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in banking and finance, clearing refers to all activities from the time a commitment is made for a transaction until it is settled. this process turns the promise of payment ( for example, in the form of a cheque or electronic payment request ) into the actual movement of money from one account to another. clearing houses were formed to facilitate such transactions among banks. = = description = = in trading, clearing is necessary because the speed of trades is much faster than the cycle time for completing the underlying transaction. it involves the management of post - trading, pre - settlement credit exposures to ensure that trades are settled in accordance with market rules, even if a buyer or seller should become insolvent prior to settlement. processes included in clearing are reporting / monitoring, risk margining, netting of trades to single positions, tax handling, and failure handling. systemically important payment systems ( sips ) are payment systems which have the characteristic that a failure of these systems could potentially endanger the operation of the whole economy. in general, these are the major payment clearing or real - time gross settlement systems of individual countries, but in the case of europe, there are certain pan - european payment systems. t2 is a pan - european sips dealing with major inter - bank payments. step2, operated by the euro banking association is a major pan - european clearing system for retail payments which has the potential to become a sips. in the united states, the federal reserve system is a sips. = = history = = = = = cheque clearing = = = the first payment method that required clearing was cheques, as cheques would have to be returned to the issuing bank for payment. though many debit cards are drawn against chequing accounts, direct deposit and point - of - purchase electronic payments are cleared through networks separate from the cheque clearing system ( in the united states, the federal reserve's automated clearing house and the private electronic payments network ). = = = securities and derivatives clearing = = = securities clearing was required to ensure payment had been received and the physical stock certificate delivered. this caused a few days'delay between the trade date and final settlement. to reduce the risk associated with failure to deliver on the trade on settlement date, a clearing agent or clearing house often sat between the trading parties. the trading parties would deliver the physical stock certificate and the payment to the clearing house, who would then ensure the certificate was handed over and the payment complete. this process is known as delivery versus payment. during the 1700s the
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between the trading parties. the trading parties would deliver the physical stock certificate and the payment to the clearing house, who would then ensure the certificate was handed over and the payment complete. this process is known as delivery versus payment. during the 1700s the amsterdam stock exchange had close links with the london stock exchange, and the two would often list each other's stocks. to clear the trades, time was required for the physical stock certificate or cash to move from amsterdam to london and back. this led to a standard settlement period of 14 days, which was the time it usually took for a courier to make the journey between the two cities. most exchanges copied the model, which was used for the next few hundred years. with the advent of the computer in the 1970s and 1980s, there was a move to reduce settlement times in most exchanges, leading by stages to a current standard of one day, known as t + 1. with the advent of electronic settlement, and a move to dematerialisation of securities, standardised clearing systems were required, as well as standardised securities depositories, custodians and registrars. until this point, many exchanges would act as their own clearing house, however the additional computer systems required to handle large volumes of trades, and the opening of new financial markets in the 1980s, such as the 1986 big bang in the uk, led to a number of exchanges separating or contracting the clearing and settlement functions to dedicated organisations. in some specialist financial markets, clearing had already been separate from trading. one example was the london clearing house ( later renamed lch. clearnet ), which, since the 1950s, cleared derivatives and commodities for a number of london exchanges. clearing houses who clear financial instruments, such as lch, are generally called central counterparties ( ccps ). in the wake of the 2008 financial crisis the g20 leaders agreed at the 2009 pittsburgh summit that all standardised derivatives contracts should be traded on exchanges or electronic trading platforms and cleared through central counterparties ( ccps ). although some derivatives were already traded on exchange and cleared, many over - the - counter derivatives that met the criteria needed to be novated to ccps as a result. = = united states clearing system = = the united states clearing system, known as chips, is the largest clearing system in the world. millions of transactions, valued in the trillions of dollars, are conducted between sellers and purchasers of goods, services, or financial assets daily. most of the payments making up the transactions flow between several banks, most
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largest clearing system in the world. millions of transactions, valued in the trillions of dollars, are conducted between sellers and purchasers of goods, services, or financial assets daily. most of the payments making up the transactions flow between several banks, most of which maintain accounts with the federal reserve banks. the federal reserve therefore performs an intermediary role, clearing and settling international bank payments. prior to the completion of the clearing, the banks settle payment transactions by debiting the accounts of the depository institutions, while crediting the accounts of depository institutions receiving the payments. the fedwire funds service provides a real - time gross settlement system in which more than 9, 500 participants are able to initiate electronic funds transfers that are immediate, final, and irrevocable. depository institutions that maintain an account with a reserve bank are eligible to use the service to send payments directly to, or receive payments from, other participants. depository institutions can also use a correspondent relationship with a fedwire participant to make or receive transfers indirectly through the system. participants generally use fedwire to handle large - value, time - critical payments, such as payments to settle interbank purchases and sales of federal funds ; to purchase, sell, or finance securities transactions ; to disburse or repay large loans ; and to settle real estate transactions. the department of the treasury, other federal agencies, and government - sponsored enterprises also use fedwire to disburse and collect funds. in 2003, the reserve banks processed 123 million fedwire payments having a total value of $ 436. 7 trillion. the fedwire securities service ( fss ) provides safekeeping, transfer, and settlement services for securities issued by the us treasury, federal agencies, government - sponsored enterprises, and certain international organizations. the reserve banks perform these services as fiscal agents for these entities. securities are safekept in the form of electronic records of securities held in custody accounts. securities are transferred according to instructions provided by parties with access to the system. access to the fss is limited to depository institutions that maintain accounts with a reserve bank and a few other organizations such as federal agencies, government - sponsored enterprises, and state government treasurer β s offices ( which are designated by the u. s. treasury to hold securities accounts ). other parties, specifically brokers and dealers, typically hold and transfer securities through depository institutions that are fedwire participants and that provide specialized government securities clearing services. in 2003, the fedwire securities service processed 20. 4 million securities transfers with a value of $
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, specifically brokers and dealers, typically hold and transfer securities through depository institutions that are fedwire participants and that provide specialized government securities clearing services. in 2003, the fedwire securities service processed 20. 4 million securities transfers with a value of $ 267. 6 trillion. the ach network is an electronic payment system, developed jointly by the private sector and the federal reserve in the early 1970s as a more efficient alternative to checks. since then, the ach has evolved into a nationwide mechanism that processes credit and debit transfers electronically. ach credit transfers are used to make direct deposit payroll payments and corporate payments to vendors. ach debit transfers are used by consumers to authorize the payment of insurance premiums, mortgages, loans, and other bills from their account. the ach is also used by businesses to concentrate funds at a primary bank and to make payments to other businesses. in 2003, the reserve banks processed 6. 5 billion ach payments with a value of $ 16. 8 trillion. = = see also = = = = further reading = = dudley p. bailey. 1890. the clearing - house system. reprint from the bankers magazine. homans publishing co. new york. = = references = = this article incorporates text from this source, which is in the public domain : the federal reserve system : purposes and functions ( pdf ). = = external links = = understanding derivatives : markets and infrastructure - chapter 2, central counterparty clearing by robert steigerwald ( federal reserve bank of chicago ) clearing and settlement of exchange - traded derivatives by john mcpartland ( federal reserve bank of chicago )
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in mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. one of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings. = = elementary examples = = the additive identity familiar from elementary mathematics is zero, denoted 0. for example, 5 + 0 = 5 = 0 + 5. { \ displaystyle 5 + 0 = 5 = 0 + 5. } in the natural numbers n { \ displaystyle \ mathbb { n } } ( if 0 is included ), the integers z, { \ displaystyle \ mathbb { z }, } the rational numbers q, { \ displaystyle \ mathbb { q }, } the real numbers r, { \ displaystyle \ mathbb { r }, } and the complex numbers c, { \ displaystyle \ mathbb { c }, } the additive identity is 0. this says that for a number n belonging to any of these sets, n + 0 = n = 0 + n. { \ displaystyle n + 0 = n = 0 + n. } = = formal definition = = let n be a group that is closed under the operation of addition, denoted +. an additive identity for n, denoted e, is an element in n such that for any element n in n, e + n = n = n + e. { \ displaystyle e + n = n = n + e. } = = further examples = = in a group, the additive identity is the identity element of the group, is often denoted 0, and is unique ( see below for proof ). a ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. this is defined to be different from the multiplicative identity 1 if the ring ( or field ) has more than one element. if the additive identity and the multiplicative identity are the same, then the ring is trivial ( proved below ). in the ring mm Γ n ( r ) of m - by - n matrices over a ring r, the additive identity is the zero matrix, denoted o or 0, and is the m - by - n matrix whose entries consist entirely of the identity element 0 in r. for example, in the 2Γ2 matrices over the integers m 2 ( z ) { \ displaystyle \ operator
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, denoted o or 0, and is the m - by - n matrix whose entries consist entirely of the identity element 0 in r. for example, in the 2Γ2 matrices over the integers m 2 ( z ) { \ displaystyle \ operatorname { m } _ { 2 } ( \ mathbb { z } ) } the additive identity is 0 = [ 0 0 0 0 ] { \ displaystyle 0 = { \ begin { bmatrix } 0 & 0 \ \ 0 & 0 \ end { bmatrix } } } in the quaternions, 0 is the additive identity. in the ring of functions from r β r { \ displaystyle \ mathbb { r } \ to \ mathbb { r } }, the function mapping every number to 0 is the additive identity. in the additive group of vectors in r n, { \ displaystyle \ mathbb { r } ^ { n }, } the origin or zero vector is the additive identity. = = properties = = = = = the additive identity is unique in a group = = = let ( g, + ) be a group and let 0 and 0'in g both denote additive identities, so for any g in g, 0 + g = g = g + 0, 0 β² + g = g = g + 0 β². { \ displaystyle 0 + g = g = g + 0, \ qquad 0'+ g = g = g + 0 '. } it then follows from the above that 0 β² = 0 β² + 0 = 0 β² + 0 = 0. { \ displaystyle { \ color { green } 0'} = { \ color { green } 0'} + 0 = 0'+ { \ color { red } 0 } = { \ color { red } 0 }. } = = = the additive identity annihilates ring elements = = = in a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in s, s Β· 0 = 0. this follows because : s β
0 = s β
( 0 + 0 ) = s β
0 + s β
0 β s β
0 = s β
0 β s β
0 β s β
0 = 0. { \ displaystyle { \ begin { aligned } s \ cdot 0 & = s \ cdot ( 0 + 0 ) = s \ cdot 0 + s \ cdot 0 \ \ \ rightarrow s \
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0 = 0. { \ displaystyle { \ begin { aligned } s \ cdot 0 & = s \ cdot ( 0 + 0 ) = s \ cdot 0 + s \ cdot 0 \ \ \ rightarrow s \ cdot 0 & = s \ cdot 0 - s \ cdot 0 \ \ \ rightarrow s \ cdot 0 & = 0. \ end { aligned } } } = = = the additive and multiplicative identities are different in a non - trivial ring = = = let r be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i. e. 0 = 1. let r be any element of r. then r = r Γ 1 = r Γ 0 = 0 { \ displaystyle r = r \ times 1 = r \ times 0 = 0 } proving that r is trivial, i. e. r = { 0 }. the contrapositive, that if r is non - trivial then 0 is not equal to 1, is therefore shown. = = see also = = 0 ( number ) additive inverse identity element multiplicative identity = = references = = = = bibliography = = david s. dummit, richard m. foote, abstract algebra, wiley ( 3rd ed. ) : 2003, isbn 0 - 471 - 43334 - 9. = = external links = = uniqueness of additive identity in a ring at planetmath.
