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disturbances became most dramatically apparent in the populations of our first series of three experiments, in which we observed the development of what we called a behavioral sink. the animals would crowd together in greatest number in one of the four interconnecting pens in which the colony was maintained. as many as 60 of the 80 rats in each experimental population would assemble in one pen during periods of feeding. individual rats would rarely eat except in the company of other rats. as a result extreme population densities developed in the pen adopted for eating, leaving the others with sparse populations. in the experiments in which the behavioral sink developed, infant mortality ran as high as 96 percent among the most disoriented groups in the population. following his earlier experiments with rats, calhoun later created his " mortality - inhibiting environment for mice " in 1968 : a 101 - by - 101 - inch ( 260 cm × 260 cm ) cage for mice with food and water replenished to support any increase in population, which took his experimental approach to its limits. in his most famous experiment in the series, " universe 25 ", population peaked at 2, 200 mice even though the habitat was built to tolerate a total population of 4000. having reached a level of high population density, the mice began exhibiting a variety of abnormal, often destructive, behaviors including refusal to engage in courtship, and females abandoning their young. by the 600th day, the population was on its way to extinction. though physically able to reproduce, the mice had lost the social skills required to mate. calhoun retired from nimh in 1984, but continued to work on his research results until his death on september 7, 1995. = = analysis = = the specific voluntary crowding of rats to which the term " behavioral sink " refers is thought to have resulted from the earlier involuntary crowding : individual rats became so used to the proximity of others while eating that they began to associate feeding with the company of other rats. calhoun eventually found a way to prevent this by changing some of the settings and thereby decreased mortality somewhat, but the overall pathological consequences of overcrowding remained. further, researchers argued that " calhoun's work was not simply about density in a physical sense, as number of individuals - per - square - unit - area, but was about degrees of social interaction. " " social density " appears to be key. = = applicability to humans = = calhoun had phrased much of his work in anthropomorphic terms, in a way that made his ideas highly accessible to a lay audience
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social interaction. " " social density " appears to be key. = = applicability to humans = = calhoun had phrased much of his work in anthropomorphic terms, in a way that made his ideas highly accessible to a lay audience. calhoun himself saw the fate of the population of mice as a metaphor for the potential fate of humankind. he characterized the social breakdown as a " spiritual death ", with reference to bodily death as the " second death " mentioned in the biblical verse revelation 2 : 11. the implications of the experiment are controversial. psychologist jonathan freedman's experiment recruited high school and university students to carry out a series of experiments that measured the effects of density on human behavior. he measured their stress, discomfort, aggression, competitiveness, and general unpleasantness. he declared to have found no appreciable negative effects in 1975. the 1962 scientific american article came at a time when overpopulation had become a subject of great public interest, and had a considerable cultural influence. however, such discussions often oversimplified the original findings in various ways. it should however be noted that the work has another message than, for example, paul ehrlich's now widely disputed book the population bomb. calhoun's worries primarily concerned a human population surge and a potentially independent increase in urbanization as an early stage of rendering much of a given society functionally sterile. under such circumstances, he hypothesized, society would move from some modality of overpopulation towards a much more irredeemable underpopulation. = = see also = = decadence population decline overpopulation rats of nimh societal collapse dysgenics = = references = = = = external links = = fessenden, marissa 2015, how 1960s mouse utopias led to grim predictions for future of humanity, smithsonian magazine. what humans can learn from calhoun's rodent utopia, victor. national library of medicine ( 2018 ). john b. calhoun film 7. 1 [ edited ], ( nimh, 1970 - 1972 ) gwamanda, paul ( may 14, 2021 ). behavioral sink : the overpopulation experiments of john b. calhoun. adams, j. & ramsden, e. ( 2017 ). the falls of 1972 : john b calhoun and urban pessimism. apex ( 4 january 2021 ). oversocialization : an introduction how socialization goes awry and the jekyll / hyde case of credentials. wiles, will ( 2011 )
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of 1972 : john b calhoun and urban pessimism. apex ( 4 january 2021 ). oversocialization : an introduction how socialization goes awry and the jekyll / hyde case of credentials. wiles, will ( 2011 ). the behavioral sink : the mouse universes of john b. calhoun. forgetting, issue 42.
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whispering is an unvoiced mode of phonation in which the vocal cords are abducted so that they do not vibrate ; air passes between the arytenoid cartilages to create audible turbulence during speech. supralaryngeal articulation remains the same as in normal speech. in normal speech, the vocal cords alternate between states of voice and voicelessness. in whispering, only the voicing segments change, so that the vocal cords alternate between whisper and voicelessness ( though the acoustic difference between the two states is minimal ). because of this, implementing speech recognition for whispered speech is more difficult, as the characteristic spectral range needed to detect syllables and words is not given through the total absence of tone. more advanced techniques such as neural networks may be used, however, as is done by amazon alexa. there is no symbol in the ipa for whispered phonation, since it is not used phonemically in any language. however, a sub - dot under phonemically voiced segments is sometimes seen in the literature, as [ ʃʊd ] for whispered should. = = social role = = whispering is generally used quietly, to limit the hearing of speech to those closest to the speaker ; for example, to convey secret information without being overheard or to avoid disturbing others in a quiet place such as a library or place of worship. loud whispering, known as a stage whisper, is generally used only for dramatic or emphatic purposes. whispering can strain the vocal cords more than regular speech in some people, for whom speaking softly is recommended instead. = = asmr = = in 2010, it was discovered that whispering is one of the many triggers of asmr, a tingling sensation caused by listening to soft, relaxing sounds. this phenomenon made news headlines after videos on youtube of people speaking up close to the camera in a soft whisper, giving the viewer tingles. people often listen to these videos to help them sleep and to relax. = = in non - humans = = the prevalence and function of low - amplitude signaling by non - humans are poorly characterized. as such, it is difficult to ascertain the existence of whispering in non - humans. this is made more difficult by the specific physiology of human whispering. by sufficiently relaxing the definition of whispering, it can be argued any number of non - human species demonstrate whisper - like behaviors. often these behaviors function to increase fitness. if whispering is more broadly defined as the " production of short - range, low - amplitude acoustic signals, " whispering is
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, it can be argued any number of non - human species demonstrate whisper - like behaviors. often these behaviors function to increase fitness. if whispering is more broadly defined as the " production of short - range, low - amplitude acoustic signals, " whispering is observed in myriad animals including non - human mammals, fish, and insects. if whispering is restricted to include only acoustic signals which are significantly different than those produced at high amplitude, whispering is still observed across biological taxa. an unlikely example is the croaking gourami. croaking gouramis produce a high - amplitude " croak " during agonistic disputes by beating specialized pectoral fins. female gouramis additionally use these fins to produce an acoustically distinct, low - amplitude " purr " during copulation. if whispering is restricted to include only creatures possessing vocal folds ( i. e., mammals and some reptiles ), whispering has been observed in species including cotton - top tamarins and a variety of bats. in captive cotton - top tamarins, whisper - like behavior is speculated to enable troop communication while not alerting predators. numerous species of bats ( e. g., spotted bats, northern long - eared bats, and western barbastelles ) alter their echolocation calls to avoid detection by prey. such a relaxed definition of whispering ( i. e., production of short - range, low - amplitude acoustic signals which are significantly different than those produced at high amplitude ) cannot be applied to humans without including vocalizations distinct from human whispering ( e. g., creaky voice, and falsetto ). further research is needed to ascertain the existence of whispering in non - humans as established in the larger article. = = notes = = = = see also = = aspiration ( phonetics ) chinese whispers cocktail party effect egressive sound vs. ingressive speech whispering campaign whispering gallery whispery voice other forms of unvoiced vocalization : gasping, sighing and panting autonomous sensory meridian response = = references = = = = external links = = functional neuroanatomy of human vocalization : an h215o pet study
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a number is a mathematical object used to count, measure, and label. the most basic examples are the natural numbers 1, 2, 3, 4, and so forth. numbers can be represented in language with number words. more universally, individual numbers can be represented by symbols, called numerals ; for example, " 5 " is a numeral that represents the number five. as only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. the most common numeral system is the hindu – arabic numeral system, which allows for the representation of any non - negative integer using a combination of ten fundamental numeric symbols, called digits. in addition to their use in counting and measuring, numerals are often used for labels ( as with telephone numbers ), for ordering ( as with serial numbers ), and for codes ( as with isbns ). in common usage, a numeral is not clearly distinguished from the number that it represents. in mathematics, the notion of number has been extended over the centuries to include zero ( 0 ), negative numbers, rational numbers such as one half ( 1 2 ) { \ displaystyle \ left ( { \ tfrac { 1 } { 2 } } \ right ) }, real numbers such as the square root of 2 ( 2 ) { \ displaystyle \ left ( { \ sqrt { 2 } } \ right ) } and π, and complex numbers which extend the real numbers with a square root of −1 ( and its combinations with real numbers by adding or subtracting its multiples ). calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers. besides their practical uses, numbers have cultural significance throughout the world. for example, in western society, the number 13 is often regarded as unlucky, and " a million " may signify " a lot " rather than an exact quantity. though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought. numerology heavily influenced the development of greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today. during the 19th century, mathematicians began to develop many
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as numerology, permeated ancient and medieval thought. numerology heavily influenced the development of greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today. during the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. in modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields, and the application of the term " number " is a matter of convention, without fundamental significance. = = history = = = = = first use of numbers = = = bones and other artifacts have been discovered with marks cut into them that many believe are tally marks. these tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. a tallying system has no concept of place value ( as in modern decimal notation ), which limits its representation of large numbers. nonetheless, tallying systems are considered the first kind of abstract numeral system. the earliest unambiguous numbers in the archaeological record are the mesopotamian base 60 system ( c. 3400 bc ) ; place value emerged in it in the 3rd millennium bce. the earliest known base 10 system dates to 3100 bc in egypt. = = = numerals = = = numbers should be distinguished from numerals, the symbols used to represent numbers. the egyptians invented the first ciphered numeral system, and the greeks followed by mapping their counting numbers onto ionian and doric alphabets. roman numerals, a system that used combinations of letters from the roman alphabet, remained dominant in europe until the spread of the superior hindu – arabic numeral system around the late 14th century, and the hindu – arabic numeral system remains the most common system for representing numbers in the world today. the key to the effectiveness of the system was the symbol for zero, which was developed by ancient indian mathematicians around 500 ad. = = = zero = = = the first known recorded use of zero dates to ad 628, and appeared in the brahmasphutasiddhanta, the main work of the indian mathematician brahmagupta. he treated 0 as a number and discussed operations involving it, including division by zero. by this time ( the 7th century ), the concept had clearly reached cambodia in the form
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##dhanta, the main work of the indian mathematician brahmagupta. he treated 0 as a number and discussed operations involving it, including division by zero. by this time ( the 7th century ), the concept had clearly reached cambodia in the form of khmer numerals, and documentation shows the idea later spreading to china and the islamic world. brahmagupta's brahmasphutasiddhanta is the first book that mentions zero as a number, hence brahmagupta is usually considered the first to formulate the concept of zero. he gave rules of using zero with negative and positive numbers, such as " zero plus a positive number is a positive number, and a negative number plus zero is the negative number ". the brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the babylonians or as a symbol for a lack of quantity as was done by ptolemy and the romans. the use of 0 as a number should be distinguished from its use as a placeholder numeral in place - value systems. many ancient texts used 0. babylonian and egyptian texts used it. egyptians used the word nfr to denote zero balance in double entry accounting. indian texts used a sanskrit word shunye or shunya to refer to the concept of void. in mathematics texts this word often refers to the number zero. in a similar vein, panini ( 5th century bc ) used the null ( zero ) operator in the ashtadhyayi, an early example of an algebraic grammar for the sanskrit language ( also see pingala ). there are other uses of zero before brahmagupta, though the documentation is not as complete as it is in the brahmasphutasiddhanta. records show that the ancient greeks seemed unsure about the status of 0 as a number : they asked themselves " how can'nothing'be something? " leading to interesting philosophical and, by the medieval period, religious arguments about the nature and existence of 0 and the vacuum. the paradoxes of zeno of elea depend in part on the uncertain interpretation of 0. ( the ancient greeks even questioned whether 1 was a number. ) the late olmec people of south - central mexico began to use a symbol for zero, a shell glyph, in the new world, possibly by the 4th century bc but certainly by 40 bc, which became an integral part of maya numerals
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olmec people of south - central mexico began to use a symbol for zero, a shell glyph, in the new world, possibly by the 4th century bc but certainly by 40 bc, which became an integral part of maya numerals and the maya calendar. maya arithmetic used base 4 and base 5 written as base 20. george i. sanchez in 1961 reported a base 4, base 5 " finger " abacus. by 130 ad, ptolemy, influenced by hipparchus and the babylonians, was using a symbol for 0 ( a small circle with a long overbar ) within a sexagesimal numeral system otherwise using alphabetic greek numerals. because it was used alone, not as just a placeholder, this hellenistic zero was the first documented use of a true zero in the old world. in later byzantine manuscripts of his syntaxis mathematica ( almagest ), the hellenistic zero had morphed into the greek letter omicron ( otherwise meaning 70 ). another true zero was used in tables alongside roman numerals by 525 ( first known use by dionysius exiguus ), but as a word, nulla meaning nothing, not as a symbol. when division produced 0 as a remainder, nihil, also meaning nothing, was used. these medieval zeros were used by all future medieval computists ( calculators of easter ). an isolated use of their initial, n, was used in a table of roman numerals by bede or a colleague about 725, a true zero symbol. = = = negative numbers = = = the abstract concept of negative numbers was recognized as early as 100 – 50 bc in china. the nine chapters on the mathematical art contains methods for finding the areas of figures ; red rods were used to denote positive coefficients, black for negative. the first reference in a western work was in the 3rd century ad in greece. diophantus referred to the equation equivalent to 4x + 20 = 0 ( the solution is negative ) in arithmetica, saying that the equation gave an absurd result. during the 600s, negative numbers were in use in india to represent debts. diophantus'previous reference was discussed more explicitly by indian mathematician brahmagupta, in brahmasphutasiddhanta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. however, in the 12th century in india, bhaskara gives negative roots for quadratic equations but
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brahmasphutasiddhanta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. however, in the 12th century in india, bhaskara gives negative roots for quadratic equations but says the negative value " is in this case not to be taken, for it is inadequate ; people do not approve of negative roots ". european mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although fibonacci allowed negative solutions in financial problems where they could be interpreted as debts ( chapter 13 of liber abaci, 1202 ) and later as losses ( in flos ). rene descartes called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. at the same time, the chinese were indicating negative numbers by drawing a diagonal stroke through the right - most non - zero digit of the corresponding positive number's numeral. the first use of negative numbers in a european work was by nicolas chuquet during the 15th century. he used them as exponents, but referred to them as " absurd numbers ". as recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. = = = rational numbers = = = it is likely that the concept of fractional numbers dates to prehistoric times. the ancient egyptians used their egyptian fraction notation for rational numbers in mathematical texts such as the rhind mathematical papyrus and the kahun papyrus. classical greek and indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. the best known of these is euclid's elements, dating to roughly 300 bc. of the indian texts, the most relevant is the sthananga sutra, which also covers number theory as part of a general study of mathematics. the concept of decimal fractions is closely linked with decimal place - value notation ; the two seem to have developed in tandem. for example, it is common for the jain math sutra to include calculations of decimal - fraction approximations to pi or the square root of 2. similarly, babylonian math texts used sexagesimal ( base 60 ) fractions with great frequency. = = = irrational numbers = = = the earliest known use of irrational numbers was in the indian sulba sutras composed between 800 and 500 bc. the first existence proofs of irrational numbers is usually attributed to pythagoras
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great frequency. = = = irrational numbers = = = the earliest known use of irrational numbers was in the indian sulba sutras composed between 800 and 500 bc. the first existence proofs of irrational numbers is usually attributed to pythagoras, more specifically to the pythagorean hippasus of metapontum, who produced a ( most likely geometrical ) proof of the irrationality of the square root of 2. the story goes that hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. however, pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. he could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced hippasus to death by drowning, to impede spreading of this disconcerting news. the 16th century brought final european acceptance of negative integral and fractional numbers. by the 17th century, mathematicians generally used decimal fractions with modern notation. it was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. it had remained almost dormant since euclid. in 1872, the publication of the theories of karl weierstrass ( by his pupil e. kossak ), eduard heine, georg cantor, and richard dedekind was brought about. in 1869, charles meray had taken the same point of departure as heine, but the theory is generally referred to the year 1872. weierstrass's method was completely set forth by salvatore pincherle ( 1880 ), and dedekind's has received additional prominence through the author's later work ( 1888 ) and endorsement by paul tannery ( 1894 ). weierstrass, cantor, and heine base their theories on infinite series, while dedekind founds his on the idea of a cut ( schnitt ) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. the subject has received later contributions at the hands of weierstrass, kronecker, and meray. the search for roots of quintic and higher degree equations was an important development, the abel – ruffini theorem ( ruffini 1799, abel 1824 ) showed that they could not be solved by radicals ( formulas involving only arithmetical operations and roots ). hence it was necessary
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##c and higher degree equations was an important development, the abel – ruffini theorem ( ruffini 1799, abel 1824 ) showed that they could not be solved by radicals ( formulas involving only arithmetical operations and roots ). hence it was necessary to consider the wider set of algebraic numbers ( all solutions to polynomial equations ). galois ( 1832 ) linked polynomial equations to group theory giving rise to the field of galois theory. simple continued fractions, closely related to irrational numbers ( and due to cataldi, 1613 ), received attention at the hands of euler, and at the opening of the 19th century were brought into prominence through the writings of joseph louis lagrange. other noteworthy contributions have been made by druckenmuller ( 1837 ), kunze ( 1857 ), lemke ( 1870 ), and gunther ( 1872 ). ramus first connected the subject with determinants, resulting, with the subsequent contributions of heine, mobius, and gunther, in the theory of kettenbruchdeterminanten. = = = transcendental numbers and reals = = = the existence of transcendental numbers was first established by liouville ( 1844, 1851 ). hermite proved in 1873 that e is transcendental and lindemann proved in 1882 that π is transcendental. finally, cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers. = = = infinity and infinitesimals = = = the earliest known conception of mathematical infinity appears in the yajur veda, an ancient indian script, which at one point states, " if you remove a part from infinity or add a part to infinity, still what remains is infinity. " infinity was a popular topic of philosophical study among the jain mathematicians c. 400 bc. they distinguished between five types of infinity : infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. the symbol ∞ { \ displaystyle { \ text { ∞ } } } is often used to represent an infinite quantity. aristotle defined the traditional western notion of mathematical infinity. he distinguished between actual infinity and potential infinity — the general consensus being that only the latter had true value. galileo galilei's two new sciences discussed the idea of one - to - one correspondences between infinite sets. but the next major advance in the theory was made
