The symbiosome membrane, or peribacteroid membrane, surrounds the bacteroid membrane, separated by a symbiosome space. This unit provides an inter-kingdom, micro-environment for the production of nitrogen for the plant, [ 3 ] [ 6 ] and the receipt of malate for energy for the bacteroid. [ 7 ]
The concept of the symbiosome was first described in 1983, by Neckelmann and Muscatine, as seen in the symbiotic relationship between Chlorella ( a class of green algae , and Hydra a cnidarian animal host. [ 1 ] Until then it had been described as a vacuole . A few years later in 1989, Lauren Roth with Gary Stacey [ 8 ] as well as Robert B Mellor [ 9 ] applied this concept to the nitrogen-fixing unit seen in the plant root nodule, [ 1 ] previously called an infection vacuole . [ 10 ]
This has since engendered a great deal of research, one result of this has been the provision of a more detailed description of the symbiosome (peribacteroid) membrane, as well as comparisons with similar structures in Vesicular Arbuscular Mycorrhizal symbioses in plants. [ 11 ] In the animal models, the symbiosome has a more complex arrangement of membranes, such that it has proved difficult to isolate, purify and study. [ 1 ]
A symbiosome is formed as a result of a complex and coordinated interaction between the symbiont host and the endosymbiont . [ 5 ] At the point of entry into a symbiont host cell , part of the cell's membrane envelops the endosymbiont and breaks off into the cytoplasm as a discrete unit, an organelle-like vacuole called the symbiosome. [ 5 ] [ 12 ] This is an endocytosis -like process that forms a symbiosome rather than an endosome . In plants this process is unique. [ 13 ]
The symbiosome membrane is separated from the endosymbiont membrane by a space known as the symbiosome space , which allows for the exchange of solutes between the symbionts. [ 14 ] [ 12 ] In the plant root nodule the symbiosome membrane is also called the peribacteroid membrane. [ 13 ]
In the legume - rhizobia symbioses the symbiosome is the nitrogen-fixing unit in the plant, formed by an interaction of plant and bacterial signals, and their cooperation. The legumes are protein-rich, and have a high demand for nitrogen that is usually available from nitrates in the soil. When these are scarce the plant secretes flavonoids that attract free-living diazotrophic (nitrogen-fixing) rhizobia to their root hairs . In turn the bacteria release Nod factors that stimulate the infection process in the plant. [ 1 ] [ 13 ]
To enable infection the tip of the root hair curls over the rhizobia and by an inward growth produces an infection thread to carry the endosymbionts into the cortical cells. At the same time the cortical cells divide to produce the tough root nodules that will house and protect the bacteria. [ 15 ] [ 13 ] The bacterial production of extracellular polymeric substance (EPS) is seen to be necessary for enabling infection. [ 13 ] The rhizobia infect the plant in large numbers, only seen to be actively dividing at the tip of the injection thread, where they are released into the cells inside symbiosomes. [ 15 ] [ 1 ] The symbiosome is formed as a result of an endocytosis-like process that produces an endosome. Typically endosomes target to lysosomes , but the symbiosome re-targets the host-cell proteins.
The changes in the plant needed to form the infection thread, the increased division of the cortical cells, the formation of the root nodule, and symbiosome, are brought about by dynamic changes in the actin cytoskeleton . [ 16 ] [ 13 ] Filamentous actin (F-actin) channels the elongation of the injection threads and short F-actin fragments are dotted around the symbiosome membrane. [ 16 ] The bacteria are released as injection drops into the host root nodule cells where the plasma membrane encloses them in the organelle-like structure of the symbiosome. In most plants a symbiosome encloses a single endosymbiont bacterium but some types may contain more than one. A negative feedback loop called the autoregulation of nodulation works to balance the need for nitrogen and thus the formation of nodules. [ 17 ] [ 18 ]
The outer host-cell derived symbiosome membrane encloses a space called the symbisome space or the peribacteroid space that surrounds the endosymbiont. In order for the symbiosome to be established as a nitrogen-fixing unit the enclosed bacterium has to be terminally differentiated
into a morphologically changed bacteroid . The bacterium in the soil is free-living and motile. In the symbiosome it has to change its gene expression to adapt to a non-motile, non-reproductive form as the bacteroid. This change is noted by an increase in the size of the bacterium and its elongation. The bacterial membrane is also made permeable. [ 19 ] [ 1 ] [ 13 ] The process of differentiation is plant-driven using peptides known as nodule specific cysteine-rich peptides ( NCR peptides).
NCRs are antimicrobial peptides that are similar to the defensin peptides used in mammals in response to invading pathogens. The NCRs are targeted to the symbiosome where they induce differentiation of the bacterium to the bacteroid. A major effect of NCR targeting is to limit the reproductive ability of the endosymbiont. These changes are controlled, since the bacterium is not killed as a result of exposure to the NCRs. Some of that control comes from the bacterium itself. [ 20 ] [ 21 ] [ 5 ] In order to survive the NCR activities, the bacteria need to produce a protein called BacA . In addition the lipopolysaccharide produced by the bacteria is modified by an unusual fatty acid that also gives protection against environmental stresses. These defensive measures help the differentiation process and ensures their survival as bacteroids. Some strains of rhizobia produce a peptidase that degrades the NCRs. [ 20 ] [ 21 ]
The established bacteroid is able to fix nitrogen into a chemically usable form of ammonium for the plant. This is an energy-demanding process fuelled by the plant's carbohydrates. [ 13 ] Transport vesicles form in the symbiosome membrane allowing the passage of ammonium into the symbiosome space from the bacteroid, and the passage of plant nutrients to the bacteroid. [ 13 ] The rhizobia infect the plant in large numbers where they are released into the cells inside symbiosomes. They are protected by the tough structure of the root nodule. [ 15 ]
The most well studied symbiosis involving an animal host is that between the cnidaria and the dinoflagellates , most commonly the single-celled zooxanthellae . The symbiosis of the Chlorella – Hydra first described the symbiosome . The coral Zoanthus robustus has been used as a model organism to study the symbiosis with its microsymbiont algal species of Symbiodinium , with a focus on the symbiosome and its membranes. Methods for isolating the symbiosome membranes have been looked for – the symbiont in the animal host has a multilayered membrane complex which has proved resistant to disruption making their isolation difficult. [ 1 ] [ 22 ]
The endosymbiont dinoflagellates are used for their ability to photosynthesise and provide energy, giving the host cnidarians such as corals , and anemones , plant properties. [ 23 ] Free-living dinoflagellates are ingested into the gastrodermal cells of the host, and their symbiosome membrane is derived from the host cell. [ 24 ] The process of symbiosome formation is often seen in the animal host to be that of phagocytosis , [ 24 ] and it is hypothesised that the symbiosome is a phagosome that has been subject to early arrest. [ 25 ]
A similar structure to the symbiosome is the parasitophorous vacuole formed within host cells infected by apicomplexan parasites . The vacuole is derived from the host cell plasma membrane. It is made safe from the host's endolysomal system by modifying-proteins released by the parasite. [ 26 ] [ 27 ] The parasitophorous vacuole membrane is greatly remodelled by the parasite. [ 28 ] | https://en.wikipedia.org/wiki/Symbiosome |
Symbiotic bacteria are bacteria living in symbiosis with another organism or each other. For example, rhizobia living in root nodules of legumes provide nitrogen fixing activity for these plants. [ 1 ]
Types of symbiotic relationships are mutualism , commensalism , parasitism , and amensalism . [ 2 ]
Endosymbionts live inside other organisms whether that be in their bodies or cells. [ 3 ] The theory of endosymbiosis, as known as symbiogenesis , provides an explanation for the evolution of eukaryotic organisms. According to the theory of endosymbiosis for the origin of eukaryotic cells, scientists believe that eukaryotes originated from the relationship between two or more prokaryotic cells approximately 2.7 billion years ago. It is suggested that specifically ancestors of mitochondria and chloroplasts entered into an endosymbiotic relationship with another prokaryotic cell, eventually evolving into the eukaryotic cells that people are familiar with today. [ 4 ]
Ectosymbiosis is defined as a symbiotic relationship in which one organism lives on the outside surface of a different organism. [ 3 ] For instance, barnacles on whales is an example of an ectosymbiotic relationship where the whale provides the barnacle with a home, a ride, and access to food. The whale is not harmed, but it also does not receive any benefits so this is also an example of commensalism. An example of ectosymbiotic bacteria is cutibacterium acnes . These bacteria are involved in a symbiotic relationship with humans on whose skin they live. Cutibacterium acnes can cause acne when the skin becomes too oily, but they also reduce the skin's susceptibility to skin diseases caused by oxidative stress. [ 5 ]
Certain plants establish a symbiotic relationship with bacteria, enabling them to produce nodules that facilitate the conversion of atmospheric nitrogen to ammonia. In this connection, cytokinins have been found to play a role in the development of root fixing nodules. [ 6 ] It appears that not only must the plant have a need for nitrogen fixing bacteria, but they must also be able to synthesize cytokinins which promote the production of root nodules, required for nitrogen fixation.
Symbiotic bacteria are able to live in or on plant or animal tissue . In digestive systems, symbiotic bacteria help break down foods that contain fiber . They also help produce vitamins . Symbiotic bacteria can live near hydrothermal vents. They usually have a mutual relationship with other bacteria. Some live in tube worms .
There are two major modes of transmission for symbiotic bacteria. The first is horizontal transmission in which microbes are acquired from the environment and either the environment or the host population serves as the inoculum for the symbiosis. [ 7 ] An example of horizontal transmission is the deep sea tube worm and its symbiont. [ 7 ] The second type of transmission is vertical transmission in which the symbiont is passed down from the parent to the offspring and there is no aposymbiotic phase. [ 7 ] An example of vertical transmission is seen in Drosophila melanogaster and its Wolbachia spp. symbionts. [ 7 ]
Corals have been found to form characteristic associations with symbiotic nitrogen-fixing bacteria. [ 8 ] Corals have evolved in oligotrophic waters which are typically poor in nitrogen. Corals must therefore form a mutualistic relationship with nitrogen fixing organism, in this case the subject of this study, namely Symbiodinium . In addition to this dinoflagellate, coral also form relationships with bacteria, archae and fungi. [ 9 ] The problem is that these dinoflagellates are also nitrogen limited and must form a symbiotic relationship with another organism; here it is suggested to be diazotrophs. In addition, cyanobacteria have been found to possess genes that enable them to undergo nitrogen fixation. [ 8 ] This particular study goes further to investigate the possibility that in addition to the named dinoflagellate and certain cyanobacteria, endosymbiotic algae and the coral contain enzymes enabling them to both undergo ammonium assimilation.
Due to the small size of the genome of most endosymbionts, they are unable to exist for any length of time outside of the host cell, thereby preventing a long-term symbiotic relationship. However, in the case of the endonuclear symbiotic bacterium Holospora, it has been discovered [ 10 ] that Holospora species can maintain their infectivity for a limited time and form a symbiotic relationship with Paramecium species.
There is a mutualistic relationship between legumes and rhizobial bacteria enabling the plants to survive in an otherwise nitrogen-poor soil environment. Co-evolution is described as a situation where two organisms evolve in response to one another. In a study reported in Functional Ecology , [ 11 ] these scientists investigated whether such a mutualistic relationship conferred an evolutionary advantage to either plant or symbiont. They did not find that the rhizobial bacteria studied had any evolutionary advantage with their host but did find great genetic variation among the populations of rhizobial bacteria studied.
Symbiotic, chemosynthetic bacteria that have been discovered associated with mussels ( Bathymodiolus ) located near hydrothermal vents have a gene that enables them to utilize hydrogen as a source of energy, in preference to sulphur or methane as their energy source for production of energy. [ 2 ]
Termites are known by many as pests that feed on wood. However, termites cannot digest the wood alone. Instead, they rely on a non-bacterial protozoan called Trichonympha to help in the digestion process. [ 12 ] Trichonympha is an endosymbiont that lives inside termites and also acts as a host to bacterial symbionts. The bacteria inside Trichonympha in termites produces cellulase. Cellulase enzymes are used to break down cellulose which is found in plants' cell walls. The termites, the gut protist Trichonympha , and the cellulase-producing bacteria are all involved in a 3-way obligate symbiotic mutualism. The termites benefit from the other two species because they transform the wood into nutrients that the termites can digest. Additionally, the Trichonympha benefit from the termites because the termites provide a place to live and access to food. The Trichonympha also benefit from the bacteria because they help break down the cellulose in wood that the protist consumes. Finally, the bacteria benefits because it gains a place to live and the nutrients it needs to survive.
The human gut contains approximately thirty-eight trillion microbes. [ 13 ] The gut is a dynamic ecosystem as it is composed of both constant and transient components, meaning some bacteria establishes itself and remains throughout the human’s lifetime and other bacteria is ingested and later leaves in feces. [ 14 ] When babies are born, they are born without any bacteria in their intestines. However, as soon as they enter the world, they begin accumulating gut bacteria through food and other means. [ 15 ] Most bacteria in the human body are actually good for us and help with carrying out necessary life processes. Gut bacteria in humans often aid in the breakdown of foods and synthesize important vitamins that could not be processed by humans alone. [ 16 ] Therefore, humans must be careful when taking antibiotics when they are sick. Antibiotics do not differentiate between the good and bad bacteria in our bodies and therefore, kill both. If not treated carefully, this can lead to issues with the gastrointestinal tract because of an imbalance of bacteria in this microbiome. [ 17 ] Therefore, some doctors recommend taking a probiotic when taking antibiotics to restore the good bacteria.
Organisms typically establish a symbiotic relationship due to their limited availability of resources in their habitat or due to a limitation of their food source. Triatomine vectors have only one host and therefore must establish a relationship with bacteria to enable them to obtain the nutrients required to maintain themselves. [ 18 ]
A use for symbiotic bacteria is in paratransgenesis for controlling important vectors for disease, such as the transmission of Chagas disease by Triatome kissing bugs .
Symbiotic bacteria in legume roots provide the plants with ammonia in exchange for the plants' carbon and a protected home. | https://en.wikipedia.org/wiki/Symbiotic_bacteria |
Symbiotic fermentation is a form of fermentation in which multiple organisms ( yeasts , acetic acid bacteria , lactic acid bacteria and others) interact in symbiosis in order to produce the desired product. For example, a yeast may produce ethanol, which is then consumed by an acetic acid bacterium. [ 1 ] Described early on as the fermentation of sugars following saccharification in a mixed fermentation process. [ 2 ]
The earliest mention of the term can be found in a lecture given by Dr. Allan Macfadyen of the Jenner Institute of Preventative Medicine in 1902. Dr. Macfadyen described symbiotic fermentation as noting "a close relationship between the organisms at work, the action of one aiding or modifying the action of the other, whilst both members are more active as a results of the partnership." [ 2 ] Fermentative microorganisms have had a deep history as seen by kefir and kumis fermentations of milk by Nomadic tribes in Russia, as well as Japanese koji fermentation (see Aspergillus oryzae ).
In 1927, Dr. Aldo Castellani defined symbiotic fermentation as "two microorganisms neither of which alone produces fermentation with gas in certain carbohydrates, may do so when living in symbiosis or when artificially mixed." [ 3 ] He based this definition on the observation that ordinary bakers yeast consisted of two or more microorganisms- Saccharomyces and Bacilli. He performed experiments to show that when two different Bacilli species were grown in culture together with maltose as the sugar, gas was produced as a result of symbiotic fermentation. Dr. Castellani also described symbiotic fermentation as a method to distinguish between Bacillus dysentariae Shiga (now Shigella dysentariae Shiga) and B. dysentariae Flexner (now Shigella flexneri ) by fermenting each of them with Bacillus morgani (now Morganella morganii ) in mannitol . The culture with Flexner would always produce gas and acid, while the culture with Shiga only produced acid. To summarize, one bacteria performs acid fermentation to produce acid from sugar, then the other bacteria performs gas fermentation using the acid products to produce gas. Thus creating a type of symbiotic relationship based on fermentation metabolism.
More recently, symbiotic fermentation is described in a traditional sense for the fermentation of food and beverage products. Biofilm aggregates of fermentative microorganisms are commonly associated with fermentation of many products including vinegar, sake , shochu , and kefir. [ 1 ] In the U.S., kombucha has become a popular fermented beverage that is also a model of symbiotic fermentation. In kombucha, bacteria create the biofilm network that initiates SCOBY formation, while the yeast produce invertase that makes sugars available to the bacteria and yeast for fermentation. [ 4 ] | https://en.wikipedia.org/wiki/Symbiotic_fermentation |
A symbol is a mark, sign , or word that indicates, signifies, or is understood as representing an idea , object , or relationship . Symbols allow people to go beyond what is known or seen by creating linkages between otherwise different concepts and experiences. All communication is achieved through the use of symbols: for example, a red octagon is a common symbol for " STOP "; on maps , blue lines often represent rivers; and a red rose often symbolizes love and compassion. Numerals are symbols for numbers ; letters of an alphabet may be symbols for certain phonemes ; and personal names are symbols representing individuals. The academic study of symbols is called semiotics .
In the arts, symbolism is the use of a concrete element to represent a more abstract idea. In cartography , an organized collection of symbols forms a legend for a map.
The word symbol derives from the late Middle French masculine noun symbole , which appeared around 1380 in a theological sense signifying a formula used in the Roman Catholic Church as a sort of synonym for 'the credo'; by extension in the early Renaissance it came to mean 'a maxim' or 'the external sign of a sacrament'; these meanings were lost in secular contexts. It was during the Renaissance in the mid-16th century that the word took on the meaning that is dominant today, that of 'a natural fact or object evoking by its form or its nature an association of ideas with something abstract or absent'; this appears, for example, in François Rabelais , Le Quart Livre , in 1552. [ 1 ] This French word derives from Latin, where both the masculine noun symbolus and the neuter noun symbolum refer to "a mark or sign as a means of recognition." [ 2 ] The Latin word derives from Ancient Greek : σύμβολον symbolon , from a verb meaning 'put together', 'compare', alluding to the Classical practice of breaking a piece of ceramic in two and giving one half to the person who would receive a future message, and one half to the person who would send it: when the two fit perfectly together, the receiver could be sure that the messenger bearing it did indeed also carry a genuine message from the intended person. [ 1 ] A literary or artistic symbol as an "outward sign" of something else is a metaphorical extension of this notion of a message from a sender to a recipient. In English, the meaning "something which stands for something else" was first recorded in 1590, in Edmund Spenser 's Faerie Queene .
Symbols are a means of complex communication that often can have multiple levels of meaning. [ 3 ] Symbols are the basis of all human understanding and serve as vehicles of conception for all human knowledge. [ 4 ] Symbols facilitate understanding of the world in which we live, thus serving as the grounds upon which we make judgments. [ 5 ] In this way, people use symbols not only to make sense of the world around them but also to identify and cooperate in society through constitutive rhetoric .
Human cultures use symbols to express specific ideologies and social structures and to represent aspects of their specific culture. Thus, symbols carry meanings that depend upon one's cultural background. As a result, the meaning of a symbol is not inherent in the symbol itself but is culturally learned. [ 3 ]
Heinrich Zimmer gives a concise overview of the nature, and perennial relevance, of symbols.
Concepts and words are symbols, just as visions, rituals, and images are; so too are the manners and customs of daily life. Through all of these, a transcendent reality is mirrored. There are so many metaphors reflecting and implying something which, though thus variously expressed, is ineffable, though thus rendered multiform, remains inscrutable. Symbols hold the mind to truth but are not themselves the truth, hence it is delusory to borrow them. Each civilisation, every age, must bring forth its own." [ 6 ]
In the book Signs and Symbols , it is stated that
A symbol ... is a visual image or sign representing an idea – a deeper indicator of universal truth. [ 7 ]
Semiotics is the study of signs, symbols, and signification as communicative behavior. Semiotics studies focus on the relationship of the signifier and the signified, also taking into account the interpretation of visual cues, body language, sound, and other contextual clues. Semiotics is linked with linguistics and psychology. Semioticians not only study what a symbol implies but also how it got its meaning and how it functions to make meaning in society. For example, symbols can cause confusion in translation when the same symbol means different things in the source and target languages. A potential error documented in survey translation is the symbol of "x" used to denote "yes" when marking a response in the English language surveys, but "x" usually means "no" in the Chinese convention. [ 8 ] Symbols allow the human brain continuously to create meaning using sensory input and decode symbols through both denotation and connotation .
An alternative definition of symbol , distinguishing it from the term sign was proposed by Swiss psychoanalyst Carl Jung . In his studies on what is now called Jungian archetypes , a sign stands for something known, as a word stands for its referent. He contrasted a sign with a symbol : something that is unknown and that cannot be made clear or precise. An example of a symbol in this sense is Christ as a symbol of the archetype called self . [ 9 ]
Kenneth Burke described Homo sapiens as a "symbol-using, symbol making, and symbol misusing animal" to suggest that a person creates symbols as well as misuses them. One example he uses to indicate what he means by the misuse of the symbol is the story of a man who, when told that a particular food item was whale blubber, could barely keep from throwing it up. Later, his friend discovered it was actually just a dumpling. But the man's reaction was a direct consequence of the symbol of "blubber" representing something inedible in his mind. In addition, the symbol of "blubber" was created by the man through various kinds of learning .
Burke goes on to describe symbols as also being derived from Sigmund Freud 's work on condensation and displacement , further stating that symbols are not just relevant to the theory of dreams but also to "normal symbol systems". He says they are related through "substitution", where one word, phrase, or symbol is substituted for another in order to change the meaning. [ clarification needed ] In other words, if one person does not understand a certain word or phrase, another person may substitute a synonym or symbol in order to get the meaning across. However, upon learning the new way of interpreting a specific symbol, the person may change his or her already-formed ideas to incorporate the new information.