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matrix mechanics is a formulation of quantum mechanics created by werner heisenberg, max born, and pascual jordan in 1925. it was the first conceptually autonomous and logically consistent formulation of quantum mechanics. its account of quantum jumps supplanted the bohr model's electron orbits. it did so by interpreting the physical properties of particles as matrices that evolve in time. it is equivalent to the schrodinger wave formulation of quantum mechanics, as manifest in dirac's bra β ket notation. in some contrast to the wave formulation, it produces spectra of ( mostly energy ) operators by purely algebraic, ladder operator methods. relying on these methods, wolfgang pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics. = = development of matrix mechanics = = in 1925, werner heisenberg, max born, and pascual jordan formulated the matrix mechanics representation of quantum mechanics. = = = epiphany at helgoland = = = in 1925 werner heisenberg was working in gottingen on the problem of calculating the spectral lines of hydrogen. by may 1925 he began trying to describe atomic systems by observables only. on june 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine, heisenberg left for the pollen - free north sea island of helgoland. while there, in between climbing and memorizing poems from goethe's west - ostlicher diwan, he continued to ponder the spectral issue and eventually realised that adopting non - commuting observables might solve the problem. he later wrote : it was about three o'clock at night when the final result of the calculation lay before me. at first i was deeply shaken. i was so excited that i could not think of sleep. so i left the house and awaited the sunrise on the top of a rock. : 275 = = = the three fundamental papers = = = after heisenberg returned to gottingen, he showed wolfgang pauli his calculations, commenting at one point : everything is still vague and unclear to me, but it seems as if the electrons will no more move on orbits. on july 9 heisenberg gave the same paper of his calculations to max born, saying that " he had written a crazy paper and did not dare to send it in for publication, and that born should read it and advise him " prior to publication. heisenberg then departed for a while, leaving born to analyse the paper. in
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" he had written a crazy paper and did not dare to send it in for publication, and that born should read it and advise him " prior to publication. heisenberg then departed for a while, leaving born to analyse the paper. in the paper, heisenberg formulated quantum theory without sharp electron orbits. hendrik kramers had earlier calculated the relative intensities of spectral lines in the sommerfeld model by interpreting the fourier coefficients of the orbits as intensities. but his answer, like all other calculations in the old quantum theory, was only correct for large orbits. heisenberg, after a collaboration with kramers, began to understand that the transition probabilities were not quite classical quantities, because the only frequencies that appear in the fourier series should be the ones that are observed in quantum jumps, not the fictional ones that come from fourier - analyzing sharp classical orbits. he replaced the classical fourier series with a matrix of coefficients, a fuzzed - out quantum analog of the fourier series. classically, the fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright - line spectrum. the quantities in heisenberg's formulation were the classical position and momentum, but now they were no longer sharply defined. each quantity was represented by a collection of fourier coefficients with two indices, corresponding to the initial and final states. when born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under jakob rosanes at breslau university. born, with the help of his assistant and former student pascual jordan, began immediately to make the transcription and extension, and they submitted their results for publication ; the paper was received for publication just 60 days after heisenberg's paper. a follow - on paper was submitted for publication before the end of the year by all three authors. ( a brief review of born's role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non - commutativity of the probability amplitudes can be found in an article by jeremy bernstein. a detailed historical and technical account can be found in mehra and rechenberg's book the historical development of quantum theory. volume 3. the formulation of matrix mechanics and its modifications 1925 β 1926. ) up until this time, matrices were seldom used by physicists ; they were
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be found in mehra and rechenberg's book the historical development of quantum theory. volume 3. the formulation of matrix mechanics and its modifications 1925 β 1926. ) up until this time, matrices were seldom used by physicists ; they were considered to belong to the realm of pure mathematics. gustav mie had used them in a paper on electrodynamics in 1912 and born had used them in his work on the lattices theory of crystals in 1921. while matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. born, however, had learned matrix algebra from rosanes, as already noted, but born had also learned hilbert's theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by born of hilbert's work grundzuge einer allgemeinen theorie der linearen integralgleichungen published in 1912. jordan, too, was well equipped for the task. for a number of years, he had been an assistant to richard courant at gottingen in the preparation of courant and david hilbert's book methoden der mathematischen physik i, which was published in 1924. this book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics. in 1926, john von neumann became assistant to david hilbert, and he would coin the term hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics. a linchpin contribution to this formulation was achieved in dirac's reinterpretation / synthesis paper of 1925, which invented the language and framework usually employed today, in full display of the noncommutative structure of the entire construction. = = = heisenberg's reasoning = = = before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum x ( t ), p ( t ), with the restriction that the time integral over one period t of the momentum times the velocity must be a positive integer multiple of the planck constant 0 t p d x d t d t = 0 t p d x = n h. { \ displaystyle \ int _ { 0 } ^ { t } p \ ; { \ frac { dx } { dt } } \ ; dt = \ int _ { 0 } ^ { t } p \ ; dx
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h. { \ displaystyle \ int _ { 0 } ^ { t } p \ ; { \ frac { dx } { dt } } \ ; dt = \ int _ { 0 } ^ { t } p \ ; dx = nh. } while this restriction correctly selects orbits with more or less the right energy values en, the old quantum mechanical formalism did not describe time dependent processes, such as the emission or absorption of radiation. when a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglected, it will emit radiation in a pattern that repeats itself every orbital period. the frequencies that make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that x ( t ) is periodic, so that its fourier representation has frequencies 2Οn / t only. x ( t ) = n = β β β e 2 Ο i n t / t x n. { \ displaystyle x ( t ) = \ sum _ { n = - \ infty } ^ { \ infty } e ^ { 2 \ pi int / t } x _ { n }. } the coefficients xn are complex numbers. the ones with negative frequencies must be the complex conjugates of the ones with positive frequencies, so that x ( t ) will always be real, x n = x β n β. { \ displaystyle x _ { n } = x _ { - n } ^ { * }. } a quantum mechanical particle, on the other hand, cannot emit radiation continuously ; it can only emit photons. assuming that the quantum particle started in orbit number n, emitted a photon, then ended up in orbit number m, the energy of the photon is en β em, which means that its frequency is en β em / h. for large n and m, but with n β m relatively small, these are the classical frequencies by bohr's correspondence principle e n β e m β h ( n β m ) t. { \ displaystyle e _ { n } - e _ { m } \ approx { \ frac { h ( n - m ) } { t } }. } in the formula above, t is the classical period of either orbit n or orbit m, since the difference between them is higher order in h. but for small n and m, or if n β m is large, the frequencies are not integer multiples of any
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above, t is the classical period of either orbit n or orbit m, since the difference between them is higher order in h. but for small n and m, or if n β m is large, the frequencies are not integer multiples of any single frequency. since the frequencies that the particle emits are the same as the frequencies in the fourier description of its motion, this suggests that something in the time - dependent description of the particle is oscillating with frequency en β em / h. heisenberg called this quantity xnm, and demanded that it should reduce to the classical fourier coefficients in the classical limit. for large values of n and m but with n β m relatively small, xnm is the ( n β m ) th fourier coefficient of the classical motion at orbit n. since xnm has opposite frequency to xmn, the condition that x is real becomes x n m = x m n β. { \ displaystyle x _ { nm } = x _ { mn } ^ { * }. } by definition, xnm only has the frequency en β em / h, so its time evolution is simple : x n m ( t ) = e 2 Ο i ( e n β e m ) t / h x n m ( 0 ) = e i ( e n β e m ) t / x n m ( 0 ). { \ displaystyle x _ { nm } ( t ) = e ^ { 2 \ pi i ( e _ { n } - e _ { m } ) t / h } x _ { nm } ( 0 ) = e ^ { i ( e _ { n } - e _ { m } ) t / \ hbar } x _ { nm } ( 0 ). } this is the original form of heisenberg's equation of motion. given two arrays xnm and pnm describing two physical quantities, heisenberg could form a new array of the same type by combining the terms xnkpkm, which also oscillate with the right frequency. since the fourier coefficients of the product of two quantities is the convolution of the fourier coefficients of each one separately, the correspondence with fourier series allowed heisenberg to deduce the rule by which the arrays should be multiplied, ( x p ) m n = k = 0 β x m k p k n. { \ displaystyle ( xp ) _ { mn } = \ sum _ { k = 0 } ^ {
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rule by which the arrays should be multiplied, ( x p ) m n = k = 0 β x m k p k n. { \ displaystyle ( xp ) _ { mn } = \ sum _ { k = 0 } ^ { \ infty } x _ { mk } p _ { kn }. } born pointed out that this is the law of matrix multiplication, so that the position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. under this multiplication rule, the product depends on the order : xp is different from px. the x matrix is a complete description of the motion of a quantum mechanical particle. because the frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as the fourier coefficients of a sharp classical trajectory. nevertheless, as matrices, x ( t ) and p ( t ) satisfy the classical equations of motion ; also see ehrenfest's theorem, below. = = = matrix basics = = = when it was introduced by werner heisenberg, max born and pascual jordan in 1925, matrix mechanics was not immediately accepted and was a source of controversy, at first. schrodinger's later introduction of wave mechanics was greatly favored. part of the reason was that heisenberg's formulation was in an odd mathematical language, for the time, while schrodinger's formulation was based on familiar wave equations. but there was also a deeper sociological reason. quantum mechanics had been developing by two paths, one led by einstein, who emphasized the wave β particle duality he proposed for photons, and the other led by bohr, that emphasized the discrete energy states and quantum jumps that bohr discovered. de broglie had reproduced the discrete energy states within einstein's framework β the quantum condition is the standing wave condition, and this gave hope to those in the einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuous wave mechanics. matrix mechanics, on the other hand, came from the bohr school, which was concerned with discrete energy states and quantum jumps. bohr's followers did not appreciate physical models that pictured electrons as waves, or as anything at all. they preferred to focus on the quantities that were directly connected to experiments. in atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. the bohr school required that only those quantities that were in principle