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and potential infinity — the general consensus being that only the latter had true value. galileo galilei's two new sciences discussed the idea of one - to - one correspondences between infinite sets. but the next major advance in the theory was made by georg cantor ; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. in the 1960s, abraham robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. the system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by newton and leibniz. a modern geometrical version of infinity is given by projective geometry, which introduces " ideal points at infinity ", one for each spatial direction. each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. this is closely related to the idea of vanishing points in perspective drawing. = = = complex numbers = = = the earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor heron of alexandria in the 1st century ad, when he considered the volume of an impossible frustum of a pyramid. they became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by italian mathematicians such as niccolo fontana tartaglia and gerolamo cardano. it was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. this was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. when rene descartes coined the term " imaginary " for these quantities in 1637, he intended it as derogatory. ( see imaginary number for a discussion of the " reality " of complex numbers. ) a further source of confusion was that the equation ( − 1 ) 2 = − 1 − 1 = − 1 { \ displaystyle \ left ( { \ sqrt { - 1 } } \ right ) ^ { 2 } = { \ sqrt { - 1 } } { \ sqrt { - 1 } } = - 1 } seemed capriciously inconsistent with the algebraic identity a b = a b, { \ displaystyle { \ sqrt { a } } { \ sqrt
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##rt { - 1 } } { \ sqrt { - 1 } } = - 1 } seemed capriciously inconsistent with the algebraic identity a b = a b, { \ displaystyle { \ sqrt { a } } { \ sqrt { b } } = { \ sqrt { ab } }, } which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. the incorrect use of this identity, and the related identity 1 a = 1 a { \ displaystyle { \ frac { 1 } { \ sqrt { a } } } = { \ sqrt { \ frac { 1 } { a } } } } in the case when both a and b are negative even bedeviled euler. this difficulty eventually led him to the convention of using the special symbol i in place of − 1 { \ displaystyle { \ sqrt { - 1 } } } to guard against this mistake. the 18th century saw the work of abraham de moivre and leonhard euler. de moivre's formula ( 1730 ) states : ( cos θ + i sin θ ) n = cos n θ + i sin n θ { \ displaystyle ( \ cos \ theta + i \ sin \ theta ) ^ { n } = \ cos n \ theta + i \ sin n \ theta } while euler's formula of complex analysis ( 1748 ) gave us : cos θ + i sin θ = e i θ. { \ displaystyle \ cos \ theta + i \ sin \ theta = e ^ { i \ theta }. } the existence of complex numbers was not completely accepted until caspar wessel described the geometrical interpretation in 1799. carl friedrich gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. the idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in wallis's de algebra tractatus. in the same year, gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. gauss studied complex numbers of the form a + bi, where a and b are integers ( now called gaussian integers ) or rational numbers. his student, gotthold eisenstein, studied the type a + bω, where ω is a complex root of x
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the form a + bi, where a and b are integers ( now called gaussian integers ) or rational numbers. his student, gotthold eisenstein, studied the type a + bω, where ω is a complex root of x3 − 1 = 0 ( now called eisenstein integers ). other such classes ( called cyclotomic fields ) of complex numbers derive from the roots of unity xk − 1 = 0 for higher values of k. this generalization is largely due to ernst kummer, who also invented ideal numbers, which were expressed as geometrical entities by felix klein in 1893. in 1850 victor alexandre puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. this eventually led to the concept of the extended complex plane. = = = prime numbers = = = prime numbers have been studied throughout recorded history. they are positive integers that are divisible only by 1 and themselves. euclid devoted one book of the elements to the theory of primes ; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the euclidean algorithm for finding the greatest common divisor of two numbers. in 240 bc, eratosthenes used the sieve of eratosthenes to quickly isolate prime numbers. but most further development of the theory of primes in europe dates to the renaissance and later eras. in 1796, adrien - marie legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. other results concerning the distribution of the primes include euler's proof that the sum of the reciprocals of the primes diverges, and the goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. yet another conjecture related to the distribution of prime numbers is the riemann hypothesis, formulated by bernhard riemann in 1859. the prime number theorem was finally proved by jacques hadamard and charles de la vallee - poussin in 1896. goldbach and riemann's conjectures remain unproven and unrefuted. = = main classification = = numbers can be classified into sets, called number sets or number systems, such as the natural numbers and the real numbers. the main number systems are as follows : each of these number systems is a subset of the next one. so, for example, a rational number is also a real number, and every real number is also a complex
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natural numbers and the real numbers. the main number systems are as follows : each of these number systems is a subset of the next one. so, for example, a rational number is also a real number, and every real number is also a complex number. this can be expressed symbolically as n ⊂ z ⊂ q ⊂ r ⊂ c { \ displaystyle \ mathbb { n } \ subset \ mathbb { z } \ subset \ mathbb { q } \ subset \ mathbb { r } \ subset \ mathbb { c } }. a more complete list of number sets appears in the following diagram. = = = natural numbers = = = the most familiar numbers are the natural numbers ( sometimes called whole numbers or counting numbers ) : 1, 2, 3, and so on. traditionally, the sequence of natural numbers started with 1 ( 0 was not even considered a number for the ancient greeks. ) however, in the 19th century, set theorists and other mathematicians started including 0 ( cardinality of the empty set, i. e. 0 elements, where 0 is thus the smallest cardinal number ) in the set of natural numbers. today, different mathematicians use the term to describe both sets, including 0 or not. the mathematical symbol for the set of all natural numbers is n, also written n { \ displaystyle \ mathbb { n } }, and sometimes n 0 { \ displaystyle \ mathbb { n } _ { 0 } } or n 1 { \ displaystyle \ mathbb { n } _ { 1 } } when it is necessary to indicate whether the set should start with 0 or 1, respectively. in the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. the radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers ( for the decimal system, the radix is 10 ). in this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right. in set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. for instance, the number 3 can be represented as the class of all sets that have exactly three
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. in set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. for instance, the number 3 can be represented as the class of all sets that have exactly three elements. alternatively, in peano arithmetic, the number 3 is represented as sss0, where s is the " successor " function ( i. e., 3 is the third successor of 0 ). many different representations are possible ; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times. = = = integers = = = the negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. negative numbers are usually written with a negative sign ( a minus sign ). as an example, the negative of 7 is written −7, and 7 + ( −7 ) = 0. when the set of negative numbers is combined with the set of natural numbers ( including 0 ), the result is defined as the set of integers, z also written z { \ displaystyle \ mathbb { z } }. here the letter z comes from german zahl'number '. the set of integers forms a ring with the operations addition and multiplication. the natural numbers form a subset of the integers. as there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as positive integers, and the natural numbers with zero are referred to as non - negative integers. = = = rational numbers = = = a rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. the fraction m / n represents m parts of a whole divided into n equal parts. two different fractions may correspond to the same rational number ; for example 1 / 2 and 2 / 4 are equal, that is : 1 2 = 2 4. { \ displaystyle { 1 \ over 2 } = { 2 \ over 4 }. } in general, a b = c d { \ displaystyle { a \ over b } = { c \ over d } } if and only if a × d = c × b
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\ over 2 } = { 2 \ over 4 }. } in general, a b = c d { \ displaystyle { a \ over b } = { c \ over d } } if and only if a × d = c × b. { \ displaystyle { a \ times d } = { c \ times b }. } if the absolute value of m is greater than n ( supposed to be positive ), then the absolute value of the fraction is greater than 1. fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. the set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. for example −7 can be written −7 / 1. the symbol for the rational numbers is q ( for quotient ), also written q { \ displaystyle \ mathbb { q } }. = = = real numbers = = = the symbol for the real numbers is r, also written as r. { \ displaystyle \ mathbb { r }. } they include all the measuring numbers. every real number corresponds to a point on the number line. the following paragraph will focus primarily on positive real numbers. the treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign, e. g. −123. 456. most real numbers can only be approximated by decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. each digit to the right of the decimal point has a place value one - tenth of the place value of the digit to its left. for example, 123. 456 represents 123456 / 1000, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. a real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. thus, for example, one half is 0. 5, one fifth is 0. 2, one - tenth is 0. 1, and one fiftieth is 0. 02. representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point.
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5, one fifth is 0. 2, one - tenth is 0. 1, and one fiftieth is 0. 02. representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. if this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. such a decimal is called a repeating decimal. thus 1 / 3 can be written as 0. 333..., with an ellipsis to indicate that the pattern continues. forever repeating 3s are also written as 0. 3. it turns out that these repeating decimals ( including the repetition of zeroes ) denote exactly the rational numbers, i. e., all rational numbers are also real numbers, but it is not the case that every real number is rational. a real number that is not rational is called irrational. a famous irrational real number is the π, the ratio of the circumference of any circle to its diameter. when pi is written as π = 3. 14159265358979 …, { \ displaystyle \ pi = 3. 14159265358979 \ dots, } as it sometimes is, the ellipsis does not mean that the decimals repeat ( they do not ), but rather that there is no end to them. it has been proved that π is irrational. another well - known number, proven to be an irrational real number, is 2 = 1. 41421356237 …, { \ displaystyle { \ sqrt { 2 } } = 1. 41421356237 \ dots, } the square root of 2, that is, the unique positive real number whose square is 2. both these numbers have been approximated ( by computer ) to trillions ( 1 trillion = 1012 = 1, 000, 000, 000, 000 ) of digits. not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. they can only be approximated by decimal numerals, denoting rounded or truncated real numbers. any rounded or truncated number is necessarily a rational number, of which there are only countably many. all measurements are, by their nature, approximations, and always have a margin of error. thus 123. 456 is considered an approximation of any real number greater or equal to 1234555 / 10000 and strictly less than 1234565 /
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all measurements are, by their nature, approximations, and always have a margin of error. thus 123. 456 is considered an approximation of any real number greater or equal to 1234555 / 10000 and strictly less than 1234565 / 10000 ( rounding to 3 decimals ), or of any real number greater or equal to 123456 / 1000 and strictly less than 123457 / 1000 ( truncation after the 3. decimal ). digits that suggest a greater accuracy than the measurement itself does, should be removed. the remaining digits are then called significant digits. for example, measurements with a ruler can seldom be made without a margin of error of at least 0. 001 m. if the sides of a rectangle are measured as 1. 23 m and 4. 56 m, then multiplication gives an area for the rectangle between 5. 614591 m2 and 5. 603011 m2. since not even the second digit after the decimal place is preserved, the following digits are not significant. therefore, the result is usually rounded to 5. 61. just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. for example, 0. 999..., 1. 0, 1. 00, 1. 000,..., all represent the natural number 1. a given real number has only the following decimal representations : an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. in this last case, the last non - zero digit may be replaced by the digit one smaller followed by an unlimited number of 9s, or the last non - zero digit may be followed by an unlimited number of zeros. thus the exact real number 3. 74 can also be written 3. 7399999999... and 3. 74000000000.... similarly, a decimal numeral with an unlimited number of 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9s can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. finally, an unlimited sequence of 0s to the right of a decimal place can be dropped. for example
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##written by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. finally, an unlimited sequence of 0s to the right of a decimal place can be dropped. for example, 6. 849999999999... = 6. 85 and 6. 850000000000... = 6. 85. finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9s, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. for example, 99. 999... = 100. the real numbers also have an important but highly technical property called the least upper bound property. it can be shown that any ordered field, which is also complete, is isomorphic to the real numbers. the real numbers are not, however, an algebraically closed field, because they do not include a solution ( often called a square root of minus one ) to the algebraic equation x 2 + 1 = 0 { \ displaystyle x ^ { 2 } + 1 = 0 }. = = = complex numbers = = = moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. this set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. this led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number : a square root of −1, denoted by i, a symbol assigned by leonhard euler, and called the imaginary unit. the complex numbers consist of all numbers of the form a + b i { \ displaystyle \, a + bi } where a and b are real numbers. because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. in the expression a + bi, the real number a is called the real part and b is called the imaginary part. if the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary ; if the imaginary part is 0, then the number is a real number. thus the real numbers are a subset of the complex numbers. if the real and imaginary parts of a complex number are both integers, then the number is
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to as purely imaginary ; if the imaginary part is 0, then the number is a real number. thus the real numbers are a subset of the complex numbers. if the real and imaginary parts of a complex number are both integers, then the number is called a gaussian integer. the symbol for the complex numbers is c or c { \ displaystyle \ mathbb { c } }. the fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a root in the complex numbers. like the reals, the complex numbers form a field, which is complete, but unlike the real numbers, it is not ordered. that is, there is no consistent meaning assignable to saying that i is greater than 1, nor is there any meaning in saying that i is less than 1. in technical terms, the complex numbers lack a total order that is compatible with field operations. = = subclasses of the integers = = = = = even and odd numbers = = = an even number is an integer that is " evenly divisible " by two, that is divisible by two without remainder ; an odd number is an integer that is not even. ( the old - fashioned term " evenly divisible " is now almost always shortened to " divisible ". ) any odd number n may be constructed by the formula n = 2k + 1, for a suitable integer k. starting with k = 0, the first non - negative odd numbers are { 1, 3, 5, 7,... }. any even number m has the form m = 2k where k is again an integer. similarly, the first non - negative even numbers are { 0, 2, 4, 6,... }. = = = prime numbers = = = a prime number, often shortened to just prime, is an integer greater than 1 that is not the product of two smaller positive integers. the first few prime numbers are 2, 3, 5, 7, and 11. there is no such simple formula as for odd and even numbers to generate the prime numbers. the primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. the study of these questions belongs to number theory. goldbach's conjecture is an example of a still unanswered question : " is every even number the sum of two primes? " one answered question, as to whether every integer greater
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the study of these questions belongs to number theory. goldbach's conjecture is an example of a still unanswered question : " is every even number the sum of two primes? " one answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed ; this proven claim is called the fundamental theorem of arithmetic. a proof appears in euclid's elements. = = = other classes of integers = = = many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. example of such sets of integers are fibonacci numbers and perfect numbers. for more examples, see integer sequence. = = subclasses of the complex numbers = = = = = algebraic, irrational and transcendental numbers = = = algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. real numbers that are not rational numbers are called irrational numbers. complex numbers which are not algebraic are called transcendental numbers. the algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers. = = = periods and exponential periods = = = a period is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. the periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π. the set of periods form a countable ring and bridge the gap between algebraic and transcendental numbers. the periods can be extended by permitting the integrand to be the product of an algebraic function and the exponential of an algebraic function. this gives another countable ring : the exponential periods. the number e as well as euler's constant are exponential periods. = = = constructible numbers = = = motivated by the classical problems of constructions with straightedge and compass, the constructible numbers are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps. = = = computable numbers = = = a computable number, also known as recursive number, is a real number such that there exists an algorithm which, given a positive number n as input, produces the first n digits of the computable number's decimal representation. equivalent definitions can be given using μ - recursive
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##ive number, is a real number such that there exists an algorithm which, given a positive number n as input, produces the first n digits of the computable number's decimal representation. equivalent definitions can be given using μ - recursive functions, turing machines or λ - calculus. the computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers. the computable numbers may be viewed as the real numbers that may be exactly represented in a computer : a computable number is exactly represented by its first digits and a program for computing further digits. however, the computable numbers are rarely used in practice. one reason is that there is no algorithm for testing the equality of two computable numbers. more precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. the set of computable numbers has the same cardinality as the natural numbers. therefore, almost all real numbers are non - computable. however, it is very difficult to produce explicitly a real number that is not computable. = = extensions of the concept = = = = = p - adic numbers = = = the p - adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. the number system that results depends on what base is used for the digits : any base is possible, but a prime number base provides the best mathematical properties. the set of the p - adic numbers contains the rational numbers, but is not contained in the complex numbers. the elements of an algebraic function field over a finite field and algebraic numbers have many similar properties ( see function field analogy ). therefore, they are often regarded as numbers by number theorists. the p - adic numbers play an important role in this analogy. = = = hypercomplex numbers = = = some number systems that are not included in the complex numbers may be constructed from the real numbers r { \ displaystyle \ mathbb { r } } in a way that generalize the construction of the complex numbers. they are sometimes called hypercomplex numbers. they include the quaternions h { \ displaystyle \ mathbb { h } }, introduced by sir william rowan hamilton, in which multiplication is not commutative, the
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the complex numbers. they are sometimes called hypercomplex numbers. they include the quaternions h { \ displaystyle \ mathbb { h } }, introduced by sir william rowan hamilton, in which multiplication is not commutative, the octonions o { \ displaystyle \ mathbb { o } }, in which multiplication is not associative in addition to not being commutative, and the sedenions s { \ displaystyle \ mathbb { s } }, in which multiplication is not alternative, neither associative nor commutative. the hypercomplex numbers include one real unit together with 2 n − 1 { \ displaystyle 2 ^ { n } - 1 } imaginary units, for which n is a non - negative integer. for example, quaternions can generally represented using the form a + b i + c j + d k, { \ displaystyle a + b \, \ mathbf { i } + c \, \ mathbf { j } + d \, \ mathbf { k }, } where the coefficients a, b, c, d are real numbers, and i, j, k are 3 different imaginary units. each hypercomplex number system is a subset of the next hypercomplex number system of double dimensions obtained via the cayley – dickson construction. for example, the 4 - dimensional quaternions h { \ displaystyle \ mathbb { h } } are a subset of the 8 - dimensional quaternions o { \ displaystyle \ mathbb { o } }, which are in turn a subset of the 16 - dimensional sedenions s { \ displaystyle \ mathbb { s } }, in turn a subset of the 32 - dimensional trigintaduonions t { \ displaystyle \ mathbb { t } }, and ad infinitum with 2 n { \ displaystyle 2 ^ { n } } dimensions, with n being any non - negative integer. including the complex and real numbers and their subsets, this can be expressed symbolically as : n ⊂ z ⊂ q ⊂ r ⊂ c ⊂ h ⊂ o ⊂ s ⊂ t ⊂ { \ displaystyle \ mathbb { n } \ subset \ mathbb { z } \ subset \ mathbb { q } \ subset \ mathbb { r } \ subset \ mathbb { c } \ subset \ mathbb { h } \ subset \ mathbb { o } \ subset \ mathbb { s
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{ z } \ subset \ mathbb { q } \ subset \ mathbb { r } \ subset \ mathbb { c } \ subset \ mathbb { h } \ subset \ mathbb { o } \ subset \ mathbb { s } \ subset \ mathbb { t } \ subset \ cdots } alternatively, starting from the real numbers r { \ displaystyle \ mathbb { r } }, which have zero complex units, this can be expressed as c 0 ⊂ c 1 ⊂ c 2 ⊂ c 3 ⊂ c 4 ⊂ c 5 ⊂ ⊂ c n { \ displaystyle { \ mathcal { c } } _ { 0 } \ subset { \ mathcal { c } } _ { 1 } \ subset { \ mathcal { c } } _ { 2 } \ subset { \ mathcal { c } } _ { 3 } \ subset { \ mathcal { c } } _ { 4 } \ subset { \ mathcal { c } } _ { 5 } \ subset \ cdots \ subset c _ { n } } with c n { \ displaystyle c _ { n } } containing 2 n { \ displaystyle 2 ^ { n } } dimensions. = = = transfinite numbers = = = for dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. the former gives the ordering of the set, while the latter gives its size. for finite sets, both ordinal and cardinal numbers are identified with the natural numbers. in the infinite case, many ordinal numbers correspond to the same cardinal number. = = = nonstandard numbers = = = hyperreal numbers are used in non - standard analysis. the hyperreals, or nonstandard reals ( usually denoted as * r ), denote an ordered field that is a proper extension of the ordered field of real numbers r and satisfies the transfer principle. this principle allows true first - order statements about r to be reinterpreted as true first - order statements about * r. superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields. = = see also = = = = notes = = = = references = = tobias dantzig, number, the language of science ; a critical survey written for the cultured non - mathematician, new york, the macmillan company, 1930. erich friedman, what's special about this number? archived 2018 - 02