Jean Dalby Clift says that people not only add their own interpretations to symbols, but they also create personal symbols that represent their own understanding of their lives: what she calls "core images" of the person. Clift argues that symbolic work with these personal symbols or core images can be as useful as working with dream symbols in psychoanalysis or counseling. [ 10 ]
William Indick suggests that the symbols that are commonly found in myth, legend, and fantasy fulfill psychological functions and hence are why archetypes such as "the hero", "the princess" and "the witch" have remained popular for centuries. [ 11 ]
Symbols can carry symbolic value in three primary forms: Ideological, comparative, and isomorphic. [ 12 ] Ideological symbols such as religious and state symbols convey complex sets of beliefs and ideas that indicate "the right thing to do". Comparative symbols such as prestigious office addresses, fine art, and prominent awards indicate answers to questions of "better or worse" and "superior or inferior". Isomorphic symbols blend in with the surrounding cultural environment such that they enable individuals and organizations to conform to their surroundings and evade social and political scrutiny. Examples of symbols with isomorphic value include wearing a professional dress during business meetings, shaking hands to greet others in the West, or bowing to greet others in the East. A single symbol can carry multiple distinct meanings such that it provides multiple types of symbolic value. [ 12 ]
Paul Tillich argued that, while signs are invented and forgotten, symbols are born and die. There are, therefore, dead and living symbols. A living symbol can reveal to an individual hidden levels of meaning and transcendent or religious realities. For Tillich a symbol always "points beyond itself" to something that is unquantifiable and mysterious; symbols open up the "depth dimension of reality itself". [ 13 ] Symbols are complex, and their meanings can evolve as the individual or culture evolves. When a symbol loses its meaning and power for an individual or culture, it becomes a dead symbol.
When a symbol becomes identified with the deeper reality to which it refers, it becomes idolatrous as the "symbol is taken for reality." The symbol itself is substituted for the deeper meaning it intends to convey. The unique nature of a symbol is that it gives access to deeper layers of reality that are otherwise inaccessible. [ 14 ]
A symbol's meaning may be modified by various factors including popular usage, history , and contextual intent .
The history of a symbol is one of many factors in determining a particular symbol's apparent meaning. Consequently, symbols with emotive power carry problems analogous to false etymologies . [ 15 ]
The context of a symbol may change its meaning. Similar five-pointed stars might signify a law enforcement officer or a member of the armed services , depending upon the uniform .
Symbols are used in cartography to communicate geographical information (generally as point, line, or area features). [ 16 ] As with other symbols, visual variables such as size, shape, orientation, texture, and pattern provide meaning to the symbol. [ 17 ] According to semiotics , map symbols are "read" by map users when they make a connection between the graphic mark on the map (the sign ), a general concept (the interpretant ), and a particular feature of the real world (the referent ). Map symbols can thus be categorized by how they suggest this connection:
A symbolic action is an action that symbolizes or signals what the actor wants or believes. The action conveys meaning to the viewers. Symbolic action may overlap with symbolic speech , such as the use of flag burning to express hostility or saluting the flag to express patriotism. [ 18 ] In response to intense public criticism, businesses, organizations, and governments may take symbolic actions rather than, or in addition to, directly addressing the identified problems. [ 19 ] | https://en.wikipedia.org/wiki/Symbol |
A logical symbol is a fundamental concept in logic , tokens of which may be marks or a configuration of marks which form a particular pattern. [ citation needed ] Although the term symbol in common use sometimes refers to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic , the term symbol refers to the idea, and the marks are considered to be a token instance of the symbol. [ dubious – discuss ] In logic, symbols build literal utility to illustrate ideas.
Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). Symbols of a formal language must be capable of being specified without any reference to any interpretation of them.
A symbol or string of symbols may comprise a well-formed formula if it is consistent with the formation rules of the language.
In a formal system a symbol may be used as a token in formal operations. The set of formal symbols in a formal language is referred to as an alphabet (hence each symbol may be referred to as a "letter") [ 1 ] [ page needed ]
A formal symbol as used in first-order logic may be a variable (member from a universe of discourse ), a constant, a function (mapping to another member of universe) or a predicate (mapping to T/F).
Formal symbols are usually thought of as purely syntactic structures, composed into larger structures using a formal grammar , though sometimes they may be associated with an interpretation or model (a formal semantics ).
The move to view units in natural language (e.g. English) as formal symbols was initiated by Noam Chomsky (it was this work that resulted in the Chomsky hierarchy in formal languages). The generative grammar model looked upon syntax as autonomous from semantics. Building on these models, the logician Richard Montague proposed that semantics could also be constructed on top of the formal structure:
This is the philosophical premise underlying Montague grammar .
However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition of cognitive linguistics , by philosophers like Stevan Harnad , and linguists like George Lakoff and Ronald Langacker . | https://en.wikipedia.org/wiki/Symbol_(formal) |
The Symbol Nomenclature For Glycans ( SNFG ) [ 1 ] is a community-curated standard for the depiction of simple monosaccharides and complex carbohydrates ( glycans ) using various colored-coded, geometric shapes, along with defined text additions. [ 2 ] [ 3 ] It is hosted by the National Center for Biotechnology Information at the NCBI-Glycans Page. [ 4 ] It is curated by an international groups of researchers in the field that are collectively called the SNFG Discussion Group. The overall goal of the SNFG is to:
The SNFG consists of a table that provides color coded symbols for various monosaccharides that are commonly found in nature. It also includes a set of footnotes that describe rules for rendering glycans, including guidelines on how to modify the base set of symbols depicted in the table. These footnotes are organized into 10 themes that provide streamlined recommendations for: i. general usage of the SNFG; ii. CMYK / RGB color codes; iii. symbol colors and shapes; iv. ring configurations; v. bond linkage presentation; vi. sialic acids ; vii. glycan modifications; viii. amino substitutions; ix. handling ambiguous or partially defined glycans; and x. depicting non-glycan entities using SNFG renderings. More details are available at the main SNFG webpage, [ 1 ] which is periodically updated with additional directions.
The monosaccharides can be linked together to describe complex carbohydrate structures or glycans . More exhaustive cases for mammalian species, other eukaryotes , plants and microbes are considered at the main SNFG page. [ 1 ]
Several software tools have been developed to support SNFG implementation by the community including:
The SNFG nomenclature has also been adopted as a standard by major databases and journals in the Biomedical Sciences.
In 1978, Stuart Kornfeld and colleagues at the Washington University School of Medicine presented a system for symbolic representation of vertebrate glycans. [ 10 ] This system gained popularity when it was implemented as a core method for glycan representation in the NCBI text book Essentials of Glycobiology edited by Ajit Varki (University of California, San Diego) and colleagues. [ 11 ] While the first edition of this text published in 1999 used black-and-white symbols similar to the Kornfeld system, color was introduced in the second edition of the text (2009). The advantage of color is that different monosaccharide stereoisomers could now be depicted using the same shape, only with different colors. The system of carbohydrate representation was adopted and widely disseminated by many including the NIGMS-funded Consortium for Functional Glycomics , and thus was often referred to as "CFG Nomenclature". This color representation was vastly expanded in the third edition of the text to include 49 new monosaccharides that appear mostly in non-vertebrates, microbes and plants. Inputs and recommendations from a number of scientists beyond the editors of the Essentials textbook was included in this implementation, and the release of the expanded glycan symbol system was coordinated with the IUPAC Carbohydrate Nomenclature committee. For long-term development of this symbol nomenclature and standardization of glycan representation in the Glycosciences, in 2015, the Essentials editors suggested that the representation be formally called SNFG ('Symbol Nomenclature For Glycans'), and future development be entrusted to a global community of scientists. To aid this development, each of the SNFG monosaccharide symbols was linked to PubChem entries at NCBI/NLM and a dedicated website at NCBI was established for future SNFG updates. [ 1 ] Thus, the development of the SNFG is currently undertaken by an international community of scientist that are called the SNFG Discussion Group. | https://en.wikipedia.org/wiki/Symbol_Nomenclature_For_Glycans |
Chickens have been widely used as national symbols, and as mascots for clubs, businesses, and other associations. | https://en.wikipedia.org/wiki/Symbolic_chickens |
In mathematics , symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence.
Because of their explicit, discrete nature, such systems are often relatively easy to characterize and understand. They form a key tool for studying topological or smooth dynamical systems , because in many important cases it is possible to reduce the dynamics of a more general dynamical system to a symbolic system. To do so, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.
The idea goes back to Jacques Hadamard 's 1898 paper on the geodesics on surfaces of negative curvature . [ 1 ] It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by Emil Artin in 1924 (for the system now called Artin billiard ), Pekka Myrberg , Paul Koebe , Jakob Nielsen , G. A. Hedlund .
The first formal treatment was developed by Morse and Hedlund in their 1938 paper. [ 2 ] George Birkhoff , Norman Levinson and the pair Mary Cartwright and J. E. Littlewood have applied similar methods to qualitative analysis of nonautonomous second order differential equations .
Claude Shannon used symbolic sequences and shifts of finite type in his 1948 paper A mathematical theory of communication that gave birth to information theory .
During the late 1960s the method of symbolic dynamics was developed to hyperbolic toral automorphisms by Roy Adler and Benjamin Weiss , [ 3 ] and to Anosov diffeomorphisms by Yakov Sinai who used the symbolic model to construct Gibbs measures . [ 4 ] In the early 1970s the theory was extended to Anosov flows by Marina Ratner , and to Axiom A diffeomorphisms and flows by Rufus Bowen .
A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about periodic orbits of a continuous map of an interval into itself (1964).
Consider the set of two-sided infinite sequences on two symbols, 0 and 1. A typical element in this set looks like: (..., 0, 1, 0, 0, 1, 0, 1, ... )
There will be exactly two fixed points under the shift map: the sequence of all zeroes, and the sequence of all ones. A periodic sequence will have a periodic orbit. For instance, the sequence (..., 0, 1, 0, 1, 0, 1, 0, 1, ...) will have period two.
More complex concepts such as heteroclinic orbits and homoclinic orbits also have simple descriptions in this system. For example, any sequence that has only a finite number of ones will have a homoclinic orbit, tending to the sequence of all zeros in forward and backward iterations.
Itinerary of point with respect to the partition is a sequence of symbols. It describes dynamic of the point. [ 5 ]
Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in data storage and transmission , linear algebra , the motions of the planets and many other areas [ citation needed ] . The distinct feature in symbolic dynamics is that time is measured in discrete intervals. So at each time interval the system is in a particular state . Each state is associated with a symbol and the evolution of the system is described by an infinite sequence of symbols—represented effectively as strings . If the system states are not inherently discrete, then the state vector must be discretized, so as to get a coarse-grained description of the system.
Recent work has generalized symbolic dynamics to layered dynamical systems, introducing the concept of symbolic conditional entropy, thus expanding symbolic dynamics to more abstract informational structures and deeper symbolic architectures. [ 6 ] | https://en.wikipedia.org/wiki/Symbolic_dynamics |
In engineering , a symbolic language is a language that uses standard symbols , marks , and abbreviations to represent concepts such as entities, aspects, attributes , and relationships. [ 1 ] [ original research? ]
Engineering symbolic language may be used for the specification , design , implementation , management , operation, and execution of engineered systems . [ 2 ] [ original research? ]
Communication using precise , concise representations of concepts is critical in engineering. [ 3 ] The Nuclear Principles in Engineering book begins with a quote on symbolic language from Erich Fromm and its power to express and depict associations. [ 4 ] [ 5 ] The engineering employs symbolic language in a way that is not purely text-based and not purely image-based to represent and communicate knowledge. [ 6 ]
Examples in chemical engineering include the symbolic languages developed for process flow diagrams and for piping and instrumentation diagrams (P&IDs) . [ 7 ]
In electrical engineering , examples include the symbolic languages developed for network diagrams used in computing. [ 8 ] [ 9 ]
Ladder logic was originally a written symbolic language for the design and construction of programmable logic control (PLC) operations in mechanical and control engineering . [ 10 ]
This design -related article is a stub . You can help Wikipedia by expanding it .
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symbolic_language_(engineering) |
In mathematics , a symbolic language is a language that uses characters or symbols to represent concepts, such as mathematical operations , expressions , and statements , and the entities or operands on which the operations are performed. [ 1 ] [ 2 ]
This mathematics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symbolic_language_(mathematics) |
In computer science , a symbolic language , or assembly language, is a language that uses characters or symbols to represent concepts, such as mathematical operations and the entities (or operands ) on which these operations are performed. [ 1 ]
Modern programming languages use symbols to represent concepts and/or data and are, therefore, examples of symbolic languages. [ 1 ]
Some programming languages (such as Lisp and Mathematica ) make it easy to represent higher-level abstractions as expressions in the language, enabling symbolic programming . [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/Symbolic_language_(programming) |
In mathematics , the symbolic method in invariant theory is an algorithm developed by Arthur Cayley , [ 1 ] Siegfried Heinrich Aronhold , [ 2 ] Alfred Clebsch , [ 3 ] and Paul Gordan [ 4 ] in the 19th century for computing invariants of algebraic forms . It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.
The symbolic method uses a compact, but rather confusing and mysterious notation for invariants, depending on the introduction of new symbols a , b , c , ... (from which the symbolic method gets its name) with apparently contradictory properties.
These symbols can be explained by the following example from Gordan. [ 5 ] Suppose that
is a binary quadratic form with an invariant given by the discriminant
The symbolic representation of the discriminant is
where a and b are the symbols. The meaning of the expression ( ab ) 2 is as follows. First of all, ( ab ) is a shorthand form for the determinant of a matrix whose rows are a 1 , a 2 and b 1 , b 2 , so
Squaring this we get
Next we pretend that
so that
and we ignore the fact that this does not seem to make sense if f is not a power of a linear form.
Substituting these values gives
More generally if
is a binary form of higher degree, then one introduces new variables a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , with the properties
What this means is that the following two vector spaces are naturally isomorphic:
The isomorphism is given by mapping a n − j 1 a j 2 , b n − j 1 b j 2 , .... to A j . This mapping does not preserve products of polynomials.
The extension to a form f in more than two variables x 1 , x 2 , x 3 ,... is similar: one introduces symbols a 1 , a 2 , a 3 and so on with the properties
The rather mysterious formalism of the symbolic method corresponds to embedding a symmetric product S n ( V ) of a vector space V into a tensor product of n copies of V , as the elements preserved by the action of the symmetric group. In fact this is done twice, because the invariants of degree n of a quantic of degree m are the invariant elements of S n S m ( V ), which gets embedded into a tensor product of mn copies of V , as the elements invariant under a wreath product of the two symmetric groups. The brackets of the symbolic method are really invariant linear forms on this tensor product, which give invariants of S n S m ( V ) by restriction.
Footnotes | https://en.wikipedia.org/wiki/Symbolic_method |
In combinatorics , the symbolic method is a technique for counting combinatorial objects . It uses the internal structure of the objects to derive formulas for their generating functions . The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick , Analytic Combinatorics , while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions.
During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of Bernoulli , Euler , Arthur Cayley , Schröder , Ramanujan , Riordan , Knuth , Comtet [ fr ] , etc.).
It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures
translates, via some isomorphisms, into noteworthy identities on the corresponding generating functions.
Following the works of Pólya , further advances were thus done in this spirit in the 1970s with generic uses of languages for specifying combinatorial classes and their generating functions, as found in works by Foata and Schützenberger [ 1 ] on permutations,
Bender and Goldman on prefabs, [ 2 ] and Joyal on combinatorial species . [ 3 ]
Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for umbral calculus .
The symbolic method in combinatorics constitutes the first step of many analyses of combinatorial structures,
which can then lead to fast computation schemes, to asymptotic properties and limit laws , to random generation , all of them being suitable to automatization via computer algebra .
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration is the sum of the weights of the objects in the slots. We will first explain how to solve this problem in the labelled and the unlabelled case and use the solution to motivate the creation of classes of combinatorial structures .
The Pólya enumeration theorem solves this problem in the unlabelled case. Let f ( z ) be the ordinary generating function (OGF) of the objects, then the OGF of the configurations is given by the substituted cycle index
In the labelled case we use an exponential generating function (EGF) g ( z ) of the objects and apply the Labelled enumeration theorem , which says that the EGF of the configurations is given by
We are able to enumerate filled slot configurations using either Pólya enumeration theorem in the unlabelled case or the labelled enumeration theorem in the labelled case. We now ask about the generating function of configurations obtained when there is more than one set of slots, with a permutation group acting on each. Clearly the orbits do not intersect and we may add the respective generating functions. Suppose, for example, that we want to enumerate unlabelled sequences of length two or three of some objects contained in a set X . There are two sets of slots, the first one containing two slots, and the second one, three slots. The group acting on the first set is the full symmetric group S 2 {\displaystyle S_{2}} , which in symbolic combinatorics is traditionally denoted E 2 {\displaystyle E_{2}} . The group acting on the second set is, analogously, S 3 = E 3 {\displaystyle S_{3}=E_{3}} . We represent this by the following formal power series in X :
where the term X n / G {\displaystyle X^{n}/G} is used to denote the set of orbits under G and X n = X × ⋯ × X {\displaystyle X^{n}=X\times \cdots \times X} , which denotes in the obvious way the process of distributing the objects from X with repetition into the n slots. Similarly, consider the labelled problem of creating cycles of arbitrary length from a set of labelled objects X . This yields the following series of actions of cyclic groups:
Clearly we can assign meaning to any such power series of quotients (orbits) with respect to permutation groups, where we restrict the groups of degree n to the conjugacy classes Cl ( S n ) {\displaystyle \operatorname {Cl} (S_{n})} of the symmetric group S n {\displaystyle S_{n}} , which form a unique factorization domain. (The orbits with respect to two groups from the same conjugacy class are isomorphic.) This motivates the following definition.
A class C ∈ N [ M ] {\displaystyle {\mathcal {C}}\in \mathbb {N} [{\mathfrak {M}}]} of combinatorial structures is a formal series
where M {\displaystyle {\mathfrak {M}}} (the "M" is for "molecules") is the set of primes of the UFD { Cl ( S n ) } n ≥ 1 {\displaystyle \{\operatorname {Cl} (S_{n})\}_{n\geq 1}} and c G ∈ N . {\displaystyle c_{G}\in \mathbb {N} .}
In the following we will simplify our notation a bit and write e.g.
for the classes mentioned above.
A theorem in the Flajolet–Sedgewick theory of symbolic combinatorics treats the enumeration problem of labelled and unlabelled combinatorial classes by means of the creation of symbolic operators that make it possible to translate equations involving combinatorial structures directly (and automatically) into equations in the generating functions of these structures.
Let C ∈ N [ A ] {\displaystyle {\mathcal {C}}\in \mathbb {N} [{\mathfrak {A}}]} be a class of combinatorial structures. The OGF F ( z ) {\displaystyle F(z)} of C ( X ) {\displaystyle {\mathcal {C}}(X)} where X has OGF f ( z ) {\displaystyle f(z)} and the EGF G ( z ) {\displaystyle G(z)} of C ( X ) {\displaystyle {\mathcal {C}}(X)} where X is labelled with EGF g ( z ) {\displaystyle g(z)} are given by
and
In the labelled case we have the additional requirement that X not contain elements of size zero. It will sometimes prove convenient to add one to G ( z ) {\displaystyle G(z)} to indicate the presence of one copy of the empty set. It is possible to assign meaning to both C ∈ Z [ A ] {\displaystyle {\mathcal {C}}\in \mathbb {Z} [{\mathfrak {A}}]} (the most common example is the case of unlabelled sets) and C ∈ Q [ A ] . {\displaystyle {\mathcal {C}}\in \mathbb {Q} [{\mathfrak {A}}].} To prove the theorem simply apply PET (Pólya enumeration theorem) and the labelled enumeration theorem.
The power of this theorem lies in the fact that it makes it possible to construct operators on generating functions that represent combinatorial classes. A structural equation between combinatorial classes thus translates directly into an equation in the corresponding generating functions. Moreover, in the labelled case it is evident from the formula that we may replace g ( z ) {\displaystyle g(z)} by the atom z and compute the resulting operator, which may then be applied to EGFs. We now proceed to construct the most important operators. The reader may wish to compare with the data on the cycle index page.
This operator corresponds to the class
and represents sequences, i.e. the slots are not being permuted and there is exactly one empty sequence. We have
and
This operator corresponds to the class
i.e., cycles containing at least one object. We have
or
and
This operator, together with the set operator SET , and their restrictions to specific degrees are used to compute random permutation statistics . There are two useful restrictions of this operator, namely to even and odd cycles.
The labelled even cycle operator CYC even is
which yields
This implies that the labelled odd cycle operator CYC odd
is given by
The series is
i.e., the symmetric group S n = E n {\displaystyle S_{n}=E_{n}} is applied to the n th slot. This creates multisets in the unlabelled case and sets in the labelled case (there are no multisets in the labelled case because the labels distinguish multiple instances of the same object from the set being put into different slots). We include the empty set in both the labelled and the unlabelled case.
The unlabelled case is done using the function
so that
Evaluating M ( f ( z ) , 1 ) {\displaystyle M(f(z),1)} we obtain
For the labelled case we have
In the labelled case we denote the operator by SET , and in the unlabelled case, by MSET . This is because in the labeled case there are no multisets (the labels distinguish the constituents of a compound combinatorial class) whereas in the unlabeled case there are multisets and sets, with the latter being given by
Typically, one starts with the neutral class E {\displaystyle {\mathcal {E}}} , containing a single object of size 0 (the neutral object , often denoted by ϵ {\displaystyle \epsilon } ), and one or more atomic classes Z {\displaystyle {\mathcal {Z}}} , each containing a single object of size 1. Next, set-theoretic relations involving various simple operations, such as disjoint unions , products , sets , sequences , and multisets define more complex classes in terms of the already defined classes. These relations may be recursive . The elegance of symbolic combinatorics lies in that the set theoretic, or symbolic , relations translate directly into algebraic relations involving the generating functions.