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they preferred to focus on the quantities that were directly connected to experiments. in atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. the bohr school required that only those quantities that were in principle measurable by spectroscopy should appear in the theory. these quantities include the energy levels and their intensities but they do not include the exact location of a particle in its bohr orbit. it is very hard to imagine an experiment that could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of the nucleus. it was a deep conviction that such questions did not have an answer. the matrix formulation was built on the premise that all physical observables are represented by matrices, whose elements are indexed by two different energy levels. the set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. since heisenberg's matrices are hermitian, the eigenvalues are real. if an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. the act of measurement in matrix mechanics collapses the state of the system. if one measures two observables simultaneously, the state of the system collapses to a common eigenvector of the two observables. since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at the same time. this is the uncertainty principle. if two matrices share their eigenvectors, they can be simultaneously diagonalized. in the basis where they are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. the uncertainty principle, by contrast, is an expression of the fact that often two matrices a and b do not always commute, i. e., that ab β ba does not necessarily equal 0. the fundamental commutation relation of matrix mechanics, k ( x n k p k m β p n k x k m ) = i Ξ΄ n m { \ displaystyle \ sum _ { k } \ left ( x _ { nk } p _ { km } - p _ { nk } x _ { km } \ right ) = i \ hbar \, \ delta _ { nm } } implies then that there are no states that
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} \ left ( x _ { nk } p _ { km } - p _ { nk } x _ { km } \ right ) = i \ hbar \, \ delta _ { nm } } implies then that there are no states that simultaneously have a definite position and momentum. this principle of uncertainty holds for many other pairs of observables as well. for example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom. = = = nobel prize = = = in 1928, albert einstein nominated heisenberg, born, and jordan for the nobel prize in physics. the announcement of the nobel prize in physics for 1932 was delayed until november 1933. it was at that time that it was announced heisenberg had won the prize for 1932 " for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen " and erwin schrodinger and paul adrien maurice dirac shared the 1933 prize " for the discovery of new productive forms of atomic theory ". it might well be asked why born was not awarded the prize in 1932, along with heisenberg, and bernstein proffers speculations on this matter. one of them relates to jordan joining the nazi party on may 1, 1933, and becoming a stormtrooper. jordan's party affiliations and jordan's links to born may well have affected born's chance at the prize at that time. bernstein further notes that when born finally won the prize in 1954, jordan was still alive, while the prize was awarded for the statistical interpretation of quantum mechanics, attributable to born alone. heisenberg's reactions to born for heisenberg receiving the prize for 1932 and for born receiving the prize in 1954 are also instructive in evaluating whether born should have shared the prize with heisenberg. on november 25, 1933, born received a letter from heisenberg in which he said he had been delayed in writing due to a " bad conscience " that he alone had received the prize " for work done in gottingen in collaboration β you, jordan and i ". heisenberg went on to say that born and jordan's contribution to quantum mechanics cannot be changed by " a wrong decision from the outside ". in 1954, heisenberg wrote an article honoring max planck for his insight in 1900. in the article, heisenberg credited born and jordan for the
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' s contribution to quantum mechanics cannot be changed by " a wrong decision from the outside ". in 1954, heisenberg wrote an article honoring max planck for his insight in 1900. in the article, heisenberg credited born and jordan for the final mathematical formulation of matrix mechanics and heisenberg went on to stress how great their contributions were to quantum mechanics, which were not " adequately acknowledged in the public eye ". = = mathematical development = = once heisenberg introduced the matrices for x and p, he could find their matrix elements in special cases by guesswork, guided by the correspondence principle. since the matrix elements are the quantum mechanical analogs of fourier coefficients of the classical orbits, the simplest case is the harmonic oscillator, where the classical position and momentum, x ( t ) and p ( t ), are sinusoidal. = = = harmonic oscillator = = = in units where the mass and frequency of the oscillator are equal to one ( see nondimensionalization ), the energy of the oscillator is h = 1 2 ( p 2 + x 2 ). { \ displaystyle h = { \ tfrac { 1 } { 2 } } \ left ( p ^ { 2 } + x ^ { 2 } \ right ). } the level sets of h are the clockwise orbits, and they are nested circles in phase space. the classical orbit with energy e is x ( t ) = 2 e cos ( t ), p ( t ) = β 2 e sin ( t ). { \ displaystyle x ( t ) = { \ sqrt { 2e } } \ cos ( t ), \ qquad p ( t ) = - { \ sqrt { 2e } } \ sin ( t ) ~. } the old quantum condition dictates that the integral of p dx over an orbit, which is the area of the circle in phase space, must be an integer multiple of the planck constant. the area of the circle of radius β2e is 2Οe. so e = n h 2 Ο = n, { \ displaystyle e = { \ frac { nh } { 2 \ pi } } = n \ hbar \,, } or, in natural units where Δ§ = 1, the energy is an integer. the fourier components of x ( t ) and p ( t ) are simple, and more so if they are combined into the quantities a ( t )
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\,, } or, in natural units where Δ§ = 1, the energy is an integer. the fourier components of x ( t ) and p ( t ) are simple, and more so if they are combined into the quantities a ( t ) = x ( t ) + i p ( t ) = 2 e e β i t, a β ( t ) = x ( t ) β i p ( t ) = 2 e e i t. { \ displaystyle a ( t ) = x ( t ) + ip ( t ) = { \ sqrt { 2e } } \, e ^ { - it }, \ quad a ^ { \ dagger } ( t ) = x ( t ) - ip ( t ) = { \ sqrt { 2e } } \, e ^ { it }. } both a and a β have only a single frequency, and x and p can be recovered from their sum and difference. since a ( t ) has a classical fourier series with only the lowest frequency, and the matrix element amn is the ( m β n ) th fourier coefficient of the classical orbit, the matrix for a is nonzero only on the line just above the diagonal, where it is equal to β2en. the matrix for a β is likewise only nonzero on the line below the diagonal, with the same elements. thus, from a and a β , reconstruction yields 2 x ( 0 ) = [ 0 1 0 0 0 1 0 2 0 0 0 2 0 3 0 0 0 3 0 4 ], { \ displaystyle { \ sqrt { 2 } } x ( 0 ) = { \ sqrt { \ hbar } } \ ; { \ begin { bmatrix } 0 & { \ sqrt { 1 } } & 0 & 0 & 0 & \ cdots \ \ { \ sqrt { 1 } } & 0 & { \ sqrt { 2 } } & 0 & 0 & \ cdots \ \ 0 & { \ sqrt { 2 } } & 0 & { \ sqrt { 3 } } & 0 & \ cdots \ \ 0 & 0 & { \ sqrt { 3 } } & 0 & { \ sqrt { 4 } } & \ cdots \ \ \ vdots & \ vdots & \ vdots & \ vdots & \ vdots & \ ddots \ \ \ end { bmatrix }
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\ sqrt { 4 } } & \ cdots \ \ \ vdots & \ vdots & \ vdots & \ vdots & \ vdots & \ ddots \ \ \ end { bmatrix } }, } and 2 p ( 0 ) = [ 0 β i 1 0 0 0 i 1 0 β i 2 0 0 0 i 2 0 β i 3 0 0 0 i 3 0 β i 4 ], { \ displaystyle { \ sqrt { 2 } } p ( 0 ) = { \ sqrt { \ hbar } } \ ; { \ begin { bmatrix } 0 & - i { \ sqrt { 1 } } & 0 & 0 & 0 & \ cdots \ \ i { \ sqrt { 1 } } & 0 & - i { \ sqrt { 2 } } & 0 & 0 & \ cdots \ \ 0 & i { \ sqrt { 2 } } & 0 & - i { \ sqrt { 3 } } & 0 & \ cdots \ \ 0 & 0 & i { \ sqrt { 3 } } & 0 & - i { \ sqrt { 4 } } & \ cdots \ \ \ vdots & \ vdots & \ vdots & \ vdots & \ vdots & \ ddots \ \ \ end { bmatrix } }, } which, up to the choice of units, are the heisenberg matrices for the harmonic oscillator. both matrices are hermitian, since they are constructed from the fourier coefficients of real quantities. finding x ( t ) and p ( t ) is direct, since they are quantum fourier coefficients so they evolve simply with time, x m n ( t ) = x m n ( 0 ) e i ( e m β e n ) t, p m n ( t ) = p m n ( 0 ) e i ( e m β e n ) t. { \ displaystyle x _ { mn } ( t ) = x _ { mn } ( 0 ) e ^ { i ( e _ { m } - e _ { n } ) t }, \ quad p _ { mn } ( t ) = p _ { mn } ( 0 ) e ^ { i ( e _ { m } - e _ { n } ) t } ~. } the matrix product of x and p is not hermitian, but
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{ mn } ( t ) = p _ { mn } ( 0 ) e ^ { i ( e _ { m } - e _ { n } ) t } ~. } the matrix product of x and p is not hermitian, but has a real and imaginary part. the real part is one half the symmetric expression xp + px, while the imaginary part is proportional to the commutator [ x, p ] = ( x p β p x ). { \ displaystyle [ x, p ] = ( xp - px ). } it is simple to verify explicitly that xp β px in the case of the harmonic oscillator, is iΔ§, multiplied by the identity. it is likewise simple to verify that the matrix h = 1 2 ( x 2 + p 2 ) { \ displaystyle h = { \ tfrac { 1 } { 2 } } \ left ( x ^ { 2 } + p ^ { 2 } \ right ) } is a diagonal matrix, with eigenvalues ei. = = = conservation of energy = = = the harmonic oscillator is an important case. finding the matrices is easier than determining the general conditions from these special forms. for this reason, heisenberg investigated the anharmonic oscillator, with hamiltonian h = 1 2 p 2 + 1 2 x 2 + Ξ΅ x 3. { \ displaystyle h = { \ tfrac { 1 } { 2 } } p ^ { 2 } + { \ tfrac { 1 } { 2 } } x ^ { 2 } + \ varepsilon x ^ { 3 } ~. } in this case, the x and p matrices are no longer simple off - diagonal matrices, since the corresponding classical orbits are slightly squashed and displaced, so that they have fourier coefficients at every classical frequency. to determine the matrix elements, heisenberg required that the classical equations of motion be obeyed as matrix equations, d x d t = p, d p d t = β x β 3 Ξ΅ x 2. { \ displaystyle { \ frac { dx } { dt } } = p ~, \ qquad { \ frac { dp } { dt } } = - x - 3 \ varepsilon x ^ { 2 } ~. } he noticed that if this could be done, then h, considered as a matrix function of x and p, will have zero time derivative. d h d t = p
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x - 3 \ varepsilon x ^ { 2 } ~. } he noticed that if this could be done, then h, considered as a matrix function of x and p, will have zero time derivative. d h d t = p β d p d t + ( x + 3 Ξ΅ x 2 ) β d x d t = 0, { \ displaystyle { \ frac { dh } { dt } } = p * { \ frac { dp } { dt } } + \ left ( x + 3 \ varepsilon x ^ { 2 } \ right ) * { \ frac { dx } { dt } } = 0 ~, } where aβb is the anticommutator, a β b = 1 2 ( a b + b a ). { \ displaystyle a * b = { \ tfrac { 1 } { 2 } } ( ab + ba ) ~. } given that all the off diagonal elements have a nonzero frequency ; h being constant implies that h is diagonal. it was clear to heisenberg that in this system, the energy could be exactly conserved in an arbitrary quantum system, a very encouraging sign. the process of emission and absorption of photons seemed to demand that the conservation of energy will hold at best on average. if a wave containing exactly one photon passes over some atoms, and one of them absorbs it, that atom needs to tell the others that they can't absorb the photon anymore. but if the atoms are far apart, any signal cannot reach the other atoms in time, and they might end up absorbing the same photon anyway and dissipating the energy to the environment. when the signal reached them, the other atoms would have to somehow recall that energy. this paradox led bohr, kramers and slater to abandon exact conservation of energy. heisenberg's formalism, when extended to include the electromagnetic field, was obviously going to sidestep this problem, a hint that the interpretation of the theory will involve wavefunction collapse. = = = differentiation trick β canonical commutation relations = = = demanding that the classical equations of motion are preserved is not a strong enough condition to determine the matrix elements. the planck constant does not appear in the classical equations, so that the matrices could be constructed for many different values of Δ§ and still satisfy the equations of motion, but with different energy levels. so, in order to implement his program, heisenberg needed to use the old quantum condition