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= = tobias dantzig, number, the language of science ; a critical survey written for the cultured non - mathematician, new york, the macmillan company, 1930. erich friedman, what's special about this number? archived 2018 - 02 - 23 at the wayback machine steven galovich, introduction to mathematical structures, harcourt brace javanovich, 1989, isbn 0 - 15 - 543468 - 3. paul halmos, naive set theory, springer, 1974, isbn 0 - 387 - 90092 - 6. morris kline, mathematical thought from ancient to modern times, oxford university press, 1990. isbn 978 - 0195061352 alfred north whitehead and bertrand russell, principia mathematica to * 56, cambridge university press, 1910. leo cory, a brief history of numbers, oxford university press, 2015, isbn 978 - 0 - 19 - 870259 - 7. = = external links = = nechaev, v. i. ( 2001 ) [ 1994 ]. " number ". encyclopedia of mathematics. ems press. tallant, jonathan. " do numbers exist ". numberphile. brady haran. archived from the original on 8 march 2016. retrieved 6 april 2013. in our time : negative numbers. bbc radio 4. 9 march 2006. archived from the original on 31 may 2022. robin wilson ( 7 november 2007 ). " 4000 years of numbers ". gresham college. archived from the original on 8 april 2022. krulwich, robert ( 22 july 2011 ). " what's the world's favorite number? ". npr. archived from the original on 18 may 2021. retrieved 17 september 2011. ; " cuddling with 9, smooching with 8, winking at 7 ". npr. 21 august 2011. archived from the original on 6 november 2018. retrieved 17 september 2011. online encyclopedia of integer sequences
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meta - selective c – h functionalization refers to the regioselective reaction of a substituted aromatic ring on the c – h bond meta to the substituent. substituted aromatic ring is an important type of substructure in pharmaceuticals and industrial compounds. thus, synthetic methods towards substituted aromatic rings are always of great interest to chemists. traditionally, regioselectivity on the aromatic ring is achieved by the electronic effect of substituents. taking the well - known friedel – craft electrophilic aromatic substitution as example, electron donating groups direct the electrophile to ortho - / para - position while electron withdrawing groups direct the electrophile to meta - position. however, with complicated systems, electronic difference between different c – h bonds can be subtle and electronic directing effect alone could become less synthetically useful. the fast development of c – h activation in the past few decades provides synthetic chemists with the powerful tools to synthesize functionalized aromatic compounds with high selectivity. the widely used approach to achieve ortho - selectivity involves metal - chelating directing groups, which forms a relatively stable 6 - or 7 - membered cyclic pre - transition state to bring the metal catalyst to the proximity of the ortho - hydrogen. however, applying the same strategy to meta - or para - c - h functionalization does not work because the corresponding cyclophane - like cyclic pre - transition state is highly strained. thus, while ortho - selectivity has been achieved by numerous catalytic systems, meta - and para - selectivity remains a challenge. in recent years, new strategies that override the electronic and steric bias have been developed to address meta - c – h functionalization. however, before these discoveries, synthesis of meta - substituted aromatic compounds could be either limited or cumbersome. for example, before the development of the c – h activation involving one - pot synthetic route to meta - substituted phenol derivatives by maleczka and co - workers, the traditional synthesis requires 10 steps from tnt. some early attempts utilize steric and electronic effects to achieve meta - selectivity. however, they are either limited to certain structure of substrates or are not highly selective. in recent years, several highly selective meta - c - h functionalization strategies have been reported which can override the intrinsic electronic and steric properties of the substrates and can apply to a wide range of substrate derivatives. the development of the modern meta - c - h functionalization strategies “ open doors for numerous possibilities
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functionalization strategies have been reported which can override the intrinsic electronic and steric properties of the substrates and can apply to a wide range of substrate derivatives. the development of the modern meta - c - h functionalization strategies “ open doors for numerous possibilities ” for synthesis and catalyst development. = = recently - developed meta - selective c – h activation strategies = = = = = copper catalyzed meta - selective c – h arylation = = = in 2009, gaunt's group reported a copper catalyzed meta - selective c – h arylation reaction on anilide derivatives. despite the intrinsic ortho - / para - selectivity of the amido group, the arylation occurs exclusively on the meta position on a variety of anilide substrates. remarkably, the regioselectivity is totally different from the earlier reported pd catalyzed c - h functionalizations, where the amido group serve as a powerful ortho - directing group. the method is robust under mild reaction conditions. it is compatible with a spectrum of substituted anilide as well as different bisaryliodonium salts. however, the meta - selectivity is lost when highly ortho / para - directing methoxy group substitutes one of the meta - hydrogen of the anilide, which marks the limitation of this method. despite the limitation, the paper was of high impact. it has been highlighted in a number of journals and news and was voted as one of the top 12 papers of 2009 by chemical and engineering news. in a more recent report from the same group, α - arylcarbonyl compounds were found to be good substrates for the copper catalyzed meta - selective c - h arylation. the power of the meta - selectivity overrides the electronic effect of different substituents, including the strong ortho / para - directing m - methoxy group. although the copper catalyzed meta - selective c – h arylation is quite successful, the mechanism behind the meta - selectivity is not completely understood. there are generally two proposed mechanisms both involving a cu ( i ) / cu ( iii ) catalyst cycle. in gaunt's original paper, he proposed a mechanism involving an anti - oxy - cupration step as the key to the meta - selectivity. first, the cu ( ii ) salt generates the active cu ( i ) species through either disproportionation or reduction by nucleophile. the active cu ( i )
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step as the key to the meta - selectivity. first, the cu ( ii ) salt generates the active cu ( i ) species through either disproportionation or reduction by nucleophile. the active cu ( i ) species undergoes oxidative addition with diphenyliodonium salt to generate a highly electrophilic cu ( iii ) species. while the cu ( iii ) species activates the aromatic ring, the amide oxygen attacks the ortho position, breaking the aromaticity and allowing cupration at the meta position. the intermediate then rearomatizes with base and undergoes reductive elimination to afford the meta - arylated product and regenerate the active cu ( i ) catalyst. alternatively, li and wu, based on dft calculations, proposed a mechanism involving a " heck - like four - membered - ring transition state ". the amide oxygen first coordinates to the cu ( iii ) species generated from oxidative addition of cu ( i ) triflate and diphenyliodonium triflate. then, the phenyl group bonded to copper interacts with the aromatic ring at the meta - position, forming a four - membered - ring transition state. according to their calculations, the aromaticity is not completely lost during the transformation. in the last step, the cu ( iii ) - c bond breaks to regenerate the cu ( i ) catalyst while the triflate ion abstract the meta - hydrogen to recover the aromaticity and gives the product. = = = meta - selective c - h functionalization assisted by a remote " end - on " template = = = in 2012, yu and co - workers reported a pioneering meta - selective c - h olefination using nitrile - containing templates to deliver the palladium to the meta - position via a macrocyclic cyclophane - like pre - transition state. the nitrile group is tethered to the aromatic ring by a removable linker. it coordinates weakly to palladium in an " end - on " fashion, which refers to the linear structure of c – cn – pd bonds. the linear coordination is proposed to help overcome the high strain in the cyclophane - like pre - transition state that brings palladium to the vicinity of the meta - hydrogen. the template is designed as such that the flat arene linker keeps the substrate aromatic ring and the nitrile group coplanar. bulky substitu
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pre - transition state that brings palladium to the vicinity of the meta - hydrogen. the template is designed as such that the flat arene linker keeps the substrate aromatic ring and the nitrile group coplanar. bulky substituents on the arene linker " lock " the nitrile tether in the desired position. the delicate design results in high regioselectivity towards the meta - c - h bond. the templates can be removed easily to give toluene derivatives or hydrocinnamic acid derivatives in high yield. in their subsequent works, yu and co - workers report the application of the same strategy in meta - selective c - h cross - coupling, meta - c - h acetoxylation and meta - c – h olefination in a broad substrate scope. it is demonstrated that the " end - on " template not only work with the pd ( 0 ) / pd ( ii ) catalytic cycle but is also compatible with the pd ( ii ) / pd ( iv ) cycle. in all three works, addition of mono - n - protected amino acid ( mpaa ) such as n - acetyl glycine improves the reaction yields and enhances the regioselectivity. = = = mechanism = = = the mechanistic study of the palladium - catalyzed meta - selective c – h bond activation with a nitrile - containing template was done by yu, wu, houk and their co - workers. the dft calculations suggest that the regioselectivity is achieved in the c – h activation step, which is the rate - determining step. it proceeds via a concerted metalation - deprotonation ( cmd ) pathway, which means that palladation and deprotonation of the c – h bond happen at the same time. surprisingly, calculations reveal that the pd – ag heterodimeric transition state leads to meta - selectivity while the pd monomeric transition state leads to ortho - selectivity. the role of mono - n - protected amino acid is proposed as a dianionic ligand which participates in the cmd step assisting the deprotonation of the c – h bond in the rate - and regio - determining step. = = = meta - alkylation achieved by remote ortho - ruthenation and electrophilic - type substitution = = = originally discovered by frost and co - workers, the meta selective sulfonation of 2 - phenylpy
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= = = meta - alkylation achieved by remote ortho - ruthenation and electrophilic - type substitution = = = originally discovered by frost and co - workers, the meta selective sulfonation of 2 - phenylpyridine using a sulfonyl chloride coupling partner, utilising a ruthenium ( ii ) catalyst. this reaction has been proposed to proceed via a similar method to that of the meta alkylation reported by frost and ackermann which involves a meta - selective c - h bond alkylation reaction with secondary and tertiary alkyl halides catalyzed by ruthenium ( ii ) carboxylate catalysts. the directing group first coordinate to the ruthenium catalyst. a reversible metalation takes place to generate the cycloruthenated complex as the key intermediates. the cycloruthenation activates the aromatic ring to undergo sear type alkylation at the position para to the c – ru bond. = = potential applications = = in gaunt's and yu's works, some derivatives of drug molecules and biologically active compounds were successfully functionalized in their meta - position. for instance, meta - arylated derivatives of anti - inflammatory drugs ( s ) - ibruprofen and ( s ) - naproxen were synthesized with copper catalyzed c – h arylation. meta - olefinated biologically important biphenyl, amino acid and baclofen derivatives have been accessed by remote c – h activation assisted by the " end - on " template. these demonstrate the potential applications of meta - selective c – h functionalization in medicinal chemistry. = = see also = = carbon – hydrogen bond activation = = references = = = = external links = = meta - selective c - h bond activation, seminar presentation by prakash kumar shee, department of chemistry, michigan state university
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a magnetogravity wave is a type of plasma wave. a magnetogravity wave is an acoustic gravity wave which is associated with fluctuations in the background magnetic field. in this context, gravity wave refers to a classical fluid wave, and is completely unrelated to the relativistic gravitational wave. = = examples = = magnetogravity waves are found in the corona of the sun. = = see also = = wave plasma magnetosonic wave helioseismology = = references = =
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construction robots are a subset of industrial robots used for building and infrastructure construction at site. despite being traditionally slow to adopt new technologies, 55 % of construction companies in the united states, europe, and china now say they use robots on job sites. most of the robots working on jobsites today are designed to remove strains on humans, e. g., excavating and lifting heavy objects. robots that survey and layout markers, tie rebar, and install drywall are also now on the market. other robots are being developed to perform tasks such as finishing the exterior, steel placement, construction of masonry wall, reinforcement concrete, etc. the main challenge to use robots in site is due to limitation in workspace. = = features = = general features include : it must be able to move. it must be able to handle components of variable size and weight. it must be able to adjust with changing environment. it must be able to interact with its surroundings. it must be able to perform multiple tasks. = = capabilities = = construction robots have been tested to carry out the followings : building walls monitor the construction progress inspection robots are used to investigate the infrastructures, mainly at dangerous locations = = notable construction by robots = = 30 storied rail city building at yokohama, japan was constructed by an automated system. concrete floor finish robot was used by kajima and tokimec companies in japan. obayashi corporation in japan has developed and used a system to lay concrete layers in dam construction. = = social impact = = use of the construction robots in the usa is rare, mainly due to opposition from labour unions. however, in japan, these robots are taken positively. = = see also = = industrial robots = = references = =
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in control theory and stability theory, the nyquist stability criterion or strecker – nyquist stability criterion, independently discovered by the german electrical engineer felix strecker at siemens in 1930 and the swedish - american electrical engineer harry nyquist at bell telephone laboratories in 1932, is a graphical technique for determining the stability of a linear dynamical system. because it only looks at the nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed - loop or open - loop system ( although the number of each type of right - half - plane singularities must be known ). as a result, it can be applied to systems defined by non - rational functions, such as systems with delays. in contrast to bode plots, it can handle transfer functions with right half - plane singularities. in addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. the nyquist stability criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. while nyquist is one of the most general stability tests, it is still restricted to linear time - invariant ( lti ) systems. nevertheless, there are generalizations of the nyquist criterion ( and plot ) for non - linear systems, such as the circle criterion and the scaled relative graph of a nonlinear operator. additionally, other stability criteria like lyapunov methods can also be applied for non - linear systems. although nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. techniques like bode plots, while less general, are sometimes a more useful design tool. = = nyquist plot = = a nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. the most common use of nyquist plots is for assessing the stability of a system with feedback. in cartesian coordinates, the real part of the transfer function is plotted on the x - axis while the imaginary part is plotted on the y - axis. the frequency is swept as a parameter, resulting in one point per frequency. the same plot can be described using polar coordinates, where gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate. the nyquist plot is named after harry nyquist, a former engineer at bell laboratories. assessment of
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be described using polar coordinates, where gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate. the nyquist plot is named after harry nyquist, a former engineer at bell laboratories. assessment of the stability of a closed - loop negative feedback system is done by applying the nyquist stability criterion to the nyquist plot of the open - loop system ( i. e. the same system without its feedback loop ). this method is easily applicable even for systems with delays and other non - rational transfer functions, which may appear difficult to analyze with other methods. stability is determined by looking at the number of encirclements of the point ( −1, 0 ). the range of gains over which the system will be stable can be determined by looking at crossings of the real axis. the nyquist plot can provide some information about the shape of the transfer function. for instance, the plot provides information on the difference between the number of zeros and poles of the transfer function by the angle at which the curve approaches the origin. when drawn by hand, a cartoon version of the nyquist plot is sometimes used, which shows the linearity of the curve, but where coordinates are distorted to show more detail in regions of interest. when plotted computationally, one needs to be careful to cover all frequencies of interest. this typically means that the parameter is swept logarithmically, in order to cover a wide range of values. = = background = = the mathematics uses the laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. we consider a system whose transfer function is g ( s ) { \ displaystyle g ( s ) } ; when placed in a closed loop with negative feedback h ( s ) { \ displaystyle h ( s ) }, the closed loop transfer function ( cltf ) then becomes : g ( s ) 1 + g ( s ) h ( s ) { \ displaystyle { \ frac { g ( s ) } { 1 + g ( s ) h ( s ) } } } stability can be determined by examining the roots of the desensitivity factor polynomial 1 + g ( s ) h ( s ) { \ displaystyle 1 + g ( s ) h ( s ) }, e. g. using the routh array, but this method is somewhat tedious. conclusions can also be reached by examining the open loop transfer function ( oltf ) g ( s ) h (
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g ( s ) h ( s ) }, e. g. using the routh array, but this method is somewhat tedious. conclusions can also be reached by examining the open loop transfer function ( oltf ) g ( s ) h ( s ) { \ displaystyle g ( s ) h ( s ) }, using its bode plots or, as here, its polar plot using the nyquist criterion, as follows. any laplace domain transfer function t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) } can be expressed as the ratio of two polynomials : t ( s ) = n ( s ) d ( s ). { \ displaystyle { \ mathcal { t } } ( s ) = { \ frac { n ( s ) } { d ( s ) } }. } the roots of n ( s ) { \ displaystyle n ( s ) } are called the zeros of t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) }, and the roots of d ( s ) { \ displaystyle d ( s ) } are the poles of t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) }. the poles of t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) } are also said to be the roots of the characteristic equation d ( s ) = 0 { \ displaystyle d ( s ) = 0 }. the stability of t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) } is determined by the values of its poles : for stability, the real part of every pole must be negative. if t ( s ) { \ displaystyle { \ mathcal { t } } ( s ) } is formed by closing a negative unity feedback loop around the open - loop transfer function, g ( s ) h ( s ) = a ( s ) b ( s ) { \ displaystyle g ( s ) h ( s ) = { \ frac { a ( s ) } { b ( s ) } } } then the roots of the characteristic equation are also the zeros of 1 + g ( s ) h ( s ) { \ displaystyle 1 + g ( s ) h ( s ) }, or simply the roots of a ( s ) + b ( s ) = 0 { \ displaystyle a ( s ) + b ( s ) = 0 }
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s ) { \ displaystyle 1 + g ( s ) h ( s ) }, or simply the roots of a ( s ) + b ( s ) = 0 { \ displaystyle a ( s ) + b ( s ) = 0 }. = = cauchy's argument principle = = from complex analysis, a contour γ s { \ displaystyle \ gamma _ { s } } drawn in the complex s { \ displaystyle s } plane, encompassing but not passing through any number of zeros and poles of a function f ( s ) { \ displaystyle f ( s ) }, can be mapped to another plane ( named f ( s ) { \ displaystyle f ( s ) } plane ) by the function f { \ displaystyle f }. precisely, each complex point s { \ displaystyle s } in the contour γ s { \ displaystyle \ gamma _ { s } } is mapped to the point f ( s ) { \ displaystyle f ( s ) } in the new f ( s ) { \ displaystyle f ( s ) } plane yielding a new contour. the nyquist plot of f ( s ) { \ displaystyle f ( s ) }, which is the contour γ f ( s ) = f ( γ s ) { \ displaystyle \ gamma _ { f ( s ) } = f ( \ gamma _ { s } ) } will encircle the point s = − 1 / k + j 0 { \ displaystyle s = { - 1 / k + j0 } } of the f ( s ) { \ displaystyle f ( s ) } plane n { \ displaystyle n } times, where n = p − z { \ displaystyle n = p - z } by cauchy's argument principle. here z { \ displaystyle z } and p { \ displaystyle p } are, respectively, the number of zeros of 1 + k f ( s ) { \ displaystyle 1 + kf ( s ) } and poles of f ( s ) { \ displaystyle f ( s ) } inside the contour γ s { \ displaystyle \ gamma _ { s } }. note that we count encirclements in the f ( s ) { \ displaystyle f ( s ) } plane in the same sense as the contour γ s { \ displaystyle \ gamma _ { s } } and that encirclements in the opposite direction are negative encirclements. that is,
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\ displaystyle f ( s ) } plane in the same sense as the contour γ s { \ displaystyle \ gamma _ { s } } and that encirclements in the opposite direction are negative encirclements. that is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. instead of cauchy's argument principle, the original paper by harry nyquist in 1932 uses a less elegant approach. the approach explained here is similar to the approach used by leroy maccoll ( fundamental theory of servomechanisms 1945 ) or by hendrik bode ( network analysis and feedback amplifier design 1945 ), both of whom also worked for bell laboratories. this approach appears in most modern textbooks on control theory. = = definition = = we first construct the nyquist contour, a contour that encompasses the right - half of the complex plane : a path traveling up the j ω { \ displaystyle j \ omega } axis, from 0 − j ∞ { \ displaystyle 0 - j \ infty } to 0 + j ∞ { \ displaystyle 0 + j \ infty }. a semicircular arc, with radius r → ∞ { \ displaystyle r \ to \ infty }, that starts at 0 + j ∞ { \ displaystyle 0 + j \ infty } and travels clock - wise to 0 − j ∞ { \ displaystyle 0 - j \ infty }. the nyquist contour mapped through the function 1 + g ( s ) { \ displaystyle 1 + g ( s ) } yields a plot of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } in the complex plane. by the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } in the right - half complex plane minus the number of poles of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } in the right - half complex plane. if instead, the contour is mapped through the open - loop transfer function g ( s ) { \ displaystyle g ( s ) }, the result is the nyquist plot of g ( s ) { \ displaystyle g ( s ) }. by counting the resulting contour's encirclements of −1, we find the difference between the number of