In this article, we will follow the convention of using script uppercase letters to denote combinatorial classes and the corresponding plain letters for the generating functions (so the class A {\displaystyle {\mathcal {A}}} has generating function A ( z ) {\displaystyle A(z)} ).
There are two types of generating functions commonly used in symbolic combinatorics— ordinary generating functions , used for combinatorial classes of unlabelled objects, and exponential generating functions , used for classes of labelled objects.
It is trivial to show that the generating functions (either ordinary or exponential) for E {\displaystyle {\mathcal {E}}} and Z {\displaystyle {\mathcal {Z}}} are E ( z ) = 1 {\displaystyle E(z)=1} and Z ( z ) = z {\displaystyle Z(z)=z} , respectively. The disjoint union is also simple — for disjoint sets B {\displaystyle {\mathcal {B}}} and C {\displaystyle {\mathcal {C}}} , A = B ∪ C {\displaystyle {\mathcal {A}}={\mathcal {B}}\cup {\mathcal {C}}} implies A ( z ) = B ( z ) + C ( z ) {\displaystyle A(z)=B(z)+C(z)} . The relations corresponding to other operations depend on whether we are talking about labelled or unlabelled structures (and ordinary or exponential generating functions).
The restriction of unions to disjoint unions is an important one; however, in the formal specification of symbolic combinatorics, it is too much trouble to keep track of which sets are disjoint. Instead, we make use of a construction that guarantees there is no intersection ( be careful, however; this affects the semantics of the operation as well ). In defining the combinatorial sum of two sets A {\displaystyle {\mathcal {A}}} and B {\displaystyle {\mathcal {B}}} , we mark members of each set with a distinct marker, for example ∘ {\displaystyle \circ } for members of A {\displaystyle {\mathcal {A}}} and ∙ {\displaystyle \bullet } for members of B {\displaystyle {\mathcal {B}}} . The combinatorial sum is then:
This is the operation that formally corresponds to addition.
With unlabelled structures, an ordinary generating function (OGF) is used. The OGF of a sequence A n {\displaystyle A_{n}} is defined as
The product of two combinatorial classes A {\displaystyle {\mathcal {A}}} and B {\displaystyle {\mathcal {B}}} is specified by defining the size of an ordered pair as the sum of the sizes of the elements in the pair. Thus we have for a ∈ A {\displaystyle a\in {\mathcal {A}}} and b ∈ B {\displaystyle b\in {\mathcal {B}}} , | ( a , b ) | = | a | + | b | {\displaystyle |(a,b)|=|a|+|b|} . This should be a fairly intuitive definition. We now note that the number of elements in A × B {\displaystyle {\mathcal {A}}\times {\mathcal {B}}} of size n is
Using the definition of the OGF and some elementary algebra, we can show that
The sequence construction , denoted by A = G { B } {\displaystyle {\mathcal {A}}={\mathfrak {G}}\{{\mathcal {B}}\}} is defined as
In other words, a sequence is the neutral element, or an element of B {\displaystyle {\mathcal {B}}} , or an ordered pair, ordered triple, etc. This leads to the relation
The set (or powerset ) construction , denoted by A = P { B } {\displaystyle {\mathcal {A}}={\mathfrak {P}}\{{\mathcal {B}}\}} is defined as
which leads to the relation
where the expansion
was used to go from line 4 to line 5.
The multiset construction , denoted A = M { B } {\displaystyle {\mathcal {A}}={\mathfrak {M}}\{{\mathcal {B}}\}} is a generalization of the set construction. In the set construction, each element can occur zero or one times. In a multiset, each element can appear an arbitrary number of times. Therefore,
This leads to the relation
where, similar to the above set construction, we expand ln ( 1 − z n ) {\displaystyle \ln(1-z^{n})} , swap the sums, and substitute for the OGF of B {\displaystyle {\mathcal {B}}} .
Other important elementary constructions are:
The derivations for these constructions are too complicated to show here. Here are the results:
Many combinatorial classes can be built using these elementary constructions. For example, the class of plane trees (that is, trees embedded in the plane, so that the order of the subtrees matters) is specified by the recursive relation
In other words, a tree is a root node of size 1 and a sequence of subtrees. This gives
we solve for G ( z ) by multiplying 1 − G ( z ) {\displaystyle 1-G(z)} to get
G ( z ) − G ( z ) 2 = z {\displaystyle G(z)-G(z)^{2}=z}
subtracting z and solving for G(z) using the quadratic formula gives
Another example (and a classic combinatorics problem) is integer partitions . First, define the class of positive integers I {\displaystyle {\mathcal {I}}} , where the size of each integer is its value:
The OGF of I {\displaystyle {\mathcal {I}}} is then
Now, define the set of partitions P {\displaystyle {\mathcal {P}}} as
The OGF of P {\displaystyle {\mathcal {P}}} is
Unfortunately, there is no closed form for P ( z ) {\displaystyle P(z)} ; however, the OGF can be used to derive a recurrence relation , or using more advanced methods of analytic combinatorics, calculate the asymptotic behavior of the counting sequence.
The elementary constructions mentioned above allow us to define the notion of specification . This specification allows us to use a set of recursive equations, with multiple combinatorial classes.
Formally, a specification for a set of combinatorial classes ( A 1 , … , A r ) {\displaystyle ({\mathcal {A}}_{1},\dots ,{\mathcal {A}}_{r})} is a set of r {\displaystyle r} equations A i = Φ i ( A 1 , … , A r ) {\displaystyle {\mathcal {A}}_{i}=\Phi _{i}({\mathcal {A}}_{1},\dots ,{\mathcal {A}}_{r})} , where Φ i {\displaystyle \Phi _{i}} is an expression, whose atoms are E , Z {\displaystyle {\mathcal {E}},{\mathcal {Z}}} and the A i {\displaystyle {\mathcal {A}}_{i}} 's, and whose operators are the elementary constructions listed above.
A class of combinatorial structures is said to be constructible or specifiable when it admits a specification.
For example, the set of trees whose leaves' depth is even (respectively, odd) can be defined using the specification with two classes A even {\displaystyle {\mathcal {A}}_{\text{even}}} and A odd {\displaystyle {\mathcal {A}}_{\text{odd}}} . Those classes should satisfy the equation A odd = Z × Seq ≥ 1 A even {\displaystyle {\mathcal {A}}_{\text{odd}}={\mathcal {Z}}\times \operatorname {Seq} _{\geq 1}{\mathcal {A}}_{\text{even}}} and A even = Z × Seq A odd {\displaystyle {\mathcal {A}}_{\text{even}}={\mathcal {Z}}\times \operatorname {Seq} {\mathcal {A}}_{\text{odd}}} .
An object is weakly labelled if each of its atoms has a nonnegative integer label, and each of these labels is distinct. An object is ( strongly or well ) labelled , if furthermore, these labels comprise the consecutive integers [ 1 … n ] {\displaystyle [1\ldots n]} . Note: some combinatorial classes are best specified as labelled structures or unlabelled structures, but some readily admit both specifications. A good example of labelled structures is the class of labelled graphs .
With labelled structures, an exponential generating function (EGF) is used. The EGF of a sequence A n {\displaystyle A_{n}} is defined as
For labelled structures, we must use a different definition for product than for unlabelled structures. In fact, if we simply used the cartesian product, the resulting structures would not even be well labelled. Instead, we use the so-called labelled product , denoted A ⋆ B . {\displaystyle {\mathcal {A}}\star {\mathcal {B}}.}
For a pair β ∈ B {\displaystyle \beta \in {\mathcal {B}}} and γ ∈ C {\displaystyle \gamma \in {\mathcal {C}}} , we wish to combine the two structures into a single structure. In order for the result to be well labelled, this requires some relabelling of the atoms in β {\displaystyle \beta } and γ {\displaystyle \gamma } . We will restrict our attention to relabellings that are consistent with the order of the original labels. Note that there are still multiple ways to do the relabelling; thus, each pair of members determines not a single member in the product, but a set of new members. The details of this construction are found on the page of the Labelled enumeration theorem .
To aid this development, let us define a function, ρ {\displaystyle \rho } , that takes as its argument a (possibly weakly) labelled object α {\displaystyle \alpha } and relabels its atoms in an order-consistent way so that ρ ( α ) {\displaystyle \rho (\alpha )} is well labelled. We then define the labelled product for two objects α {\displaystyle \alpha } and β {\displaystyle \beta } as
Finally, the labelled product of two classes A {\displaystyle {\mathcal {A}}} and B {\displaystyle {\mathcal {B}}} is
The EGF can be derived by noting that for objects of size k {\displaystyle k} and n − k {\displaystyle n-k} , there are ( n k ) {\displaystyle {n \choose k}} ways to do the relabelling. Therefore, the total number of objects of size n {\displaystyle n} is
This binomial convolution relation for the terms is equivalent to multiplying the EGFs,
The sequence construction A = G { B } {\displaystyle {\mathcal {A}}={\mathfrak {G}}\{{\mathcal {B}}\}} is defined similarly to the unlabelled case:
and again, as above,
In labelled structures, a set of k {\displaystyle k} elements corresponds to exactly k ! {\displaystyle k!} sequences. This is different from the unlabelled case, where some of the permutations may coincide. Thus for A = P { B } {\displaystyle {\mathcal {A}}={\mathfrak {P}}\{{\mathcal {B}}\}} , we have
Cycles are also easier than in the unlabelled case. A cycle of length k {\displaystyle k} corresponds to k {\displaystyle k} distinct sequences. Thus for A = C { B } {\displaystyle {\mathcal {A}}={\mathfrak {C}}\{{\mathcal {B}}\}} , we have
In labelled structures, the min-boxed product A min = B ◻ ⋆ C {\displaystyle {\mathcal {A}}_{\min }={\mathcal {B}}^{\square }\star {\mathcal {C}}} is a variation of the original product which requires the element of B {\displaystyle {\mathcal {B}}} in the product with the minimal label. Similarly, we can also define a max-boxed product, denoted by A max = B ◼ ⋆ C {\displaystyle {\mathcal {A}}_{\max }={\mathcal {B}}^{\blacksquare }\star {\mathcal {C}}} , by the same manner. Then we have,
or equivalently,
An increasing Cayley tree is a labelled non-plane and rooted tree whose labels along any branch stemming from the root form an increasing sequence. Then, let L {\displaystyle {\mathcal {L}}} be the class of such trees. The recursive specification is now L = Z ◻ ⋆ SET ( L ) . {\displaystyle {\mathcal {L}}={\mathcal {Z}}^{\square }\star \operatorname {SET} ({\mathcal {L}}).}
The operators CYC even ,
CYC odd ,
SET even , and SET odd represent cycles of even and odd length, and sets of even and odd cardinality.
Stirling numbers of the second kind may be derived and analyzed using the structural decomposition
The decomposition
is used to study unsigned Stirling numbers of the first kind , and in the derivation of the statistics of random permutations . A detailed examination of the exponential generating functions associated to Stirling numbers within symbolic combinatorics may be found on the page on Stirling numbers and exponential generating functions in symbolic combinatorics . | https://en.wikipedia.org/wiki/Symbolic_method_(combinatorics) |
In algebra and algebraic geometry , given a commutative Noetherian ring R {\displaystyle R} and an ideal I {\displaystyle I} in it, the n -th symbolic power of I {\displaystyle I} is the ideal
where R P {\displaystyle R_{P}} is the localization of R {\displaystyle R} at P {\displaystyle P} , we set φ P : R → R P {\displaystyle \varphi _{P}:R\to R_{P}} is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of R / I {\displaystyle R/I} .
Though this definition does not require I {\displaystyle I} to be prime , this assumption is often worked with because in the case of a prime ideal , the symbolic power can be equivalently defined as the I {\displaystyle I} - primary component of I n {\displaystyle I^{n}} . Very roughly, it consists of functions with zeros of order n along the variety defined by I {\displaystyle I} . We have: I ( 1 ) = I {\displaystyle I^{(1)}=I} and if I {\displaystyle I} is a maximal ideal , then I ( n ) = I n {\displaystyle I^{(n)}=I^{n}} .
Symbolic powers induce the following chain of ideals:
The study and use of symbolic powers has a long history in commutative algebra . Krull’s famous proof of his principal ideal theorem uses them in an essential way. They first arose after primary decompositions were proved for Noetherian rings . Zariski used symbolic powers in his study of the analytic normality of algebraic varieties . Chevalley's famous lemma comparing topologies states that in a complete local domain the symbolic powers topology of any prime is finer than the m -adic topology . A crucial step in the vanishing theorem on local cohomology of Hartshorne and Lichtenbaum uses that for a prime I {\displaystyle I} defining a curve in a complete local domain , the powers of I {\displaystyle I} are cofinal with the symbolic powers of I {\displaystyle I} . This important property of being cofinal was further developed by Schenzel in the 1970s. [ 1 ]
Though generators for ordinary powers of I {\displaystyle I} are well understood when I {\displaystyle I} is given in terms of its generators as I = ( f 1 , … , f k ) {\displaystyle I=(f_{1},\ldots ,f_{k})} , it is still very difficult in many cases to determine the generators of symbolic powers of I {\displaystyle I} . But in the geometric setting, there is a clear geometric interpretation in the case when I {\displaystyle I} is a radical ideal over an algebraically closed field of characteristic zero .
If X {\displaystyle X} is an irreducible variety whose ideal of vanishing is I {\displaystyle I} , then the differential power of I {\displaystyle I} consists of all the functions in R {\displaystyle R} that vanish
to order ≥ n on X {\displaystyle X} , i.e.
Or equivalently, if m p {\displaystyle \mathbf {m} _{p}} is the maximal ideal for a point p ∈ X {\displaystyle p\in X} , I ⟨ n ⟩ = ⋂ p ∈ X m p n {\displaystyle I^{\langle n\rangle }=\bigcap _{p\in X}\mathbf {m} _{p}^{n}} .
Theorem (Nagata, Zariski) [ 2 ] Let I {\displaystyle I} be a prime ideal in a polynomial ring K [ x 1 , … , x N ] {\displaystyle K[x_{1},\ldots ,x_{N}]} over an algebraically closed field. Then
This result can be extended to any radical ideal . [ 3 ] This formulation is very useful because, in characteristic zero , we can compute the differential powers in terms of generators as:
For another formulation, we can consider the case when the base ring is a polynomial ring over a field . In this case, we can interpret the n -th symbolic power as the sheaf of all function germs over X = Spec ( R ) vanishing to order ≥ n at Z = V ( I ) {\displaystyle X=\operatorname {Spec} (R){\text{ vanishing to order}}\geq n{\text{ at }}Z=V(I)} In fact, if X {\displaystyle X} is a smooth variety over a perfect field , then
It is natural to consider whether or not symbolic powers agree with ordinary powers, i.e. does I n = I ( n ) {\displaystyle I^{n}=I^{(n)}} hold? In general this is not the case. One example of this is the prime ideal P = ( x 4 − y z , y 2 − x z , x 3 y − z 2 ) ⊆ K [ x , y , z ] {\displaystyle P=(x^{4}-yz,\,y^{2}-xz,\,x^{3}y-z^{2})\subseteq K[x,y,z]} . Here we have that P 2 ≠ P ( 2 ) {\displaystyle P^{2}\neq P^{(2)}} . [ 1 ] However, P 2 ⊂ P ( 2 ) {\displaystyle P^{2}\subset P^{(2)}} does hold and the generalization of this inclusion is well understood. Indeed, the containment I n ⊆ I ( n ) {\displaystyle I^{n}\subseteq I^{(n)}} follows from the definition. Further, it is known that I r ⊆ I ( m ) {\displaystyle I^{r}\subseteq I^{(m)}} if and only if m ≤ r {\displaystyle m\leq r} . The proof follows from Nakayama's lemma . [ 4 ]
There has been extensive study into the other containment, when symbolic powers are contained in ordinary powers of ideals, referred to as the Containment Problem. Once again this has an easily stated answer summarized in the following theorem. It was developed by Ein, Lazarfeld, and Smith in characteristic zero [ 5 ] and was expanded to positive characteristic by Hochster and Huneke. [ 6 ] Their papers both build upon the results of Irena Swanson in Linear Equivalence of Ideal Topologies (2000). [ 7 ]
Theorem (Ein, Lazarfeld, Smith; Hochster, Huneke) Let I ⊂ K [ x 1 , x 2 , … , x N ] {\displaystyle I\subset K[x_{1},x_{2},\ldots ,x_{N}]} be a homogeneous ideal . Then the inclusion
It was later verified that the bound of N {\displaystyle N} in the theorem cannot be tightened for general ideals. [ 8 ] However, following a question posed [ 8 ] by Bocci, Harbourne, and Huneke, it was discovered that a better bound exists in some cases.
Theorem The inclusion I ( m ) ⊆ I r {\displaystyle I^{(m)}\subseteq I^{r}} for all m ≥ N r − N + 1 {\displaystyle m\geq Nr-N+1} holds | https://en.wikipedia.org/wiki/Symbolic_power_of_an_ideal |
Symbolic regression ( SR ) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity.
No particular model is provided as a starting point for symbolic regression. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators , analytic functions , constants , and state variables . Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming , [ 1 ] as well as more recent methods utilizing Bayesian methods [ 2 ] and neural networks . [ 3 ] Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy. [ 4 ] Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function . [ 5 ]
By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge . It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, [ 6 ] thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system, as well as improving generalisability and extrapolation behaviour by preventing overfitting . Accuracy and simplicity may be left as two separate objectives of the regression—in which case the optimum solutions form a Pareto front —or they may be combined into a single objective by means of a model selection principle such as minimum description length .
It has been proven that symbolic regression is an NP-hard problem, in the sense that one cannot always find the best possible mathematical expression to fit to a given dataset in polynomial time . [ 7 ] Nevertheless, if the sought-for equation is not too complex it is possible to solve the symbolic regression problem exactly by generating every possible function (built from some predefined set of operators) and evaluating them on the dataset in question. [ 8 ]
While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters.
This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system.
Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity.
In 2021, SRBench [ 9 ] was proposed as a large benchmark for symbolic regression.
In its inception, SRBench featured 14 symbolic regression methods, 7 other ML methods, and 252 datasets from PMLB .
The benchmark intends to be a living project: it encourages the submission of improvements, new datasets, and new methods, to keep track of the state of the art in SR.
In 2022, SRBench announced the competition Interpretable Symbolic Regression for Data Science, which was held at the GECCO conference in Boston, MA. The competition pitted nine leading symbolic regression algorithms against each other on a novel set of data problems and considered different evaluation criteria. The competition was organized in two tracks, a synthetic track and a real-world data track. [ 10 ]
In the synthetic track, methods were compared according to five properties: re-discovery of exact expressions; feature selection; resistance to local optima; extrapolation; and sensitivity to noise. Rankings of the methods were:
In the real-world track, methods were trained to build interpretable predictive models for 14-day forecast counts of COVID-19 cases, hospitalizations, and deaths in New York State. These models were reviewed by a subject expert and assigned trust ratings and evaluated for accuracy and simplicity. The ranking of the methods was:
Most symbolic regression algorithms prevent combinatorial explosion by implementing evolutionary algorithms that iteratively improve the best-fit expression over many generations. Recently, researchers have proposed algorithms utilizing other tactics in AI .
Silviu-Marian Udrescu and Max Tegmark developed the "AI Feynman" algorithm, [ 11 ] [ 12 ] which attempts symbolic regression by training a neural network to represent the mystery function, then runs tests against the neural network to attempt to break up the problem into smaller parts. For example, if f ( x 1 , . . . , x i , x i + 1 , . . . , x n ) = g ( x 1 , . . . , x i ) + h ( x i + 1 , . . . , x n ) {\displaystyle f(x_{1},...,x_{i},x_{i+1},...,x_{n})=g(x_{1},...,x_{i})+h(x_{i+1},...,x_{n})} , tests against the neural network can recognize the separation and proceed to solve for g {\displaystyle g} and h {\displaystyle h} separately and with different variables as inputs. This is an example of divide and conquer , which reduces the size of the problem to be more manageable. AI Feynman also transforms the inputs and outputs of the mystery function in order to produce a new function which can be solved with other techniques, and performs dimensional analysis to reduce the number of independent variables involved. The algorithm was able to "discover" 100 equations from The Feynman Lectures on Physics , while a leading software using evolutionary algorithms, Eureqa , solved only 71. AI Feynman, in contrast to classic symbolic regression methods, requires a very large dataset in order to first train the neural network and is naturally biased towards equations that are common in elementary physics. | https://en.wikipedia.org/wiki/Symbolic_regression |
The modern numerical digit 0 is usually written as a circle, an ellipse or a rounded square or rectangle.
In most modern typefaces , the height of the 0 character is the same as the other digits. However, in typefaces with text figures , the character is often shorter ( x-height ).
Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0. [ 1 ] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays . [ 1 ]
The digit 0 with a dot in the centre seems to have originated as an option on IBM 3270 displays. Its appearance has continued with Taligent 's command line typeface Andalé Mono . One variation used a short vertical bar instead of the dot. This could be confused with the Greek letter Theta on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character and very little used anyway.
An alternative, the slashed zero (looking similar to the letter O except for the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the Teletype Model 33 ASR. This form is similar to the symbol ∅ {\displaystyle \emptyset } , or "∅" ( Unicode character U+2205), representing the empty set , as well as to the letter Ø used in several Scandinavian languages . Some Burroughs / Unisys equipment displays a digit 0 with a reversed slash.
The opposing convention that has the letter O with a slash and the digit 0 without was advocated by SHARE , a prominent IBM user group, [ 1 ] and recommended by IBM for writing FORTRAN programs, [ 2 ] and by a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Others advocated the opposite convention, [ 1 ] including IBM for writing Algol programs. [ 2 ] Another convention used on some early line printers left digit 0 unornamented but added a tail or hook to the capital O so that it resembled an inverted Q (like U+213A ℺ ) or cursive capital letter-O ( O {\displaystyle {\mathcal {O}}} ). [ 1 ]
Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). The TI-99/4A computer has a more angular capital O and a more rounded digit 0, whereas others made the choice the other way around.