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the classical equations, so that the matrices could be constructed for many different values of Δ§ and still satisfy the equations of motion, but with different energy levels. so, in order to implement his program, heisenberg needed to use the old quantum condition to fix the energy levels, then fill in the matrices with fourier coefficients of the classical equations, then alter the matrix coefficients and the energy levels slightly to make sure the classical equations are satisfied. this is clearly not satisfactory. the old quantum conditions refer to the area enclosed by the sharp classical orbits, which do not exist in the new formalism. the most important thing that heisenberg discovered is how to translate the old quantum condition into a simple statement in matrix mechanics. to do this, he investigated the action integral as a matrix quantity, 0 t k p m k ( t ) d x k n d t d t β? j m n. { \ displaystyle \ int _ { 0 } ^ { t } \ sum _ { k } p _ { mk } ( t ) { \ frac { dx _ { kn } } { dt } } dt \, \, { \ stackrel { \ scriptstyle? } { \ approx } } \, \, j _ { mn } ~. } there are several problems with this integral, all stemming from the incompatibility of the matrix formalism with the old picture of orbits. which period t should be used? semiclassically, it should be either m or n, but the difference is order Δ§, and an answer to order Δ§ is sought. the quantum condition tells us that jmn is 2Οn on the diagonal, so the fact that j is classically constant tells us that the off - diagonal elements are zero. his crucial insight was to differentiate the quantum condition with respect to n. this idea only makes complete sense in the classical limit, where n is not an integer but the continuous action variable j, but heisenberg performed analogous manipulations with matrices, where the intermediate expressions are sometimes discrete differences and sometimes derivatives. in the following discussion, for the sake of clarity, the differentiation will be performed on the classical variables, and the transition to matrix mechanics will be done afterwards, guided by the correspondence principle. in the classical setting, the derivative is the derivative with respect to j of the integral which defines j, so it is tautologically equal to 1. d d j 0 t p d x = 1 = 0 t d t ( d p d j d
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, the derivative is the derivative with respect to j of the integral which defines j, so it is tautologically equal to 1. d d j 0 t p d x = 1 = 0 t d t ( d p d j d x d t + p d d j d x d t ) = 0 t d t ( d p d j d x d t β d p d t d x d j ) { \ displaystyle { \ begin { aligned } { } { \ frac { d } { dj } } \ int _ { 0 } ^ { t } pdx & = 1 \ \ & = \ int _ { 0 } ^ { t } dt \ left ( { \ frac { dp } { dj } } { \ frac { dx } { dt } } + p { \ frac { d } { dj } } { \ frac { dx } { dt } } \ right ) \ \ & = \ int _ { 0 } ^ { t } dt \ left ( { \ frac { dp } { dj } } { \ frac { dx } { dt } } - { \ frac { dp } { dt } } { \ frac { dx } { dj } } \ right ) \ end { aligned } } } where the derivatives dp / dj and dx / dj should be interpreted as differences with respect to j at corresponding times on nearby orbits, exactly what would be obtained if the fourier coefficients of the orbital motion were differentiated. ( these derivatives are symplectically orthogonal in phase space to the time derivatives dp / dt and dx / dt ). the final expression is clarified by introducing the variable canonically conjugate to j, which is called the angle variable ΞΈ : the derivative with respect to time is a derivative with respect to ΞΈ, up to a factor of 2Οt, 2 Ο t 0 t d t ( d p d j d x d ΞΈ β d p d ΞΈ d x d j ) = 1. { \ displaystyle { \ frac { 2 \ pi } { t } } \ int _ { 0 } ^ { t } dt \ left ( { \ frac { dp } { dj } } { \ frac { dx } { d \ theta } } - { \ frac { dp } { d \ theta } } { \ frac { dx } { dj } } \ right ) = 1 \
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dj } } { \ frac { dx } { d \ theta } } - { \ frac { dp } { d \ theta } } { \ frac { dx } { dj } } \ right ) = 1 \,. } so the quantum condition integral is the average value over one cycle of the poisson bracket of x and p. an analogous differentiation of the fourier series of p dx demonstrates that the off - diagonal elements of the poisson bracket are all zero. the poisson bracket of two canonically conjugate variables, such as x and p, is the constant value 1, so this integral really is the average value of 1 ; so it is 1, as we knew all along, because it is dj / dj after all. but heisenberg, born and jordan, unlike dirac, were not familiar with the theory of poisson brackets, so, for them, the differentiation effectively evaluated { x, p } in j, ΞΈ coordinates. the poisson bracket, unlike the action integral, does have a simple translation to matrix mechanics β it normally corresponds to the imaginary part of the product of two variables, the commutator. to see this, examine the ( antisymmetrized ) product of two matrices a and b in the correspondence limit, where the matrix elements are slowly varying functions of the index, keeping in mind that the answer is zero classically. in the correspondence limit, when indices m, n are large and nearby, while k, r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the j derivative of the corresponding classical quantity. so it is possible to shift any matrix element diagonally through the correspondence, a ( m + r ) ( n + r ) β a m n β r ( d a d j ) m n { \ displaystyle a _ { ( m + r ) ( n + r ) } - a _ { mn } \ approx r \ ; \ left ( { \ frac { da } { dj } } \ right ) _ { mn } } where the right hand side is really only the ( m β n ) th fourier component of da / dj at the orbit near m to this semiclassical order, not a full well - defined matrix. the semiclassical time derivative of a matrix element is obtained up to a factor of i by multiplying by the distance from the diagonal, i k a m ( m + k ) β ( t 2 Ο d
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full well - defined matrix. the semiclassical time derivative of a matrix element is obtained up to a factor of i by multiplying by the distance from the diagonal, i k a m ( m + k ) β ( t 2 Ο d a d t ) m ( m + k ) = ( d a d ΞΈ ) m ( m + k ). { \ displaystyle ika _ { m ( m + k ) } \ approx \ left ( { \ frac { t } { 2 \ pi } } { \ frac { da } { dt } } \ right ) _ { m ( m + k ) } = \ left ( { \ frac { da } { d \ theta } } \ right ) _ { m ( m + k ) } \,. } since the coefficient am ( m + k ) is semiclassically the kth fourier coefficient of the mth classical orbit. the imaginary part of the product of a and b can be evaluated by shifting the matrix elements around so as to reproduce the classical answer, which is zero. the leading nonzero residual is then given entirely by the shifting. since all the matrix elements are at indices which have a small distance from the large index position ( m, m ), it helps to introduce two temporary notations : a [ r, k ] = a ( m + r ) ( m + k ) for the matrices, and da / dj [ r ] for the rth fourier components of classical quantities, ( a b β b a ) [ 0, k ] = r = β β β ( a [ 0, r ] b [ r, k ] β a [ r, k ] b [ 0, r ] ) = r ( a [ β r + k, k ] + ( r β k ) d a d j [ r ] ) ( b [ 0, k β r ] + r d b d j [ r β k ] ) β r a [ r, k ] b [ 0, r ]. { \ displaystyle { \ begin { aligned } ( ab - ba ) [ 0, k ] & = \ sum _ { r = - \ infty } ^ { \ infty } { \ bigl ( } a [ 0, r ] b [ r, k ] - a [ r, k ] b [ 0, r ] { \ bigr ) } \ \ & = \ sum _ { r } \ left ( a [ - r + k
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a [ 0, r ] b [ r, k ] - a [ r, k ] b [ 0, r ] { \ bigr ) } \ \ & = \ sum _ { r } \ left ( a [ - r + k, k ] + ( r - k ) { \ frac { da } { dj } } [ r ] \ right ) \ left ( b [ 0, k - r ] + r { \ frac { db } { dj } } [ r - k ] \ right ) - \ sum _ { r } a [ r, k ] b [ 0, r ] \,. \ end { aligned } } } flipping the summation variable in the first sum from r to r β² = k β r, the matrix element becomes, r β² ( a [ r β², k ] β r β² d a d j [ k β r β² ] ) ( b [ 0, r β² ] + ( k β r β² ) d b d j [ r β² ] ) β r a [ r, k ] b [ 0, r ] { \ displaystyle \ sum _ { r'} \ left ( a [ r ', k ] - r'{ \ frac { da } { dj } } [ k - r'] \ right ) \ left ( b [ 0, r'] + ( k - r') { \ frac { db } { dj } } [ r'] \ right ) - \ sum _ { r } a [ r, k ] b [ 0, r ] } and it is clear that the principal ( classical ) part cancels. the leading quantum part, neglecting the higher order product of derivatives in the residual expression, is then equal to r β² ( d b d j [ r β² ] ( k β r β² ) a [ r β², k ] β d a d j [ k β r β² ] r β² b [ 0, r β² ] ) { \ displaystyle \ sum _ { r'} \ left ( { \ frac { db } { dj } } [ r'] ( k - r') a [ r ', k ] - { \ frac { da } { dj } } [ k - r'] r'b [ 0, r'] \ right ) } so that, finally, ( a b β b a ) [ 0, k ] = r β² ( d b d j [ r β² ] i d a
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- r'] r'b [ 0, r'] \ right ) } so that, finally, ( a b β b a ) [ 0, k ] = r β² ( d b d j [ r β² ] i d a d ΞΈ [ k β r β² ] β d a d j [ k β r β² ] i d b d ΞΈ [ r β² ] ) { \ displaystyle ( ab - ba ) [ 0, k ] = \ sum _ { r'} \ left ( { \ frac { db } { dj } } [ r'] i { \ frac { da } { d \ theta } } [ k - r'] - { \ frac { da } { dj } } [ k - r'] i { \ frac { db } { d \ theta } } [ r'] \ right ) } which can be identified with i times the kth classical fourier component of the poisson bracket. heisenberg's original differentiation trick was eventually extended to a full semiclassical derivation of the quantum condition, in collaboration with born and jordan. once they were able to establish that i { x, p } p b [ x, p ] β‘ x p β p x = i, { \ displaystyle i \ hbar \ { x, p \ } _ { \ mathrm { pb } } \ qquad \ longmapsto \ qquad [ x, p ] \ equiv xp - px = i \ hbar \,, } this condition replaced and extended the old quantization rule, allowing the matrix elements of p and x for an arbitrary system to be determined simply from the form of the hamiltonian. the new quantization rule was assumed to be universally true, even though the derivation from the old quantum theory required semiclassical reasoning. ( a full quantum treatment, however, for more elaborate arguments of the brackets, was appreciated in the 1940s to amount to extending poisson brackets to moyal brackets. ) = = = state vectors and the heisenberg equation = = = to make the transition to standard quantum mechanics, the most important further addition was the quantum state vector, now written | Ο β©, which is the vector that the matrices act on. without the state vector, it is not clear which particular motion the heisenberg matrices are describing, since they include all the motions somewhere. the interpretation of the state vector, whose components are written Οm, was furnished
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that the matrices act on. without the state vector, it is not clear which particular motion the heisenberg matrices are describing, since they include all the motions somewhere. the interpretation of the state vector, whose components are written Οm, was furnished by born. this interpretation is statistical : the result of a measurement of the physical quantity corresponding to the matrix a is random, with an average value equal to m n Ο m β a m n Ο n. { \ displaystyle \ sum _ { mn } \ psi _ { m } ^ { * } a _ { mn } \ psi _ { n } \,. } alternatively, and equivalently, the state vector gives the probability amplitude Οn for the quantum system to be in the energy state n. once the state vector was introduced, matrix mechanics could be rotated to any basis, where the h matrix need no longer be diagonal. the heisenberg equation of motion in its original form states that amn evolves in time like a fourier component, a m n ( t ) = e i ( e m β e n ) t a m n ( 0 ), { \ displaystyle a _ { mn } ( t ) = e ^ { i ( e _ { m } - e _ { n } ) t } a _ { mn } ( 0 ) ~, } which can be recast in differential form d a m n d t = i ( e m β e n ) a m n, { \ displaystyle { \ frac { da _ { mn } } { dt } } = i ( e _ { m } - e _ { n } ) a _ { mn } ~, } and it can be restated so that it is true in an arbitrary basis, by noting that the h matrix is diagonal with diagonal values em, d a d t = i ( h a β a h ). { \ displaystyle { \ frac { da } { dt } } = i ( ha - ah ) ~. } this is now a matrix equation, so it holds in any basis. this is the modern form of the heisenberg equation of motion. its formal solution is : a ( t ) = e i h t a ( 0 ) e β i h t. { \ displaystyle a ( t ) = e ^ { iht } a ( 0 ) e ^ { - iht } ~. } all these forms of the equation of motion above say the same thing, that a ( t ) is equivalent to
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\ displaystyle a ( t ) = e ^ { iht } a ( 0 ) e ^ { - iht } ~. } all these forms of the equation of motion above say the same thing, that a ( t ) is equivalent to a ( 0 ), through a basis rotation by the unitary matrix eiht, a systematic picture elucidated by dirac in his bra β ket notation. conversely, by rotating the basis for the state vector at each time by eiht, the time dependence in the matrices can be undone. the matrices are now time independent, but the state vector rotates, | Ο ( t ) β© = e β i h t | Ο ( 0 ) β©, d | Ο β© d t = β i h | Ο β©. { \ displaystyle | \ psi ( t ) \ rangle = e ^ { - iht } | \ psi ( 0 ) \ rangle, \ qquad { \ frac { d | \ psi \ rangle } { dt } } = - ih | \ psi \ rangle \,. } this is the schrodinger equation for the state vector, and this time - dependent change of basis amounts to transformation to the schrodinger picture, with β¨ x | Ο β© = Ο ( x ). in quantum mechanics in the heisenberg picture the state vector, | Ο β© does not change with time, while an observable a satisfies the heisenberg equation of motion, the extra term is for operators such as a = ( x + t 2 p ) { \ displaystyle a = \ left ( x + t ^ { 2 } p \ right ) } which have an explicit time dependence, in addition to the time dependence from the unitary evolution discussed. the heisenberg picture does not distinguish time from space, so it is better suited to relativistic theories than the schrodinger equation. moreover, the similarity to classical physics is more manifest : the hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator above by the poisson bracket ( see also below ). by the stone β von neumann theorem, the heisenberg picture and the schrodinger picture must be unitarily equivalent, as detailed below. = = further results = = matrix mechanics rapidly developed into modern quantum mechanics, and gave interesting physical results on the spectra of atoms. = = = wave mechanics = = = jordan noted that the commutation relations ensure that