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s ) }, the result is the nyquist plot of g ( s ) { \ displaystyle g ( s ) }. by counting the resulting contour's encirclements of −1, we find the difference between the number of poles and zeros in the right - half complex plane of 1 + g ( s ) { \ displaystyle 1 + g ( s ) }. recalling that the zeros of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } are the poles of the closed - loop system, and noting that the poles of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } are same as the poles of g ( s ) { \ displaystyle g ( s ) }, we now state the nyquist criterion : given a nyquist contour γ s { \ displaystyle \ gamma _ { s } }, let p { \ displaystyle p } be the number of poles of g ( s ) { \ displaystyle g ( s ) } encircled by γ s { \ displaystyle \ gamma _ { s } }, and z { \ displaystyle z } be the number of zeros of 1 + g ( s ) { \ displaystyle 1 + g ( s ) } encircled by γ s { \ displaystyle \ gamma _ { s } }. alternatively, and more importantly, if z { \ displaystyle z } is the number of poles of the closed loop system in the right half plane, and p { \ displaystyle p } is the number of poles of the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } in the right half plane, the resultant contour in the g ( s ) { \ displaystyle g ( s ) } - plane, γ g ( s ) { \ displaystyle \ gamma _ { g ( s ) } } shall encircle ( clockwise ) the point ( − 1 + j 0 ) { \ displaystyle ( - 1 + j0 ) } n { \ displaystyle n } times such that n = z − p { \ displaystyle n = z - p }. if the system is originally open - loop unstable, feedback is necessary to stabilize the system. right - half - plane ( rhp ) poles represent that instability. for closed - loop stability of a system, the number of closed - loop roots in the right half of the s - plane must be zero. hence, the number of counter - clockwise encirclements
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( rhp ) poles represent that instability. for closed - loop stability of a system, the number of closed - loop roots in the right half of the s - plane must be zero. hence, the number of counter - clockwise encirclements about − 1 + j 0 { \ displaystyle - 1 + j0 } must be equal to the number of open - loop poles in the rhp. any clockwise encirclements of the critical point by the open - loop frequency response ( when judged from low frequency to high frequency ) would indicate that the feedback control system would be destabilizing if the loop were closed. ( using rhp zeros to " cancel out " rhp poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed - loop roots travel between open - loop poles and zeros in the presence of feedback. in fact, the rhp zero can make the unstable pole unobservable and therefore not stabilizable through feedback. ) = = the nyquist criterion for systems with poles on the imaginary axis = = the above consideration was conducted with an assumption that the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } does not have any pole on the imaginary axis ( i. e. poles of the form 0 + j ω { \ displaystyle 0 + j \ omega } ). this results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. the most common case are systems with integrators ( poles at zero ). to be able to analyze systems with poles on the imaginary axis, the nyquist contour can be modified to avoid passing through the point 0 + j ω { \ displaystyle 0 + j \ omega }. one way to do it is to construct a semicircular arc with radius r → 0 { \ displaystyle r \ to 0 } around 0 + j ω { \ displaystyle 0 + j \ omega }, that starts at 0 + j ( ω − r ) { \ displaystyle 0 + j ( \ omega - r ) } and travels anticlockwise to 0 + j ( ω + r ) { \ displaystyle 0 + j ( \ omega + r ) }. such a modification implies that the phasor g ( s ) { \ displaystyle g ( s ) } travels along an arc of infinite radius by − l π { \ displaystyle - l \ pi
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j ( \ omega + r ) }. such a modification implies that the phasor g ( s ) { \ displaystyle g ( s ) } travels along an arc of infinite radius by − l π { \ displaystyle - l \ pi }, where l { \ displaystyle l } is the multiplicity of the pole on the imaginary axis. = = mathematical derivation = = our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain k, which is given by t ( s ) = k g ( s ) 1 + k g ( s ) { \ displaystyle t ( s ) = { \ frac { kg ( s ) } { 1 + kg ( s ) } } } that is, we would like to check whether the characteristic equation of the above transfer function, given by d ( s ) = 1 + k g ( s ) = 0 { \ displaystyle d ( s ) = 1 + kg ( s ) = 0 } has zeros outside the open left - half - plane ( commonly initialized as olhp ). we suppose that we have a clockwise ( i. e. negatively oriented ) contour γ s { \ displaystyle \ gamma _ { s } } enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function g ( s ) { \ displaystyle g ( s ) }. cauchy's argument principle states that − 1 2 π i γ s d ′ ( s ) d ( s ) d s = n = z − p { \ displaystyle - { \ frac { 1 } { 2 \ pi i } } \ oint _ { \ gamma _ { s } } { d'( s ) \ over d ( s ) } \, ds = n = z - p } where z { \ displaystyle z } denotes the number of zeros of d ( s ) { \ displaystyle d ( s ) } enclosed by the contour and p { \ displaystyle p } denotes the number of poles of d ( s ) { \ displaystyle d ( s ) } by the same contour. rearranging, we have z = n + p { \ displaystyle z = n + p }, which is to say z = − 1 2 π i γ s d ′ ( s ) d ( s ) d s + p { \ displaystyle z = - { \ frac { 1 }
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displaystyle z = n + p }, which is to say z = − 1 2 π i γ s d ′ ( s ) d ( s ) d s + p { \ displaystyle z = - { \ frac { 1 } { 2 \ pi i } } \ oint _ { \ gamma _ { s } } { d'( s ) \ over d ( s ) } \, ds + p } we then note that d ( s ) = 1 + k g ( s ) { \ displaystyle d ( s ) = 1 + kg ( s ) } has exactly the same poles as g ( s ) { \ displaystyle g ( s ) }. thus, we may find p { \ displaystyle p } by counting the poles of g ( s ) { \ displaystyle g ( s ) } that appear within the contour, that is, within the open right half plane ( orhp ). we will now rearrange the above integral via substitution. that is, setting u ( s ) = d ( s ) { \ displaystyle u ( s ) = d ( s ) }, we have n = − 1 2 π i γ s d ′ ( s ) d ( s ) d s = − 1 2 π i u ( γ s ) 1 u d u { \ displaystyle n = - { \ frac { 1 } { 2 \ pi i } } \ oint _ { \ gamma _ { s } } { d'( s ) \ over d ( s ) } \, ds = - { \ frac { 1 } { 2 \ pi i } } \ oint _ { u ( \ gamma _ { s } ) } { 1 \ over u } \, du } we then make a further substitution, setting v ( u ) = u − 1 k { \ displaystyle v ( u ) = { \ frac { u - 1 } { k } } }. this gives us n = − 1 2 π i u ( γ s ) 1 u d u = − 1 2 π i v ( u ( γ s ) ) 1 v + 1 / k d v { \ displaystyle n = - { \ frac { 1 } { 2 \ pi i } } \ oint _ { u ( \ gamma _ { s } ) } { 1 \ over u } \, du = - { { 1 } \ over { 2 \ pi i } } \ oint _ { v ( u
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i } } \ oint _ { u ( \ gamma _ { s } ) } { 1 \ over u } \, du = - { { 1 } \ over { 2 \ pi i } } \ oint _ { v ( u ( \ gamma _ { s } ) ) } { 1 \ over { v + 1 / k } } \, dv } we now note that v ( u ( γ s ) ) = d ( γ s ) − 1 k = g ( γ s ) { \ displaystyle v ( u ( \ gamma _ { s } ) ) = { { d ( \ gamma _ { s } ) - 1 } \ over { k } } = g ( \ gamma _ { s } ) } gives us the image of our contour under g ( s ) { \ displaystyle g ( s ) }, which is to say our nyquist plot. we may further reduce the integral n = − 1 2 π i g ( γ s ) ) 1 v + 1 / k d v { \ displaystyle n = - { \ frac { 1 } { 2 \ pi i } } \ oint _ { g ( \ gamma _ { s } ) ) } { \ frac { 1 } { v + 1 / k } } \, dv } by applying cauchy's integral formula. in fact, we find that the above integral corresponds precisely to the number of times the nyquist plot encircles the point − 1 / k { \ displaystyle - 1 / k } clockwise. thus, we may finally state that z = n + p = ( number of times the nyquist plot encircles − 1 / k clockwise ) + ( number of poles of g ( s ) in orhp ) { \ displaystyle { \ begin { aligned } z = { } & n + p \ \ [ 6pt ] = { } & { \ text { ( number of times the nyquist plot encircles } } { - 1 / k } { \ text { clockwise ) } } \ \ & { } + { \ text { ( number of poles of } } g ( s ) { \ text { in orhp ) } } \ end { aligned } } } we thus find that t ( s ) { \ displaystyle t ( s ) } as defined above corresponds to a stable unity - feedback system when z { \ displaystyle z }, as evaluated above, is equal to 0. = = importance
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} we thus find that t ( s ) { \ displaystyle t ( s ) } as defined above corresponds to a stable unity - feedback system when z { \ displaystyle z }, as evaluated above, is equal to 0. = = importance = = the nyquist stability criterion is a graphical technique that determines the stability of a dynamical system, such as a feedback control system. it is based on the argument principle and the nyquist plot of the open - loop transfer function of the system. it can be applied to systems that are not defined by rational functions, such as systems with delays. it can also handle transfer functions with singularities in the right half - plane, unlike bode plots. the nyquist stability criterion can also be used to find the phase and gain margins of a system, which are important for frequency domain controller design. = = summary = = if the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } has a zero pole of multiplicity l { \ displaystyle l }, then the nyquist plot has a discontinuity at ω = 0 { \ displaystyle \ omega = 0 }. during further analysis it should be assumed that the phasor travels l { \ displaystyle l } times clockwise along a semicircle of infinite radius. after applying this rule, the zero poles should be neglected, i. e. if there are no other unstable poles, then the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } should be considered stable. if the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } is stable, then the closed - loop system is unstable, if and only if, the nyquist plot encircle the point −1 at least once. if the open - loop transfer function g ( s ) { \ displaystyle g ( s ) } is unstable, then for the closed - loop system to be stable, there must be one counter - clockwise encirclement of −1 for each pole of g ( s ) { \ displaystyle g ( s ) } in the right - half of the complex plane. the number of surplus encirclements ( n + p greater than 0 ) is exactly the number of unstable poles of the closed - loop system. however, if the graph happens to pass through the point − 1 + j 0 { \ displaystyle - 1 + j0 }, then deciding upon even the marginal stability of the system becomes difficult
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number of unstable poles of the closed - loop system. however, if the graph happens to pass through the point − 1 + j 0 { \ displaystyle - 1 + j0 }, then deciding upon even the marginal stability of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the j ω { \ displaystyle j \ omega } axis. = = see also = = bibo stability bode plot routh – hurwitz stability criterion root locus analysis gain margin nichols plot hall circles phase margin barkhausen stability criterion circle criterion control engineering hankel singular value = = references = = = = further reading = = faulkner, e. a. ( 1969 ) : introduction to the theory of linear systems ; chapman & hall ; isbn 0 - 412 - 09400 - 2 pippard, a. b. ( 1985 ) : response & stability ; cambridge university press ; isbn 0 - 521 - 31994 - 3 gessing, r. ( 2004 ) : control fundamentals ; silesian university of technology ; isbn 83 - 7335 - 176 - 0 franklin, g. ( 2002 ) : feedback control of dynamic systems ; prentice hall, isbn 0 - 13 - 032393 - 4 = = external links = = applets with modifiable parameters eis spectrum analyser - a freeware program for analysis and simulation of impedance spectra matlab function for creating a nyquist plot of a frequency response of a dynamic system model. pid nyquist plot shaping - free interactive virtual tool, control loop simulator mathematica function for creating the nyquist plot the nyquist diagram for electrical circuits
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in molecular biology, an intrinsically disordered protein ( idp ) is a protein that lacks a fixed or ordered three - dimensional structure, typically in the absence of its macromolecular interaction partners, such as other proteins or rna. idps range from fully unstructured to partially structured and include random coil, molten globule - like aggregates, or flexible linkers in large multi - domain proteins. they are sometimes considered as a separate class of proteins along with globular, fibrous and membrane proteins. idps are a very large and functionally important class of proteins. they are most numerous in eukaryotes, with an estimated 30 - 40 % of residues in the eukaryotic proteome located in disordered regions. disorder is present in around 70 % of proteins, either in the form of disordered tails or flexible linkers. proteins can also be entirely disordered and lack a defined secondary and / or tertiary structure. their discovery has disproved the idea that three - dimensional structures of proteins must be fixed to accomplish their biological functions. for example, idps have been identified to participate in weak multivalent interactions that are highly cooperative and dynamic, lending them importance in dna regulation and in cell signaling. many idps can also adopt a fixed three - dimensional structure after binding to other macromolecules. overall, idps are different from structured proteins in many ways and tend to have distinctive function, structure, sequence, interactions, evolution and regulation. = = history = = in the 1930s - 1950s, the first protein structures were solved by protein crystallography. these early structures suggested that a fixed three - dimensional structure might be generally required to mediate biological functions of proteins. these publications solidified the central dogma of molecular biology in that the amino acid sequence of a protein determines its structure which, in turn, determines its function. in 1950, karush wrote about'configurational adaptability'contradicting this assumption. he was convinced that proteins have more than one configuration at the same energy level and can choose one when binding to other substrates. in the 1960s, levinthal's paradox suggested that the systematic conformational search of a long polypeptide is unlikely to yield a single folded protein structure on biologically relevant timescales ( i. e. microseconds to minutes ). curiously, for many ( small ) proteins or protein domains, relatively rapid and efficient refolding can be observed in vitro. as stated in anfin
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on biologically relevant timescales ( i. e. microseconds to minutes ). curiously, for many ( small ) proteins or protein domains, relatively rapid and efficient refolding can be observed in vitro. as stated in anfinsen's dogma from 1973, the fixed 3d structure of these proteins is uniquely encoded in its primary structure ( the amino acid sequence ), is kinetically accessible and stable under a range of ( near ) physiological conditions, and can therefore be considered as the native state of such " ordered " proteins. during the subsequent decades, however, many large protein regions could not be assigned in x - ray datasets, indicating that they occupy multiple positions, which average out in electron density maps. the lack of fixed, unique positions relative to the crystal lattice suggested that these regions were " disordered ". nuclear magnetic resonance spectroscopy of proteins also demonstrated the presence of large flexible linkers and termini in many solved structural ensembles. in 2001, dunker questioned whether the newly found information was ignored for 50 years with more quantitative analyses becoming available in the 2000s. in the 2010s it became clear that idps are common among disease - related proteins, such as alpha - synuclein and tau. = = abundance = = it is now generally accepted that proteins exist as an ensemble of similar structures with some regions more constrained than others. idps occupy the extreme end of this spectrum of flexibility and include proteins of considerable local structure tendency or flexible multidomain assemblies. intrinsic disorder is particularly elevated among proteins that regulate chromatin and transcription, and bioinformatic predictions indicate that is more common in genomes and proteomes than in known structures in the protein database. based on disopred2 prediction, long ( > 30 residue ) disordered segments occur in 2. 0 % of archaean, 4. 2 % of eubacterial and 33. 0 % of eukaryotic proteins, including certain disease - related proteins. = = biological roles = = highly dynamic disordered regions of proteins have been linked to functionally important phenomena such as allosteric regulation and enzyme catalysis. many disordered proteins have the binding affinity with their receptors regulated by post - translational modification, thus it has been proposed that the flexibility of disordered proteins facilitates the different conformational requirements for binding the modifying enzymes as well as their receptors. intrinsic disorder is particularly enriched in proteins implicated in cell signaling and transcription, as well as chromatin remodeling functions. genes that have recently
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##ed proteins facilitates the different conformational requirements for binding the modifying enzymes as well as their receptors. intrinsic disorder is particularly enriched in proteins implicated in cell signaling and transcription, as well as chromatin remodeling functions. genes that have recently been born de novo tend to have higher disorder. in animals, genes with high disorder are lost at higher rates during evolution. = = = flexible linkers = = = disordered regions are often found as flexible linkers or loops connecting domains. linker sequences vary greatly in length but are typically rich in polar uncharged amino acids. flexible linkers allow the connecting domains to freely twist and rotate to recruit their binding partners via protein domain dynamics. they also allow their binding partners to induce larger scale conformational changes by long - range allostery. the flexible linker of fbp25 which connects two domains of fkbp25 is important for the binding of fkbp25 with dna. = = = linear motifs = = = linear motifs are short disordered segments of proteins that mediate functional interactions with other proteins or other biomolecules ( rna, dna, sugars etc. ). many roles of linear motifs are associated with cell regulation, for instance in control of cell shape, subcellular localisation of individual proteins and regulated protein turnover. often, post - translational modifications such as phosphorylation tune the affinity ( not rarely by several orders of magnitude ) of individual linear motifs for specific interactions. relatively rapid evolution and a relatively small number of structural restraints for establishing novel ( low - affinity ) interfaces make it particularly challenging to detect linear motifs but their widespread biological roles and the fact that many viruses mimick / hijack linear motifs to efficiently recode infected cells underlines the timely urgency of research on this very challenging and exciting topic. = = = pre - structured motifs = = = unlike globular proteins, idps do not have spatially - disposed active pockets. fascinatingly, 80 % of target - unbound idps ( ~ 4 dozens ) subjected to detailed structural characterization by nmr possess linear motifs termed presmos ( pre - structured motifs ) that are transient secondary structural elements primed for target recognition. in several cases it has been demonstrated that these transient structures become full and stable secondary structures, e. g., helices, upon target binding. hence, presmos are the putative active sites in idps. = = = coupled folding and binding = = = many unstructured proteins undergo transitions to more ordered states upon
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, e. g., helices, upon target binding. hence, presmos are the putative active sites in idps. = = = coupled folding and binding = = = many unstructured proteins undergo transitions to more ordered states upon binding to their targets ( e. g. molecular recognition features ( morfs ) ). the coupled folding and binding may be local, involving only a few interacting residues, or it might involve an entire protein domain. it was recently shown that the coupled folding and binding allows the burial of a large surface area that would be possible only for fully structured proteins if they were much larger. moreover, certain disordered regions might serve as " molecular switches " in regulating certain biological function by switching to ordered conformation upon molecular recognition like small molecule - binding, dna / rna binding, ion interactions etc. the ability of disordered proteins to bind, and thus to exert a function, shows that stability is not a required condition. many short functional sites, for example short linear motifs are over - represented in disordered proteins. disordered proteins and short linear motifs are particularly abundant in many rna viruses such as hendra virus, hcv, hiv - 1 and human papillomaviruses. this enables such viruses to overcome their informationally limited genomes by facilitating binding, and manipulation of, a large number of host cell proteins. = = = disorder in the bound state ( fuzzy complexes ) = = = intrinsically disordered proteins can retain their conformational freedom even when they bind specifically to other proteins. the structural disorder in bound state can be static or dynamic. in fuzzy complexes structural multiplicity is required for function and the manipulation of the bound disordered region changes activity. the conformational ensemble of the complex is modulated via post - translational modifications or protein interactions. specificity of dna binding proteins often depends on the length of fuzzy regions, which is varied by alternative splicing. some fuzzy complexes may exhibit high binding affinity, although other studies showed different affinity values for the same system in a different concentration regime. = = structural aspects = = intrinsically disordered proteins adapt a dynamic range of rapidly interchanging conformations in vivo according to the cell's conditions, creating a structural or conformational ensemble. therefore, their structures are strongly function - related. however, only few proteins are fully disordered in their native state. disorder is mostly found in intrinsically disordered regions ( idrs ) within an otherwise well - structured protein. the term intrinsically disordered protein (
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are strongly function - related. however, only few proteins are fully disordered in their native state. disorder is mostly found in intrinsically disordered regions ( idrs ) within an otherwise well - structured protein. the term intrinsically disordered protein ( idp ) therefore includes proteins that contain idrs as well as fully disordered proteins. the existence and kind of protein disorder is encoded in its amino acid sequence. in general, idps are characterized by a low content of bulky hydrophobic amino acids and a high proportion of polar and charged amino acids, usually referred to as low hydrophobicity. this property leads to good interactions with water. furthermore, high net charges promote disorder because of electrostatic repulsion resulting from equally charged residues. thus disordered sequences cannot sufficiently bury a hydrophobic core to fold into stable globular proteins. in some cases, hydrophobic clusters in disordered sequences provide the clues for identifying the regions that undergo coupled folding and binding ( refer to biological roles ). many disordered proteins reveal regions without any regular secondary structure. these regions can be termed as flexible, compared to structured loops. while the latter are rigid and contain only one set of ramachandran angles, idps involve multiple sets of angles. the term flexibility is also used for well - structured proteins, but describes a different phenomenon in the context of disordered proteins. flexibility in structured proteins is bound to an equilibrium state, while it is not so in idps. many disordered proteins also reveal low complexity sequences, i. e. sequences with over - representation of a few residues. while low complexity sequences are a strong indication of disorder, the reverse is not necessarily true, that is, not all disordered proteins have low complexity sequences. disordered proteins have a low content of predicted secondary structure. due to the disordered nature of these proteins, topological approaches have been developed to search for conformational patterns in their dynamics. for instance, circuit topology has been applied to track the dynamics of disordered protein domains. by employing a topological approach, one can categorize motifs according to their topological buildup and the timescale of their formation. a common aspect of idp structural ensembles is the ability or tendency to fold upon an interaction to a binding partner in the cell. examples of idp folding in a binding context are binding - coupled folding, and formation of fuzzy complexes. however, it is also possible for proteins to remain entirely disordered in a binding scenario. = = experimental validation = = idps can be validated in several contexts.