The typeface used on most European vehicle registration plates distinguishes the two symbols partially in this manner (having a more rectangular or wider shape for the capital O than the digit 0), but in several countries a further distinction is made by slitting open the digit 0 on the upper right side (as in German plates using the fälschungserschwerende Schrift , "forgery-impeding typeface").
Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. For example, confirmation numbers [ 3 ] [ better source needed ] used by Southwest Airlines use only the capital letters O and I instead of the digits 0 and 1, while Canadian postal codes use only the digits 1 and 0 and never the capital letters O and I, although letters and numbers always alternate.
On the seven-segment displays of calculators, watches, and household appliances, 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments.
The international maritime signal flag has five plus signs in an X arrangement. | https://en.wikipedia.org/wiki/Symbols_for_zero |
Symbols of death are the motifs, images and concepts associated with death throughout different cultures, religions and societies.
Various images are used traditionally to symbolize death; these rank from blunt depictions of cadavers and their parts to more allusive suggestions that time is fleeting and all men are mortals.
The human skull is an obvious and frequent symbol of death, found in many cultures and religious traditions. [ 1 ] Human skeletons and sometimes non-human animal skeletons and skulls can also be used as blunt images of death; the traditional figure of the Grim Reaper – a black-hooded skeleton with a scythe – is one use of such symbolism. [ 2 ] Within the Grim Reaper itself, the skeleton represents the decayed body whereas the robe symbolizes those worn by religious people conducting funeral services. [ 2 ] The skull and crossbones motif (☠) has been used among Europeans as a symbol of both piracy and poison . [ 3 ] The skull is also important as it remains the only "recognizable" aspect of a person once they have died. [ 3 ]
Decayed cadavers can also be used to depict death; in medieval Europe, they were often featured in artistic depictions of the danse macabre , or in cadaver tombs which depicted the living and decomposed body of the person entombed. Coffins also serve as blunt reminders of mortality. [ 4 ] Europeans were also seen to use coffins and cemeteries to symbolize the wealth and status of the person who has died, serving as a reminder to the living and the deceased as well. [ 4 ] Less blunt symbols of death frequently allude to the passage of time and the fragility of life , and can be described as memento mori ; [ 5 ] that is, an artistic or symbolic reminder of the inevitability of death. Clocks , hourglasses , sundials , and other timepieces both call to mind that time is passing. [ 3 ] Similarly, a candle both marks the passage of time, and bears witness that it will eventually burn itself out as well as a symbol of hope of salvation. [ 3 ] These sorts of symbols were often incorporated into vanitas paintings, a variety of early still life .
Throughout history and across many cultures, some animals have come to symbolize death and dying. This symbolism is often demonstrated in the legends and folklore of the culture. The specific animals and the details of their symbolism vary widely from culture to culture. [ 6 ] However, some animals tend to appear more frequently than others; such as dogs, bats, owls and crows.
Several societies, associate a type of dog with death. Dogs often serve as companions or guides to humans. Unsurprisingly, these animals that are so much a part of human life would have a role in death as well. In Mexico, the Xoloitzcuintli, a hairless dog, is thought to guide the spirits of the deceased and are associated with Dia de Los Muertas. [ 7 ] In Greece, Cerberus is a three-headed dog which guards the entrance to the underworld. In Welsh mythology, there is also a dog that guards the underworld. [ 8 ] In England, the black dog, black shuck, is associate with death or misfortune.
Bats, as a nocturnal animal, are often associated with darkness and death. In Christianity, bats are considered to be the bird of the devil and connections between the physiology of bats and demons are made. [ 6 ] In New Zealand, bats are associated with the Hokioi, a mythical nocturnal bird that foretells death. [ 9 ] The discover of the vampire bats in North America and the exaggeration of their qualities made lasting associations between bats and death that eventually lead to Dracula and vampire stories. [ 8 ] Bats are often connected to both Halloween and witches.
Owls, another nocturnal animal, are also tied to death. Some Mediterranean folklore tells of women who turn into to owls at night to suck the breath away from babies. [ 6 ] The hoot of an owl, according to Roman Mythology, is said to be an omen of imminent death or demise. It is said that the sound of an owl was heard shortly before the death of several Roman Emperors. [ 9 ] Sri Lankan folklore tells of an owl-like creature whose human sounding shrieks are heard across the jungle at night. Like in Roman mythology, they are said to foretell of death. [ 8 ]
Another common animal found to symbolize death across many cultures are crows. Crows are scavenging birds. This might explain why so many cultures have associated them with death as they were often seen near dead bodies. [ 7 ] Irish folklore tells of Badb, one of a trio of war goddesses, took the form of a crow. Badb is said to foreshadow bloodshed. [ 9 ] Traditional Swedish folklore says that crows are the ghost of those who did not receive a proper burial. [ 7 ] Some literature uses crows circling in the air above a specific place to foretell the death in that area. [ 6 ] All over the world crows are commonly associated with death. In fact, a group of crows is called a murder.
Religious symbols of death and depictions of the afterlife will vary with the religion practiced by the people who use them.
Tombs , tombstones , and other items of funeral architecture are obvious candidates for symbols of death. [ 3 ] In ancient Egypt , the gods Osiris and Ptah were typically depicted as mummies ; these gods governed the Egyptian afterlife . In Christianity , the Christian cross is frequently used on graves , and is meant to call to mind the crucifixion of Jesus . [ 3 ] Some Christians also erect temporary crosses along public highways as memorials for those who died in accidents. In Buddhism , the symbol of a wheel represents the perpetual cycle of death and rebirth that happens in samsara. [ 10 ] The symbol of a grave or tomb, especially one in a picturesque or unusual location, can be used to represent death, as in Nicolas Poussin 's famous painting Et in Arcadia ego .
Images of life in the afterlife are also symbols of death. Here, again, the ancient Egyptians produced detailed pictorial representations of the life enjoyed by the dead. In Christian folk religion , the spirits of the dead are often depicted as winged angels or angel-like creatures, dwelling among the clouds; this imagery of the afterlife is frequently used in comic depictions of life after death. [ 3 ] In the Islamic view of the Afterlife, death is symbolised by a black and white ram which in turn will be slain to symbolise the Death of Death .
The Banshee also symbolizes the coming of death in Irish Mythology. [ 3 ] This is typically represented by an older woman who is seen sobbing to symbolize the suffering of a person before their death. [ 3 ]
Black is the color of mourning in many European cultures. Black clothing is typically worn at funerals to show mourning for the death of the person. In East Asia , white is similarly associated with mourning ; it represents the purity and perfection of the deceased person's spirit. [ 11 ] Hindus similarly also wear white during mourning and funerals . During the Victorian era , purple and grey were considered to be mourning colors in addition to black . [ 12 ] Furthermore, in Revelation 6 in The Bible , Death is one of the four horsemen; and he rides a pale horse. [ 13 ] | https://en.wikipedia.org/wiki/Symbols_of_death |
In mathematics and related subjects, understanding a mathematical expression depends on an understanding of symbols of grouping, such as parentheses (), square brackets [], and braces {} [ 1 ] (see note on terminology below). These same symbols are also used in ways where they are not symbols of grouping. For example, in the expression 3(x+y) the parentheses are symbols of grouping, but in the expression (3, 5) the parentheses may indicate an open interval .
The most common symbols of grouping are the parentheses and the square brackets, and the latter are usually used to avoid too many repeated parentheses. For example, to indicate the product of binomials, parentheses are usually used, thus: ( 2 x + 3 ) ( 3 x + 4 ) {\displaystyle (2x+3)(3x+4)} . But if one of the binomials itself contains parentheses, as in ( 2 ( a + b ) + 3 ) {\displaystyle (2(a+b)+3)} one or more pairs of () may be replaced by [], thus: [ ( 2 ( a + b ) + 3 ] [ 3 x + 4 ] {\displaystyle [(2(a+b)+3][3x+4]} . Beyond elementary mathematics, [] are mostly used for other purposes, e.g. to denote a closed interval , or an equivalence class , so they appear rarely for grouping.
The usage of the word "brackets" varies from country. In the United States, the term denotes [], known elsewhere as "square brackets". In the United Kingdom and many other English-speaking countries, "brackets" means (), known in the US as "parentheses" (singular "parenthesis"). That said, the specific terms "parentheses" and "square brackets" are generally understood everywhere and may be used to avoid ambiguity.
The symbol of grouping knows as "braces" has two major uses. If two of these symbols are used, one on the left and the mirror image of it on the right, it almost always indicates a set , as in { a , b , c } {\displaystyle \{a,b,c\}} , the set containing three members, a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . But if it is used only on the left, it groups two or more simultaneous equations or the cases of a piecewise-defined function .
There are other symbols of grouping. One is the bar above an expression, as in the square root sign in which the bar is a symbol of grouping. For example √ p + q is the square root of the sum. The bar is also a symbol of grouping in repeated decimal digits. A decimal point followed by one or more digits with a bar over them, for example 0. 123 , represents the repeating decimal 0.123123123... . [ 2 ] Another symbol of grouping is the horizontal bar of a fraction, for example 1 2 + 3 {\textstyle {\frac {1}{2+3}}} , which is thus evaluated to 1 5 {\textstyle {\frac {1}{5}}} .
A superscript is understood to be grouped as long as it continues in the form of a superscript. For example if an x has a superscript of the form a + b , the sum is the exponent. For example: x 2 + 3 , it is understood that the 2+3 is grouped, and that the exponent is the sum of 2 and 3.
These rules are understood by all mathematicians.
In most mathematics, the operations of addition and multiplication are associative .
The associative law for addition, for example, states that ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} . This means that once the associative law is stated, the parentheses are unnecessary and are usually omitted. More generally, any sum , of any number of terms , can be written without parentheses and any product , of any number of factors , can be written without parentheses.
The "hierarchy of operations", also called the " order of operations " is a rule that saves needing an excessive number of symbols of grouping. In its simplest form, if a number had a plus sign on one side and a multiplication sign on the other side, the multiplication acts first. If we were to express this idea using symbols of grouping, the factors in a product. Example: 2+3×4 = 2 +(3×4)=2+12=14.
In understanding expressions without symbols of grouping, it is useful to think of subtraction as addition of the opposite, and to think of division as multiplication by the reciprocal. | https://en.wikipedia.org/wiki/Symbols_of_grouping |
A symlink race is a kind of software security vulnerability that results from a program creating files in an insecure manner. [ 1 ] A malicious user can create a symbolic link to a file not otherwise accessible to them. When the privileged program creates a file of the same name as the symbolic link, it actually creates the linked-to file instead, possibly inserting content desired by the malicious user (see example below), or even provided by the malicious user (as input to the program).
It is called a " race " because in its typical manifestation, the program checks to see if a file by that name already exists; if it does not exist, the program then creates the file. An attacker must create the link in the interval between the check and when the file is created .
A symlink race can happen with antivirus products that decide they will quarantine or delete a suspicious file, and then go ahead and do that. During the interval between decision and action, malicious software can replace the suspicious file with a system or antivirus file that the malicious software wants overwritten. [ 2 ]
In this naive example, the Unix program foo is setuid . Its function is to retrieve information for the accounts specified by the user. For "efficiency", it sorts the requested accounts into a temporary file ( /tmp/foo naturally) before making the queries.
The directory /tmp is world-writable. Malicious user Mallory creates a symbolic link to the file /root/.rhosts named /tmp/foo . Then, Mallory invokes foo with user as the requested account. The program creates the (temporary) file /tmp/foo (really creating /root/.rhosts ) and puts information about the requested account (e.g. user password ) in it. It removes the temporary file (merely removing the symbolic link).
Now the /root/.rhosts contains password information, which (if it even happens to be in the proper format) is the incantation necessary to allow anyone to use rlogin to log into the computer as the superuser .
In some Unix-systems there is a special flag O_NOFOLLOW for open(2) to prevent opening a file via a symbolic-link (dangling or otherwise) and has become standardized in POSIX.1-2008 .
The POSIX C standard library function mkstemp can be used to safely create temporary files. For shell scripts, the system utility mktemp(1) does the same thing.
This Unix -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symlink_race |
In mathematics , the symmetric algebra S ( V ) (also denoted Sym( V )) on a vector space V over a field K is a commutative algebra over K that contains V , and is, in some sense, minimal for this property. Here, "minimal" means that S ( V ) satisfies the following universal property : for every linear map f from V to a commutative algebra A , there is a unique algebra homomorphism g : S ( V ) → A such that f = g ∘ i , where i is the inclusion map of V in S ( V ) .
If B is a basis of V , the symmetric algebra S ( V ) can be identified, through a canonical isomorphism , to the polynomial ring K [ B ] , where the elements of B are considered as indeterminates. Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V .
The symmetric algebra S ( V ) can be built as the quotient of the tensor algebra T ( V ) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x .
All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
It is possible to use the tensor algebra T ( V ) to describe the symmetric algebra S ( V ) . In fact, S ( V ) can be defined as the quotient algebra of T ( V ) by the two-sided ideal generated by the commutators v ⊗ w − w ⊗ v . {\displaystyle v\otimes w-w\otimes v.}
It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction. Because of the universal property of the tensor algebra, a linear map f from V to a commutative algebra A extends to an algebra homomorphism T ( V ) → A {\displaystyle T(V)\rightarrow A} , which factors through S(V) because A is commutative. The extension of f to an algebra homomorphism S ( V ) → A {\displaystyle S(V)\rightarrow A} is unique because V generates S(V) as a K -algebra.
This results also directly from a general result of category theory , which asserts that the composition of two left adjoint functors is also a left adjoint functor. Here, the forgetful functor from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property.
The symmetric algebra S ( V ) can also be built from polynomial rings .
If V is a K -vector space or a free K -module , with a basis B , let K [ B ] be the polynomial ring that has the elements of B as indeterminates. The homogeneous polynomials of degree one form a vector space or a free module that can be identified with V . It is straightforward to verify that this makes K [ B ] a solution to the universal problem stated in the introduction. This implies that K [ B ] and S ( V ) are canonically isomorphic, and can therefore be identified. This results also immediately from general considerations of category theory , since free modules and polynomial rings are free objects of their respective categories.
If V is a module that is not free, it can be written V = L / M , {\displaystyle V=L/M,} where L is a free module, and M is a submodule of L . In this case, one has
where ⟨ M ⟩ {\displaystyle \langle M\rangle } is the ideal generated by M . (Here, equals signs mean equality up to a canonical isomorphism.) Again this can be proved by showing that one has a solution of the universal property, and this can be done either by a straightforward but boring computation, or by using category theory, and more specifically, the fact that a quotient is the solution of the universal problem for morphisms that map to zero a given subset. (Depending on the case, the kernel is a normal subgroup , a submodule or an ideal, and the usual definition of quotients can be viewed as a proof of the existence of a solution of the universal problem.)
The symmetric algebra is a graded algebra . That is, it is a direct sum
where S n ( V ) , {\displaystyle S^{n}(V),} called the n th symmetric power of V , is the vector subspace or submodule generated by the products of n elements of V . (The second symmetric power S 2 ( V ) {\displaystyle S^{2}(V)} is sometimes called the symmetric square of V ).
This can be proved by various means. One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by a homogeneous ideal : the ideal generated by all x ⊗ y − y ⊗ x , {\displaystyle x\otimes y-y\otimes x,} where x and y are in V , that is, homogeneous of degree one.
In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the total degree . A non-free module can be written as L / M , where L is a free module of base B ; its symmetric algebra is the quotient of the (graded) symmetric algebra of L (a polynomial ring) by the homogeneous ideal generated by the elements of M , which are homogeneous of degree one.
One can also define S n ( V ) {\displaystyle S^{n}(V)} as the solution of the universal problem for n -linear symmetric functions from V into a vector space or a module, and then verify that the direct sum of all S n ( V ) {\displaystyle S^{n}(V)} satisfies the universal problem for the symmetric algebra.
As the symmetric algebra of a vector space is a quotient of the tensor algebra, an element of the symmetric algebra is not a tensor, and, in particular, is not a symmetric tensor . However, symmetric tensors are strongly related to the symmetric algebra.
A symmetric tensor of degree n is an element of T n ( V ) that is invariant under the action of the symmetric group S n . {\displaystyle {\mathcal {S}}_{n}.} More precisely, given σ ∈ S n , {\displaystyle \sigma \in {\mathcal {S}}_{n},} the transformation v 1 ⊗ ⋯ ⊗ v n ↦ v σ ( 1 ) ⊗ ⋯ ⊗ v σ ( n ) {\displaystyle v_{1}\otimes \cdots \otimes v_{n}\mapsto v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}} defines a linear endomorphism of T n ( V ) . A symmetric tensor is a tensor that is invariant under all these endomorphisms. The symmetric tensors of degree n form a vector subspace (or module) Sym n ( V ) ⊂ T n ( V ) . The symmetric tensors are the elements of the direct sum ⨁ n = 0 ∞ Sym n ( V ) , {\displaystyle \textstyle \bigoplus _{n=0}^{\infty }\operatorname {Sym} ^{n}(V),} which is a graded vector space (or a graded module ). It is not an algebra, as the tensor product of two symmetric tensors is not symmetric in general.
Let π n {\displaystyle \pi _{n}} be the restriction to Sym n ( V ) of the canonical surjection T n ( V ) → S n ( V ) . {\displaystyle T^{n}(V)\to S^{n}(V).} If n ! is invertible in the ground field (or ring), then π n {\displaystyle \pi _{n}} is an isomorphism . This is always the case with a ground field of characteristic zero. The inverse isomorphism is the linear map defined (on products of n vectors) by the symmetrization
The map π n {\displaystyle \pi _{n}} is not injective if the characteristic is less than n +1; for example π n ( x ⊗ y + y ⊗ x ) = 2 x y {\displaystyle \pi _{n}(x\otimes y+y\otimes x)=2xy} is zero in characteristic two. Over a ring of characteristic zero, π n {\displaystyle \pi _{n}} can be non surjective; for example, over the integers, if x and y are two linearly independent elements of V = S 1 ( V ) that are not in 2 V , then x y ∉ π n ( Sym 2 ( V ) ) , {\displaystyle xy\not \in \pi _{n}(\operatorname {Sym} ^{2}(V)),} since 1 2 ( x ⊗ y + y ⊗ x ) ∉ Sym 2 ( V ) . {\displaystyle {\frac {1}{2}}(x\otimes y+y\otimes x)\not \in \operatorname {Sym} ^{2}(V).}
In summary, over a field of characteristic zero, the symmetric tensors and the symmetric algebra form two isomorphic graded vector spaces. They can thus be identified as far as only the vector space structure is concerned, but they cannot be identified as soon as products are involved. Moreover, this isomorphism does not extend to the cases of fields of positive characteristic and rings that do not contain the rational numbers .
Given a module V over a commutative ring K , the symmetric algebra S ( V ) can be defined by the following universal property :
As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, up to a canonical isomorphism . It follows that all properties of the symmetric algebra can be deduced from the universal property. This section is devoted to the main properties that belong to category theory .
The symmetric algebra is a functor from the category of K -modules to the category of K -commutative algebra, since the universal property implies that every module homomorphism f : V → W {\displaystyle f:V\to W} can be uniquely extended to an algebra homomorphism S ( f ) : S ( V ) → S ( W ) . {\displaystyle S(f):S(V)\to S(W).}
The universal property can be reformulated by saying that the symmetric algebra is a left adjoint to the forgetful functor that sends a commutative algebra to its underlying module.
One can analogously construct the symmetric algebra on an affine space . The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra : one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.
For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).
The S k are functors comparable to the exterior powers ; here, though, the dimension grows with k ; it is given by
where n is the dimension of V . This binomial coefficient is the number of n -variable monomials of degree k .
In fact, the symmetric algebra and the exterior algebra appear as the isotypical components of the trivial and sign representation of the action of S n {\displaystyle S_{n}} acting on the tensor product V ⊗ n {\displaystyle V^{\otimes n}} (for example over the complex field) [ citation needed ]
The symmetric algebra can be given the structure of a Hopf algebra . See Tensor algebra for details.
The symmetric algebra S ( V ) is the universal enveloping algebra of an abelian Lie algebra , i.e. one in which the Lie bracket is identically 0. | https://en.wikipedia.org/wiki/Symmetric_algebra |
In mathematics , the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. [ 1 ]
Given a measurable set , A , {\displaystyle A,} in R n , {\displaystyle \mathbb {R} ^{n},} one defines the symmetric rearrangement of A , {\displaystyle A,} called A ∗ , {\displaystyle A^{*},} as the ball centered at the origin, whose volume ( Lebesgue measure ) is the same as that of the set A . {\displaystyle A.}
An equivalent definition is A ∗ = { x ∈ R n : ω n ⋅ | x | n < | A | } , {\displaystyle A^{*}=\left\{x\in \mathbb {R} ^{n}:\,\omega _{n}\cdot |x|^{n}<|A|\right\},} where ω n {\displaystyle \omega _{n}} is the volume of the unit ball and where | A | {\displaystyle |A|} is the volume of A . {\displaystyle A.}
The rearrangement of a non-negative, measurable real-valued function f {\displaystyle f} whose level sets f − 1 ( y ) {\displaystyle f^{-1}(y)} (for y ∈ R ≥ 0 {\displaystyle y\in \mathbb {R} _{\geq 0}} ) have finite measure is f ∗ ( x ) = ∫ 0 ∞ I { y : f ( y ) > t } ∗ ( x ) d t , {\displaystyle f^{*}(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:f(y)>t\}^{*}}(x)\,dt,} where I A {\displaystyle \mathbb {I} _{A}} denotes the indicator function of the set A . {\displaystyle A.} In words, the value of f ∗ ( x ) {\displaystyle f^{*}(x)} gives the height t {\displaystyle t} for which the radius of the symmetric
rearrangement of { y : f ( y ) > t } {\displaystyle \{y:f(y)>t\}} is equal to x . {\displaystyle x.} We have the following motivation for this definition. Because the identity g ( x ) = ∫ 0 ∞ I { y : g ( y ) > t } ( x ) d t , {\displaystyle g(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:g(y)>t\}}(x)\,dt,} holds for any non-negative function g , {\displaystyle g,} the above definition is the unique definition that forces the identity I A ∗ = I A ∗ {\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*}}} to hold.