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equivalent, as detailed below. = = further results = = matrix mechanics rapidly developed into modern quantum mechanics, and gave interesting physical results on the spectra of atoms. = = = wave mechanics = = = jordan noted that the commutation relations ensure that p acts as a differential operator. the operator identity [ a, b c ] = a b c β b c a = a b c β b a c + b a c β b c a = [ a, b ] c + b [ a, c ] { \ displaystyle [ a, bc ] = abc - bca = abc - bac + bac - bca = [ a, b ] c + b [ a, c ] } allows the evaluation of the commutator of p with any power of x, and it implies that [ p, x n ] = β i n x n β 1 { \ displaystyle \ left [ p, x ^ { n } \ right ] = - in ~ x ^ { n - 1 } } which, together with linearity, implies that a p - commutator effectively differentiates any analytic matrix function of x. assuming limits are defined sensibly, this extends to arbitrary functionsβbut the extension need not be made explicit until a certain degree of mathematical rigor is required, since x is a hermitian matrix, it should be diagonalizable, and it will be clear from the eventual form of p that every real number can be an eigenvalue. this makes some of the mathematics subtle, since there is a separate eigenvector for every point in space. in the basis where x is diagonal, an arbitrary state can be written as a superposition of states with eigenvalues x, | Ο β© = x Ο ( x ) | x β©, { \ displaystyle | \ psi \ rangle = \ int _ { x } \ psi ( x ) | x \ rangle \,, } so that Ο ( x ) = β¨ x | Ο β©, and the operator x multiplies each eigenvector by x, x | Ο β© = x x Ο ( x ) | x β©. { \ displaystyle x | \ psi \ rangle = \ int _ { x } x \ psi ( x ) | x \ rangle ~. } define a linear operator d which differentiates Ο, d x Ο ( x ) | x β© = x Ο β² ( x ) | x β©, { \ displaystyle d
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\ psi ( x ) | x \ rangle ~. } define a linear operator d which differentiates Ο, d x Ο ( x ) | x β© = x Ο β² ( x ) | x β©, { \ displaystyle d \ int _ { x } \ psi ( x ) | x \ rangle = \ int _ { x } \ psi'( x ) | x \ rangle \,, } and note that ( d x β x d ) | Ο β© = x [ ( x Ο ( x ) ) β² β x Ο β² ( x ) ] | x β© = x Ο ( x ) | x β© = | Ο β©, { \ displaystyle ( dx - xd ) | \ psi \ rangle = \ int _ { x } \ left [ \ left ( x \ psi ( x ) \ right )'- x \ psi'( x ) \ right ] | x \ rangle = \ int _ { x } \ psi ( x ) | x \ rangle = | \ psi \ rangle \,, } so that the operator βid obeys the same commutation relation as p. thus, the difference between p and βid must commute with x, [ p + i d, x ] = 0, { \ displaystyle [ p + id, x ] = 0 \,, } so it may be simultaneously diagonalized with x : its value acting on any eigenstate of x is some function f of the eigenvalue x. this function must be real, because both p and βid are hermitian, ( p + i d ) | x β© = f ( x ) | x β©, { \ displaystyle ( p + id ) | x \ rangle = f ( x ) | x \ rangle \,, } rotating each state | x β© by a phase f ( x ), that is, redefining the phase of the wavefunction : Ο ( x ) β e β i f ( x ) Ο ( x ). { \ displaystyle \ psi ( x ) \ rightarrow e ^ { - if ( x ) } \ psi ( x ) \,. } the operator id is redefined by an amount : i d β i d + f ( x ), { \ displaystyle id \ rightarrow id + f ( x ) \,, } which means that, in the rotated basis, p is equal to βid. hence,
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amount : i d β i d + f ( x ), { \ displaystyle id \ rightarrow id + f ( x ) \,, } which means that, in the rotated basis, p is equal to βid. hence, there is always a basis for the eigenvalues of x where the action of p on any wavefunction is known : p x Ο ( x ) | x β© = x β i Ο β² ( x ) | x β©, { \ displaystyle p \ int _ { x } \ psi ( x ) | x \ rangle = \ int _ { x } - i \ psi'( x ) | x \ rangle \,, } and the hamiltonian in this basis is a linear differential operator on the state - vector components, [ p 2 2 m + v ( x ) ] x Ο x | x β© = x [ β 1 2 m β 2 β x 2 + v ( x ) ] Ο x | x β© { \ displaystyle \ left [ { \ frac { p ^ { 2 } } { 2m } } + v ( x ) \ right ] \ int _ { x } \ psi _ { x } | x \ rangle = \ int _ { x } \ left [ - { \ frac { 1 } { 2m } } { \ frac { \ partial ^ { 2 } } { \ partial x ^ { 2 } } } + v ( x ) \ right ] \ psi _ { x } | x \ rangle } thus, the equation of motion for the state vector is but a celebrated differential equation, since d is a differential operator, in order for it to be sensibly defined, there must be eigenvalues of x which neighbors every given value. this suggests that the only possibility is that the space of all eigenvalues of x is all real numbers, and that p is id, up to a phase rotation. to make this rigorous requires a sensible discussion of the limiting space of functions, and in this space this is the stone β von neumann theorem : any operators x and p which obey the commutation relations can be made to act on a space of wavefunctions, with p a derivative operator. this implies that a schrodinger picture is always available. matrix mechanics easily extends to many degrees of freedom in a natural way. each degree of freedom has a separate x operator and a separate effective differential operator p, and the wavefu
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operator. this implies that a schrodinger picture is always available. matrix mechanics easily extends to many degrees of freedom in a natural way. each degree of freedom has a separate x operator and a separate effective differential operator p, and the wavefunction is a function of all the possible eigenvalues of the independent commuting x variables. [ x i, x j ] = 0 [ p i, p j ] = 0 [ x i, p j ] = i Ξ΄ i j. { \ displaystyle { \ begin { aligned } \ left [ x _ { i }, x _ { j } \ right ] & = 0 \ \ [ 1ex ] \ left [ p _ { i }, p _ { j } \ right ] & = 0 \ \ [ 1ex ] \ left [ x _ { i }, p _ { j } \ right ] & = i \ delta _ { ij } \,. \ end { aligned } } } in particular, this means that a system of n interacting particles in 3 dimensions is described by one vector whose components in a basis where all the x are diagonal is a mathematical function of 3n - dimensional space describing all their possible positions, effectively a much bigger collection of values than the mere collection of n three - dimensional wavefunctions in one physical space. schrodinger came to the same conclusion independently, and eventually proved the equivalence of his own formalism to heisenberg's. since the wavefunction is a property of the whole system, not of any one part, the description in quantum mechanics is not entirely local. the description of several quantum particles has them correlated, or entangled. this entanglement leads to strange correlations between distant particles which violate the classical bell's inequality. even if the particles can only be in just two positions, the wavefunction for n particles requires 2n complex numbers, one for each total configuration of positions. this is exponentially many numbers in n, so simulating quantum mechanics on a computer requires exponential resources. conversely, this suggests that it might be possible to find quantum systems of size n which physically compute the answers to problems which classically require 2n bits to solve. this is the aspiration behind quantum computing. = = = ehrenfest theorem = = = for the time - independent operators x and p, βa / βt = 0 so the heisenberg equation above reduces to : i d a d t = [ a, h ] = a
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= = ehrenfest theorem = = = for the time - independent operators x and p, βa / βt = 0 so the heisenberg equation above reduces to : i d a d t = [ a, h ] = a h β h a, { \ displaystyle i \ hbar { \ frac { da } { dt } } = [ a, h ] = ah - ha, } where the square brackets [, ] denote the commutator. for a hamiltonian which is p2 / 2m + v ( x ), the x and p operators satisfy : d x d t = p m, d p d t = β β v, { \ displaystyle { \ frac { dx } { dt } } = { \ frac { p } { m } }, \ quad { \ frac { dp } { dt } } = - \ nabla v, } where the first is classically the velocity, and second is classically the force, or potential gradient. these reproduce hamilton's form of newton's laws of motion. in the heisenberg picture, the x and p operators satisfy the classical equations of motion. you can take the expectation value of both sides of the equation to see that, in any state | Ο β© : d d t β¨ x β© = d d t β¨ Ο | x | Ο β© = 1 m β¨ Ο | p | Ο β© = 1 m β¨ p β© d d t β¨ p β© = d d t β¨ Ο | p | Ο β© = β¨ Ο | ( β β v ) | Ο β© = β β¨ β v β©. { \ displaystyle { \ begin { aligned } { \ frac { d } { dt } } \ langle x \ rangle & = { \ frac { d } { dt } } \ langle \ psi | x | \ psi \ rangle = { \ frac { 1 } { m } } \ langle \ psi | p | \ psi \ rangle = { \ frac { 1 } { m } } \ langle p \ rangle \ \ [ 1. 5ex ] { \ frac { d } { dt } } \ langle p \ rangle & = { \ frac { d } { dt } } \ langle \ psi | p | \ psi \ rangle = \ langle \ psi | ( - \ nabla v ) | \ psi \ rangle = - \ langle
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= { \ frac { d } { dt } } \ langle \ psi | p | \ psi \ rangle = \ langle \ psi | ( - \ nabla v ) | \ psi \ rangle = - \ langle \ nabla v \ rangle \,. \ end { aligned } } } so newton's laws are exactly obeyed by the expected values of the operators in any given state. this is ehrenfest's theorem, which is an obvious corollary of the heisenberg equations of motion, but is less trivial in the schrodinger picture, where ehrenfest discovered it. = = = transformation theory = = = in classical mechanics, a canonical transformation of phase space coordinates is one which preserves the structure of the poisson brackets. the new variables x β², p β² have the same poisson brackets with each other as the original variables x, p. time evolution is a canonical transformation, since the phase space at any time is just as good a choice of variables as the phase space at any other time. the hamiltonian flow is the canonical transformation : x β x + d x = x + β h β p d t p β p + d p = p β β h β x d t. { \ displaystyle { \ begin { aligned } x & \ rightarrow x + dx = x + { \ frac { \ partial h } { \ partial p } } dt \ \ [ 1ex ] p & \ rightarrow p + dp = p - { \ frac { \ partial h } { \ partial x } } dt ~. \ end { aligned } } } since the hamiltonian can be an arbitrary function of x and p, there are such infinitesimal canonical transformations corresponding to every classical quantity g, where g serves as the hamiltonian to generate a flow of points in phase space for an increment of time s, d x = β g β p d s = { g, x } d s d p = β β g β x d s = { g, p } d s. { \ displaystyle { \ begin { aligned } dx & = { \ frac { \ partial g } { \ partial p } } ds = \ left \ { g, x \ right \ } ds \ \ [ 1ex ] dp & = - { \ frac { \ partial g } { \ partial x } } ds = \ left \ { g, p \ right \
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\ left \ { g, x \ right \ } ds \ \ [ 1ex ] dp & = - { \ frac { \ partial g } { \ partial x } } ds = \ left \ { g, p \ right \ } ds \,. \ end { aligned } } } for a general function a ( x, p ) on phase space, its infinitesimal change at every step ds under this map is d a = β a β x d x + β a β p d p = { a, g } d s. { \ displaystyle da = { \ frac { \ partial a } { \ partial x } } dx + { \ frac { \ partial a } { \ partial p } } dp = \ { a, g \ } ds \,. } the quantity g is called the infinitesimal generator of the canonical transformation. in quantum mechanics, the quantum analog g is now a hermitian matrix, and the equations of motion are given by commutators, d a = i [ g, a ] d s. { \ displaystyle da = i [ g, a ] ds \,. } the infinitesimal canonical motions can be formally integrated, just as the heisenberg equation of motion were integrated, a β² = u β a u { \ displaystyle a'= u ^ { \ dagger } au } where u = eigs and s is an arbitrary parameter. the definition of a quantum canonical transformation is thus an arbitrary unitary change of basis on the space of all state vectors. u is an arbitrary unitary matrix, a complex rotation in phase space, u β = u β 1. { \ displaystyle u ^ { \ dagger } = u ^ { - 1 } \,. } these transformations leave the sum of the absolute square of the wavefunction components invariant, while they take states which are multiples of each other ( including states which are imaginary multiples of each other ) to states which are the same multiple of each other. the interpretation of the matrices is that they act as generators of motions on the space of states. for example, the motion generated by p can be found by solving the heisenberg equation of motion using p as a hamiltonian, d x = i [ x, p ] d s = d s d p = i [ p, p ] d s = 0. { \ displaystyle { \ begin { aligned } dx & = i [ x, p ] ds = ds \ \