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in a binding context are binding - coupled folding, and formation of fuzzy complexes. however, it is also possible for proteins to remain entirely disordered in a binding scenario. = = experimental validation = = idps can be validated in several contexts. most approaches for experimental validation of idps are restricted to extracted or purified proteins while some new experimental strategies aim to explore in vivo conformations and structural variations of idps inside intact living cells and systematic comparisons between their dynamics in vivo and in vitro. = = = in vivo approaches = = = the first direct evidence for in vivo persistence of intrinsic disorder has been achieved by in - cell nmr upon electroporation of a purified idp and recovery of cells to an intact state. larger - scale in vivo validation of idr predictions is now possible using biotin'painting '. = = = in vitro approaches = = = intrinsically unfolded proteins, once purified, can be identified by various experimental methods. the primary method to obtain information on disordered regions of a protein is nmr spectroscopy. the lack of electron density in x - ray crystallographic studies may also be a sign of disorder. folded proteins have a high density ( partial specific volume of 0. 72 - 0. 74 ml / g ) and commensurately small radius of gyration. hence, unfolded proteins can be detected by methods that are sensitive to molecular size, density or hydrodynamic drag, such as size exclusion chromatography, analytical ultracentrifugation, small angle x - ray scattering ( saxs ), and measurements of the diffusion constant. unfolded proteins are also characterized by their lack of secondary structure, as assessed by far - uv ( 170 – 250 nm ) circular dichroism ( esp. a pronounced minimum at ~ 200 nm ) or infrared spectroscopy. unfolded proteins also have exposed backbone peptide groups exposed to solvent, so that they are readily cleaved by proteases, undergo rapid hydrogen - deuterium exchange and exhibit a small dispersion ( < 1 ppm ) in their 1h amide chemical shifts as measured by nmr. ( folded proteins typically show dispersions as large as 5 ppm for the amide protons. ) recently, new methods including fast parallel proteolysis ( fastpp ) have been introduced, which allow to determine the fraction folded / disordered without the need for purification. even subtle differences in the stability of missense mutations, protein partner binding and ( self ) polymerisation - induced folding of (
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##ysis ( fastpp ) have been introduced, which allow to determine the fraction folded / disordered without the need for purification. even subtle differences in the stability of missense mutations, protein partner binding and ( self ) polymerisation - induced folding of ( e. g. ) coiled - coils can be detected using fastpp as recently demonstrated using the tropomyosin - troponin protein interaction. fully unstructured protein regions can be experimentally validated by their hypersusceptibility to proteolysis using short digestion times and low protease concentrations. bulk methods to study idp structure and dynamics include saxs for ensemble shape information, nmr for atomistic ensemble refinement, fluorescence for visualising molecular interactions and conformational transitions, x - ray crystallography to highlight more mobile regions in otherwise rigid protein crystals, cryo - em to reveal less fixed parts of proteins, light scattering to monitor size distributions of idps or their aggregation kinetics, nmr chemical shift and circular dichroism to monitor secondary structure of idps. single - molecule methods to study idps include spfret to study conformational flexibility of idps and the kinetics of structural transitions, optical tweezers for high - resolution insights into the ensembles of idps and their oligomers or aggregates, nanopores to reveal global shape distributions of idps, magnetic tweezers to study structural transitions for long times at low forces, high - speed atomic force microscopy ( afm ) to visualise the spatio - temporal flexibility of idps directly. = = disorder annotation = = intrinsic disorder can be either annotated from experimental information or predicted with specialized software. disorder prediction algorithms can predict intrinsic disorder ( id ) propensity with high accuracy ( approaching around 80 % ) based on primary sequence composition, similarity to unassigned segments in protein x - ray datasets, flexible regions in nmr studies and physico - chemical properties of amino acids. = = = disorder databases = = = databases have been established to annotate protein sequences with intrinsic disorder information. the disprot database contains a collection of manually curated protein segments which have been experimentally determined to be disordered. mobidb is a database combining experimentally curated disorder annotations ( e. g. from disprot ) with data derived from missing residues in x - ray crystallographic structures and flexible regions in nmr structures. = = = predicting idps by sequence =
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database combining experimentally curated disorder annotations ( e. g. from disprot ) with data derived from missing residues in x - ray crystallographic structures and flexible regions in nmr structures. = = = predicting idps by sequence = = = separating disordered from ordered proteins is essential for disorder prediction. one of the first steps to find a factor that distinguishes idps from non - idps is to specify biases within the amino acid composition. the following hydrophilic, charged amino acids a, r, g, q, s, p, e and k have been characterized as disorder - promoting amino acids, while order - promoting amino acids w, c, f, i, y, v, l, and n are hydrophobic and uncharged. the remaining amino acids h, m, t and d are ambiguous, found in both ordered and unstructured regions. a more recent analysis ranked amino acids by their propensity to form disordered regions as follows ( order promoting to disorder promoting ) : w, f, y, i, m, l, v, n, c, t, a, g, r, d, h, q, k, s, e, p. as it can be seen from the list, small, charged, hydrophilic residues often promote disorder, while large and hydrophobic residues promote order. this information is the basis of most sequence - based predictors. regions with little to no secondary structure, also known as nors ( no regular secondary structure ) regions, and low - complexity regions can easily be detected. however, not all disordered proteins contain such low complexity sequences. = = = prediction methods = = = determining disordered regions from biochemical methods is very costly and time - consuming. due to the variable nature of idps, only certain aspects of their structure can be detected, so that a full characterization requires a large number of different methods and experiments. this further increases the expense of idp determination. in order to overcome this obstacle, computer - based methods are created for predicting protein structure and function. it is one of the main goals of bioinformatics to derive knowledge by prediction. predictors for idp function are also being developed, but mainly use structural information such as linear motif sites. there are different approaches for predicting idp structure, such as neural networks or matrix calculations, based on different structural and / or biophysical properties. many computational methods exploit sequence information to predict whether a protein is disordered. notable examples of such software include
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there are different approaches for predicting idp structure, such as neural networks or matrix calculations, based on different structural and / or biophysical properties. many computational methods exploit sequence information to predict whether a protein is disordered. notable examples of such software include iupred and disopred. different methods may use different definitions of disorder. meta - predictors show a new concept, combining different primary predictors to create a more competent and exact predictor. due to the different approaches of predicting disordered proteins, estimating their relative accuracy is fairly difficult. for example, neural networks are often trained on different datasets. the disorder prediction category is a part of biannual casp experiment that is designed to test methods according accuracy in finding regions with missing 3d structure ( marked in pdb files as remark465, missing electron densities in x - ray structures ). = = disorder and disease = = intrinsically unstructured proteins have been implicated in a number of diseases. aggregation of misfolded proteins is the cause of many synucleinopathies and toxicity as those proteins start binding to each other randomly and can lead to cancer or cardiovascular diseases. thereby, misfolding can happen spontaneously because millions of copies of proteins are made during the lifetime of an organism. the aggregation of the intrinsically unstructured protein α - synuclein is thought to be responsible. the structural flexibility of this protein together with its susceptibility to modification in the cell leads to misfolding and aggregation. genetics, oxidative and nitrative stress as well as mitochondrial impairment impact the structural flexibility of the unstructured α - synuclein protein and associated disease mechanisms. many key tumour suppressors have large intrinsically unstructured regions, for example p53 and brca1. these regions of the proteins are responsible for mediating many of their interactions. taking the cell's native defense mechanisms as a model drugs can be developed, trying to block the place of noxious substrates and inhibiting them, and thus counteracting the disease. = = computer simulations = = owing to high structural heterogeneity, nmr / saxs experimental parameters obtained will be an average over a large number of highly diverse and disordered states ( an ensemble of disordered states ). hence, to understand the structural implications of these experimental parameters, there is a necessity for accurate representation of these ensembles by computer simulations. all - atom molecular dynamic simulations can be used for this purpose but their use is limited by the accuracy of current force
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. hence, to understand the structural implications of these experimental parameters, there is a necessity for accurate representation of these ensembles by computer simulations. all - atom molecular dynamic simulations can be used for this purpose but their use is limited by the accuracy of current force - fields in representing disordered proteins. nevertheless, some force - fields have been explicitly developed for studying disordered proteins by optimising force - field parameters using available nmr data for disordered proteins. ( examples are charmm 22 *, charmm 32, amber ff03 * etc. ) md simulations restrained by experimental parameters ( restrained - md ) have also been used to characterise disordered proteins. in principle, one can sample the whole conformational space given an md simulation ( with accurate force - field ) is run long enough. because of very high structural heterogeneity, the time scales that needs to be run for this purpose are very large and are limited by computational power. however, other computational techniques such as accelerated - md simulations, replica exchange simulations, metadynamics, multicanonical md simulations, or methods using coarse - grained representation with implicit and explicit solvents have been used to sample broader conformational space in smaller time scales. moreover, various protocols and methods of analyzing idps, such as studies based on quantitative analysis of gc content in genes and their respective chromosomal bands, have been used to understand functional idp segments. = = see also = = idpbynmr disprot database mobidb database molten globule prion random coil dark proteome = = references = = = = external links = = intrinsically disordered protein at proteopedia mobidb : a comprehensive database of intrinsic protein disorder annotations ideal - intrinsically disordered proteins with extensive annotations and literature archived 2020 - 05 - 02 at the wayback machine d2p2 database of disordered protein predictions gallery of images of intrinsically disordered proteins first idp journal covering all topics of idp research idp journal database of experimentally validated idps idp ensemble database archived 2018 - 03 - 10 at the wayback machine
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ambiguity resolution is used to find the value of a measurement that requires modulo sampling. this is required for pulse - doppler radar signal processing. = = measurements = = some types of measurements introduce an unavoidable modulo operation in the measurement process. this happens with all radar systems. radar aliasing happens when : pulse repetition frequency ( prf ) is too low to sample doppler frequency directly prf is too high to sample range directly pulse doppler sonar uses similar principles to measure position and velocity involving liquids. = = = radar systems = = = radar systems operating at a prf below about 3 khz pulse rate produce true range, but produce ambiguous target speed. radar systems operating at a prf above 30 khz produce true target speed, but produce ambiguous target range. medium prf systems produce both ambiguous range measurement and ambiguous radial speed measurement using prf from 3 khz to 30 khz. ambiguity resolution finds true range and true speed by using ambiguous range and ambiguous speed measurements with multiple prf. = = = doppler measurements = = = doppler systems involve velocity measurements similar to the kind of measurements made using a strobe light. for example, a strobe light can be used as a tachometer to measure rotational velocity for rotating machinery. strobe light measurements can be inaccurate because the light may be flashing 2 or 3 times faster than shaft rotation speed. the user can only produce an accurate measurement by increasing the pulse rate starting near zero until pulses are fast enough to make the rotating object appear stationary. radar and sonar systems use the same phenomenon to detect target speed. = = operation = = the ambiguity region is shown graphically in this image. the x axis is range ( left - right ). the y axis is radial speed. the z axis is amplitude ( up - down ). the shape of the rectangles changes when the prf changes. the unambiguous zone is in the lower left corner. all of the other blocks have ambiguous range or ambiguous radial velocity. pulse doppler radar relies on medium pulse repetition frequency ( prf ) from about 3 khz to 30 khz. each transmit pulse is separated by between 5 km and 50 km of distance. = = = range ambiguity resolution = = = the received signals from multiple prf are compared using the range ambiguity resolution process. each range sample is converted from time domain i / q samples into frequency domain. older systems use individual filters for frequency filtering. newer systems use digital sampling and a fast fourier transform or discrete
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from multiple prf are compared using the range ambiguity resolution process. each range sample is converted from time domain i / q samples into frequency domain. older systems use individual filters for frequency filtering. newer systems use digital sampling and a fast fourier transform or discrete fourier transform instead of physical filters. each filter converts time samples into a frequency spectrum. each spectrum frequency corresponds with a different speed. these samples are thresholded to obtain ambiguous range for several different prf. = = = frequency ambiguity resolution = = = the received signals are also compared using the frequency ambiguity resolution process. a blind velocity occurs when doppler frequency falls close to the prf. this folds the return signal into the same filter as stationary clutter reflections. rapidly alternating different prf while scanning eliminates blind frequencies. = = further reading = = george w. stimson ; david adamy ; christopher baker ( 30 june 2013 ). stimson's introduction to airborne radar. scitech publishing, incorporated. isbn 978 - 1 - 61353 - 022 - 1. = = references = =
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drop - in replacement is a term used in computer science and other fields. it refers to the ability to replace one hardware or software component with another, without any other code or configuration changes being required and resulting in no negative impacts. usually, the replacement has some benefits including one or more of the following : increased security increased speed increased feature set increased compatibility ( e. g. with other components or standards support ) increased support ( e. g. the old component may no longer be supported, maintained, or manufactured ) = = see also = = pin compatibility plug compatible clone ( computing ) backward compatibility kludge
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matrix isolation is an experimental technique used in chemistry and physics. it generally involves a material being trapped within an unreactive matrix. a host matrix is a continuous solid phase in which guest particles ( atoms, molecules, ions, etc. ) are embedded. the guest is said to be isolated within the host matrix. initially the term matrix - isolation was used to describe the placing of a chemical species in any unreactive material, often polymers or resins, but more recently has referred specifically to gases in low - temperature solids. a typical matrix isolation experiment involves a guest sample being diluted in the gas phase with the host material, usually a noble gas or nitrogen. this mixture is then deposited on a window that is cooled to below the melting point of the host gas. the sample may then be studied using various spectroscopic procedures. = = experimental setup = = the transparent window, on to which the sample is deposited, is usually cooled using a compressed helium or similar refrigerant. experiments must be performed under a high vacuum to prevent contaminants from unwanted gases freezing to the cold window. lower temperatures are preferred, due to the improved rigidity and " glassiness " of the matrix material. noble gases such as argon are used not just because of their unreactivity but also because of their broad optical transparency in the solid state. mono - atomic gases have relatively simple face - centered cubic ( fcc ) crystal structure, which can make interpretations of the site occupancy and crystal - field splitting of the guest easier. in some cases a reactive material, for example, methane, hydrogen or ammonia, may be used as the host material so that the reaction of the host with the guest species may be studied. using the matrix isolation technique, short - lived, highly - reactive species such as radical ions and reaction intermediates may be observed and identified by spectroscopic means. for example, the solid noble gas krypton can be used to form an inert matrix within which a reactive f3− ion can sit in chemical isolation. the reactive species can either be generated outside ( before deposition ) the apparatus and then be condensed, inside the matrix ( after deposition ) by irradiating or heating a precursor, or by bringing together two reactants on the growing matrix surface. for the deposition of two species it can be crucial to control the contact time and temperature. in twin jet deposition the two species have a much shorter contact time ( and lower temperature ) than in merged jet. with concentric jet the contact
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matrix surface. for the deposition of two species it can be crucial to control the contact time and temperature. in twin jet deposition the two species have a much shorter contact time ( and lower temperature ) than in merged jet. with concentric jet the contact time is adjustable. = = spectroscopy = = within the host matrix, the rotation and translation of the guest particle is usually inhibited. therefore, the matrix isolation technique may be used to simulate a spectrum of a species in the gas phase without rotational and translational interference. the low temperatures also help to produce simpler spectra, since only the lower electronic and vibrational quantum states are populated. especially infrared ( ir ) spectroscopy, which is used to investigate molecular vibration, benefits from the matrix isolation technique. for example, in the gas - phase ir spectrum of fluoroethane some spectral regions are very difficult to interpret, as vibrational quantum states heavily overlap with multiple rotational - vibrational quantum states. when fluoroethane is isolated in argon or neon matrices at low temperatures, the rotation of the fluoroethane molecule is inhibited. because rotational - vibrational quantum states are quenched in the matrix isolation ir spectrum of fluoroethane, all vibrational quantum states can be identified. this is especially useful for the validation of simulated infrared spectra that can be obtained from computational chemistry. = = history = = matrix isolation has its origins in the first half of the 20th century with the experiments by photo - chemists and physicists freezing samples in liquefied gases. the earliest isolation experiments involved the freezing of species in transparent, low temperature organic glasses, such as epa ( ether / isopentane / ethanol 5 : 5 : 2 ). the modern matrix isolation technique was developed extensively during the 1950s, in particular by george c. pimentel. he initially used higher - boiling inert gases like xenon and nitrogen as the host material, and is often said to be the " father of matrix isolation ". laser vaporization in matrix isolation spectroscopy was first brought about in 1969 by schaeffer and pearson using a yttrium aluminum garnet ( yag ) laser to vaporize carbon which reacted with hydrogen to produce acetylene. they also showed that laser - vaporized boron would react with hcl to create bcl3. in the 1970s, koerner von gustorf's lab used the technique to produce free metal atoms which were then deposited with organic substrates for use in organometallic chemistry. spectroscopic studies were done on reactive intermediates
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##l3. in the 1970s, koerner von gustorf's lab used the technique to produce free metal atoms which were then deposited with organic substrates for use in organometallic chemistry. spectroscopic studies were done on reactive intermediates in around the early 1980s by bell labs. they used laser - induced fluorescence to characterize multiple molecules like snbi and sic2. smalley's group employed the use of this method with time - of - flight mass spectrometry by analyzing al clusters. with the work of chemists like these, laser - vaporization in matrix isolation spectroscopy rose in popularity due to its ability to generate transients involving metals, alloys and semi - conductor molecules and clusters. = = see also = = host – guest chemistry inert gas van der waals interactions radicals = = references = = = = further reading = = dunkin, iain r ( 1998 ). matrix - isolation techniques – a practical approach. oxford : oxford university press. isbn 0 - 19 - 855863 - 5. daintith, john ( senior editor ) ( 2004 ). oxford dictionary of chemistry. oxford : oxford university press. isbn 0 - 19 - 860918 - 3. { { cite book } } : | author = has generic name ( help ) ball, david w., zakya h. kafafi, et al., a bibliography of matrix isolation spectroscopy, 1954 - 1985, rice university press, houston, 1988
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the miriam registry, a by - product of the miriam guidelines, is a database of namespaces and associated information that is used in the creation of uniform resource identifiers. it contains the set of community - approved namespaces for databases and resources serving, primarily, the biological sciences domain. these shared namespaces, when combined with'data collection'identifiers, can be used to create globally unique identifiers for knowledge held in data repositories. for more information on the use of uris to annotate models, see the specification of sbml level 2 version 2 ( and above ). a'data collection'is defined as a set of data which is generated by a provider. a'resource'is defined as a distributor of that data. such a description allows numerous resources to be associated with a single collection, allowing accurate representation of how biological information is available on the world wide web ; often the same information, from a single data collection, may be mirrored by different resources, or the core information may be supplemented with other data. data collection name : gene ontology data collection identifier : mir : 00000022 data collection synonyms : go data collection identifier pattern : ^ go : \ d { 7 } $ data collection namespace : urn : miriam : obo. go data collection'root url': http : / / identifiers. org / obo. go / data collection'root urn': urn : miriam : obo. go : collection resources : resource # 1 resource identifier : mir : 00100012 resource location website : http : / / www. ebi. ac. uk / ego / resource access url ( tokenised ) : http : / / www. ebi. ac. uk / ego / displaygoterm? selected = $ 1 resource description : quickgo ( gene ontology browser ) resource institution : european bioinformatics institute, united kingdom resource # 2 [... ] the miriam registry is a curated resource, which is freely available and open to all. submissions for new collections can be made through the website. = = identifiers using the miriam system = = the miriam guidelines require the use of uniform resource identifiers in the annotation of model components. these are created using the shared list of namespaces defined in the miriam registry. = = = miriam uris = = = using the namespaces defined in the miriam registry, it
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identifiers in the annotation of model components. these are created using the shared list of namespaces defined in the miriam registry. = = = miriam uris = = = using the namespaces defined in the miriam registry, it is possible to create identifiers in both a urn and a url forms. this requires a unique collection - specific identifier, as well as a namespace to globally constrain the information space. both the namespace and the root of each uri form are given for each data collection in the registry. both forms are derived from the same namespace. for example : urn form : urn : miriam : pubmed : 16333295 url form : http : / / identifiers. org / pubmed / 16333295 in this example, the collection - specific identifier is 16333295, and the namespace is pubmed. the urn form of identifiers requires the use of web services or programmatic means to access the referenced record. this means that one cannot simply put the urn form into a browser window and arrive at the referenced information. the url form is directly resolvable, and relies on a resolving layer provided by identifiers. org. = = = supporting features and availability = = = to enable efficient use of the miriam registry and the rapid adoption of the annotation scheme, a number of supporting features are provided. these include web services, a website interface to access the registry itself, and a java library the miriam registry is developed by the proteomics services team at the european bioinformatics institute. the source code for the entire project, including supporting features, is available from sourceforge. net. the miriam registry is used by several worldwide projects such as biomodels database, sabio - rk, copasi, a more thorough listing can be found on the website. = = see also = = minimum information standards miriam identifiers. org metadata standards computational systems biology biomodels database sbml cellml = = references = =
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categorization is a type of cognition involving conceptual differentiation between characteristics of conscious experience, such as objects, events, or ideas. it involves the abstraction and differentiation of aspects of experience by sorting and distinguishing between groupings, through classification or typification on the basis of traits, features, similarities or other criteria that are universal to the group. categorization is considered one of the most fundamental cognitive abilities, and it is studied particularly by psychology and cognitive linguistics. categorization is sometimes considered synonymous with classification ( cf., classification synonyms ). categorization and classification allow humans to organize things, objects, and ideas that exist around them and simplify their understanding of the world. categorization is something that humans and other organisms do : " doing the right thing with the right kind of thing. " the activity of categorizing things can be nonverbal or verbal. for humans, both concrete objects and abstract ideas are recognized, differentiated, and understood through categorization. objects are usually categorized for some adaptive or pragmatic purposes. categorization is grounded in the features that distinguish the category's members from nonmembers. categorization is important in learning, prediction, inference, decision making, language, and many forms of organisms'interaction with their environments. = = overview = = categories are distinct collections of concrete or abstract instances ( category members ) that are considered equivalent by the cognitive system. using category knowledge requires one to access mental representations that define the core features of category members ( cognitive psychologists refer to these category - specific mental representations as concepts ). to categorization theorists, the categorization of objects is often considered using taxonomies with three hierarchical levels of abstraction. for example, a plant could be identified at a high level of abstraction by simply labeling it a flower, a medium level of abstraction by specifying that the flower is a rose, or a low level of abstraction by further specifying this particular rose as a dog rose. categories in a taxonomy are related to one another via class inclusion, with the highest level of abstraction being the most inclusive and the lowest level of abstraction being the least inclusive. the three levels of abstraction are as follows : superordinate level, genus ( e. g., flower ) - the highest and most inclusive level of abstraction. exhibits the highest degree of generality and the lowest degree of within - category similarity. basic level, species ( e. g., rose ) - the middle level of abstraction. rosch and colleagues ( 1976
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highest and most inclusive level of abstraction. exhibits the highest degree of generality and the lowest degree of within - category similarity. basic level, species ( e. g., rose ) - the middle level of abstraction. rosch and colleagues ( 1976 ) suggest the basic level to be the most cognitively efficient. basic level categories exhibit high within - category similarities and high between - category dissimilarities. furthermore, the basic level is the most inclusive level at which category exemplars share a generalized identifiable shape. adults most - often use basic level object names, and children learn basic object names first. subordinate level ( e. g., dog rose ) - the lowest level of abstraction. exhibits the highest degree of specificity and within - category similarity. = = beginning of categorization = = the essential issue in studying categorization is how conceptual differentiation between characteristics of conscious experience begins in young, inexperienced organisms. growing experimental data show evidence of differentiation between characteristics of objects and events in newborns and even in foetuses during the prenatal period. this development succeeds in organisms that only demonstrate simple reflexes ( see articles on the binding problem, cognition, cognitive development, infant cognitive development, multisensory integration, and perception ). for their nervous systems, the environment is a cacophony of sensory stimuli : electromagnetic waves, chemical interactions, and pressure fluctuations. categorization thought involves the abstraction and differentiation of aspects of experience that rely upon such power of mind as intentionality and perception. the problem is that these young organisms should already grasp the abilities of intentionality and perception to categorize the environment. intentionality and perception already require their ability to recognise objects ( or events ), i. e., to identify objects by the sensory system. this is a vicious circle : categorization needs intentionality and perception, which only appear in the categorized environment. so, the young, inexperienced organism does not have abstract thinking and cannot independently accomplish conceptual differentiation between characteristics of conscious experience if it solves the categorization problem alone. studying the origins of social cognition in child development, developmental psychologist michael tomasello developed the notion of shared intentionality to account for unaware processes during social learning after birth to explain processes in shaping intentionality. further, latvian professor igor val danilov expanded this concept to the intrauterine period by introducing a mother - fetus neurocognitive model : a hypothesis of neurophysiological processes occurring during shared intentionality. the hypothesis attempts to explain the beginning of