The function f ∗ {\displaystyle f^{*}} is a symmetric and decreasing function whose level sets have the same measure as the level sets of f , {\displaystyle f,} that is, | { x : f ∗ ( x ) > t } | = | { x : f ( x ) > t } | . {\displaystyle |\{x:f^{*}(x)>t\}|=|\{x:f(x)>t\}|.}
If f {\displaystyle f} is a function in L p , {\displaystyle L^{p},} then ‖ f ‖ L p = ‖ f ∗ ‖ L p . {\displaystyle \|f\|_{L^{p}}=\|f^{*}\|_{L^{p}}.}
The Hardy–Littlewood inequality holds, that is, ∫ f g ≤ ∫ f ∗ g ∗ . {\displaystyle \int fg\leq \int f^{*}g^{*}.}
Further, the Pólya–Szegő inequality holds. This says that if 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } and if f ∈ W 1 , p {\displaystyle f\in W^{1,p}} then ‖ ∇ f ∗ ‖ p ≤ ‖ ∇ f ‖ p . {\displaystyle \|\nabla f^{*}\|_{p}\leq \|\nabla f\|_{p}.}
The symmetric decreasing rearrangement is order preserving and decreases L p {\displaystyle L^{p}} distance, that is, f ≤ g implies f ∗ ≤ g ∗ {\displaystyle f\leq g{\text{ implies }}f^{*}\leq g^{*}} and ‖ f − g ‖ L p ≥ ‖ f ∗ − g ∗ ‖ L p . {\displaystyle \|f-g\|_{L^{p}}\geq \|f^{*}-g^{*}\|_{L^{p}}.}
The Pólya–Szegő inequality yields, in the limit case, with p = 1 , {\displaystyle p=1,} the isoperimetric inequality . Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality .
We can also define f ∗ {\displaystyle f^{*}} as a function on the nonnegative real numbers rather than on all of R n . {\displaystyle \mathbb {R} ^{n}.} [ 2 ] Let ( E , μ ) {\displaystyle (E,\mu )} be a σ-finite measure space , and let f : E → [ − ∞ , ∞ ] {\displaystyle f:E\to [-\infty ,\infty ]} be a measurable function that takes only finite (that is, real) values μ-a.e. (where " μ {\displaystyle \mu } -a.e." means except possibly on a set of μ {\displaystyle \mu } -measure zero). We define the distribution function μ f : [ 0 , ∞ ] → [ 0 , ∞ ] {\displaystyle \mu _{f}:[0,\infty ]\to [0,\infty ]} by the rule μ f ( s ) = μ { x ∈ E : | f ( x ) | > s } . {\displaystyle \mu _{f}(s)=\mu \{x\in E:\vert f(x)\vert >s\}.} We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement ) of f {\displaystyle f} as the function f ∗ : [ 0 , ∞ ) → [ 0 , ∞ ] {\displaystyle f^{*}:[0,\infty )\to [0,\infty ]} by the rule f ∗ ( t ) = inf { s ∈ [ 0 , ∞ ] : μ f ( s ) ≤ t } . {\displaystyle f^{*}(t)=\inf\{s\in [0,\infty ]:\mu _{f}(s)\leq t\}.} Note that this version of the decreasing rearrangement is not symmetric, as it is only defined on the nonnegative real numbers. However, it inherits many of the same properties listed above as the symmetric version, namely:
The (nonsymmetric) decreasing rearrangement function arises often in the theory of rearrangement-invariant Banach function spaces. Especially important is the following:
Note that the definitions of all the terminology in the above theorem (that is, Banach function norms, rearrangement-invariant Banach function spaces, and resonant measure spaces) can be found in sections 1 and 2 of Bennett and Sharpley's book (cf. the references below). | https://en.wikipedia.org/wiki/Symmetric_decreasing_rearrangement |
In mathematics , the symmetric derivative is an operation generalizing the ordinary derivative .
It is defined as: [ 1 ] [ 2 ] lim h → 0 f ( x + h ) − f ( x − h ) 2 h . {\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x-h)}{2h}}.}
The expression under the limit is sometimes called the symmetric difference quotient . [ 3 ] [ 4 ] A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.
If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f ( x ) = | x | , which is not differentiable at x = 0 , but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. [ 3 ]
The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [ 1 ] [ 2 ] : 6
Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
For the absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} , using the notation f s ( x ) {\displaystyle f_{s}(x)} for the symmetric derivative, we have at x = 0 {\displaystyle x=0} that f s ( 0 ) = lim h → 0 f ( 0 + h ) − f ( 0 − h ) 2 h = lim h → 0 f ( h ) − f ( − h ) 2 h = lim h → 0 | h | − | − h | 2 h = lim h → 0 | h | − | h | 2 h = lim h → 0 0 2 h = 0. {\displaystyle {\begin{aligned}f_{s}(0)&=\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {f(h)-f(-h)}{2h}}\\&=\lim _{h\to 0}{\frac {|h|-|{-h}|}{2h}}\\&=\lim _{h\to 0}{\frac {|h|-|h|}{2h}}\\&=\lim _{h\to 0}{\frac {0}{2h}}=0.\\\end{aligned}}}
Hence the symmetric derivative of the absolute value function exists at x = 0 {\displaystyle x=0} and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at x = 0 {\displaystyle x=0} ).
Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected.
For the function f ( x ) = 1 / x 2 {\displaystyle f(x)=1/x^{2}} , at x = 0 {\displaystyle x=0} we have f s ( 0 ) = lim h → 0 f ( 0 + h ) − f ( 0 − h ) 2 h = lim h → 0 f ( h ) − f ( − h ) 2 h = lim h → 0 1 / h 2 − 1 / ( − h ) 2 2 h = lim h → 0 1 / h 2 − 1 / h 2 2 h = lim h → 0 0 2 h = 0. {\displaystyle {\begin{aligned}f_{s}(0)&=\lim _{h\to 0}{\frac {f(0+h)-f(0-h)}{2h}}=\lim _{h\to 0}{\frac {f(h)-f(-h)}{2h}}\\[1ex]&=\lim _{h\to 0}{\frac {1/h^{2}-1/(-h)^{2}}{2h}}=\lim _{h\to 0}{\frac {1/h^{2}-1/h^{2}}{2h}}=\lim _{h\to 0}{\frac {0}{2h}}=0.\end{aligned}}}
Again, for this function the symmetric derivative exists at x = 0 {\displaystyle x=0} , while its ordinary derivative does not exist at x = 0 {\displaystyle x=0} due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an essential discontinuity .
The Dirichlet function , defined as: f ( x ) = { 1 , if x is rational 0 , if x is irrational {\displaystyle f(x)={\begin{cases}1,&{\text{if }}x{\text{ is rational}}\\0,&{\text{if }}x{\text{ is irrational}}\end{cases}}} has a symmetric derivative at every x ∈ Q {\displaystyle x\in \mathbb {Q} } , but is not symmetrically differentiable at any x ∈ R ∖ Q {\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} } ; i.e. the symmetric derivative exists at rational numbers but not at irrational numbers .
The symmetric derivative does not obey the usual mean-value theorem (of Lagrange). As a counterexample, the symmetric derivative of f ( x ) = | x | has the image {−1, 0, 1} , but secants for f can have a wider range of slopes; for instance, on the interval [−1, 2] , the mean-value theorem would mandate that there exist a point where the (symmetric) derivative takes the value | 2 | − | − 1 | 2 − ( − 1 ) = 1 3 {\displaystyle {\frac {|2|-|-1|}{2-(-1)}}={\frac {1}{3}}} . [ 5 ]
A theorem somewhat analogous to Rolle's theorem but for the symmetric derivative was established in 1967 by C. E. Aull, who named it quasi-Rolle theorem. If f is continuous on the closed interval [ a , b ] and symmetrically differentiable on the open interval ( a , b ) , and f ( a ) = f ( b ) = 0 , then there exist two points x , y in ( a , b ) such that f s ( x ) ≥ 0 , and f s ( y ) ≤ 0 . A lemma also established by Aull as a stepping stone to this theorem states that if f is continuous on the closed interval [ a , b ] and symmetrically differentiable on the open interval ( a , b ) , and additionally f ( b ) > f ( a ) , then there exist a point z in ( a , b ) where the symmetric derivative is non-negative, or with the notation used above, f s ( z ) ≥ 0 . Analogously, if f ( b ) < f ( a ) , then there exists a point z in ( a , b ) where f s ( z ) ≤ 0 . [ 5 ]
The quasi-mean-value theorem for a symmetrically differentiable function states that if f is continuous on the closed interval [ a , b ] and symmetrically differentiable on the open interval ( a , b ) , then there exist x , y in ( a , b ) such that [ 5 ] [ 2 ] : 7
f s ( x ) ≤ f ( b ) − f ( a ) b − a ≤ f s ( y ) . {\displaystyle f_{s}(x)\leq {\frac {f(b)-f(a)}{b-a}}\leq f_{s}(y).}
As an application, the quasi-mean-value theorem for f ( x ) = | x | on an interval containing 0 predicts that the slope of any secant of f is between −1 and 1.
If the symmetric derivative of f has the Darboux property , then the (form of the) regular mean-value theorem (of Lagrange) holds, i.e. there exists z in ( a , b ) such that [ 5 ] f s ( z ) = f ( b ) − f ( a ) b − a . {\displaystyle f_{s}(z)={\frac {f(b)-f(a)}{b-a}}.}
As a consequence, if a function is continuous and its symmetric derivative is also continuous (thus has the Darboux property), then the function is differentiable in the usual sense. [ 5 ]
The notion generalizes to higher-order symmetric derivatives and also to n -dimensional Euclidean spaces .
The second symmetric derivative is defined as [ 6 ] [ 2 ] : 1 lim h → 0 f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 . {\displaystyle \lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}
If the (usual) second derivative exists, then the second symmetric derivative exists and is equal to it. [ 6 ] The second symmetric derivative may exist, however, even when the (ordinary) second derivative does not. As example, consider the sign function sgn ( x ) {\displaystyle \operatorname {sgn}(x)} , which is defined by sgn ( x ) = { − 1 if x < 0 , 0 if x = 0 , 1 if x > 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}
The sign function is not continuous at zero, and therefore the second derivative for x = 0 {\displaystyle x=0} does not exist. But the second symmetric derivative exists for x = 0 {\displaystyle x=0} : lim h → 0 sgn ( 0 + h ) − 2 sgn ( 0 ) + sgn ( 0 − h ) h 2 = lim h → 0 sgn ( h ) − 2 ⋅ 0 + ( − sgn ( h ) ) h 2 = lim h → 0 0 h 2 = 0. {\displaystyle \lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.} | https://en.wikipedia.org/wiki/Symmetric_derivative |
In mathematics , the symmetric difference of two sets , also known as the disjunctive union and set sum , is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 } {\displaystyle \{1,2,4\}} .
The symmetric difference of the sets A and B is commonly denoted by A Δ B {\displaystyle A\operatorname {\Delta } B} (alternatively, A △ B {\displaystyle A\operatorname {\vartriangle } B} ), A ⊕ B {\displaystyle A\oplus B} , or A ⊖ B {\displaystyle A\ominus B} .
It can be viewed as a form of addition modulo 2 .
The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse . The power set of any set becomes a Boolean ring , with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
The symmetric difference is equivalent to the union of both relative complements , that is: [ 1 ]
The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation :
The same fact can be stated as the indicator function (denoted here by χ {\displaystyle \chi } ) of the symmetric difference, being the XOR (or addition mod 2 ) of the indicator functions of its two arguments: χ ( A Δ B ) = χ A ⊕ χ B {\displaystyle \chi _{(A\,\Delta \,B)}=\chi _{A}\oplus \chi _{B}} or using the Iverson bracket notation [ x ∈ A Δ B ] = [ x ∈ A ] ⊕ [ x ∈ B ] {\displaystyle [x\in A\,\Delta \,B]=[x\in A]\oplus [x\in B]} .
The symmetric difference can also be expressed as the union of the two sets, minus their intersection :
In particular, A Δ B ⊆ A ∪ B {\displaystyle A\mathbin {\Delta } B\subseteq A\cup B} ; the equality in this non-strict inclusion occurs if and only if A {\displaystyle A} and B {\displaystyle B} are disjoint sets . Furthermore, denoting D = A Δ B {\displaystyle D=A\mathbin {\Delta } B} and I = A ∩ B {\displaystyle I=A\cap B} , then D {\displaystyle D} and I {\displaystyle I} are always disjoint, so D {\displaystyle D} and I {\displaystyle I} partition A ∪ B {\displaystyle A\cup B} . Consequently, assuming intersection and symmetric difference as primitive operations, the union of two sets can be well defined in terms of symmetric difference by the right-hand side of the equality
The symmetric difference is commutative and associative :
The empty set is neutral , and every set is its own inverse:
Thus, the power set of any set X becomes an abelian group under the symmetric difference operation. (More generally, any field of sets forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has order 2) is sometimes called a Boolean group ; [ 2 ] [ 3 ] the symmetric difference provides a prototypical example of such groups. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set. [ 4 ] In the case where X has only two elements, the group thus obtained is the Klein four-group .
Equivalently, a Boolean group is an elementary abelian 2-group . Consequently, the group induced by the symmetric difference is in fact a vector space over the field with 2 elements Z 2 . If X is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of X . This construction is used in graph theory , to define the cycle space of a graph.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed. In particular:
This implies triangle inequality: [ 5 ] the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C .
Intersection distributes over symmetric difference:
and this shows that the power set of X becomes a ring , with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a Boolean ring .
Further properties of the symmetric difference include:
The symmetric difference can be defined in any Boolean algebra , by writing
This operation has the same properties as the symmetric difference of sets.
Repeated symmetric difference is in a sense equivalent to an operation on a multitude of sets (possibly with multiple appearances of the same set) giving the set of elements which are in an odd number of sets.
The symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection: Δ M = { a ∈ ⋃ M : | { A ∈ M : a ∈ A } | is odd } . {\displaystyle \Delta M=\left\{a\in \bigcup M:\left|\{A\in M:a\in A\}\right|{\text{ is odd}}\right\}.}
Evidently, this is well-defined only when each element of the union ⋃ M {\textstyle \bigcup M} is contributed by a finite number of elements of M {\displaystyle M} .
Suppose M = { M 1 , M 2 , … , M n } {\displaystyle M=\left\{M_{1},M_{2},\ldots ,M_{n}\right\}} is a multiset and n ≥ 2 {\displaystyle n\geq 2} . Then there is a formula for | Δ M | {\displaystyle |\Delta M|} , the number of elements in Δ M {\displaystyle \Delta M} , given solely in terms of intersections of elements of M {\displaystyle M} : | Δ M | = ∑ l = 1 n ( − 2 ) l − 1 ∑ 1 ≤ i 1 < i 2 < … < i l ≤ n | M i 1 ∩ M i 2 ∩ … ∩ M i l | . {\displaystyle |\Delta M|=\sum _{l=1}^{n}(-2)^{l-1}\sum _{1\leq i_{1}<i_{2}<\ldots <i_{l}\leq n}\left|M_{i_{1}}\cap M_{i_{2}}\cap \ldots \cap M_{i_{l}}\right|.}
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are.
First consider a finite set S and the counting measure on subsets given by their size. Now consider two subsets of S and set their distance apart as the size of their symmetric difference. This distance is in fact a metric , which makes the power set on S a metric space . If S has n elements, then the distance from the empty set to S is n , and this is the maximum distance for any pair of subsets. [ 6 ]
Using the ideas of measure theory , the separation of measurable sets can be defined to be the measure of their symmetric difference. If μ is a σ-finite measure defined on a σ-algebra Σ, the function
is a pseudometric on Σ. d μ becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if μ ( X Δ Y ) = 0 {\displaystyle \mu (X\,\Delta \,Y)=0} . It is sometimes called Fréchet - Nikodym metric. The resulting metric space is separable if and only if L 2 (μ) is separable.
If μ ( X ) , μ ( Y ) < ∞ {\displaystyle \mu (X),\mu (Y)<\infty } , we have: | μ ( X ) − μ ( Y ) | ≤ μ ( X Δ Y ) {\displaystyle |\mu (X)-\mu (Y)|\leq \mu (X\,\Delta \,Y)} . Indeed,
If S = ( Ω , A , μ ) {\displaystyle S=\left(\Omega ,{\mathcal {A}},\mu \right)} is a measure space and F , G ∈ A {\displaystyle F,G\in {\mathcal {A}}} are measurable sets, then their symmetric difference is also measurable: F Δ G ∈ A {\displaystyle F\Delta G\in {\mathcal {A}}} . One may define an equivalence relation on measurable sets by letting F {\displaystyle F} and G {\displaystyle G} be related if μ ( F Δ G ) = 0 {\displaystyle \mu \left(F\Delta G\right)=0} . This relation is denoted F = G [ A , μ ] {\displaystyle F=G\left[{\mathcal {A}},\mu \right]} .
Given D , E ⊆ A {\displaystyle {\mathcal {D}},{\mathcal {E}}\subseteq {\mathcal {A}}} , one writes D ⊆ E [ A , μ ] {\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]} if to each D ∈ D {\displaystyle D\in {\mathcal {D}}} there's some E ∈ E {\displaystyle E\in {\mathcal {E}}} such that D = E [ A , μ ] {\displaystyle D=E\left[{\mathcal {A}},\mu \right]} . The relation " ⊆ [ A , μ ] {\displaystyle \subseteq \left[{\mathcal {A}},\mu \right]} " is a partial order on the family of subsets of A {\displaystyle {\mathcal {A}}} .
We write D = E [ A , μ ] {\displaystyle {\mathcal {D}}={\mathcal {E}}\left[{\mathcal {A}},\mu \right]} if D ⊆ E [ A , μ ] {\displaystyle {\mathcal {D}}\subseteq {\mathcal {E}}\left[{\mathcal {A}},\mu \right]} and E ⊆ D [ A , μ ] {\displaystyle {\mathcal {E}}\subseteq {\mathcal {D}}\left[{\mathcal {A}},\mu \right]} . The relation " = [ A , μ ] {\displaystyle =\left[{\mathcal {A}},\mu \right]} " is an equivalence relationship between the subsets of A {\displaystyle {\mathcal {A}}} .
The symmetric closure of D {\displaystyle {\mathcal {D}}} is the collection of all A {\displaystyle {\mathcal {A}}} -measurable sets that are = [ A , μ ] {\displaystyle =\left[{\mathcal {A}},\mu \right]} to some D ∈ D {\displaystyle D\in {\mathcal {D}}} . The symmetric closure of D {\displaystyle {\mathcal {D}}} contains D {\displaystyle {\mathcal {D}}} . If D {\displaystyle {\mathcal {D}}} is a sub- σ {\displaystyle \sigma } -algebra of A {\displaystyle {\mathcal {A}}} , so is the symmetric closure of D {\displaystyle {\mathcal {D}}} .
F = G [ A , μ ] {\displaystyle F=G\left[{\mathcal {A}},\mu \right]} iff | 1 F − 1 G | = 0 {\displaystyle \left|\mathbf {1} _{F}-\mathbf {1} _{G}\right|=0} [ A , μ ] {\displaystyle \left[{\mathcal {A}},\mu \right]} almost everywhere .
The Hausdorff distance and the (area of the) symmetric difference are both pseudo-metrics on the set of measurable geometric shapes. However, they behave quite differently. The figure at the right shows two sequences of shapes, "Red" and "Red ∪ Green". When the Hausdorff distance between them becomes smaller, the area of the symmetric difference between them becomes larger, and vice versa. By continuing these sequences in both directions, it is possible to get two sequences such that the Hausdorff distance between them converges to 0 and the symmetric distance between them diverges, or vice versa. | https://en.wikipedia.org/wiki/Symmetric_difference |
In mathematics , a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments . For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2 , x 1 ) {\displaystyle f\left(x_{1},x_{2}\right)=f\left(x_{2},x_{1}\right)} for all x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} such that ( x 1 , x 2 ) {\displaystyle \left(x_{1},x_{2}\right)} and ( x 2 , x 1 ) {\displaystyle \left(x_{2},x_{1}\right)} are in the domain of f . {\displaystyle f.} The most commonly encountered symmetric functions are polynomial functions , which are given by the symmetric polynomials .
A related notion is alternating polynomials , which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector space V {\displaystyle V} is isomorphic to the space of homogeneous polynomials of degree k {\displaystyle k} on V . {\displaystyle V.} Symmetric functions should not be confused with even and odd functions , which have a different sort of symmetry.
Given any function f {\displaystyle f} in n {\displaystyle n} variables with values in an abelian group , a symmetric function can be constructed by summing values of f {\displaystyle f} over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations . These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions f . {\displaystyle f.} The only general case where f {\displaystyle f} can be recovered if both its symmetrization and antisymmetrization are known is when n = 2 {\displaystyle n=2} and the abelian group admits a division by 2 (inverse of doubling); then f {\displaystyle f} is equal to half the sum of its symmetrization and its antisymmetrization.