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i [ x, p ] d s = d s d p = i [ p, p ] d s = 0. { \ displaystyle { \ begin { aligned } dx & = i [ x, p ] ds = ds \ \ [ 1ex ] dp & = i [ p, p ] ds = 0 \,. \ end { aligned } } } these are translations of the matrix x by a multiple of the identity matrix, x β x + s i. { \ displaystyle x \ rightarrow x + si ~. } this is the interpretation of the derivative operator d : eips = ed, the exponential of a derivative operator is a translation ( so lagrange's shift operator ). the x operator likewise generates translations in p. the hamiltonian generates translations in time, the angular momentum generates rotations in physical space, and the operator x2 + p2 generates rotations in phase space. when a transformation, like a rotation in physical space, commutes with the hamiltonian, the transformation is called a symmetry ( behind a degeneracy ) of the hamiltonian β the hamiltonian expressed in terms of rotated coordinates is the same as the original hamiltonian. this means that the change in the hamiltonian under the infinitesimal symmetry generator l vanishes, d h d s = i [ l, h ] = 0. { \ displaystyle { \ frac { dh } { ds } } = i [ l, h ] = 0 \,. } it then follows that the change in the generator under time translation also vanishes, d l d t = i [ h, l ] = 0 { \ displaystyle { \ frac { dl } { dt } } = i [ h, l ] = 0 } so that the matrix l is constant in time : it is conserved. the one - to - one association of infinitesimal symmetry generators and conservation laws was discovered by emmy noether for classical mechanics, where the commutators are poisson brackets, but the quantum - mechanical reasoning is identical. in quantum mechanics, any unitary symmetry transformation yields a conservation law, since if the matrix u has the property that u β 1 h u = h { \ displaystyle u ^ { - 1 } hu = h } so it follows that u h = h u { \ displaystyle uh = hu } and that the time derivative of u is zero β it is conserved. the eigenvalues of unitary matrices are pure phases, so that
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} hu = h } so it follows that u h = h u { \ displaystyle uh = hu } and that the time derivative of u is zero β it is conserved. the eigenvalues of unitary matrices are pure phases, so that the value of a unitary conserved quantity is a complex number of unit magnitude, not a real number. another way of saying this is that a unitary matrix is the exponential of i times a hermitian matrix, so that the additive conserved real quantity, the phase, is only well - defined up to an integer multiple of 2Ο. only when the unitary symmetry matrix is part of a family that comes arbitrarily close to the identity are the conserved real quantities single - valued, and then the demand that they are conserved become a much more exacting constraint. symmetries which can be continuously connected to the identity are called continuous, and translations, rotations, and boosts are examples. symmetries which cannot be continuously connected to the identity are discrete, and the operation of space - inversion, or parity, and charge conjugation are examples. the interpretation of the matrices as generators of canonical transformations is due to paul dirac. the correspondence between symmetries and matrices was shown by eugene wigner to be complete, if antiunitary matrices which describe symmetries which include time - reversal are included. = = = selection rules = = = it was physically clear to heisenberg that the absolute squares of the matrix elements of x, which are the fourier coefficients of the oscillation, would yield the rate of emission of electromagnetic radiation. in the classical limit of large orbits, if a charge with position x ( t ) and charge q is oscillating next to an equal and opposite charge at position 0, the instantaneous dipole moment is q x ( t ), and the time variation of this moment translates directly into the space - time variation of the vector potential, which yields nested outgoing spherical waves. for atoms, the wavelength of the emitted light is about 10, 000 times the atomic radius, and the dipole moment is the only contribution to the radiative field, while all other details of the atomic charge distribution can be ignored. ignoring back - reaction, the power radiated in each outgoing mode is a sum of separate contributions from the square of each independent time fourier mode of d, p ( Ο ) = 2 3 Ο 4 | d i | 2. { \ displaystyle p ( \ omega ) = { \ tfrac { 2 }
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a sum of separate contributions from the square of each independent time fourier mode of d, p ( Ο ) = 2 3 Ο 4 | d i | 2. { \ displaystyle p ( \ omega ) = { \ tfrac { 2 } { 3 } } { \ omega ^ { 4 } } | d _ { i } | ^ { 2 } ~. } now, in heisenberg's representation, the fourier coefficients of the dipole moment are the matrix elements of x. this correspondence allowed heisenberg to provide the rule for the transition intensities, the fraction of the time that, starting from an initial state i, a photon is emitted and the atom jumps to a final state j, p i j = 2 3 ( e i β e j ) 4 | x i j | 2. { \ displaystyle p _ { ij } = { \ tfrac { 2 } { 3 } } \ left ( e _ { i } - e _ { j } \ right ) ^ { 4 } \ left | x _ { ij } \ right | ^ { 2 } \,. } this then allowed the magnitude of the matrix elements to be interpreted statistically : they give the intensity of the spectral lines, the probability for quantum jumps from the emission of dipole radiation. since the transition rates are given by the matrix elements of x, wherever xij is zero, the corresponding transition should be absent. these were called the selection rules, which were a puzzle until the advent of matrix mechanics. an arbitrary state of the hydrogen atom, ignoring spin, is labelled by | n ; l, m β©, where the value of l is a measure of the total orbital angular momentum and m is its z - component, which defines the orbit orientation. the components of the angular momentum pseudovector are l i = Ξ΅ i j k x j p k { \ displaystyle l _ { i } = \ varepsilon _ { ijk } x ^ { j } p ^ { k } } where the products in this expression are independent of order and real, because different components of x and p commute. the commutation relations of l with all three coordinate matrices x, y, z ( or with any vector ) are easy to find, [ l i, x j ] = i Ξ΅ i j k x k, { \ displaystyle \ left [ l _ { i }, x _ { j } \ right ] = i \ varepsilon _ { i
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to find, [ l i, x j ] = i Ξ΅ i j k x k, { \ displaystyle \ left [ l _ { i }, x _ { j } \ right ] = i \ varepsilon _ { ijk } x _ { k } \,, } which confirms that the operator l generates rotations between the three components of the vector of coordinate matrices x. from this, the commutator of lz and the coordinate matrices x, y, z can be read off, [ l z, x ] = i y, [ l z, y ] = β i x. { \ displaystyle { \ begin { aligned } \ left [ l _ { z }, x \ right ] & = iy \,, \ \ [ 1ex ] \ left [ l _ { z }, y \ right ] & = - ix \,. \ end { aligned } } } this means that the quantities x + iy and x β iy have a simple commutation rule, [ l z, x + i y ] = ( x + i y ), [ l z, x β i y ] = β ( x β i y ). { \ displaystyle { \ begin { aligned } \ left [ l _ { z }, x + iy \ right ] & = ( x + iy ) \,, \ \ [ 1ex ] \ left [ l _ { z }, x - iy \ right ] & = - ( x - iy ) \,. \ end { aligned } } } just like the matrix elements of x + ip and x β ip for the harmonic oscillator hamiltonian, this commutation law implies that these operators only have certain off diagonal matrix elements in states of definite m, l z ( ( x + i y ) | m β© ) = ( x + i y ) l z | m β© + ( x + i y ) | m β© = ( m + 1 ) ( x + i y ) | m β© { \ displaystyle l _ { z } { \ bigl ( } ( x + iy ) | m \ rangle { \ bigr ) } = ( x + iy ) l _ { z } | m \ rangle + ( x + iy ) | m \ rangle = ( m + 1 ) ( x + iy ) | m \ rangle } meaning that the matrix ( x + iy ) takes an e
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z } | m \ rangle + ( x + iy ) | m \ rangle = ( m + 1 ) ( x + iy ) | m \ rangle } meaning that the matrix ( x + iy ) takes an eigenvector of lz with eigenvalue m to an eigenvector with eigenvalue m + 1. similarly, ( x β iy ) decrease m by one unit, while z does not change the value of m. so, in a basis of | l, m β© states where l2 and lz have definite values, the matrix elements of any of the three components of the position are zero, except when m is the same or changes by one unit. this places a constraint on the change in total angular momentum. any state can be rotated so that its angular momentum is in the z - direction as much as possible, where m = l. the matrix element of the position acting on | l, m β© can only produce values of m which are bigger by one unit, so that if the coordinates are rotated so that the final state is | l β², l β² β©, the value of l β² can be at most one bigger than the biggest value of l that occurs in the initial state. so l β² is at most l + 1. the matrix elements vanish for l β² > l + 1, and the reverse matrix element is determined by hermiticity, so these vanish also when l β² < l β 1 : dipole transitions are forbidden with a change in angular momentum of more than one unit. = = = sum rules = = = the heisenberg equation of motion determines the matrix elements of p in the heisenberg basis from the matrix elements of x. p i j = m d d t x i j = i m ( e i β e j ) x i j, { \ displaystyle p _ { ij } = m { \ frac { d } { dt } } x _ { ij } = im \ left ( e _ { i } - e _ { j } \ right ) x _ { ij } \,, } which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements : j p i j x j i β x i j p j i = i j 2 m ( e i β e j ) | x i j | 2 = i. { \ displaystyle \ sum _ { j } p _ {
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j p i j x j i β x i j p j i = i j 2 m ( e i β e j ) | x i j | 2 = i. { \ displaystyle \ sum _ { j } p _ { ij } x _ { ji } - x _ { ij } p _ { ji } = i \ sum _ { j } 2m \ left ( e _ { i } - e _ { j } \ right ) \ left | x _ { ij } \ right | ^ { 2 } = i \,. } this yields a relation for the sum of the spectroscopic intensities to and from any given state, although to be absolutely correct, contributions from the radiative capture probability for unbound scattering states must be included in the sum : j 2 m ( e i β e j ) | x i j | 2 = 1. { \ displaystyle \ sum _ { j } 2m \ left ( e _ { i } - e _ { j } \ right ) \ left | x _ { ij } \ right | ^ { 2 } = 1 \,. } = = see also = = interaction picture bra β ket notation introduction to quantum mechanics heisenberg's entryway to matrix mechanics = = references = = = = further reading = = bernstein, jeremy ( 2005 ). " max born and the quantum theory ". american journal of physics. 73 ( 11 ). american association of physics teachers ( aapt ) : 999 β 1008. bibcode : 2005amjph.. 73.. 999b. doi : 10. 1119 / 1. 2060717. issn 0002 - 9505. max born the statistical interpretation of quantum mechanics. nobel lecture β december 11, 1954. nancy thorndike greenspan, " the end of the certain world : the life and science of max born " ( basic books, 2005 ) isbn 0 - 7382 - 0693 - 8. also published in germany : max born - baumeister der quantenwelt. eine biographie ( spektrum akademischer verlag, 2005 ), isbn 3 - 8274 - 1640 - x. max jammer the conceptual development of quantum mechanics ( mcgraw - hill, 1966 ) jagdish mehra and helmut rechenberg the historical development of quantum theory. volume 3. the formulation of matrix mechanics and its modifications 1925 β 1926. ( springer, 2001 )
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conceptual development of quantum mechanics ( mcgraw - hill, 1966 ) jagdish mehra and helmut rechenberg the historical development of quantum theory. volume 3. the formulation of matrix mechanics and its modifications 1925 β 1926. ( springer, 2001 ) isbn 0 - 387 - 95177 - 6 b. l. van der waerden, editor, sources of quantum mechanics ( dover publications, 1968 ) isbn 0 - 486 - 61881 - 1 aitchison, ian j. r. ; macmanus, david a. ; snyder, thomas m. ( 2004 ). " understanding heisenberg's " magical " paper of july 1925 : a new look at the calculational details ". american journal of physics. 72 ( 11 ). american association of physics teachers ( aapt ) : 1370 β 1379. arxiv : quant - ph / 0404009. doi : 10. 1119 / 1. 1775243. issn 0002 - 9505. s2cid 53118117. thomas f. jordan, quantum mechanics in simple matrix form, ( dover publications, 2005 ) isbn 978 - 0486445304 merzbacher, e ( 1968 ). " matrix methods in quantum mechanics ". am. j. phys. 36 ( 9 ) : 814 β 821. doi : 10. 1119 / 1. 1975154. = = external links = = an overview of matrix mechanics matrix methods in quantum mechanics heisenberg quantum mechanics archived 2010 - 02 - 16 at the wayback machine ( the theory's origins and its historical developing 1925 β 27 ) werner heisenberg 1970 cbc radio interview on matrix mechanics at mathpages