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igor val danilov expanded this concept to the intrauterine period by introducing a mother - fetus neurocognitive model : a hypothesis of neurophysiological processes occurring during shared intentionality. the hypothesis attempts to explain the beginning of cognitive development in organisms at different levels of bio - system complexity, from interpersonal dynamics to neuronal interactions. evidence in neuroscience supports the hypothesis. hyperscanning research studies observed inter - brain activity under conditions without communication in pairs while subjects were solving the shared cognitive problem, and they registered an increased inter - brain activity in contrast to the condition when subjects solved a similar problem alone. these data show that collaborative interaction without sensory cues can emerge in mother - child dyads, providing shared intentionality. it shows the mode to cognize at the stage without communication and abstract thinking. the significance of this knowledge is that it can reveal the new direction to study consciousness since the latter refers to awareness of internal and external existence relying on intentionality, perception and categorization of the environment. = = theories = = = = = classical view = = = the classical theory of categorization, is a term used in cognitive linguistics to denote the approach to categorization that appears in plato and aristotle and that has been highly influential and dominant in western culture, particularly in philosophy, linguistics and psychology. aristotle's categorical method of analysis was transmitted to the scholastic medieval university through porphyry's isagoge. the classical view of categories can be summarized into three assumptions : a category can be described as a list of necessary and sufficient features that its membership must have, categories are discrete in that they have clearly defined boundaries ( either an element belongs to one or not, with no possibilities in between ), and all the members of a category have the same status. ( there are no members of the category which belong more than others ). in the classical view, categories need to be clearly defined, mutually exclusive and collectively exhaustive ; this way, any entity in the given classification universe belongs unequivocally to one, and only one, of the proposed categories. the classical view of categories first appeared in the context of western philosophy in the work of plato, who, in his statesman dialogue, introduces the approach of grouping objects based on their similar properties. this approach was further explored and systematized by aristotle in his categories treatise, where he analyzes the differences between classes and objects. aristotle also applied intensively the classical categorization scheme in his approach to the classification of living
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their similar properties. this approach was further explored and systematized by aristotle in his categories treatise, where he analyzes the differences between classes and objects. aristotle also applied intensively the classical categorization scheme in his approach to the classification of living beings ( which uses the technique of applying successive narrowing questions such as " is it an animal or vegetable? ", " how many feet does it have? ", " does it have fur or feathers? ", " can it fly? "... ), establishing this way the basis for natural taxonomy. examples of the use of the classical view of categories can be found in the western philosophical works of descartes, blaise pascal, spinoza and john locke, and in the 20th century in bertrand russell, g. e. moore, the logical positivists. it has been a cornerstone of analytic philosophy and its conceptual analysis, with more recent formulations proposed in the 1990s by frank cameron jackson and christopher peacocke. at the beginning of the 20th century, the question of categories was introduced into the empirical social sciences by durkheim and mauss, whose pioneering work has been revisited in contemporary scholarship. the classical model of categorization has been used at least since the 1960s from linguists of the structural semantics paradigm, by jerrold katz and jerry fodor in 1963, which in turn have influenced its adoption also by psychologists like allan m. collins and m. ross quillian. modern versions of classical categorization theory study how the brain learns and represents categories by detecting the features that distinguish members from nonmembers. = = = prototype theory = = = the pioneering research by psychologist eleanor rosch and colleagues since 1973, introduced the prototype theory, according to which categorization can also be viewed as the process of grouping things based on prototypes. this approach has been highly influential, particularly for cognitive linguistics. it was in part based on previous insights, in particular the formulation of a category model based on family resemblance by wittgenstein ( 1953 ), and by roger brown's how shall a thing be called? ( 1958 ). prototype theory has been then adopted by cognitive linguists like george lakoff. the prototype theory is an example of a similarity - based approach to categorization, in which a stored category representation is used to assess the similarity of candidate category members. under the prototype theory, this stored representation consists of a summary representation of the category's members. this prototype stimulus can take various forms. it might be a central
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in which a stored category representation is used to assess the similarity of candidate category members. under the prototype theory, this stored representation consists of a summary representation of the category's members. this prototype stimulus can take various forms. it might be a central tendency that represents the category's average member, a modal stimulus representing either the most frequent instance or a stimulus composed of the most common category features, or, lastly, the " ideal " category member, or a caricature that emphasizes the distinct features of the category. an important consideration of this prototype representation is that it does not necessarily reflect the existence of an actual instance of the category in the world. furthermore, prototypes are highly sensitive to context. for example, while one's prototype for the category of beverages may be soda or seltzer, the context of brunch might lead them to select mimosa as a prototypical beverage. the prototype theory claims that members of a given category share a family resemblance, and categories are defined by sets of typical features ( as opposed to all members possessing necessary and sufficient features ). = = = exemplar theory = = = another instance of the similarity - based approach to categorization, the exemplar theory likewise compares the similarity of candidate category members to stored memory representations. under the exemplar theory, all known instances of a category are stored in memory as exemplars. when evaluating an unfamiliar entity's category membership, exemplars from potentially relevant categories are retrieved from memory, and the entity's similarity to those exemplars is summed to formulate a categorization decision. medin and schaffer's ( 1978 ) context model employs a nearest neighbor approach which, rather than summing an entity's similarities to relevant exemplars, multiplies them to provide weighted similarities that reflect the entity's proximity to relevant exemplars. this effectively biases categorization decisions towards exemplars most similar to the entity to be categorized. = = = conceptual clustering = = = conceptual clustering is a machine learning paradigm for unsupervised classification that was defined by ryszard s. michalski in 1980. it is a modern variation of the classical approach of categorization, and derives from attempts to explain how knowledge is represented. in this approach, classes ( clusters or entities ) are generated by first formulating their conceptual descriptions and then classifying the entities according to the descriptions. conceptual clustering developed
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of categorization, and derives from attempts to explain how knowledge is represented. in this approach, classes ( clusters or entities ) are generated by first formulating their conceptual descriptions and then classifying the entities according to the descriptions. conceptual clustering developed mainly during the 1980s, as a machine paradigm for unsupervised learning. it is distinguished from ordinary data clustering by generating a concept description for each generated category. conceptual clustering is closely related to fuzzy set theory, in which objects may belong to one or more groups, in varying degrees of fitness. a cognitive approach accepts that natural categories are graded ( they tend to be fuzzy at their boundaries ) and inconsistent in the status of their constituent members. the idea of necessary and sufficient conditions is almost never met in categories of naturally occurring things. = = category learning = = while an exhaustive discussion of category learning is beyond the scope of this article, a brief overview of category learning and its associated theories is useful in understanding formal models of categorization. if categorization research investigates how categories are maintained and used, the field of category learning seeks to understand how categories are acquired in the first place. to accomplish this, researchers often employ novel categories of arbitrary objects ( e. g., dot matrices ) to ensure that participants are entirely unfamiliar with the stimuli. category learning researchers have generally focused on two distinct forms of category learning. classification learning tasks participants with predicting category labels for a stimulus based on its provided features. classification learning is centered around learning between - category information and the diagnostic features of categories. in contrast, inference learning tasks participants with inferring the presence / value of a category feature based on a provided category label and / or the presence of other category features. inference learning is centered on learning within - category information and the category's prototypical features. category learning tasks can generally be divided into two categories, supervised and unsupervised learning. supervised learning tasks provide learners with category labels. learners then use information extracted from labeled example categories to classify stimuli into the appropriate category, which may involve the abstraction of a rule or concept relating observed object features to category labels. unsupervised learning tasks do not provide learners with category labels. learners must therefore recognize inherent structures in a data set and group stimuli together by similarity into classes. unsupervised learning is thus a process of generating a classification structure. tasks used to study category learning take various forms : rule - based tasks present categories that participants can learn through explicit reasoning processes. in these kinds of tasks,
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into classes. unsupervised learning is thus a process of generating a classification structure. tasks used to study category learning take various forms : rule - based tasks present categories that participants can learn through explicit reasoning processes. in these kinds of tasks, classification of stimuli is accomplished via the use of an acquired rule ( i. e., if stimulus is large on dimension x, respond a ). information - integration tasks require learners to synthesize perceptual information from multiple stimulus dimensions prior to making categorization decisions. unlike rule - based tasks, information - integration tasks do not afford rules that are easily articulable. reading an x - ray and trying to determine if a tumor is present can be thought of as a real - world instantiation of an information - integration task. prototype distortion tasks require learners to generate a prototype for a category. candidate exemplars for the category are then produced by randomly manipulating the features of the prototype, which learners must classify as either belonging to the category or not. = = = category learning theories = = = category learning researchers have proposed various theories for how humans learn categories. prevailing theories of category learning include the prototype theory, the exemplar theory, and the decision bound theory. the prototype theory suggests that to learn a category, one must learn the category's prototype. subsequent categorization of novel stimuli is then accomplished by selecting the category with the most similar prototype. the exemplar theory suggests that to learn a category, one must learn about the exemplars that belong to that category. subsequent categorization of a novel stimulus is then accomplished by computing its similarity to the known exemplars of potentially relevant categories and selecting the category that contains the most similar exemplars. decision bound theory suggests that to learn a category, one must either learn the regions of a stimulus space associated with particular responses or the boundaries ( the decision bounds ) that divide these response regions. categorization of a novel stimulus is then accomplished by determining which response region it is contained within. = = formal models = = computational models of categorization have been developed to test theories about how humans represent and use category information. to accomplish this, categorization models can be fit to experimental data to see how well the predictions afforded by the model line up with human performance. based on the model's success at explaining the data, theorists are able to draw conclusions about the accuracy of their theories and their theory's relevance to human category representations. to effectively capture
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the predictions afforded by the model line up with human performance. based on the model's success at explaining the data, theorists are able to draw conclusions about the accuracy of their theories and their theory's relevance to human category representations. to effectively capture how humans represent and use category information, categorization models generally operate under variations of the same three basic assumptions. first, the model must make some kind of assumption about the internal representation of the stimulus ( e. g., representing the perception of a stimulus as a point in a multi - dimensional space ). second, the model must make an assumption about the specific information that needs to be accessed in order to formulate a response ( e. g., exemplar models require the collection of all available exemplars for each category ). third, the model must make an assumption about how a response is selected given the available information. though all categorization models make these three assumptions, they distinguish themselves by the ways in which they represent and transform an input into a response representation. the internal knowledge structures of various categorization models reflect the specific representation ( s ) they use to perform these transformations. typical representations employed by models include exemplars, prototypes, and rules. exemplar models store all distinct instances of stimuli with their corresponding category labels in memory. categorization of subsequent stimuli is determined by the stimulus'collective similarity to all known exemplars. prototype models store a summary representation of all instances in a category. categorization of subsequent stimuli is determined by selecting the category whose prototype is most similar to the stimulus. rule - based models define categories by storing summary lists of the necessary and sufficient features required for category membership. boundary models can be considered as atypical rule models, as they do not define categories based on their content. rather, boundary models define the edges ( boundaries ) between categories, which subsequently serve as determinants for how a stimulus gets categorized. = = examples = = = = = prototype models = = = weighted features prototype model an early instantiation of the prototype model was produced by reed in the early 1970s. reed ( 1972 ) conducted a series of experiments to compare the performance of 18 models on explaining data from a categorization task that required participants to sort faces into one of two categories. results suggested that the prevailing model was the weighted features prototype model, which belonged to the family of average distance models. unlike traditional average distance models, however, this model differentially weighted the most distinguishing features of the two
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faces into one of two categories. results suggested that the prevailing model was the weighted features prototype model, which belonged to the family of average distance models. unlike traditional average distance models, however, this model differentially weighted the most distinguishing features of the two categories. given this model's performance, reed ( 1972 ) concluded that the strategy participants used during the face categorization task was to construct prototype representations for each of the two categories of faces and categorize test patterns into the category associated with the most similar prototype. furthermore, results suggested that similarity was determined by each categories most discriminating features. = = = exemplar models = = = generalized context model medin and schaffer's ( 1978 ) context model was expanded upon by nosofsky ( 1986 ) in the mid - 1980s, resulting in the production of the generalized context model ( gcm ). the gcm is an exemplar model that stores exemplars of stimuli as exhaustive combinations of the features associated with each exemplar. by storing these combinations, the model establishes contexts for the features of each exemplar, which are defined by all other features with which that feature co - occurs. the gcm computes the similarity of an exemplar and a stimulus in two steps. first, the gcm computes the psychological distance between the exemplar and the stimulus. this is accomplished by summing the absolute values of the dimensional difference between the exemplar and the stimulus. for example, suppose an exemplar has a value of 18 on dimension x and the stimulus has a value of 42 on dimension x ; the resulting dimensional difference would be 24. once psychological distance has been evaluated, an exponential decay function determines the similarity of the exemplar and the stimulus, where a distance of 0 results in a similarity of 1 ( which begins to decrease exponentially as distance increases ). categorical responses are then generated by evaluating the similarity of the stimulus to each category's exemplars, where each exemplar provides a " vote " to their respective categories that varies in strength based on the exemplar's similarity to the stimulus and the strength of the exemplar's association with the category. this effectively assigns each category a selection probability that is determined by the proportion of votes it receives, which can then be fit to data. = = = rule - based models = = = rulex ( rule - plus - exception ) model while
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category. this effectively assigns each category a selection probability that is determined by the proportion of votes it receives, which can then be fit to data. = = = rule - based models = = = rulex ( rule - plus - exception ) model while simple logical rules are ineffective at learning poorly defined category structures, some proponents of the rule - based theory of categorization suggest that an imperfect rule can be used to learn such category structures if exceptions to that rule are also stored and considered. to formalize this proposal, nosofsky and colleagues ( 1994 ) designed the rulex model. the rulex model attempts to form a decision tree composed of sequential tests of an object's attribute values. categorization of the object is then determined by the outcome of these sequential tests. the rulex model searches for rules in the following ways : exact search for a rule that uses a single attribute to discriminate between classes without error. imperfect search for a rule that uses a single attribute to discriminate between classes with few errors conjunctive search for a rule that uses multiple attributes to discriminate between classes with few errors. exception search for exceptions to the rule. the method that rulex uses to perform these searches is as follows : first, rulex attempts an exact search. if successful, then rulex will continuously apply that rule until misclassification occurs. if the exact search fails to identify a rule, either an imperfect or conjunctive search will begin. a sufficient, though imperfect, rule acquired during one of these search phases will become permanently implemented and the rulex model will then begin to search for exceptions. if no rule is acquired, then the model will attempt the search it did not perform in the previous phase. if successful, rulex will permanently implement the rule and then begin an exception search. if none of the previous search methods are successful rulex will default to only searching for exceptions, despite lacking an associated rule, which equates to acquiring a random rule. = = = hybrid models = = = sustain ( supervised and unsupervised stratified adaptive incremental network ) it is often the case that learned category representations vary depending on the learner's goals, as well as how categories are used during learning. thus, some categorization researchers suggest that a proper model of categorization needs to be able to account for the variability present in the learner's goals, tasks, and strategies. this proposal was realized by love and colleagues ( 2004 ) through the creation of sustain
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##rization researchers suggest that a proper model of categorization needs to be able to account for the variability present in the learner's goals, tasks, and strategies. this proposal was realized by love and colleagues ( 2004 ) through the creation of sustain, a flexible clustering model capable of accommodating both simple and complex categorization problems through incremental adaptation to the specifics of problems. in practice, the sustain model first converts a stimulus'perceptual information into features that are organized along a set of dimensions. the representational space that encompasses these dimensions is then distorted ( e. g., stretched or shrunk ) to reflect the importance of each feature based on inputs from an attentional mechanism. a set of clusters ( specific instances grouped by similarity ) associated with distinct categories then compete to respond to the stimulus, with the stimulus being subsequently assigned to the cluster whose representational space is closest to the stimulus '. the unknown stimulus dimension value ( e. g., category label ) is then predicted by the winning cluster, which, in turn, informs the categorization decision. the flexibility of the sustain model is realized through its ability to employ both supervised and unsupervised learning at the cluster level. if sustain incorrectly predicts a stimulus as belonging to a particular cluster, corrective feedback ( i. e., supervised learning ) would signal sustain to recruit an additional cluster that represents the misclassified stimulus. therefore, subsequent exposures to the stimulus ( or a similar alternative ) would be assigned to the correct cluster. sustain will also employ unsupervised learning to recruit an additional cluster if the similarity between the stimulus and the closest cluster does not exceed a threshold, as the model recognizes the weak predictive utility that would result from such a cluster assignment. sustain also exhibits flexibility in how it solves both simple and complex categorization problems. outright, the internal representation of sustain contains only a single cluster, thus biasing the model towards simple solutions. as problems become increasingly complex ( e. g., requiring solutions consisting of multiple stimulus dimensions ), additional clusters are incrementally recruited so sustain can handle the rise in complexity. = = social categorization = = social categorization consists of putting human beings into groups in order to identify them based on different criteria. categorization is a process studied by scholars in cognitive science but can also be studied as a social activity. social categorization is different from the categorization of other things because it implies that people
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order to identify them based on different criteria. categorization is a process studied by scholars in cognitive science but can also be studied as a social activity. social categorization is different from the categorization of other things because it implies that people create categories for themselves and others as human beings. groups can be created based on ethnicity, country of origin, religion, sexual identity, social privileges, economic privileges, etc. various ways to sort people exist according to one's schemas. people belong to various social groups because of their ethnicity, religion, or age. social categories based on age, race, and gender are used by people when they encounter a new person. because some of these categories refer to physical traits, they are often used automatically when people do not know each other. these categories are not objective and depend on how people see the world around them. they allow people to identify themselves with similar people, and to identify people who are different. they are useful in one's identity formation with the people around them. one can build their own identity by identifying themselves in a group or by rejecting another group. social categorization is similar to other types of categorization since it aims at simplifying the understanding of people. however, creating social categories implies that people will position themselves in relation to other groups. a hierarchy in group relations can appear as a result of social categorization. scholars argue that the categorization process starts at a young age when children start to learn about the world and the people around them. children learn how to know people according to categories based on similarities and differences. social categories made by adults also impact their understanding of the world. they learn about social groups by hearing generalities about these groups from their parents. they can then develop prejudices about people as a result of these generalities. another aspect of social categorization is mentioned by stephen reicher and nick hopkins and is related to political domination. they argue that political leaders use social categories to influence political debates. = = = negative aspects = = = the activity of sorting people according to subjective or objective criteria can be seen as a negative process because of its tendency to lead to violence from a group to another. indeed, similarities gather people who share common traits but differences between groups can lead to tensions and then the use of violence between those groups. the creation of social groups by people is responsible of a hierarchization of relations between groups. these hierarchical relations participate in the promotion of stereotypes about people and groups, sometimes based on
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lead to tensions and then the use of violence between those groups. the creation of social groups by people is responsible of a hierarchization of relations between groups. these hierarchical relations participate in the promotion of stereotypes about people and groups, sometimes based on subjective criteria. social categories can encourage people to associate stereotypes to groups of people. associating stereotypes to a group, and to people who belong to this group, can lead to forms of discrimination towards people of this group. the perception of a group and the stereotypes associated with it have an impact on social relations and activities. some social categories have more weight than others in society. for instance, in history and still today, the category of " race " is one of the first categories used to sort people. however, only a few categories of race are commonly used such as " black ", " white ", " asian " etc. it participates in the reduction of the multitude of ethnicities to a few categories based mostly on people's skin color. the process of sorting people creates a vision of the other as'different ', leading to the dehumanization of people. scholars talk about intergroup relations with the concept of social identity theory developed by h. tajfel. indeed, in history, many examples of social categorization have led to forms of domination or violence from a dominant group to a dominated group. periods of colonisation are examples of times when people from a group chose to dominate and control other people belonging to other groups because they considered them as inferior. racism, discrimination and violence are consequences of social categorization and can occur because of it. when people see others as different, they tend to develop hierarchical relation with other groups. = = miscategorization = = there cannot be categorization without the possibility of miscategorization. to do " the right thing with the right kind of thing. ", there has to be both a right and a wrong thing to do. not only does a category of which " everything " is a member lead logically to the russell paradox ( " is it or is it not a member of itself? " ), but without the possibility of error, there is no way to detect or define what distinguishes category members from nonmembers. an example of the absence of nonmembers is the problem of the poverty of the stimulus in language learning by the child : children learning the language do not hear or make errors in the rules of universal grammar ( ug ). hence they never get corrected for
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the absence of nonmembers is the problem of the poverty of the stimulus in language learning by the child : children learning the language do not hear or make errors in the rules of universal grammar ( ug ). hence they never get corrected for errors in ug. yet children's speech obeys the rules of ug, and speakers can immediately detect that something is wrong if a linguist generates ( deliberately ) an utterance that violates ug. hence speakers can categorize what is ug - compliant and ug - noncompliant. linguists have concluded from this that the rules of ug must be somehow encoded innately in the human brain. ordinary categories, however, such as " dogs, " have abundant examples of nonmembers ( cats, for example ). so it is possible to learn, by trial and error, with error - correction, to detect and define what distinguishes dogs from non - dogs, and hence to correctly categorize them. this kind of learning, called reinforcement learning in the behavioral literature and supervised learning in the computational literature, is fundamentally dependent on the possibility of error, and error - correction. miscategorization — examples of nonmembers of the category — must always exist, not only to make the category learnable, but for the category to exist and be definable at all. = = see also = = categorical perception characterization ( mathematics ) classification ( general theory ) knolling library classification multi - label classification pattern recognition shared intentionality statistical classification symbol grounding problem = = references = = = = external links = = to cognize is to categorize : cognition is categorization archived 2012 - 02 - 08 at the wayback machine wikipedia categories visualizer interdisciplinary introduction to categorization : interview with dvora yanov ( political sciences ), amie thomasson ( philosophy ) and thomas serre ( artificial intelligence ) " category ". encyclopædia britannica. vol. 5 ( 11th ed. ). 1911. pp. 508 – 510.