In statistics , an n {\displaystyle n} -sample statistic (a function in n {\displaystyle n} variables) that is obtained by bootstrapping symmetrization of a k {\displaystyle k} -sample statistic, yielding a symmetric function in n {\displaystyle n} variables, is called a U-statistic . Examples include the sample mean and sample variance . | https://en.wikipedia.org/wiki/Symmetric_function |
In the mathematical field of graph theory , a graph G is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices ( u 1 , v 1 ) {\displaystyle (u_{1},v_{1})} and ( u 2 , v 2 ) {\displaystyle (u_{2},v_{2})} of G , there is an automorphism
such that
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). [ 2 ] Such a graph is sometimes also called 1-arc -transitive [ 2 ] or flag-transitive . [ 3 ]
By definition (ignoring u 1 and u 2 ), a symmetric graph without isolated vertices must also be vertex-transitive . [ 1 ] Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive . However, an edge-transitive graph need not be symmetric, since a—b might map to c—d , but not to d—c . Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular , but not vertex-transitive.
Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree. [ 3 ] However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric. [ 4 ] Such graphs are called half-transitive . [ 5 ] The smallest connected half-transitive graph is Holt's graph , with degree 4 and 27 vertices. [ 1 ] [ 6 ] Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above.
A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. [ 1 ]
A t -arc is defined to be a sequence of t + 1 vertices, such that any two consecutive vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t -transitive graph is a graph such that the automorphism group acts transitively on t -arcs , but not on ( t + 1 )-arcs . Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t -transitive for some t , and the value of t can be used to further classify symmetric graphs. The cube is 2-transitive , for example. [ 1 ]
Note that conventionally the term "symmetric graph" is not complementary to the term " asymmetric graph ," as the latter refers to a graph that has no nontrivial symmetries at all.
Two basic families of symmetric graphs for any number of vertices are the cycle graphs (of degree 2) and the complete graphs . Further symmetric graphs are formed by the vertices and edges of the regular and quasiregular polyhedra: the cube , octahedron , icosahedron , dodecahedron , cuboctahedron , and icosidodecahedron . Extension of the cube to n dimensions gives the hypercube graphs (with 2 n vertices and degree n). Similarly extension of the octahedron to n dimensions gives the graphs of the cross-polytopes , this family of graphs (with 2n vertices and degree 2n − 2) are sometimes referred to as the cocktail party graphs - they are complete graphs with a set of edges making a perfect matching removed. Additional families of symmetric graphs with an even number of vertices 2n, are the evenly split complete bipartite graphs K n,n and the crown graphs on 2n vertices. Many other symmetric graphs can be classified as circulant graphs (but not all).
The Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree.
Combining the symmetry condition with the restriction that graphs be cubic (i.e. all vertices have degree 3) yields quite a strong condition, and such graphs are rare enough to be listed. They all have an even number of vertices. The Foster census and its extensions provide such lists. [ 7 ] The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs , [ 8 ] and in 1988 (when Foster was 92 [ 1 ] ) the then current Foster census (listing all cubic symmetric graphs up to 512 vertices) was published in book form. [ 9 ] The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices [ 10 ] [ 11 ] (ten of these are also distance-transitive ; the exceptions are as indicated):
Other well known cubic symmetric graphs are the Dyck graph , the Foster graph and the Biggs–Smith graph . The ten distance-transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph , are the only cubic distance-transitive graphs.
The vertex-connectivity of a symmetric graph is always equal to the degree d . [ 3 ] In contrast, for vertex-transitive graphs in general, the vertex-connectivity is bounded below by 2( d + 1)/3. [ 2 ]
A t -transitive graph of degree 3 or more has girth at least 2( t − 1). However, there are no finite t -transitive graphs of degree 3 or more for t ≥ 8. In the case of the degree being exactly 3 (cubic symmetric graphs), there are none for t ≥ 6. | https://en.wikipedia.org/wiki/Symmetric_graph |
In abstract algebra , the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions . In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols consists of the permutations that can be performed on the n {\displaystyle n} symbols. [ 1 ] Since there are n ! {\displaystyle n!} ( n {\displaystyle n} factorial ) such permutation operations, the order (number of elements) of the symmetric group S n {\displaystyle \mathrm {S} _{n}} is n ! {\displaystyle n!} .
Although symmetric groups can be defined on infinite sets , this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes , a finite presentation , their subgroups , their automorphism groups , and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.
The symmetric group is important to diverse areas of mathematics such as Galois theory , invariant theory , the representation theory of Lie groups , and combinatorics . Cayley's theorem states that every group G {\displaystyle G} is isomorphic to a subgroup of the symmetric group on (the underlying set of) G {\displaystyle G} .
The symmetric group on a finite set X {\displaystyle X} is the group whose elements are all bijective functions from X {\displaystyle X} to X {\displaystyle X} and whose group operation is that of function composition . [ 1 ] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree n {\displaystyle n} is the symmetric group on the set X = { 1 , 2 , … , n } {\displaystyle X=\{1,2,\ldots ,n\}} .
The symmetric group on a set X {\displaystyle X} is denoted in various ways, including S X {\displaystyle \mathrm {S} _{X}} , S X {\displaystyle {\mathfrak {S}}_{X}} , Σ X {\displaystyle \Sigma _{X}} , X ! {\displaystyle X!} , and Sym ( X ) {\displaystyle \operatorname {Sym} (X)} . [ 1 ] If X {\displaystyle X} is the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} then the name may be abbreviated to S n {\displaystyle \mathrm {S} _{n}} , S n {\displaystyle {\mathfrak {S}}_{n}} , Σ n {\displaystyle \Sigma _{n}} , or Sym ( n ) {\displaystyle \operatorname {Sym} (n)} . [ 1 ]
Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in ( Scott 1987 , Ch. 11), ( Dixon & Mortimer 1996 , Ch. 8), and ( Cameron 1999 ).
The symmetric group on a set of n {\displaystyle n} elements has order n ! {\displaystyle n!} (the factorial of n {\displaystyle n} ). [ 2 ] It is abelian if and only if n {\displaystyle n} is less than or equal to 2. [ 3 ] For n = 0 {\displaystyle n=0} and n = 1 {\displaystyle n=1} (the empty set and the singleton set ), the symmetric groups are trivial (they have order 0 ! = 1 ! = 1 {\displaystyle 0!=1!=1} ). The group S n is solvable if and only if n ≤ 4 {\displaystyle n\leq 4} . This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every n > 4 {\displaystyle n>4} there are polynomials of degree n {\displaystyle n} which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.
The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory . In invariant theory , the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions . In the representation theory of Lie groups , the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors .
In the theory of Coxeter groups , the symmetric group is the Coxeter group of type A n and occurs as the Weyl group of the general linear group . In combinatorics , the symmetric groups, their elements ( permutations ), and their representations provide a rich source of problems involving Young tableaux , plactic monoids , and the Bruhat order . Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions , homogeneous spaces , and automorphism groups of graphs , such as the Higman–Sims group and the Higman–Sims graph .
The elements of the symmetric group on a set X are the permutations of X .
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The composition f ∘ g of permutations f and g , pronounced " f of g ", maps any element x of X to f ( g ( x )) . Concretely, let (see permutation for an explanation of notation): f = ( 1 3 ) ( 2 ) ( 4 5 ) = ( 1 2 3 4 5 3 2 1 5 4 ) , {\displaystyle f=(1~3)(2)(4~5)={\begin{pmatrix}1&2&3&4&5\\3&2&1&5&4\end{pmatrix}},} g = ( 1 2 5 ) ( 3 4 ) = ( 1 2 3 4 5 2 5 4 3 1 ) . {\displaystyle g=(1~2~5)(3~4)={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}.} Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So, composing f and g gives f g = f ∘ g = ( 1 2 4 ) ( 3 5 ) = ( 1 2 3 4 5 2 4 5 1 3 ) . {\displaystyle fg=f\circ g=(1\ 2\ 4)(3\ 5)={\begin{pmatrix}1&2&3&4&5\\2&4&5&1&3\end{pmatrix}}.}
A cycle of length L = k · m , taken to the k th power, will decompose into k cycles of length m : For example, ( k = 2 , m = 3 ), ( 1 2 3 4 5 6 ) 2 = ( 1 3 5 ) ( 2 4 6 ) . {\displaystyle (1~2~3~4~5~6)^{2}=(1~3~5)(2~4~6).}
To check that the symmetric group on a set X is indeed a group , it is necessary to verify the group axioms of closure, associativity, identity, and inverses. [ 4 ]
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 2)(2 5)(3 4). Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation , whereas f is an even permutation.
The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.
The product of two even permutations is even, the product of two odd permutations is even, and the product of one of each is odd. Thus we can define the sign of a permutation:
With this definition,
is a group homomorphism ({+1, −1} is a group under multiplication, where +1 is e, the neutral element ). The kernel of this homomorphism, that is, the set of all even permutations, is called the alternating group A n . It is a normal subgroup of S n , and for n ≥ 2 it has n !/2 elements. The group S n is the semidirect product of A n and any subgroup generated by a single transposition.
Furthermore, every permutation can be written as a product of adjacent transpositions , that is, transpositions of the form ( a a +1) . For instance, the permutation g from above can also be written as g = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5) . The sorting algorithm bubble sort is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.
A cycle of length k is a permutation f for which there exists an element x in {1, ..., n } such that x , f ( x ), f 2 ( x ), ..., f k ( x ) = x are the only elements moved by f ; it conventionally is required that k ≥ 2 since with k = 1 the element x itself would not be moved either. The permutation h defined by
is a cycle of length three, since h (1) = 4 , h (4) = 3 and h (3) = 1 , leaving 2 and 5 untouched. We denote such a cycle by (1 4 3) , but it could equally well be written (4 3 1) or (3 1 4) by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are disjoint if they have disjoint subsets of elements. Disjoint cycles commute : for example, in S 6 there is the equality (4 1 3)(2 5 6) = (2 5 6)(4 1 3) . Every element of S n can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.
Cycles admit the following conjugation property with any permutation σ {\displaystyle \sigma } , this property is often used to obtain its generators and relations .
Certain elements of the symmetric group of {1, 2, ..., n } are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).
The order reversing permutation is the one given by:
This is the unique maximal element with respect to the Bruhat order and the longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions ( i i +1) , 1 ≤ i ≤ n − 1 .
This is an involution, and consists of ⌊ n / 2 ⌋ {\displaystyle \lfloor n/2\rfloor } (non-adjacent) transpositions
so it thus has sign:
which is 4-periodic in n .
In S 2 n , the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them. Its sign is also ( − 1 ) ⌊ n / 2 ⌋ . {\displaystyle (-1)^{\lfloor n/2\rfloor }.}
Note that the reverse on n elements and perfect shuffle on 2 n elements have the same sign; these are important to the classification of Clifford algebras , which are 8-periodic.
The conjugacy classes of S n correspond to the cycle types of permutations; that is, two elements of S n are conjugate in S n if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S 5 , (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S n can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example, k = ( 1 2 3 4 5 1 4 3 2 5 ) , {\displaystyle k={\begin{pmatrix}1&2&3&4&5\\1&4&3&2&5\end{pmatrix}},} which can be written as the product of cycles as (2 4).
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is, ( 2 4 ) ∘ ( 1 2 3 ) ( 4 5 ) ∘ ( 2 4 ) = ( 1 4 3 ) ( 2 5 ) . {\displaystyle (2~4)\circ (1~2~3)(4~5)\circ (2~4)=(1~4~3)(2~5).} It is clear that such a permutation is not unique.
Conjugacy classes of S n correspond to integer partitions of n : to the partition μ = ( μ 1 , μ 2 , ..., μ k ) with n = ∑ i = 1 k μ i {\textstyle n=\sum _{i=1}^{k}\mu _{i}} and μ 1 ≥ μ 2 ≥ ... ≥ μ k , is associated the set C μ of permutations with cycles of lengths μ 1 , μ 2 , ..., μ k . Then C μ is a conjugacy class of S n , whose elements are said to be of cycle-type μ {\displaystyle \mu } .
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
Other than the trivial map S n → C 1 ≅ S 0 ≅ S 1 and the sign map S n → S 2 , the most notable homomorphisms between symmetric groups, in order of relative dimension , are:
There are also a host of other homomorphisms S m → S n where m < n .
For n ≥ 5 , the alternating group A n is simple , and the induced quotient is the sign map: A n → S n → S 2 which is split by taking a transposition of two elements. Thus S n is the semidirect product A n ⋊ S 2 , and has no other proper normal subgroups, as they would intersect A n in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A n (and thus themselves be A n or S n ).
S n acts on its subgroup A n by conjugation, and for n ≠ 6 , S n is the full automorphism group of A n : Aut(A n ) ≅ S n . Conjugation by even elements are inner automorphisms of A n while the outer automorphism of A n of order 2 corresponds to conjugation by an odd element. For n = 6 , there is an exceptional outer automorphism of A n so S n is not the full automorphism group of A n .
Conversely, for n ≠ 6 , S n has no outer automorphisms, and for n ≠ 2 it has no center, so for n ≠ 2, 6 it is a complete group , as discussed in automorphism group , below.
For n ≥ 5 , S n is an almost simple group , as it lies between the simple group A n and its group of automorphisms.
S n can be embedded into A n +2 by appending the transposition ( n + 1, n + 2) to all odd permutations, while embedding into A n +1 is impossible for n > 1 .
The symmetric group on n letters is generated by the adjacent transpositions σ i = ( i , i + 1 ) {\displaystyle \sigma _{i}=(i,i+1)} that swap i and i + 1 . [ 6 ] The collection σ 1 , … , σ n − 1 {\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}} generates S n subject to the following relations: [ 7 ]
where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a Coxeter group (and so also a reflection group ).
Other possible generating sets include the set of transpositions that swap 1 and i for 2 ≤ i ≤ n , [ 8 ] or more generally any set of transpositions that forms a connected graph, [ 9 ] and a set containing any n -cycle and a 2 -cycle of adjacent elements in the n -cycle. [ 10 ] [ 11 ]
A subgroup of a symmetric group is called a permutation group .
The normal subgroups of the finite symmetric groups are well understood. If n ≤ 2 , S n has at most 2 elements, and so has no nontrivial proper subgroups. The alternating group of degree n is always a normal subgroup, a proper one for n ≥ 2 and nontrivial for n ≥ 3 ; for n ≥ 3 it is in fact the only nontrivial proper normal subgroup of S n , except when n = 4 where there is one additional such normal subgroup, which is isomorphic to the Klein four group .
The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali (1915 [ 12 ] ) proved that each permutation can be written as a product of three squares. (Any squared element must belong to the hypothesized subgroup of index 2, hence so must the product of any number of squares.) However it contains the normal subgroup S of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of S that are products of an even number of transpositions form a subgroup of index 2 in S , called the alternating subgroup A . Since A is even a characteristic subgroup of S , it is also a normal subgroup of the full symmetric group of the infinite set. The groups A and S are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Onofri (1929 [ 13 ] ) and independently Schreier – Ulam (1934 [ 14 ] ). For more details see ( Scott 1987 , Ch. 11.3). That result, often called the Schreier-Ulam theorem, is superseded by a stronger one which says that the nontrivial normal subgroups of the symmetric group on a set X {\displaystyle X} are 1) the even permutations with finite support and 2) for every cardinality ℵ 0 ≤ κ ≤ | X | {\displaystyle \aleph _{0}\leq \kappa \leq |X|} the group of permutations with support less than κ {\displaystyle \kappa } ( Dixon & Mortimer 1996 , Ch. 8.1).
The maximal subgroups of S n fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form S k × S n – k for 1 ≤ k < n /2 . The imprimitive maximal subgroups are exactly those of the form S k wr S n / k , where 2 ≤ k ≤ n /2 is a proper divisor of n and "wr" denotes the wreath product . The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups , ( Liebeck, Praeger & Saxl 1988 ) gave a fairly satisfactory description of the maximal subgroups of this type, according to ( Dixon & Mortimer 1996 , p. 268).
The Sylow subgroups of the symmetric groups are important examples of p -groups . They are more easily described in special cases first:
The Sylow p -subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p -cycles. There are ( p − 1)!/( p − 1) = ( p − 2)! such subgroups simply by counting generators . The normalizer therefore has order p ⋅( p − 1) and is known as a Frobenius group F p ( p −1) (especially for p = 5 ), and is the affine general linear group , AGL(1, p ) .
The Sylow p -subgroups of the symmetric group of degree p 2 are the wreath product of two cyclic groups of order p . For instance, when p = 3 , a Sylow 3-subgroup of Sym(9) is generated by a = (1 4 7)(2 5 8)(3 6 9) and the elements x = (1 2 3), y = (4 5 6), z = (7 8 9) , and every element of the Sylow 3-subgroup has the form a i x j y k z l for 0 ≤ i , j , k , l ≤ 2 {\displaystyle 0\leq i,j,k,l\leq 2} .
The Sylow p -subgroups of the symmetric group of degree p n are sometimes denoted W p ( n ), and using this notation one has that W p ( n + 1) is the wreath product of W p ( n ) and W p (1).
In general, the Sylow p -subgroups of the symmetric group of degree n are a direct product of a i copies of W p ( i ), where 0 ≤ a i ≤ p − 1 and n = a 0 + p ⋅ a 1 + ... + p k ⋅ a k (the base p expansion of n ).
For instance, W 2 (1) = C 2 and W 2 (2) = D 8 , the dihedral group of order 8 , and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by { (1,3)(2,4), (1,2), (3,4), (5,6) } and is isomorphic to D 8 × C 2 .
These calculations are attributed to ( Kaloujnine 1948 ) and described in more detail in ( Rotman 1995 , p. 176). Note however that ( Kerber 1971 , p. 26) attributes the result to an 1844 work of Cauchy , and mentions that it is even covered in textbook form in ( Netto 1882 , §39–40).
A transitive subgroup of S n is a subgroup whose action on {1, 2, ,..., n } is transitive . For example, the Galois group of a ( finite ) Galois extension is a transitive subgroup of S n , for some n .
A subgroup of S n that is generated by transpositions is called a Young subgroup . They are all of the form S a 1 × ⋯ × S a ℓ {\displaystyle S_{a_{1}}\times \cdots \times S_{a_{\ell }}} where ( a 1 , … , a ℓ ) {\displaystyle (a_{1},\ldots ,a_{\ell })} is an integer partition of n . These groups may also be characterized as the parabolic subgroups of S n when it is viewed as a reflection group .
Cayley's theorem states that every group G is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements of G , since every group acts on itself faithfully by (left or right) multiplication.
Cyclic groups are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the least common multiple of the lengths of its cycles. For example, in S 5 , one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S 5 are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size. [ citation needed ] For example, S 5 has no subgroup of order 15 (a divisor of the order of S 5 ), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S n ) is given by Landau's function .
For n ≠ 2, 6 , S n is a complete group : its center and outer automorphism group are both trivial.
For n = 2 , the automorphism group is trivial, but S 2 is not trivial: it is isomorphic to C 2 , which is abelian, and hence the center is the whole group.
For n = 6 , it has an outer automorphism of order 2: Out(S 6 ) = C 2 , and the automorphism group is a semidirect product Aut(S 6 ) = S 6 ⋊ C 2 .
In fact, for any set X of cardinality other than 6, every automorphism of the symmetric group on X is inner, a result first due to ( Schreier & Ulam 1936 ) according to ( Dixon & Mortimer 1996 , p. 259).
The group homology of S n is quite regular and stabilizes: the first homology (concretely, the abelianization ) is:
The first homology group is the abelianization, and corresponds to the sign map S n → S 2 which is the abelianization for n ≥ 2; for n < 2 the symmetric group is trivial. This homology is easily computed as follows: S n is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps S n → C p are to S 2 and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps S n → S 2 ≅ {±1} send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S n .
The second homology (concretely, the Schur multiplier ) is:
This was computed in ( Schur 1911 ), and corresponds to the double cover of the symmetric group , 2 · S n .
Note that the exceptional low-dimensional homology of the alternating group ( H 1 ( A 3 ) ≅ H 1 ( A 4 ) ≅ C 3 , {\displaystyle H_{1}(\mathrm {A} _{3})\cong H_{1}(\mathrm {A} _{4})\cong \mathrm {C} _{3},} corresponding to non-trivial abelianization, and H 2 ( A 6 ) ≅ H 2 ( A 7 ) ≅ C 6 , {\displaystyle H_{2}(\mathrm {A} _{6})\cong H_{2}(\mathrm {A} _{7})\cong \mathrm {C} _{6},} due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map A 4 ↠ C 3 {\displaystyle \mathrm {A} _{4}\twoheadrightarrow \mathrm {C} _{3}} extends to S 4 ↠ S 3 , {\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3},} and the triple covers of A 6 and A 7 extend to triple covers of S 6 and S 7 – but these are not homological – the map S 4 ↠ S 3 {\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3}} does not change the abelianization of S 4 , and the triple covers do not correspond to homology either.
The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map S n → S n +1 , and for fixed k , the induced map on homology H k (S n ) → H k (S n +1 ) is an isomorphism for sufficiently high n . This is analogous to the homology of families Lie groups stabilizing.
The homology of the infinite symmetric group is computed in ( Nakaoka 1961 ), with the cohomology algebra forming a Hopf algebra .
The representation theory of the symmetric group is a particular case of the representation theory of finite groups , for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles .
The symmetric group S n has order n !. Its conjugacy classes are labeled by partitions of n . Therefore, according to the representation theory of a finite group, the number of inequivalent irreducible representations , over the complex numbers , is equal to the number of partitions of n . Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n .
Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram.
Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra K S n is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).
However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called Specht modules , and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general.