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folded optics is an optical system in which the beam is bent in a way to make the optical path much longer than the size of the system. this allows the resulting focal length of the objective to be greater than the physical length of the optical device. prismatic binoculars are a well - known example. an early conventional film camera ( 35 mm ) was designed by tessina that used the concept of folded optics. fold mirrors are used to direct infrared light within the optical path of the james webb space telescope. these optical fold mirrors are not to be confused with the observatory's deployable primary mirrors, which are folded inward to fit the telescope within the launch vehicle's payload fairing ; when deployed, these segments are part of the three - mirror anastigmat design's primary element and don't serve as fold mirrors in the optical sense. = = see also = = periscope lens also called " folded lens " = = references = = = = external links = = " origami lens slims high resolution cameras ". retrieved 28 july 2016. β origami lens
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in geology and related fields, a stratum ( pl. : strata ) is a layer of rock or sediment characterized by certain lithologic properties or attributes that distinguish it from adjacent layers from which it is separated by visible surfaces known as either bedding surfaces or bedding planes. prior to the publication of the international stratigraphic guide, older publications have defined a stratum as being either equivalent to a single bed or composed of a number of beds ; as a layer greater than 1 cm in thickness and constituting a part of a bed ; or a general term that includes both bed and lamina. related terms are substrate and substratum ( pl. substrata ), a stratum underlying another stratum. = = characteristics = = typically, a stratum is generally one of a number of parallel layers that lie one upon another to form enormous thicknesses of strata. the bedding surfaces ( bedding planes ) that separate strata represent episodic breaks in deposition associated either with periodic erosion, cessation of deposition, or some combination of the two. stacked together with other strata, individual stratum can form composite stratigraphic units that can extend over hundreds of thousands of square kilometers of the earth's surface. individual stratum can cover similarly large areas. strata are typically seen as bands of different colored or differently structured material exposed in cliffs, road cuts, quarries, and river banks. individual bands may vary in thickness from a few millimeters to several meters or more. a band may represent a specific mode of deposition : river silt, beach sand, coal swamp, sand dune, lava bed, etc. = = types = = in the study of rock and sediment strata, geologists have recognized a number of different types of strata, including bed, flow, band, and key bed. a bed is a single stratum that is lithologically distinguishable from other layers above and below it. in the classification hierarchy of sedimentary lithostratigraphic units, a bed is the smallest formal unit. however, only beds that are distinctive enough to be useful for stratigraphic correlation and geological mapping are customarily given formal names and considered formal lithostratigraphic units. the volcanic equivalent of a bed, a flow, is a discrete extrusive volcanic stratum or body distinguishable by texture, composition, or other objective criteria. as in case of a bed, a flow should only be designated and named as a formal lit
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a bed, a flow, is a discrete extrusive volcanic stratum or body distinguishable by texture, composition, or other objective criteria. as in case of a bed, a flow should only be designated and named as a formal lithostratigraphic unit when it is distinctive, widespread, and useful for stratigraphic correlation. a band is a thin stratum that is distinguishable by a distinctive lithology or color and is useful in correlating strata. finally, a key bed, also called a marker bed, is a well - defined, easily identifiable stratum or body of strata that has sufficiently distinctive characteristics, such as lithology or fossil content, to be recognized and correlated during geologic field or subsurface mapping. = = gallery = = = = see also = = archaeological horizon bed ( geology ) geological formation geologic unit lamination ( geology ) law of superposition = = references = = = = external links = = geowhen database
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surroundings, or environs is an area around a given physical or geographical point or place. the exact definition depends on the field. surroundings can also be used in geography ( when it is more precisely known as vicinity, or vicinage ) and mathematics, as well as philosophy, with the literal or metaphorically extended definition. in thermodynamics, the term ( and its synonym, environment ) is used in a more restricted sense, meaning everything outside the thermodynamic system. often, the simplifying assumptions are that energy and matter may move freely within the surroundings, and that the surroundings have a uniform composition. = = see also = = the dictionary definition of surroundings at wiktionary distance environment ( biophysical ) environment ( systems ) neighbourhood ( mathematics ) social environment proxemics
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chinese computational linguistics is a subset of computational linguistics ; it is the scientific study and information processing of the chinese language by means of computers. the purpose is to obtain a better understanding of how the language works and to bring more convenience to language applications. the term chinese computational linguistics is often employed interchangeably with chinese information processing, though the former may sound more theoretical while the latter more technical. rather than introducing computational linguistics in a general sense, this article will focus on the unique issues involved with implementing the chinese language compared to other languages. the contents include chinese character information processing, word segmentation, proper noun recognition, natural language understanding and generation, corpus linguistics, and machine translation. = = chinese character information processing = = chinese character information technology ( it ) is the technology of computer processing of chinese characters. while the english writing system makes use of a few dozen different characters, chinese language needs a much larger character set. there are over ten thousand characters in the xinhua dictionary. in the unicode multilingual character set of 149, 813 characters, 98, 682 ( about 2 / 3 ) are chinese characters. this means that computer processing of chinese characters is the most intensive among all languages. = = = chinese character input = = = computer input of chinese characters is more complicated than languages which have simpler character systems. for example, the english language is written with 26 letters and a handful of other characters, and each character is assigned to a key on the keyboard. theoretically, chinese characters could be input in a similar way, but this approach is impractical for most applications due to the number of characters ; it would require a massive keyboard with thousands of keys, and the user would find it difficult and time - consuming to locate individual characters on the keyboard. an alternative method is to use the english keyboard layout, and encode each chinese character in the english characters ; this is the predominant method of chinese character input today. sound - based encoding is normally based on an existing latin character scheme for chinese phonetics, such as the pinyin scheme for mandarin chinese or putonghua, and the jyutping scheme for the cantonese dialect. the input code of a chinese character is its pinyin letter string followed by an optional number representing the tone. for example, the putonghua pinyin input code of ι¦ ( hong kong ) is " xianggang " or " xiang1gang3 ", and the cantonese jyutping code is " hoenggong " or " hoeng1gong2 ", all of which can be easily
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code of ι¦ ( hong kong ) is " xianggang " or " xiang1gang3 ", and the cantonese jyutping code is " hoenggong " or " hoeng1gong2 ", all of which can be easily input via an english keyboard. a chinese character can alternatively be input by form - based encoding. most chinese characters can be divided into a sequence of components each of which is in turn composed of a sequence of strokes in writing order. there are a few hundred basic components, much less than the number of characters. by representing each component with an english letter and putting them in writing order of the character, the input method creator can get a letter string ready to be used as an input code on the english keyboard. of course the creator can also design a rule to select representative letters from the string if it is too long. for example, in the cangjie input method, character ( border ) is encoded as " ngmwm " corresponding to components " ε δΈ η° δΈ ", with some components omitted. popular form - based encoding methods include wubi ( δΊ ) in the mainland and cangjie ( ) in taiwan and hong kong. the most important feature of intelligent input is the application of contextual constraints for candidate character selection. for example, on microsoft pinyin, when the user types input code " daxuejiaoshou ", he / she will get " 倧 ε¦ / 倧 " ( university professor ), when types " daxuepiaopiao " the computer will suggest " 倧 / 倧 " ( heavy snow flying ). though the non - toned pinyin letters of 倧 ε¦ and 倧 are both " daxue ", the computer can make a reasonable selection based on the subsequent words. = = = chinese character encoding for information interchange = = = inside the computer each character is represented by an internal code. when a character is sent between two machines, it is in information interchange code. nowadays, information interchange codes, such as ascii and unicode, are often directly employed as internal codes. the first gb chinese character encoding standard is gb2312, which was released by the prc in 1980. it includes 6, 763 chinese characters, with 3, 755 frequently - used ones sorted by pinyin, and the rest by radicals ( indexing components ). gb2312 was designed for simplified chinese characters. traditional characters which have been simplified are not covered. the code of a
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characters, with 3, 755 frequently - used ones sorted by pinyin, and the rest by radicals ( indexing components ). gb2312 was designed for simplified chinese characters. traditional characters which have been simplified are not covered. the code of a character is represented by a two - byte hexadecimal number, for instance, the gb codes of ι¦ ( hong kong ) are cfe3 and b8db respectively. gb2312 is still in use on some computers and the www, though newer versions with extended character sets, such as gb13000. 1 and gb18030, have been released. the latest version of gb encoding is gb18030, which supports both simplified and traditional chinese characters, and is consistent with the unicode character set. the standard of big5 encoding was designed by five big it companies in taiwan in the early 1980s, and has been the de facto standard for representing traditional chinese in computers ever since. big5 is popularly used in taiwan, hong kong and macau. the original big5 standard included 13, 053 chinese characters, with no simplified characters of the mainland. each character is encoded with a two byte hexadecimal code, for example, ι¦ ( adbb ) ( b4e4 ) ιΎ ( c073 ). chinese characters in the big5 character set are arranged in radical order. extended versions of big5 include big - 5e and big5 - 2003, which include some simplified characters and hong kong cantonese characters. the full version of the unicode standard represents a character with a 4 - byte digital code, providing a huge encoding space to cover all characters of all languages in the world. the basic multilingual plane ( bmp ) is a 2 - byte kernel version of unicode with 2 ^ 16 = 65, 536 code points for important characters of many languages. there are 27, 522 characters in the cjkv ( china, japan, korea and vietnam ) ideographs area, including all the simplified and traditional chinese characters in gb2312 and big5 traditional. in unicode 15. 0, there is a multilingual character set of 149, 813 characters, among which overs 98, 682 ( about 2 / 3 ) are chinese sorted by kangxi radicals. even very rarely - used characters are available. for example : h ( 0048 ) k ( 004b ), ι¦ ( 9999 ), ( 6e2f ), ιΎ ( 9f8d ), ( 9