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in computer science and quantum physics, the church – turing – deutsch principle ( ctd principle ) is a stronger, physical form of the church – turing thesis formulated by david deutsch in 1985. the principle states that a universal computing device can simulate every physical process. = = history = = the principle was stated by deutsch in 1985 with respect to finitary machines and processes. he observed that classical physics, which makes use of the concept of real numbers, cannot be simulated by a turing machine, which can only represent computable reals. deutsch proposed that quantum computers may actually obey the ctd principle, assuming that the laws of quantum physics can completely describe every physical process. an earlier version of this thesis for classical computers was stated by alan turing's friend and student robin gandy in 1980. a similar thesis was stated by michael freedman in an early review of topological quantum computing with alexei kitaev, michael j. larsen, and zhenghan wang, known as the freedman - church - turing thesis : " all'reasonable'computational models which add the resources of quantum mechanics ( or quantum field theory ) to classical computation yield ( efficiently ) inter - simulable classes : there is one quantum theory of computation. " this thesis differs from the church - turing - deutsch thesis insofar as it is a statement about computational complexity, and not computability. = = see also = = bekenstein bound digital physics holographic principle quantum complexity theory laplace's demon = = notes = = = = references = = deutsch, d. ( 1985 ). " quantum theory, the church – turing principle and the universal quantum computer " ( pdf ). proceedings of the royal society. 400 ( 1818 ) : 97 – 117. bibcode : 1985rspsa. 400... 97d. citeseerx 10. 1. 1. 41. 2382. doi : 10. 1098 / rspa. 1985. 0070. s2cid 1438116. archived from the original ( pdf ) on 2016 - 03 - 09. retrieved 2011 - 08 - 17. = = further reading = = deutsch, d. ( 1997 ). " 6 : universality and the limits of computation ". the fabric of reality. new york : allan lane. isbn 978 - 0 - 14 - 027541 - 4. christopher g. timpson quantum computers : the church - turing hypothesis versus the turing principle in
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and the limits of computation ". the fabric of reality. new york : allan lane. isbn 978 - 0 - 14 - 027541 - 4. christopher g. timpson quantum computers : the church - turing hypothesis versus the turing principle in christof teuscher, douglas hofstadter ( eds. ) alan turing : life and legacy of a great thinker, springer, 2004, isbn 3 - 540 - 20020 - 7, pp. 213 – 240 = = external links = = nielsen, michael ( 2004 - 04 - 16 ). " interesting problems : the church – turing – deutsch principle ".
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in linguistics, a subsective modifier is an expression which modifies another by delivering a subset of its denotation. for instance, the english adjective " skilled " is subsective since being a skilled surgeon entails being a surgeon. by contrast, the english adjective " alleged " is non - subsective since an " alleged spy " need not be an actual spy. [ [ skilled surgeon ] ] ⊆ [ [ surgeon ] ] { \ displaystyle [ \! [ { \ text { skilled surgeon } } ] \! ] \ subseteq [ \! [ { \ text { surgeon } } ] \! ] } a modifier can be subsective without being intersective. for instance, calling someone an " old friend " entails that they are a friend but does not entail that they are elderly. the term " subsective " is most often applied to modifiers which are not intersective and non - intersectivity is sometimes treated as part of its definition. there is no standard analysis for the semantics of ( non - intersective ) subsective modifiers. early work such as montague ( 1970 ) took subsective adjectives as evidence that adjectives do not denote properties which compose intersectively but rather functions which take and return a property which may or may not make an intersective semantic contribution. however, subsequent work has shown that variants of the property - based analysis can in fact account for the data. for instance, vague predicates often pass standard tests for nonintersectivity, e. g. " neutrons are big subatomic particles " doesn't entail that neutrons are actually big but have in fact be analyzed as intersective using degree semantics. current work tends to assume that the phenomenon of subsectivity is not a natural class. = = adverbial readings = = subsectivity can arise when an adjective receives an adverbial reading. for instance, the subsective modifiers in the examples below do not express intrinsic qualities of the subject but rather the manner in which the subject typically performs a particular action. ( without the parenthetical, these examples would be ambiguous between an adverbial reading and a garden variety intersective reading. ) oleg is a beautiful dancer ( even though he himself is ugly ). vanessa is a meticulous experimentalist ( even though she's a slob ). shaggy is a fierce advocate of gluttony ( even though he's a coward ). examples of
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( even though he himself is ugly ). vanessa is a meticulous experimentalist ( even though she's a slob ). shaggy is a fierce advocate of gluttony ( even though he's a coward ). examples of this sort have been analyzed within a davidsonian semantics as modifying an event variable introduced by the noun. in this analysis, an agentive noun such as " dancer " is formed by applying a generic quantifier gen to a predicate ) which is true of dancing events. the quantifier gen provides a habitual - like meaning, taking a predicate of events and returning a predicate ) which is true of an individual if they are the agent of the typical such event. [ [ dance ] ] = { e | e is a dancing event } { \ displaystyle [ \! [ { \ text { dance } } ] \! ] = \ { e \, | \, e { \ text { is a dancing event } } \ } } [ [ { \ displaystyle [ \! [ } gen dance ] ] = { x | x is the agent of all contextually relevant dancing events } { \ displaystyle \! { \ text { dance } } ] \! ] = \ { x \, | \, x { \ text { is the agent of all contextually relevant dancing events } } \ } } in this analysis, adjectives such as " beautiful ", " meticulous ", and " fierce " can denote properties either of events or of individuals. [ [ beautiful 1 ] ] = { e | e is beautiful } { \ displaystyle [ \! [ { \ text { beautiful } } _ { 1 } ] \! ] = \ { e \, | \, e { \ text { is beautiful } } \ } } [ [ beautiful 2 ] ] = { x | x is beautiful } { \ displaystyle [ \! [ { \ text { beautiful } } _ { 2 } ] \! ] = \ { x \, | \, x { \ text { is beautiful } } \ } } when the adjective takes scope above gen it must be interpreted as a predicate of individuals ; when it scopes below gen it must be interpreted as a predicate of events. in this latter case, the denotation of the adjective can still compose intersectively. [ [ beautiful 1 dance ] ] = { e | e is beautiful } ∩ { e | e is a dancing event } { \ displays
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##dicate of events. in this latter case, the denotation of the adjective can still compose intersectively. [ [ beautiful 1 dance ] ] = { e | e is beautiful } ∩ { e | e is a dancing event } { \ displaystyle [ \! [ { \ text { beautiful } } _ { 1 } { \ text { dance } } ] \! ] = \ { e \, | \, e { \ text { is beautiful } } \ } \ cap \ { e \, | \, e { \ text { is a dancing event } } \ } } thus, on this analysis, to say that oleg is a beautiful dancer is to say that he is the typical agent of typical beautiful dancing events. this is technically an intersective reading since it is derived by intersecting the modifier with the noun. however, it does not look like a typical intersective meaning since it does not require that oleg himself be an element of that intersection — rather that he be the agent of certain events in that intersection. = = see also = = adjective grammatical modifier intersective modifier privative adjective = = references = =
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in quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters ( such as atoms ), between a state containing few electromagnetic excitations ( as in the electromagnetic vacuum ) and a superradiant state with many electromagnetic excitations trapped inside the emitters. the superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters. the superradiant phase transition was originally predicted by the dicke model of superradiance, which assumes that atoms have only two energetic levels and that these interact with only one mode of the electromagnetic field. the phase transition occurs when the strength of the interaction between the atoms and the field is greater than the energy of the non - interacting part of the system. ( this is similar to the case of superconductivity in ferromagnetism, which leads to the dynamic interaction between ferromagnetic atoms and the spontaneous ordering of excitations below the critical temperature. ) the collective lamb shift, relating to the system of atoms interacting with the vacuum fluctuations, becomes comparable to the energies of atoms alone, and the vacuum fluctuations cause the spontaneous self - excitation of matter. the transition can be readily understood by the use of the holstein - primakoff transformation applied to a two - level atom. as a result of this transformation, the atoms become lorentz harmonic oscillators with frequencies equal to the difference between the energy levels. the whole system then simplifies to a system of interacting harmonic oscillators of atoms, and the field known as hopfield dielectric which further predicts in the normal state polarons for photons or polaritons. if the interaction with the field is so strong that the system collapses in the harmonic approximation and complex polariton frequencies ( soft modes ) appear, then the physical system with nonlinear terms of the higher order becomes the system with the mexican hat - like potential, and will undergo ferroelectric - like phase transition. in this model, the system is mathematically equivalent for one mode of excitation to the trojan wave packet, when the circularly polarized field intensity corresponds to the electromagnetic coupling constant. above the critical value, it changes to the unstable motion of the ionization. the superradiant phase transition was the subject of a wide discussion as to whether or not it is only a result of the simplified model of the matter - field interaction ; and if it can occur for the
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the unstable motion of the ionization. the superradiant phase transition was the subject of a wide discussion as to whether or not it is only a result of the simplified model of the matter - field interaction ; and if it can occur for the real physical parameters of physical systems ( a no - go theorem ). however, both the original derivation and the later corrections leading to nonexistence of the transition – due to thomas – reiche – kuhn sum rule canceling for the harmonic oscillator the needed inequality to impossible negativity of the interaction – were based on the assumption that the quantum field operators are commuting numbers, and the atoms do not interact with the static coulomb forces. this is generally not true like in case of bohr – van leeuwen theorem and the classical non - existence of landau diamagnetism. the negating results were also the consequence of using the simple quantum optics models of the electromagnetic field - matter interaction but not the more realistic condensed matter models like for example the superconductivity model of the bcs but with the phonons replaced by photons to first obtain the collective polaritons. the return of the transition basically occurs because the inter - atom dipole - dipole or generally the electron - electron coulomb interactions are never negligible in the condensed and even more in the superradiant matter density regime and the power - zienau unitary transformation eliminating the quantum vector potential in the minimum - coupling hamiltonian transforms the hamiltonian exactly to the form used when it was discovered and without the square of the vector potential which was later claimed to prevent it. alternatively within the full quantum mechanics including the electromagnetic field the generalized bohr – van leeuwen theorem does not work and the electromagnetic interactions cannot be eliminated while they only change the p ⋅ a { \ displaystyle \ mathbf { p } \ cdot \ mathbf { a } } vector potential coupling to the electric field x ⋅ e { \ displaystyle \ mathbf { x } \ cdot \ mathbf { e } } coupling and alter the effective electrostatic interactions. it can be observed in model systems like bose – einstein condensates and artificial atoms. = = theory = = = = = criticality of linearized jaynes - cummings model = = = a superradiant phase transition is formally predicted by the critical behavior of the resonant jaynes - cummings model, describing the interaction of only one atom with one mode of the electromagnetic field.