The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory. | https://en.wikipedia.org/wiki/Symmetric_group |
A symmetric hydrogen bond is a special type of hydrogen bond in which the proton is spaced exactly halfway between two identical atoms. The strength of the bond to each of those atoms is equal. It is an example of a 3-center 4-electron bond . This type of bond is much stronger than "normal" hydrogen bonds, in fact, its strength is comparable to a covalent bond . It is seen in ice at high pressure ( Ice X ), and also in the solid phase of many anhydrous acids such as hydrofluoric acid and formic acid at high pressure. It is also seen in the bifluoride ion [F−H−F] − . Much has been done to explain the symmetric hydrogen bond quantum-mechanically, as it seems to violate the duet rule for the first shell: The proton is effectively surrounded by four electrons . Because of this problem, some consider it to be an ionic bond . | https://en.wikipedia.org/wiki/Symmetric_hydrogen_bond |
The level-index ( LI ) representation of numbers, and its algorithms for arithmetic operations, were introduced by Charles Clenshaw and Frank Olver in 1984. [ 1 ]
The symmetric form of the LI system and its arithmetic operations were presented by Clenshaw and Peter Turner in 1987. [ 2 ]
Michael Anuta, Daniel Lozier, Nicolas Schabanel and Turner developed the algorithm for symmetric level-index ( SLI ) arithmetic, and a parallel implementation of it. There has been extensive work on developing the SLI arithmetic algorithms and extending them to complex and vector arithmetic operations.
The idea of the level-index system is to represent a non-negative real number X as
where 0 ≤ f < 1 {\displaystyle 0\leq f<1} , and the process of exponentiation is performed ℓ times, with ℓ ≥ 0 {\displaystyle \ell \geq 0} . ℓ and f are the level and index of X respectively. x = ℓ + f is the LI image of X . For example,
so its LI image is
The symmetric form is used to allow negative exponents, if the magnitude of X is less than 1. One takes sgn (log( X )) or sgn(| X | − | X | −1 ) and stores it (after substituting +1 for 0 for the reciprocal sign; since for X = 1 = e 0 the LI image is x = 1.0 and uniquely defines X = 1 , we can do away without a third state and use only one bit for the two states −1 and +1 [ clarification needed ] ) as the reciprocal sign r X . Mathematically, this is equivalent to taking the reciprocal (multiplicative inverse) of a small-magnitude number, and then finding the SLI image for the reciprocal. Using one bit for the reciprocal sign enables the representation of extremely small numbers.
A sign bit may also be used to allow negative numbers. One takes sgn ( X ) and stores it (after substituting +1 for 0 for the sign; since for X = 0 the LI image is x = 0.0 and uniquely defines X = 0 , we can do away without a third state and use only one bit for the two states −1 and +1 [ clarification needed ] ) as the sign s X . Mathematically, this is equivalent to taking the inverse (additive inverse) of a negative number, and then finding the SLI image for the inverse. Using one bit for the sign enables the representation of negative numbers.
The mapping function is called the generalized logarithm function . It is defined as
and it maps [ 0 , ∞ ) {\displaystyle [0,\infty )} onto itself monotonically, thus being invertible on this interval. The inverse, the generalized exponential function , is defined by
The density of values X represented by x has no discontinuities as we go from level ℓ to ℓ + 1 (a very desirable property) since
The generalized logarithm function is closely related to the iterated logarithm used in computer science analysis of algorithms.
Formally, we can define the SLI representation for an arbitrary real X (not 0 or 1) as
where s X is the sign (additive inversion or not) of X , and r X is the reciprocal sign (multiplicative inversion or not) as in the following equations:
whereas for X = 0 or 1, we have
For example,
and its SLI representation is | https://en.wikipedia.org/wiki/Symmetric_level-index_arithmetic |
In mathematics , a symmetric polynomial is a polynomial P ( X 1 , X 2 , ..., X n ) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P ( X σ(1) , X σ(2) , ..., X σ( n ) ) = P ( X 1 , X 2 , ..., X n ) .
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients , since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous , power sum , and Schur polynomials play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions , are of great importance in combinatorics and in representation theory .
The following polynomials in two variables X 1 and X 2 are symmetric:
as is the following polynomial in three variables X 1 , X 2 , X 3 :
There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant ).
On the other hand, the polynomial in two variables
is not symmetric, since if one exchanges X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} one gets a different polynomial, X 2 − X 1 {\displaystyle X_{2}-X_{1}} . Similarly in three variables
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field . These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots?
This leads to studying solutions of polynomials using the permutation group of the roots, originally in the form of Lagrange resolvents , later developed in Galois theory .
Consider a monic polynomial in t of degree n
with coefficients a i in some field K . There exist n roots x 1 ,..., x n of P in some possibly larger field (for instance if K is the field of real numbers , the roots will exist in the field of complex numbers ); some of the roots might be equal, but the fact that one has all roots is expressed by the relation
By comparing coefficients one finds that
These are in fact just instances of Vieta's formulas . They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression : although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field K or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.
Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P , and considering them as indeterminates rather than as constants in an appropriate field; the coefficients a i then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign ( − 1 ) n − i {\displaystyle (-1)^{n-i}} , are known as the elementary symmetric polynomials in x 1 , ..., x n . A basic fact, known as the fundamental theorem of symmetric polynomials , states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field K that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than K in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities , which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
There are a few types of symmetric polynomials in the variables X 1 , X 2 , ..., X n that are fundamental.
For each nonnegative integer k , the elementary symmetric polynomial e k ( X 1 , ..., X n ) is the sum of all distinct products of k distinct variables. (Some authors denote it by σ k instead.) For k = 0 there is only the empty product so e 0 ( X 1 , ..., X n ) = 1, while for k > n , no products at all can be formed, so e k ( X 1 , X 2 , ..., X n ) = 0 in these cases. The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:
For example, for n = 2, the relevant elementary symmetric polynomials are e 1 ( X 1 , X 2 ) = X 1 + X 2 , and e 2 ( X 1 , X 2 ) = X 1 X 2 . The first polynomial in the list of examples above can then be written as
(for a proof that this is always possible see the fundamental theorem of symmetric polynomials ).
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial , with only those copies required to obtain symmetry. Any monomial in X 1 , ..., X n can be written as X 1 α 1 ... X n α n where the exponents α i are natural numbers (possibly zero); writing α = (α 1 ,...,α n ) this can be abbreviated to X α . The monomial symmetric polynomial m α ( X 1 , ..., X n ) is defined as the sum of all monomials x β where β ranges over all distinct permutations of (α 1 ,...,α n ). For instance one has
Clearly m α = m β when β is a permutation of α, so one usually considers only those m α for which α 1 ≥ α 2 ≥ ... ≥ α n , in other words for which α is a partition of an integer .
These monomial symmetric polynomials form a vector space basis : every symmetric polynomial P can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in P . In particular if P has integer coefficients, then so will the linear combination.
The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has
For each integer k ≥ 1, the monomial symmetric polynomial m ( k ,0,...,0) ( X 1 , ..., X n ) is of special interest. It is the power sum symmetric polynomial, defined as
All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,
In particular, the remaining power sum polynomials p k ( X 1 , ..., X n ) for k > n can be so expressed in the first n power sum polynomials; for example
In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials.
For an example, for n = 2, the symmetric polynomial
has the expression
Using three variables one gets a different expression
The corresponding expression was valid for two variables as well (it suffices to set X 3 to zero), but since it involves p 3 , it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n . But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e 1 which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n , which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic ; however, it is valid with coefficients in any ring containing the rational numbers.
For each nonnegative integer k , the complete homogeneous symmetric polynomial h k ( X 1 , ..., X n ) is the sum of all distinct monomials of degree k in the variables X 1 , ..., X n . For instance
The polynomial h k ( X 1 , ..., X n ) is also the sum of all distinct monomial symmetric polynomials of degree k in X 1 , ..., X n , for instance for the given example
All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X 1 , ..., X n can be obtained from the complete homogeneous symmetric polynomials h 1 ( X 1 , ..., X n ), ..., h n ( X 1 , ..., X n ) via multiplications and additions. More precisely:
For example, for n = 2, the relevant complete homogeneous symmetric polynomials are h 1 ( X 1 , X 2 ) = X 1 + X 2 and h 2 ( X 1 , X 2 ) = X 1 2 + X 1 X 2 + X 2 2 . The first polynomial in the list of examples above can then be written as
As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond h n ( X 1 , ..., X n ), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.
An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities
Since e 0 ( X 1 , ..., X n ) and h 0 ( X 1 , ..., X n ) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the h k ( X 1 , ..., X n ) with 1 ≤ k ≤ n : one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.
Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to representation theory . They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.
Symmetric polynomials are important to linear algebra , representation theory , and Galois theory . They are also important in combinatorics , where they are mostly studied through the ring of symmetric functions , which avoids having to carry around a fixed number of variables all the time.
Analogous to symmetric polynomials are alternating polynomials : polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation .
These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant. | https://en.wikipedia.org/wiki/Symmetric_polynomial |
In mathematics , the n -fold symmetric product of an algebraic curve C is the quotient space of the n -fold cartesian product
or C n by the group action of the symmetric group S n on n letters permuting the factors. It exists as a smooth algebraic variety denoted by Σ n C . If C is a compact Riemann surface , Σ n C is therefore a complex manifold . Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n , that is, formal sums of points with non-negative integer coefficients.
For C the projective line (say the Riemann sphere C {\displaystyle \mathbb {C} } ∪ {∞} ≈ S 2 ), its nth symmetric product Σ n C can be identified with complex projective space C P n {\displaystyle \mathbb {CP} ^{n}} of dimension n .
If G has genus g ≥ 1 then the Σ n C are closely related to the Jacobian variety J of C . More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor ) have dimension n and fill up J , with some identifications caused by special divisors .
For g = n we have Σ g C actually birationally equivalent to J ; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C , and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil 's technique of constructing J as an abstract variety from 'birational data'. Other ways of constructing J , for example as a Picard variety , are preferred now [ 1 ] but this does mean that for any rational function F on C
makes sense as a rational function on J , for the x i staying away from the poles of F .
For n > g the mapping from Σ n C to J by addition fibers it over J ; when n is large enough (around twice g ) this becomes a projective space bundle (the Picard bundle ). It has been studied in detail, for example by Kempf and Mukai.
Let C be a smooth projective curve of genus g over the complex numbers C . The Betti numbers b i (Σ n C) of the symmetric products Σ n C for all n = 0, 1, 2, ... are given by the generating function
and their Euler characteristics e (Σ n C) are given by the generating function
Here we have set u = -1 and y = - p in the previous formula. | https://en.wikipedia.org/wiki/Symmetric_product_of_an_algebraic_curve |
In mathematics , a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold ) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry , leading to consequences in the theory of holonomy ; or algebraically through Lie theory , which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry , representation theory and harmonic analysis .
In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold ( M , g ) is said to be symmetric if and only if, for each point p of M , there exists an isometry of M fixing p and acting on the tangent space T p M {\displaystyle T_{p}M} as minus the identity (every symmetric space is complete , since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds .
From the point of view of Lie theory, a symmetric space is the quotient G / H of a connected Lie group G by a Lie subgroup H that is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and reduces to it when H is compact.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by Marcel Berger . They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
Let M be a connected Riemannian manifold and p a point of M . A diffeomorphism f of a neighborhood of p is said to be a geodesic symmetry if it fixes the point p and reverses geodesics through that point, i.e. if γ is a geodesic with γ ( 0 ) = p {\displaystyle \gamma (0)=p} then f ( γ ( t ) ) = γ ( − t ) . {\displaystyle f(\gamma (t))=\gamma (-t).} It follows that the derivative of the map f at p is minus the identity map on the tangent space of p . On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M .
M is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor.
A locally symmetric space is said to be a (globally) symmetric space if in addition its geodesic symmetries can be extended to isometries on all of M .
The Cartan–Ambrose–Hicks theorem implies that M is locally Riemannian symmetric if and only if its curvature tensor is covariantly constant , and furthermore that every simply connected , complete locally Riemannian symmetric space is actually Riemannian symmetric.
Every Riemannian symmetric space M is complete and Riemannian homogeneous (meaning that the isometry group of M acts transitively on M ). In fact, already the identity component of the isometry group acts transitively on M (because M is connected).
Locally Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.
Basic examples of Riemannian symmetric spaces are Euclidean space , spheres , projective spaces , and hyperbolic spaces , each with their standard Riemannian metrics. More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric.
Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space.
Every lens space is locally symmetric but not symmetric, with the exception of L ( 2 , 1 ) {\displaystyle L(2,1)} , which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points.
An example of a non-Riemannian symmetric space is anti-de Sitter space .
Let G be a connected Lie group . Then a symmetric space for G is a homogeneous space G / H where the stabilizer H of a typical point is an open subgroup of the fixed point set of an involution σ in Aut( G ). Thus σ is an automorphism of G with σ 2 = id G and H is an open subgroup of the invariant set
Because H is open, it is a union of components of G σ (including, of course, the identity component).
As an automorphism of G , σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra g {\displaystyle {\mathfrak {g}}} of G , also denoted by σ , whose square is the identity. It follows that the eigenvalues of σ are ±1. The +1 eigenspace is the Lie algebra h {\displaystyle {\mathfrak {h}}} of H (since this is the Lie algebra of G σ ), and the −1 eigenspace will be denoted m {\displaystyle {\mathfrak {m}}} . Since σ is an automorphism of g {\displaystyle {\mathfrak {g}}} , this gives a direct sum decomposition
with
The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer h {\displaystyle {\mathfrak {h}}} is a Lie subalgebra of g {\displaystyle {\mathfrak {g}}} . The second condition means that m {\displaystyle {\mathfrak {m}}} is an h {\displaystyle {\mathfrak {h}}} -invariant complement to h {\displaystyle {\mathfrak {h}}} in g {\displaystyle {\mathfrak {g}}} . Thus any symmetric space is a reductive homogeneous space , but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that m {\displaystyle {\mathfrak {m}}} brackets into h {\displaystyle {\mathfrak {h}}} .
Conversely, given any Lie algebra g {\displaystyle {\mathfrak {g}}} with a direct sum decomposition satisfying these three conditions, the linear map σ , equal to the identity on h {\displaystyle {\mathfrak {h}}} and minus the identity on m {\displaystyle {\mathfrak {m}}} , is an involutive automorphism.
If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie group acting transitively on M (that is, M is Riemannian homogeneous). Therefore, if we fix some point p of M , M is diffeomorphic to the quotient G/K , where K denotes the isotropy group of the action of G on M at p . By differentiating the action at p we obtain an isometric action of K on T p M . This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its 1-jet at any point) and so K is a subgroup of the orthogonal group of T p M , hence compact. Moreover, if we denote by s p : M → M the geodesic symmetry of M at p , the map
is an involutive Lie group automorphism such that the isotropy group K is contained between the fixed point group G σ {\displaystyle G^{\sigma }} and its identity component (hence an open subgroup) ( G σ ) o , {\displaystyle (G^{\sigma })_{o}\,,} see the definition and following proposition on page 209, chapter IV, section 3 in Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information.
To summarize, M is a symmetric space G / K with a compact isotropy group K . Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a K -invariant inner product on the tangent space to G / K at the identity coset eK : such an inner product always exists by averaging, since K is compact, and by acting with G , we obtain a G -invariant Riemannian metric g on G / K .
To show that G / K is Riemannian symmetric, consider any point p = hK (a coset of K , where h ∈ G ) and define
where σ is the involution of G fixing K . Then one can check that s p is an isometry with (clearly) s p ( p ) = p and (by differentiating) d s p equal to minus the identity on T p M . Thus s p is a geodesic symmetry and, since p was arbitrary, M is a Riemannian symmetric space.
If one starts with a Riemannian symmetric space M , and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" ( G , K , σ , g ) completely describe the structure of M .
The algebraic description of Riemannian symmetric spaces enabled Élie Cartan to obtain a complete classification of them in 1926.
For a given Riemannian symmetric space M let ( G , K , σ , g ) be the algebraic data associated to it. To classify the possible isometry classes of M , first note that the universal cover of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group G of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality that M is simply connected. (This implies K is connected by the long exact sequence of a fibration , because G is connected by assumption.)
A simply connected Riemannian symmetric space is said to be irreducible if it is not the product of two or more Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian symmetric spaces.
The next step is to show that any irreducible, simply connected Riemannian symmetric space M is of one of the following three types:
A more refined invariant is the rank , which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it is of noncompact type. The spaces of Euclidean type have rank equal to their dimension and are isometric to a Euclidean space of that dimension. Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type. In both cases there are two classes.
A. G is a (real) simple Lie group;
B. G is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type).
The examples in class B are completely described by the classification of simple Lie groups . For compact type, M is a compact simply connected simple Lie group, G is M × M and K is the diagonal subgroup. For non-compact type, G is a simply connected complex simple Lie group and K is its maximal compact subgroup. In both cases, the rank is the rank of G .
The compact simply connected Lie groups are the universal covers of the classical Lie groups SO( n ), SU( n ), Sp( n ) and the five exceptional Lie groups E 6 , E 7 , E 8 , F 4 , G 2 .
The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type, G is such a group and K is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of G that contains K . More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups G (up to conjugation). Such involutions extend to involutions of the complexification of G , and these in turn classify non-compact real forms of G .
In both class A and class B there is thus a correspondence between symmetric spaces of compact type and non-compact type. This is known as duality for Riemannian symmetric spaces.
Specializing to the Riemannian symmetric spaces of class A and compact type, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces G / K . They are here given in terms of G and K , together with a geometric interpretation, if readily available. The labelling of these spaces is the one given by Cartan.
A more modern classification ( Huang & Leung 2010 ) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a Freudenthal magic square construction. The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian , or a double Lagrangian Grassmannian of subspaces of ( A ⊗ B ) n , {\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{n},} for normed division algebras A and B . A similar construction produces the irreducible non-compact Riemannian symmetric spaces.
An important class of symmetric spaces generalizing the Riemannian symmetric spaces are pseudo-Riemannian symmetric spaces , in which the Riemannian metric is replaced by a pseudo-Riemannian metric (nondegenerate instead of positive definite on each tangent space). In particular, Lorentzian symmetric spaces , i.e., n dimensional pseudo-Riemannian symmetric spaces of signature ( n − 1,1), are important in general relativity , the most notable examples being Minkowski space , De Sitter space and anti-de Sitter space (with zero, positive and negative curvature respectively). De Sitter space of dimension n may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension n + 1.
Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If M = G / H is a symmetric space, then Nomizu showed that there is a G -invariant torsion-free affine connection (i.e. an affine connection whose torsion tensor vanishes) on M whose curvature is parallel . Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.
The classification of Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric space G / H with Lie algebra
is said to be irreducible if m {\displaystyle {\mathfrak {m}}} is an irreducible representation of h {\displaystyle {\mathfrak {h}}} . Since h {\displaystyle {\mathfrak {h}}} is not semisimple (or even reductive) in general, it can have indecomposable representations which are not irreducible.
However, the irreducible symmetric spaces can be classified. As shown by Katsumi Nomizu , there is a dichotomy: an irreducible symmetric space G / H is either flat (i.e., an affine space) or g {\displaystyle {\mathfrak {g}}} is semisimple. This is the analogue of the Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with g {\displaystyle {\mathfrak {g}}} semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if g {\displaystyle {\mathfrak {g}}} is simple, G / H might not be irreducible.
As in the Riemannian case there are semisimple symmetric spaces with G = H × H . Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that g {\displaystyle {\mathfrak {g}}} is simple. It remains to describe the latter case. For this, one needs to classify involutions σ of a (real) simple Lie algebra g {\displaystyle {\mathfrak {g}}} . If g c {\displaystyle {\mathfrak {g}}^{c}} is not simple, then g {\displaystyle {\mathfrak {g}}} is a complex simple Lie algebra, and the corresponding symmetric spaces have the form G / H , where H is a real form of G : these are the analogues of the Riemannian symmetric spaces G / K with G a complex simple Lie group, and K a maximal compact subgroup.
Thus we may assume g c {\displaystyle {\mathfrak {g}}^{c}} is simple. The real subalgebra g {\displaystyle {\mathfrak {g}}} may be viewed as the fixed point set of a complex antilinear involution τ of g c {\displaystyle {\mathfrak {g}}^{c}} , while σ extends to a complex antilinear involution of g c {\displaystyle {\mathfrak {g}}^{c}} commuting with τ and hence also a complex linear involution σ ∘ τ .
The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite σ ∘ τ determines a complex symmetric space, while τ determines a real form. From this it is easy to construct tables of symmetric spaces for any given g c {\displaystyle {\mathfrak {g}}^{c}} , and furthermore, there is an obvious duality given by exchanging σ and τ . This extends the compact/non-compact duality from the Riemannian case, where either σ or τ is a Cartan involution , i.e., its fixed point set is a maximal compact subalgebra.
The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.
For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing σ to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case kl = 0 .
In the 1950s Atle Selberg extended Cartan's definition of symmetric space to that of weakly symmetric Riemannian space , or in current terminology weakly symmetric space . These are defined as Riemannian manifolds M with a transitive connected Lie group of isometries G and an isometry σ normalising G such that given x , y in M there is an isometry s in G such that sx = σy and sy = σx . (Selberg's assumption that σ 2 should be an element of G was later shown to be unnecessary by Ernest Vinberg .) Selberg proved that weakly symmetric spaces give rise to Gelfand pairs , so that in particular the unitary representation of G on L 2 ( M ) is multiplicity free.
Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every point x in M and tangent vector X at x , there is an isometry s of M , depending on x and X , such that
When s is independent of X , M is a symmetric space.
An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex semisimple Lie algebras , is given in Wolf (2007) .
Some properties and forms of symmetric spaces can be noted.
The metric tensor on the Riemannian manifold M can be lifted to a scalar product on G by combining it with the Killing form . This is done by defining
Here, ⟨ ⋅ , ⋅ ⟩ p {\displaystyle \langle \cdot ,\cdot \rangle _{p}} is the Riemannian metric defined on T p M {\displaystyle T_{p}M} , and B ( X , Y ) = trace ( ad X ∘ ad Y ) {\displaystyle B(X,Y)=\operatorname {trace} (\operatorname {ad} X\circ \operatorname {ad} Y)} is the Killing form . The minus sign appears because the Killing form is negative-definite on h ; {\displaystyle {\mathfrak {h}}~;} this makes ⟨ ⋅ , ⋅ ⟩ g {\displaystyle \langle \cdot ,\cdot \rangle _{\mathfrak {g}}} positive-definite.