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linearized jaynes - cummings model = = = a superradiant phase transition is formally predicted by the critical behavior of the resonant jaynes - cummings model, describing the interaction of only one atom with one mode of the electromagnetic field. starting from the exact hamiltonian of the jaynes - cummings model at resonance h ^ jc = ω a ^ † a ^ + ω σ ^ z 2 + ω 2 ( a ^ σ ^ + + a ^ † σ ^ − + a ^ σ ^ − + a ^ † σ ^ + ), { \ displaystyle { \ hat { h } } _ { \ text { jc } } = \ hbar \ omega { \ hat { a } } ^ { \ dagger } { \ hat { a } } + \ hbar \ omega { \ frac { { \ hat { \ sigma } } _ { z } } { 2 } } + { \ frac { \ hbar \ omega } { 2 } } \ left ( { \ hat { a } } { \ hat { \ sigma } } _ { + } + { \ hat { a } } ^ { \ dagger } { \ hat { \ sigma } } _ { - } + { \ hat { a } } { \ hat { \ sigma } } _ { - } + { \ hat { a } } ^ { \ dagger } { \ hat { \ sigma } } _ { + } \ right ), } applying the holstein - primakoff transformation for two spin levels, replacing the spin raising and lowering operators by those for the harmonic oscillators σ ^ − ≈ b ^ { \ displaystyle { \ hat { \ sigma } } _ { - } \ approx { \ hat { b } } } σ ^ + ≈ b ^ † { \ displaystyle { \ hat { \ sigma } } _ { + } \ approx { \ hat { b } } ^ { \ dagger } } σ ^ z ≈ 2 b ^ † b ^ { \ displaystyle { \ hat { \ sigma } } _ { z } \ approx 2 { \ hat { b } } ^ { \ dagger } { \ hat { b } } } one gets the hamiltonian of two coupled harmonic - oscillators : h ^ jc = ω a ^ † a ^ + ω b ^ † b ^ + ω 2 ( a ^ b ^ † + a ^ † b ^ + a ^ b ^ + a ^ † b ^ † ), {
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h ^ jc = ω a ^ † a ^ + ω b ^ † b ^ + ω 2 ( a ^ b ^ † + a ^ † b ^ + a ^ b ^ + a ^ † b ^ † ), { \ displaystyle { \ hat { h } } _ { \ text { jc } } = \ hbar \ omega { \ hat { a } } ^ { \ dagger } { \ hat { a } } + \ hbar \ omega { \ hat { b } } ^ { \ dagger } { \ hat { b } } + { \ frac { \ hbar \ omega } { 2 } } \ left ( { \ hat { a } } { \ hat { b } } ^ { \ dagger } + { \ hat { a } } ^ { \ dagger } { \ hat { b } } + { \ hat { a } } { \ hat { b } } + { \ hat { a } } ^ { \ dagger } { \ hat { b } } ^ { \ dagger } \ right ), } which readily can be diagonalized. postulating its normal form h ^ jc = ω + a + ^ † a + ^ + ω − a − ^ † a − ^ + c { \ displaystyle { \ hat { h } } _ { \ text { jc } } = \ omega _ { + } { \ hat { a _ { + } } } ^ { \ dagger } { \ hat { a _ { + } } } + \ omega _ { - } { \ hat { a _ { - } } } ^ { \ dagger } { \ hat { a _ { - } } } + c } where a ± ^ = c ± 1 a ^ + c ± 2 a ^ † + c ± 3 b ^ + c ± 4 b ^ † { \ displaystyle { \ hat { a _ { \ pm } } } = c _ { \ pm 1 } { \ hat { a } } + c _ { \ pm 2 } { \ hat { a } } ^ { \ dagger } + c _ { \ pm 3 } { \ hat { b } } + c _ { \ pm 4 } { \ hat { b } } ^ { \ dagger } } one gets the eigenvalue equation [ a ± ^, h ^ jc ] = ω ± a { \ displaystyle [ { \ hat { a _ { \ pm } } }, { \ hat { h } } _
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dagger } } one gets the eigenvalue equation [ a ± ^, h ^ jc ] = ω ± a { \ displaystyle [ { \ hat { a _ { \ pm } } }, { \ hat { h } } _ { \ text { jc } } ] = \ omega _ { \ pm } a } with the solutions ω ± = ω 1 ± ω ω { \ displaystyle \ omega _ { \ pm } = \ omega { \ sqrt { 1 \ pm { \ frac { \ omega } { \ omega } } } } } the system collapses when one of the frequencies becomes imaginary, i. e. when ω > ω { \ displaystyle \ omega > \ omega } or when the atom - field coupling is stronger than the frequency of the mode and atom oscillators. while there are physically higher terms in the true system, the system in this regime will therefore undergo the phase transition. = = = criticality of jaynes - cummings model = = = the simplified hamiltonian of the jaynes - cummings model, neglecting the counter - rotating terms, is h ^ jc = ω a ^ † a ^ + ω σ ^ z 2 + ω 2 ( a ^ σ ^ + + a ^ † σ ^ − ), { \ displaystyle { \ hat { h } } _ { \ text { jc } } = \ hbar \ omega { \ hat { a } } ^ { \ dagger } { \ hat { a } } + \ hbar \ omega { \ frac { { \ hat { \ sigma } } _ { z } } { 2 } } + { \ frac { \ hbar \ omega } { 2 } } \ left ( { \ hat { a } } { \ hat { \ sigma } } _ { + } + { \ hat { a } } ^ { \ dagger } { \ hat { \ sigma } } _ { - } \ right ), } and the energies for the case of zero detuning are e ± ( n ) = ω ( n + 1 2 ) ± 1 2 ω ( n ), { \ displaystyle e _ { \ pm } ( n ) = \ hbar \ omega \ left ( n + { \ frac { 1 } { 2 } } \ right ) \ pm { \ frac { 1 } { 2 } } \ hbar \ omega ( n ), } ω ( n ) = ω n + 1 { \ displaystyle
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{ \ frac { 1 } { 2 } } \ right ) \ pm { \ frac { 1 } { 2 } } \ hbar \ omega ( n ), } ω ( n ) = ω n + 1 { \ displaystyle \ omega ( n ) = \ omega { \ sqrt { n + 1 } } } where ω { \ displaystyle \ omega } is the rabi frequency. one can approximately calculate the canonical partition function z = ±, n e − β e ± ( n ) ≈ ± e − β e ± ( n ) d n = e φ ( n ) d n { \ displaystyle z = \ sum _ { \ pm, n } \ mathrm { e } ^ { - \ beta e _ { \ pm } ( n ) } \ approx \ sum _ { \ pm } \ int \ mathrm { e } ^ { - \ beta e _ { \ pm } ( n ) } dn = \ int \ mathrm { e } ^ { \ phi ( n ) } dn }, where the discrete sum was replaced by the integral. the normal approach is that the latter integral is calculated by the gaussian approximation around the maximum of the exponent : ∂ φ ( n ) ∂ n = 0 { \ displaystyle { \ frac { \ partial \ phi ( n ) } { \ partial n } } = 0 } φ ( n ) = − β ω ( n + 1 2 ) + log 2 cosh ω ( n ) β 2 { \ displaystyle \ phi ( n ) = - \ beta \ hbar \ omega \ left ( n + { \ frac { 1 } { 2 } } \ right ) + \ log 2 \ cosh { \ frac { \ hbar \ omega ( n ) \ beta } { 2 } } } this leads to the critical equation tanh ω ( n ) β 2 = 4 ω ω n + 1 { \ displaystyle \ tanh { \ frac { \ hbar \ omega ( n ) \ beta } { 2 } } = 4 { \ frac { \ omega } { \ omega } } { \ sqrt { n + 1 } } } this has the solution only if ω > 4 ω { \ displaystyle \ omega > 4 \ omega } which means that the normal, and the superradiant phase, exist only if the field - atom coupling is significantly stronger than the energy difference between the atom levels. when
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ω > 4 ω { \ displaystyle \ omega > 4 \ omega } which means that the normal, and the superradiant phase, exist only if the field - atom coupling is significantly stronger than the energy difference between the atom levels. when the condition is fulfilled, the equation gives the solution for the order parameter n { \ displaystyle n } depending on the inverse of the temperature 1 / β { \ displaystyle 1 / \ beta }, which means non - vanishing ordered field mode. similar considerations can be done in true thermodynamic limit of the infinite number of atoms. = = = instability of the classical electrostatic model = = = the better insight on the nature of the superradiant phase transition as well on the physical value of the critical parameter which must be exceeded in order for the transition to occur may be obtained by studying the classical stability of the system of the charged classical harmonic oscillators in the 3d space interacting only with the electrostatic repulsive forces for example between electrons in the locally harmonic oscillator potential. despite the original model of the superradiance the quantum electromagnetic field is totally neglected here. the oscillators may be assumed to be placed for example on the cubic lattice with the lattice constant a { \ displaystyle a } in the analogy to the crystal system of the condensed matter. the worse scenario of the defect of the absence of the two out - of - the - plane motion - stabilizing electrons from the 6 - th nearest neighbors of a chosen electron is assumed while the four nearest electrons are first assumed to be rigid in space and producing the anti - harmonic potential in the direction perpendicular to the plane of the all five electrons. the condition of the instability of motion of the chosen electron is that the net potential being the superposition of the harmonic oscillator potential and the quadratically expanded coulomb potential from the four electrons is negative i. e. m ω 2 2 − 1 2 × 4 × e 2 4 π 0 1 a 3 < 0 { \ displaystyle { \ frac { m \ omega ^ { 2 } } { 2 } } - { \ frac { 1 } { 2 } } \ times 4 \ times { \ frac { e ^ { 2 } } { 4 \ pi \ epsilon _ { 0 } } } { \ frac { 1 } { a ^ { 3 } } } < 0 } or e 2 π 0 m ω 2 1 a 3 > 1 { \ displaystyle { \
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{ 4 \ pi \ epsilon _ { 0 } } } { \ frac { 1 } { a ^ { 3 } } } < 0 } or e 2 π 0 m ω 2 1 a 3 > 1 { \ displaystyle { \ frac { e ^ { 2 } } { \ pi \ epsilon _ { 0 } m \ omega ^ { 2 } } } { \ frac { 1 } { a ^ { 3 } } } > 1 } making it artificially quantum by multiplying the numerator and the denominator of the fraction by the { \ displaystyle \ hbar } one obtains the condition 2 π | d 12 | 2 e 12 0 ( n v ) > 1 { \ displaystyle { \ frac { 2 } { \ pi } } { \ frac { | d _ { 12 } | ^ { 2 } } { e _ { 12 } \ epsilon _ { 0 } } } \ left ( { \ frac { n } { v } } \ right ) > 1 } where | d 12 | 2 = e 2 2 m ω { \ displaystyle | d _ { 12 } | ^ { 2 } = { \ frac { e ^ { 2 } \ hbar } { 2m \ omega } } } is the square of the dipole transition strength between the ground state and the first excited state of the quantum harmonic oscillator, e 12 = ω { \ displaystyle e _ { 12 } = \ hbar \ omega } is the energy gap between consecutive levels and it is also noticed that 1 a 3 = n v { \ displaystyle { \ frac { 1 } { a ^ { 3 } } } = { \ frac { n } { v } } } is the spatial density of the oscillators. the condition is almost identical to this obtained in the original discovery of the superradiant phase transition when replacing the harmonic oscillators with two level atoms with the same distance between the energy levels, dipole transition strength, and the density which means that it occurs in the regime when the coulomb interactions between electrons dominate over locally harmonic oscillatory influence of the atoms. it that sense the free electron gas with ω = 0 { \ displaystyle \ omega = 0 } is also purely superradiant. the critical inequality rewritten yet differently ω < e 2 m π 0 n v ≈ e 2 m 0 n v { \ displaystyle
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with ω = 0 { \ displaystyle \ omega = 0 } is also purely superradiant. the critical inequality rewritten yet differently ω < e 2 m π 0 n v ≈ e 2 m 0 n v { \ displaystyle \ omega < { \ sqrt { { \ frac { e ^ { 2 } } { m \ pi \ epsilon _ { 0 } } } { \ frac { n } { v } } } } \ approx { \ sqrt { { \ frac { e ^ { 2 } } { m \ epsilon _ { 0 } } } { \ frac { n } { v } } } } } expresses the fact that superradiant phase transition occurs when the frequency of the binding atomic oscillators is lower than so called electron gas plasma frequency. = = references = = = = external links = = old warsaw school of " no - go " of the superradiant phase transition, the talk by the 2022 wigner medal recipient iwo bialynicki - birula's former phd student k. rzazewski
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flowability, also known as powder flow is a property that defines an ability of a powdered material to flow, related to cohesion. powder flowability depends on many traits : the shape and size of the powder particles due to intermolecular force, porosity electrostatic activity hygroscopy bulk density angle of repose presence of glidants oxidation rate ( of a metallic powder ) humidity iso 4490 : 2018 norm ( and its precedent, iso 4490 : 2014 ) standardizes a method for determining the flow rate of metallic powders. it uses a normalized / calibrated funnel, named hall flowmeter. = = see also = = fluid mechanics soil mechanics cohesion ( geology ) angle of repose = = references = =
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in cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality ( encryption ). typically, a cryptosystem consists of three algorithms : one for key generation, one for encryption, and one for decryption. the term cipher ( sometimes cypher ) is often used to refer to a pair of algorithms, one for encryption and one for decryption. therefore, the term cryptosystem is most often used when the key generation algorithm is important. for this reason, the term cryptosystem is commonly used to refer to public key techniques ; however both " cipher " and " cryptosystem " are used for symmetric key techniques. = = formal definition = = mathematically, a cryptosystem or encryption scheme can be defined as a tuple ( p, c, k, e, d ) { \ displaystyle ( { \ mathcal { p } }, { \ mathcal { c } }, { \ mathcal { k } }, { \ mathcal { e } }, { \ mathcal { d } } ) } with the following properties. p { \ displaystyle { \ mathcal { p } } } is a set called the " plaintext space ". its elements are called plaintexts. c { \ displaystyle { \ mathcal { c } } } is a set called the " ciphertext space ". its elements are called ciphertexts. k { \ displaystyle { \ mathcal { k } } } is a set called the " key space ". its elements are called keys. e = { e k : k ∈ k } { \ displaystyle { \ mathcal { e } } = \ { e _ { k } : k \ in { \ mathcal { k } } \ } } is a set of functions e k : p → c { \ displaystyle e _ { k } : { \ mathcal { p } } \ rightarrow { \ mathcal { c } } }. its elements are called " encryption functions ". d = { d k : k ∈ k } { \ displaystyle { \ mathcal { d } } = \ { d _ { k } : k \ in { \ mathcal { k } } \ } } is a set of functions d k : c → p { \ displaystyle d _ { k } : { \ mathcal { c } } \ rightarrow {
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} : k \ in { \ mathcal { k } } \ } } is a set of functions d k : c → p { \ displaystyle d _ { k } : { \ mathcal { c } } \ rightarrow { \ mathcal { p } } }. its elements are called " decryption functions ". for each e ∈ k { \ displaystyle e \ in { \ mathcal { k } } }, there is d ∈ k { \ displaystyle d \ in { \ mathcal { k } } } such that d d ( e e ( p ) ) = p { \ displaystyle d _ { d } ( e _ { e } ( p ) ) = p } for all p ∈ p { \ displaystyle p \ in { \ mathcal { p } } }. note ; typically this definition is modified in order to distinguish an encryption scheme as being either a symmetric - key or public - key type of cryptosystem. = = examples = = a classical example of a cryptosystem is the caesar cipher. a more contemporary example is the rsa cryptosystem. another example of a cryptosystem is the advanced encryption standard ( aes ). aes is a widely used symmetric encryption algorithm that has become the standard for securing data in various applications. paillier cryptosystem is another example used to preserve and maintain privacy and sensitive information. it is featured in electronic voting, electronic lotteries and electronic auctions. = = see also = = list of cryptosystems semantic security = = references = =
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the international organisation for world urbanism day, also known as " world town planning day ", was founded in 1949 by the late professor carlos maria della paolera of the university of buenos aires, a graduate at the institut d'urbanisme in paris, to advance public and professional interest in planning. it is celebrated in more than 30 countries on four continents each november 8. = = see also = = urbanism urban planning new urbanism institut d'urbanisme de paris ( french wikipedia ) = = references = = = = external links = = american planning association : world town planning day world urbanism day from wn network
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in web analytics, a session, or visit is a unit of measurement of a user's actions taken within a period of time or with regard to completion of a task. sessions are also used in operational analytics and provision of user - specific recommendations. there are two primary methods used to define a session : time - oriented approaches based on continuity in user activity and navigation - based approaches based on continuity in a chain of requested pages. = = definition = = the definition of " session " varies, particularly when applied to search engines. generally, a session is understood to consist of " a sequence of requests made by a single end - user during a visit to a particular site ". in the context of search engines, " sessions " and " query sessions " have at least two definitions. a session or query session may be all queries made by a user in a particular time period or it may also be a series of queries or navigations with a consistent underlying user need. = = uses = = sessions per user can be used as a measurement of website usage. other metrics used within research and applied web analytics include session length, and user actions per session. session length is seen as a more accurate alternative to measuring page views. reconstructed sessions have also been used to measure total user input, including to measure the number of labour hours taken to construct wikipedia. sessions are also used for operational analytics, data anonymization, identifying networking anomalies, and synthetic workload generation for testing servers with artificial traffic. = = session reconstruction = = essential to the use of sessions in web analytics is being able to identify them. this is known as " session reconstruction ". approaches to session reconstruction can be divided into two main categories : time - oriented, and navigation - oriented. = = = time - oriented approaches = = = time - oriented approaches to session reconstruction look for a set period of user inactivity commonly called an " inactivity threshold. " once this period of inactivity is reached, the user is assumed to have left the site or stopped using the browser entirely and the session is ended. further requests from the same user are considered a second session. a common value for the inactivity threshold is 30 minutes and sometimes described as the industry standard. some have argued that a threshold of 30 minutes produces artifacts around naturally long sessions and have experimented with other thresholds. others simply state : " no time threshold is effective at identifying [ sessions ] ". one alternative that has been proposed is using user - specific thresholds rather
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a threshold of 30 minutes produces artifacts around naturally long sessions and have experimented with other thresholds. others simply state : " no time threshold is effective at identifying [ sessions ] ". one alternative that has been proposed is using user - specific thresholds rather than a single, global threshold for the entire dataset. this has the problem of assuming that the thresholds follow a bimodal distribution, and is not suitable for datasets that cover a long period of time. = = = navigation - oriented approaches = = = navigation - oriented approaches exploit the structure of websites - specifically, the presence of hyperlinks and the tendency of users to navigate between pages on the same website by clicking on them, rather than typing the full url into their browser. one way of identifying sessions by looking at this data is to build a map of the website : if the user's first page can be identified, the " session " of actions lasts until they land on a page which cannot be accessed from any of the previously - accessed pages. this takes into account backtracking, where a user will retrace their steps before opening a new page. a simpler approach, which does not take backtracking into account, is to simply require that the http referer of each request be a page that is already in the session. if it is not, a new session is created. this class of heuristics " exhibits very poor performance " on websites that contain framesets. = = references = = = = bibliography = = arlitt, martin ( 2000 ). " characterizing web user sessions " ( pdf ). sigmetrics performance evaluation review. 28 ( 2 ) : 50 – 63. doi : 10. 1145 / 362883. 362920. s2cid 2946044. berendt, bettina ; mobasher, bamshad ; nakagawa, miki ; spiliopoulou, myra ( 2003 ). " the impact of site structure and user environment on session reconstruction in web usage analysis " ( pdf ). webkdd 2002 - mining web data for discovering usage patterns and profiles. lecture notes in computer science. vol. 2703. springer. pp. 159 – 179. doi : 10. 1007 / 978 - 3 - 540 - 39663 - 5 _ 10. isbn 978 - 3 - 540 - 39663 - 5. catledge, l. ; pitkow, j. ( 1995 ). " characterizing browsing
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##7 / 978 - 3 - 540 - 39663 - 5 _ 10. isbn 978 - 3 - 540 - 39663 - 5. catledge, l. ; pitkow, j. ( 1995 ). " characterizing browsing strategies in the world - wide web " ( pdf ). computer networks and isdn systems. 27 ( 6 ) : 1065 – 1073. doi : 10. 1016 / 0169 - 7552 ( 95 ) 00043 - 7. s2cid 14313721. cooley, robert ; mobasher, bamshad ; srivastava, jaideep ( 1999 ). " data preparation for mining world wide web browsing patterns " ( pdf ). knowledge and information systems. 1 ( 1 ) : 5 – 32. citeseerx 10. 1. 1. 33. 2792. doi : 10. 1007 / bf03325089. issn 0219 - 3116. s2cid 1165622. donato, debora ; bonchi, francesco ; chi, tom ( 2010 ). " do you want to take notes? : identifying research missions in yahoo! search pad " ( pdf ). proceedings of the 19th international conference on world wide web. acm. pp. 321 – 330. doi : 10. 1145 / 1772690. 1772724. isbn 9781605587998. s2cid 6951065. eickhoff, carsten ; teevan, jaime ; white, ryen ; dumais, susan. ( 2014 ). " lessons from the journey ". proceedings of the 7th acm international conference on web search and data mining ( pdf ). acm. pp. 223 – 232. doi : 10. 1145 / 2556195. 2556217. isbn 9781450323512. s2cid 14666769. gayo - avello, daniel ( 2009 ). " a survey on session detection methods in query logs and a proposal for future evaluation " ( pdf ). information sciences. 179 ( 12 ) : 1822 – 1843. doi : 10. 1016 / j. ins. 2009. 01. 026. hdl : 10651 / 8686. issn 0020 - 0255. archived from the original ( pdf ) on 2016 - 03 - 04. retrieved 2015 - 02 - 18. geiger, r. s. ; halfaker, a.
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hdl : 10651 / 8686. issn 0020 - 0255. archived from the original ( pdf ) on 2016 - 03 - 04. retrieved 2015 - 02 - 18. geiger, r. s. ; halfaker, a. ( 2014 ). " using edit sessions to measure participation in wikipedia ". proceedings of the 2013 conference on computer supported cooperative work ( pdf ). acm. pp. 861 – 870. doi : 10. 1145 / 2441776. 2441873. isbn 9781450313315. s2cid 7166943. he, daqing ; goker, ayse ; harper, david j. ( 2002 ). " combining evidence for automatic web session identification ". information processing and management. 38 ( 5 ) : 727 – 742. doi : 10. 1016 / s0306 - 4573 ( 01 ) 00060 - 7. issn 0306 - 4573. heer, jeffrey ; chi, ed h. ( 2002 ). " separating the swarm : categorization methods for user sessions on the web " ( pdf ). proceedings of the sigchi conference on human factors in computing systems. vol. 4. acm. pp. 243 – 250. doi : 10. 1145 / 503376. 503420. isbn 1581134533. s2cid 14018957. huang, chien - kang ; chien, lee - feng ; oyang, yen - jen ( 2003 ). " relevant term suggestion in interactive web search based on contextual information in query session logs ". journal of the american society for information science and technology. 54 ( 7 ) : 638 – 649. citeseerx 10. 1. 1. 105. 5584. doi : 10. 1002 / asi. 10256. jansen, bernard j. ; spink, amanda ; saracevic, tefko ( 2000 ). " real life, real users, and real needs : a study and analysis of user queries on the web " ( pdf ). information processing and management. 36 ( 2 ) : 207 – 227. citeseerx 10. 1. 1. 155. 1383. doi : 10. 1016 / s0306 - 4573 ( 99 ) 00056 - 4. issn 0306 - 4573. jansen, bernard j. ; spink, amanda ( 2006 ). " how
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