The tangent space m {\displaystyle {\mathfrak {m}}} can be further factored into eigenspaces classified by the Killing form. [ 1 ] This is accomplished by defining an adjoint map m → m {\displaystyle {\mathfrak {m}}\to {\mathfrak {m}}} taking Y ↦ Y # {\displaystyle Y\mapsto Y^{\#}} as
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the Riemannian metric on m {\displaystyle {\mathfrak {m}}} and B ( ⋅ , ⋅ ) {\displaystyle B(\cdot ,\cdot )} is the Killing form. This map is sometimes called the generalized transpose , as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis Y 1 , … , Y n {\displaystyle Y_{1},\ldots ,Y_{n}} of m {\displaystyle {\mathfrak {m}}} with
These are orthogonal with respect to the metric, in that
since the Killing form is symmetric. This factorizes m {\displaystyle {\mathfrak {m}}} into eigenspaces
with
for i ≠ j {\displaystyle i\neq j} . For the case of g {\displaystyle {\mathfrak {g}}} semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes:
In certain practical applications, this factorization can be interpreted as the spectrum of operators, e.g. the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital ( i.e. the Killing form being a Casimir operator that can classify the different representations under which different orbitals transform.)
Classification of symmetric spaces proceeds based on whether or not the Killing form is definite.
If the identity component of the holonomy group of a Riemannian manifold at a point acts irreducibly on the tangent space, then either the manifold is a locally Riemannian symmetric space, or it is in one of 7 families .
A Riemannian symmetric space that is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called a Hermitian symmetric space . Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
An irreducible symmetric space G / K is Hermitian if and only if K contains a central circle. A quarter turn by this circle acts as multiplication by i on the tangent space at the identity coset. Thus the Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with p = 2 , DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.
A Riemannian symmetric space that is additionally equipped with a parallel subbundle of End(T M ) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called quaternion-Kähler symmetric space .
An irreducible symmetric space G / K is quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space . Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p = 2 or q = 2 (these are isomorphic), BDI with p = 4 or q = 4, CII with p = 1 or q = 1, EII, EVI, EIX, FI and G.
In the Bott periodicity theorem , the loop spaces of the stable orthogonal group can be interpreted as reductive symmetric spaces. | https://en.wikipedia.org/wiki/Symmetric_space |
In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group Σ n {\displaystyle \Sigma _{n}} on X n {\displaystyle X_{n}} such that the composition of structure maps
is equivariant with respect to Σ p × Σ n {\displaystyle \Sigma _{p}\times \Sigma _{n}} . A morphism between symmetric spectra is a morphism of spectra that is equivariant with respect to the actions of symmetric groups.
The technical advantage of the category S p Σ {\displaystyle {\mathcal {S}}p^{\Sigma }} of symmetric spectra is that it has a closed symmetric monoidal structure (with respect to smash product ). It is also a simplicial model category . A symmetric ring spectrum is a monoid in S p Σ {\displaystyle {\mathcal {S}}p^{\Sigma }} ; if the monoid is commutative, it's a commutative ring spectrum . The possibility of this definition of "ring spectrum" was one of motivations behind the category.
A similar technical goal is also achieved by May's theory of S-modules , a competing theory.
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symmetric_spectrum |
In mathematics , a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments:
for every permutation σ of the symbols {1, 2, ..., r }. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies
The space of symmetric tensors of order r on a finite-dimensional vector space V is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V . Over fields of characteristic zero , the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V . A related concept is that of the antisymmetric tensor or alternating form . Symmetric tensors occur widely in engineering , physics and mathematics .
Let V be a vector space and
a tensor of order k . Then T is a symmetric tensor if
for the braiding maps associated to every permutation σ on the symbols {1,2,..., k } (or equivalently for every transposition on these symbols).
Given a basis { e i } of V , any symmetric tensor T of rank k can be written as
for some unique list of coefficients T i 1 i 2 ⋯ i k {\displaystyle T_{i_{1}i_{2}\cdots i_{k}}} (the components of the tensor in the basis) that are symmetric on the indices. That is to say
for every permutation σ .
The space of all symmetric tensors of order k defined on V is often denoted by S k ( V ) or Sym k ( V ). It is itself a vector space, and if V has dimension N then the dimension of Sym k ( V ) is the binomial coefficient
We then construct Sym( V ) as the direct sum of Sym k ( V ) for k = 0,1,2,...
There are many examples of symmetric tensors. Some include, the metric tensor , g μ ν {\displaystyle g_{\mu \nu }} , the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} and the Ricci tensor , R μ ν {\displaystyle R_{\mu \nu }} .
Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress , strain , and anisotropic conductivity . Also, in diffusion MRI one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.
Ellipsoids are examples of algebraic varieties ; and so, for general rank, symmetric tensors, in the guise of homogeneous polynomials , are used to define projective varieties , and are often studied as such.
Given a Riemannian manifold ( M , g ) {\displaystyle (M,g)} equipped with its Levi-Civita connection ∇ {\displaystyle \nabla } , the covariant curvature tensor is a symmetric order 2 tensor over the vector space V = Ω 2 ( M ) = ⋀ 2 T ∗ M {\textstyle V=\Omega ^{2}(M)=\bigwedge ^{2}T^{*}M} of differential 2-forms. This corresponds to the fact that, viewing R i j k ℓ ∈ ( T ∗ M ) ⊗ 4 {\displaystyle R_{ijk\ell }\in (T^{*}M)^{\otimes 4}} , we have the symmetry R i j k ℓ = R k ℓ i j {\displaystyle R_{ij\,k\ell }=R_{k\ell \,ij}} between the first and second pairs of arguments in addition to antisymmetry within each pair: R j i k ℓ = − R i j k ℓ = R i j ℓ k {\displaystyle R_{jik\ell }=-R_{ijk\ell }=R_{ij\ell k}} . [ 1 ]
Suppose V {\displaystyle V} is a vector space over a field of characteristic 0. If T ∈ V ⊗ k is a tensor of order k {\displaystyle k} , then the symmetric part of T {\displaystyle T} is the symmetric tensor defined by
the summation extending over the symmetric group on k symbols. In terms of a basis, and employing the Einstein summation convention , if
then
The components of the tensor appearing on the right are often denoted by
with parentheses () around the indices being symmetrized. Square brackets [] are used to indicate anti-symmetrization.
If T is a simple tensor, given as a pure tensor product
then the symmetric part of T is the symmetric product of the factors:
In general we can turn Sym( V ) into an algebra by defining the commutative and associative product ⊙. [ 2 ] Given two tensors T 1 ∈ Sym k 1 ( V ) and T 2 ∈ Sym k 2 ( V ) , we use the symmetrization operator to define:
It can be verified (as is done by Kostrikin and Manin [ 2 ] ) that the resulting product is in fact commutative and associative. In some cases the operator is omitted: T 1 T 2 = T 1 ⊙ T 2 .
In some cases an exponential notation is used:
Where v is a vector.
Again, in some cases the ⊙ is left out:
In analogy with the theory of symmetric matrices , a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor T ∈ Sym 2 ( V ), there is an integer r , non-zero unit vectors v 1 ,..., v r ∈ V and weights λ 1 ,..., λ r such that
The minimum number r for which such a decomposition is possible is the (symmetric) rank of T . The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of inertia. Also see Sylvester's law of inertia .
For symmetric tensors of arbitrary order k , decompositions
are also possible. The minimum number r for which such a decomposition is possible is the symmetric rank of T . [ 3 ] This minimal decomposition is called a Waring decomposition; it is a symmetric form of the tensor rank decomposition . For second-order tensors this corresponds to the rank of the matrix representing the tensor in any basis, and it is well known that the maximum rank is equal to the dimension of the underlying vector space. However, for higher orders this need not hold: the rank can be higher than the number of dimensions in the underlying vector space. Moreover, the rank and symmetric rank of a symmetric tensor may differ. [ 4 ] | https://en.wikipedia.org/wiki/Symmetric_tensor |
In electrical engineering , the method of symmetrical components simplifies analysis of unbalanced three-phase power systems under both normal and abnormal conditions. The basic idea is that an asymmetrical set of N phasors can be expressed as a linear combination of N symmetrical sets of phasors by means of a complex linear transformation . [ 1 ] Fortescue's theorem (symmetrical components) is based on superposition principle , [ 2 ] so it is applicable to linear power systems only, or to linear approximations of non-linear power systems.
In the most common case of three-phase systems, the resulting "symmetrical" components are referred to as direct (or positive ), inverse (or negative ) and zero (or homopolar ). The analysis of power system is much simpler in the domain of symmetrical components, because the resulting equations are mutually linearly independent if the circuit itself is balanced . [ 3 ]
In 1918 Charles Legeyt Fortescue presented a paper [ 4 ] which demonstrated that any set of N unbalanced phasors (that is, any such polyphase signal) could be expressed as the sum of N symmetrical sets of balanced phasors, for values of N that are prime. Only a single frequency component is represented by the phasors.
In 1943 Edith Clarke published a textbook giving a method of use of symmetrical components for three-phase systems that greatly simplified calculations over the original Fortescue paper. [ 5 ] In a three-phase system, one set of phasors has the same phase sequence as the system under study (positive sequence; say ABC), the second set has the reverse phase sequence (negative sequence; ACB), and in the third set the phasors A, B and C are in phase with each other ( zero sequence , the common-mode signal ). Essentially, this method converts three unbalanced phases into three independent sources, which makes asymmetric fault analysis more tractable.
By expanding a one-line diagram to show the positive sequence, negative sequence, and zero sequence impedances of generators , transformers and other devices including overhead lines and cables , analysis of such unbalanced conditions as a single line to ground short-circuit fault is greatly simplified. The technique can also be extended to higher order phase systems.
Physically, in a three phase system, a positive sequence set of currents produces a normal rotating field, a negative sequence set produces a field with the opposite rotation, and the zero sequence set produces a field that oscillates but does not rotate between phase windings. Since these effects can be detected physically with sequence filters, the mathematical tool became the basis for the design of protective relays , which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip circuit breakers or take other steps to protect electrical systems.
The analytical technique was adopted and advanced by engineers at General Electric and Westinghouse , and after World War II it became an accepted method for asymmetric fault analysis.
As shown in the figure to the above right, the three sets of symmetrical components (positive, negative, and zero sequence) add up to create the system of three unbalanced phases as pictured in the bottom of the diagram. The imbalance between phases arises because of the difference in magnitude and phase shift between the sets of vectors. Notice that the colors (red, blue, and yellow) of the separate sequence vectors correspond to three different phases (A, B, and C, for example). To arrive at the final plot, the sum of vectors of each phase is calculated. This resulting vector is the effective phasor representation of that particular phase. This process, repeated, produces the phasor for each of the three phases.
Symmetrical components are most commonly used for analysis of three-phase electrical power systems . The voltage or current of a three-phase system at some point can be indicated by three phasors, called the three components of the voltage or the current.
This article discusses voltage; however, the same considerations also apply to current. In a perfectly balanced three-phase power system, the voltage phasor components have equal magnitudes but are 120 degrees apart. In an unbalanced system, the magnitudes and phases of the voltage phasor components are different.
Decomposing the voltage phasor components into a set of symmetrical components helps analyze the system as well as visualize any imbalances.
If the three voltage components are expressed as phasors (which are complex numbers), a complex vector can be formed in which the three phase components are the components of the vector. A vector for three phase voltage components can be written as
and decomposing the vector into three symmetrical components gives
where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components (NSC). The sequence components differ only by their phase angles, which are symmetrical and so are 2 3 π {\displaystyle \scriptstyle {\frac {2}{3}}\pi } radians or 120°.
Define a phasor rotation operator α {\displaystyle \alpha } , which rotates a phasor vector counterclockwise by 120 degrees when multiplied by it:
Note that α 3 = 1 {\displaystyle \alpha ^{3}=1} so that α − 1 = α 2 {\displaystyle \alpha ^{-1}=\alpha ^{2}} .
The zero sequence components have equal magnitude and are in phase with each other, therefore:
and the other sequence components have the same magnitude, but their phase angles differ by 120°. If the original unbalanced set of voltage phasors have positive or abc phase sequence, then:
meaning that
Thus,
where
If instead the original unbalanced set of voltage phasors have negative or acb phase sequence, the following matrix can be similarly derived:
The sequence components are derived from the analysis equation
where
The above two equations tell how to derive symmetrical components corresponding to an asymmetrical set of three phasors:
Visually, if the original components are symmetrical, sequences 0 and 2 will each form a triangle, summing to zero, and sequence 1 components will sum to a straight line.
The phasors V ( a b ) = V ( a ) − V ( b ) ; V ( b c ) = V ( b ) − V ( c ) ; V ( c a ) = V ( c ) − V ( a ) {\displaystyle \scriptstyle V_{(ab)}=V_{(a)}-V_{(b)};\;V_{(bc)}=V_{(b)}-V_{(c)};\;V_{(ca)}=V_{(c)}-V_{(a)}} form a closed triangle (e.g., outer voltages or line to line voltages). To find the synchronous and inverse components of the phases, take any side of the outer triangle and draw the two possible equilateral triangles sharing the selected side as base. These two equilateral triangles represent a synchronous and an inverse system.
If the phasors V were a perfectly synchronous system, the vertex of the outer triangle not on the base line would be at the same position as the corresponding vertex of the equilateral triangle representing the synchronous system. Any amount of inverse component would mean a deviation from this position. The deviation is exactly 3 times the inverse phase component.
The synchronous component is in the same manner 3 times the deviation from the "inverse equilateral triangle". The directions of these components are correct for the relevant phase. It seems counter intuitive that this works for all three phases regardless of the side chosen but that is the beauty of this illustration. The graphic is from Napoleon's Theorem , which matches a graphical calculation technique that sometimes appears in older references books. [ 6 ]
It can be seen that the transformation matrix A above is a DFT matrix , and as such, symmetrical components can be calculated for any poly-phase system.
Harmonics often occur in power systems as a consequence of non-linear loads. Each order of harmonics contributes to different sequence components. The fundamental and harmonics of order 3 n + 1 {\displaystyle \scriptstyle 3n+1} will contribute to the positive sequence component. Harmonics of order 3 n − 1 {\displaystyle \scriptstyle 3n-1} will contribute to the negative sequence. Harmonics of order 3 n {\displaystyle \scriptstyle 3n} contribute to the zero sequence.
Note that the rules above are only applicable if the phase values (or distortion) in each phase are exactly the same. Please further note that even harmonics are not common in power systems.
The zero sequence represents the component of the unbalanced phasors that is equal in magnitude and phase. Because they are in phase, zero sequence currents flowing through an n-phase network will sum to n times the magnitude of the individual zero sequence currents components. Under normal operating conditions this sum is small enough to be negligible. However, during large zero sequence events such as lightning strikes, this nonzero sum of currents can lead to a larger current flowing through the neutral conductor than the individual phase conductors. Because neutral conductors are typically not larger than individual phase conductors, and are often smaller than these conductors, a large zero sequence component can lead to overheating of neutral conductors and to fires.
One way to prevent large zero sequence currents is to use a delta connection, which appears as an open circuit to zero sequence currents. For this reason, most transmission, and much sub-transmission is implemented using delta. Much distribution is also implemented using delta, although "old work" distribution systems have occasionally been "wyed-up" (converted from delta to wye ) so as to increase the line's capacity at a low converted cost, but at the expense of a higher central station protective relay cost. | https://en.wikipedia.org/wiki/Symmetrical_components |
In radio technology , symmetrical double-sided two-way ranging (SDS-TWR) is a ranging method that uses two delays that naturally occur in signal transmission to determine the range between two stations: [ 1 ]
This method is called symmetrical double-sided two-way ranging because:
A special type of packet (test packets) is transmitted from station A (node A) to station B (node B). As time the packet travels through space per meter is known (from physical laws), the difference in time from when it was sent from the transmitter and received at the receiver can be calculated. This time delay is known as the signal propagation delay .
Station A now expects an acknowledgement from Station B. A station takes a known amount of time to process the incoming test packet, generate an acknowledgement (ack packet), and prepare it for transmission. The sum of time taken to process this acknowledgement is known as processing delay .
The acknowledgement sent back to station A includes in its header those two delay values – the signal propagation delay and the processing delay. A further signal propagation delay can be calculated by Station A on the received acknowledgement, even as this delay was calculated on the test packet. These three values can then be used by an algorithm to calculate the range between these two stations.
To verify that the range calculation was accurate, the same procedure is repeated by station B sending a test packet to station A and station A sending an acknowledgement to station B. At the end of this procedure, two range values are determined and an average of the two can be used to achieve a fairly accurate distance measurement between these two stations. | https://en.wikipedia.org/wiki/Symmetrical_double-sided_two-way_ranging |
In mathematics , a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is symmetrically continuous at a point x if
The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function x − 2 {\displaystyle x^{-2}} is symmetrically continuous at x = 0 {\displaystyle x=0} , but not continuous.
Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability.
The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over R {\displaystyle \mathbb {R} } , similarly to the usually continuous functions, which form a linear subspace within it.
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symmetrically_continuous_function |
Symmetries of Culture: Theory and Practice of Plane Pattern Analysis is a book by anthropologist Dorothy K. Washburn and mathematician Donald W. Crowe published in 1988 by the University of Washington Press . The book is about the identification of patterns on cultural objects.
The book is divided into seven chapters. Chapter 1 reviews the historical application of symmetry analysis to the discovery and enumeration of patterns in the plane , otherwise known as tessellations or tilings , and the application of geometry to design and the decorative arts .
Chapters 2 to 6 describe how to identify and classify patterns on cultural objects such as ceramics, textiles and surface designs. Chapter 2 establishes the mathematical tools required to perform the symmetry analysis of patterns. Chapter 3 introduces the concept of color symmetry, for two-colored and multicolored patterns. Chapters 4 and 5 describe the one-dimensional (frieze) designs and the two-dimensional (plane) designs respectively; flow charts are used to help the reader to identify patterns. Chapter 6 describes finite designs, for example circular designs, which are those without translations or glide refections . Chapter 7 discusses problems that may arise in symmetry classification, for example pattern irregularities.
The benefit of the flow charts is that they allow the reader to analyse the design of any cultural object in order to assign it to a specific pattern. The number of distinct patterns in one or two dimensions, with one or two colors, is shown in the table.
The book, which was 10 years in development, has over 500 illustrations, and includes a mathematical appendix, a 270-entry bibliography, and an index.
The authors describe their book as a "handbook for the non-mathematician" of the theory and practice of plane pattern analysis.
Reviewers of the book identified the audience for the book in various ways. Roger Neich writing in Man said "[The authors'] aim is to make symmetry analysis accessible to all researchers, regardless of any mathematical training, and in this aim they succeed admirably, provided the reader is prepared to invest some considerable effort." [ 1 ]
Doris Schattschneider writing in The American Mathematical Monthly commented: "[The book] was written for archaeologists, anthropologists, and art historians, but the authors have taken care in their presentation of the geometry of symmetry and color symmetry analysis." [ 2 ] H.C. Williams reviewing the book for The Mathematical Gazette said: "This interesting book is written by a mathematician and an anthropologist and is aimed primarily at the non-mathematician. That said, it is well worth the attention of mathematicians, particularly teachers, who have an interest in pattern." [ 3 ]
Contemporary reviews of the book were mostly positive. The book was reviewed by journals in the fields of anthropology, archaeology, the arts, and mathematics.
Mary Frame in African Arts said: "a solid and attractive book that takes the reader in logical stages toward an understanding of the symmetrical basis of pattern repeats." [...] "I believe that Symmetries of Culture is a landmark work that will furnish the impetus and method for many studies in this fertile area." [ 4 ] Owen Lindauer in American Anthropologist commented: "Question-answer flowcharts enable the reader to correctly classify designs using a standard notation. The book is extensively illustrated with carvings, textiles, basketry, tiles, and pottery, which are used as examples of various symmetry patterns." [ 5 ]
Dwight W. Read in Antiquity : " Symmetries of Culture is an impressive book - both in
terms of its physical appearance and its content. [...] will undoubtedly become the major reference on the analysis of patterns in terms of symmetry properties." [ 6 ] Jon Muller writing in American Antiquity : " ... a fine book that achieves its goals in a straight-forward and clear fashion. It presents a set of methods that can be applied consistently and usefully in looking at symmetrical plane designs." [ 7 ] and Roger Neich in Man : "... wide use of this book will certainly contribute to a great improvement in the systematic study of material culture." [ 1 ]
The reviewer in African Arts pointed out the existence of cultural patterns, such as in ancient Peruvian art, that are not included in the crystallographic symmetry approach to patterns used in the book. This criticism was echoed by the reviewer in American Antiquity who had some reservations about the potential dangers of limiting design analysis to certain convenient classes of design.
George Kubler , an art historian writing in Winterthur Portfolio criticised the book: "The authors' present method is non-historical. The objects illustrated are mostly undatable, and nowhere is concern shown for their seriation or place in time." [ 8 ] Kubler criticises the authors' entire approach as being non-historical, because it analyses each object individually rather than considering them in chronological order.
In 2021 the book was praised by Palaguta and Starkova in Terra Artis. Art and Design . In their review, they stated that the problem of creating a basis for systematizing patterns on the principles of symmetry was solved in Symmetries of Culture . They give three reasons for continuing to value the book: firstly, despite the passage of time, the book is still valid and useful; secondly, since the release of the book, the authors have done a great deal to attract new workers into the field; and thirdly, in recent years, interdisciplinary research on symmetry and ornamentation has increased, and the interest in this topic has grown among both anthropologists and art historians, which greatly broadens the readership of the book. [ 9 ] | https://en.wikipedia.org/wiki/Symmetries_of_Culture:_Theory_and_Practice_of_Plane_Pattern_Analysis |
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