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. even very rarely - used characters are available. for example : h ( 0048 ) k ( 004b ), ι¦ ( 9999 ), ( 6e2f ), ιΎ ( 9f8d ), ( 9f99 ), ( 9f96 ), ( 9f98 ), ( 2a6a5 ). unicode is becoming more and more popular. it is reported that utf - 8 ( unicode ) is used by 98. 1 % of all the websites. it is widely believed that unicode will ultimately replace all other information interchange codes and internal codes, and there will be no more code confusing. = = = chinese character output = = = like english and other languages, chinese characters are output on printers and screens in different fonts and styles. the most popular chinese fonts are the song ( ), kai ( ), hei ( ) and fangsong ( ) families. fonts appear in different sizes. in addition to the international measurement system of points, chinese characters are also measured by size numbers ( called zihao, ) invented by an american for chinese printing in 1859. = = word segmentation = = it is straightforward to recognize words in english text because they are separated by spaces. however, chinese words are not separated by any boundary markers. hence, word segmentation is the first step for text analysis of chinese. for example, δΈ ζ δΏ‘ ε¦ ( chinese original text ) δΈ ζ δΏ‘ ε¦ ( word - segmented text ) chinese information journal ( word - by - word english translation ) journal of chinese information processing ( english name ) chinese word segmentation on a computer is carried out by matching characters in the chinese text against a lexicon ( list of chinese words ) forwardly from the beginning of the sentence or backwardly from the end. there are two kinds of segmentation ambiguities : the intersection - type ( ) and polynomial type ( ) ). typically an intersection ambiguity is in the format of abc, where a, ab, bc and c are all words in the lexicon. it is possible to divide the original character string into word ab followed by c, or a followed by bc. for example β ηΎ ε½ δΌ β may mean β ηΎ ε½ δΌ β ( the us parliament ) or β ηΎ ε½ δΌ β ( the us can / will ). the most common form
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into word ab followed by c, or a followed by bc. for example β ηΎ ε½ δΌ β may mean β ηΎ ε½ δΌ β ( the us parliament ) or β ηΎ ε½ δΌ β ( the us can / will ). the most common form of polynomial segmentation ambiguity is ab, where a, b, and ab are all words. that means the character string can be regarded as one single word or be divided into two. for example, string β β in the following sentences : ( 1 ) δΈ γ you can sit down. you can sit down. ( 2 ) γ you can take them as example. you can take them as an example. word segmentation ambiguities can be resolved with contextual information, using linguistic rules and probability of word co - locations derived from chinese corpora. usually longer words matching are more reliable. the correctness rate of automatic word segmentation has reached 95 %. however there will be no guarantee of 100 % percent correctness in the foreseeable future, because that will involve a complete understanding of the text. an alternative solution is to encourage people to write in a word segmented way, like the case in english. but that does not means computer word segmentation will no longer be needed, because even in english, word segmentation is required for speech analysis. = = proper noun recognition = = a proper noun is the name of a person, a place, an institution, etc. and is written in english with the initial letter of each word capitalized, for example, " mr. john nealon ", " america " and " cambridge university ". however, chinese proper nouns are usually not marked in any style. recognition of names of people and place in chinese text can be supported by a list of names. however such a list can never be complete, considering the huge number of places and people all over the world, not to mention their dynamic feature of coming, changing and going. and there are names similar to non - proper nouns. for example, there is a town named ζ° ( minzhong ) in southern china, which is also a common noun meaning " people ". therefore, recognition of names of people and place has to make use of their distinguishing features in internal composition and external context. corpora with proper nouns annotated can also serve as useful reference. a people's name not found in the dictionary can be recognized with a list of surnames and titles, for example "
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their distinguishing features in internal composition and external context. corpora with proper nouns annotated can also serve as useful reference. a people's name not found in the dictionary can be recognized with a list of surnames and titles, for example " 倧 ζΉ η ", ζ ", where ( zhang ) and ζ ( li ) are chinese surnames, and η ( mr. ) and ( manager ) are titles. in 倧 ζΉ, 倧 ζΉ can be successfully recognized as a person's name by the rule that a chinese given name normally follow the surname and consists of 1 or 2 characters, and the fact that people can speak ( ). names of place also have characteristics useful for computer recognition. for example, in " εΉΏ η δΈ ε±± ζ° ι ", component words η ( province ), ( city ) and ι ( town ) are end markers of place names, while ( in, at, on ) is a preposition frequently appearing in front of a location. the correctness rate of computer recognition has reached around 90 % for persons'names and 95 % for place names. = = journals and proceedings = = journal of chinese information processing ( http : / / jcip. cipsc. org. cn / cn / home ) international journal of computational linguistics and chinese language processing ( ijclclp ) ( https : / / www. aclclp. org. tw / journal / index. php ) china national conference on chinese computational linguistics ( https : / / link. springer. com / conference / cncl ) rocling proceedings ( https : / / www. aclclp. org. tw / pub _ proce. php ) = = see also = = computational linguistics natural language processing chinese language chinese characters chinese character it = = references = = = = = citations = = = = = = works cited = = =
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in computational complexity theory, a language b ( or a complexity class b ) is said to be low for a complexity class a ( with some reasonable relativized version of a ) if ab = a ; that is, a with an oracle for b is equal to a. such a statement implies that an abstract machine which solves problems in a achieves no additional power if it is given the ability to solve problems in b at unit cost. in particular, this means that if b is low for a then b is contained in a. informally, lowness means that problems in b are not only solvable by machines which can solve problems in a, but are β easy to solve β. an a machine can simulate many oracle queries to b without exceeding its resource bounds. results and relationships that establish one class as low for another are often called lowness results. the set of languages low for a complexity class a is denoted low ( a ). = = classes that are low for themselves = = several natural complexity classes are known to be low for themselves. such a class is sometimes called self - low. scott aaronson calls such a class a physical complexity class. note that being self - low is a stronger condition than being closed under complement. informally, a class being low for itself means a problem can use other problems in the class as unit - cost subroutines without exceeding the power of the complexity class. the following classes are known to be self - low : p is self - low ( that is, pp = p ) because polynomial - time algorithms are closed under composition : a polynomial - time algorithm can make polynomially many queries to other polynomial - time algorithms, while retaining a polynomial running time. pspace ( with restricted oracle access mechanism ) is also self - low, and this can be established by exactly the same argument. l is self - low because it can simulate log space oracle queries in log space, reusing the same space for each query. nc is also self - low for the same reason. zpp is also low for itself and the same arguments almost work for bpp, but one has to account for errors, making it slightly harder to show that bpp is low for itself. similarly, the argument for bpp almost goes through for bqp, but we have to additionally show that quantum queries can be performed in coherent superposition. both parity p ( β p { \ displaystyle { \ oplus } { \ hbox { p } } } ) and bp
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bqp, but we have to additionally show that quantum queries can be performed in coherent superposition. both parity p ( β p { \ displaystyle { \ oplus } { \ hbox { p } } } ) and bpp are low for themselves. these were important in showing toda's theorem. np β© conp is low for itself. every class which is low for itself is closed under complement, provided that it is powerful enough to negate the boolean result. this implies that np isn't low for itself unless np = co - np, which is considered unlikely because it implies that the polynomial hierarchy collapses to the first level, whereas it is widely believed that the hierarchy is infinite. the converse to this statement is not true. if a class is closed under complement, it does not mean that the class is low for itself. an example of such a class is exp, which is closed under complement, but is not low for itself. = = classes that are low for other complexity classes = = some of the more complex and famous results regarding lowness of classes include : bqp is low for pp in other words, a program based around taking the majority decision of an unbounded number of iterations of a poly - time randomized algorithm can easily solve all the problems that a quantum computer can solve efficiently. the graph isomorphism problem is low for parity p ( β p { \ displaystyle { \ oplus } { \ hbox { p } } } ). this means that if we can determine whether an np machine has an even or odd number of accepting paths, we can easily solve graph isomorphism. in fact, it was later shown that graph isomorphism is low for zppnp. amplified pp is low for pp. np β© conp is equal to the set of languages low for np, i. e., low ( np ) = np β© conp. am β© coam is low for zppnp. = = applications = = lowness is particularly valuable in relativization arguments, where it can be used to establish that the power of a class does not change in the " relativized universe " where a particular oracle machine is available for free. this allows us to reason about it in the same manner we normally would. for example, in the relativized universe of bqp, pp is still closed under union and intersection. it's also useful when seeking to expand the power of a machine with oracle
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about it in the same manner we normally would. for example, in the relativized universe of bqp, pp is still closed under union and intersection. it's also useful when seeking to expand the power of a machine with oracles, because lowness results determine when the machine's power remains the same. = = see also = = low ( computability ) = = references = =
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social preferences describe the human tendency to not only care about one's own material payoff, but also the reference group's payoff or / and the intention that leads to the payoff. social preferences are studied extensively in behavioral and experimental economics and social psychology. types of social preferences include altruism, fairness, reciprocity, and inequity aversion. the field of economics originally assumed that humans were rational economic actors, and as it became apparent that this was not the case, the field began to change. the research of social preferences in economics started with lab experiments in 1980, where experimental economists found subjects'behavior deviated systematically from self - interest behavior in economic games such as ultimatum game and dictator game. these experimental findings then inspired various new economic models to characterize agent's altruism, fairness and reciprocity concern between 1990 and 2010. more recently, there are growing amounts of field experiments that study the shaping of social preference and its applications throughout society. = = determinants : nature vs. nurture = = social preferences are thought to come about by two different methods : nature and nurture. whilst nature encompasses biological makeup and genetics, nurture refers to the social environment in which one develops. the majority of literature would support that β nature β influences social preferences more strongly whereas there is still research to support the heavy influence of sociocultural factors. some of these factors include social distance between economic agents, the distribution of economic resources, social norms, religion and ethnicity. = = importance = = an understanding of social preferences and the disparity that occurs across individuals and groups can help create models that better represent reality. within the financial sector, research supports the existence of a positive relationship between the elements of trust and reciprocity to economic growth as observed in a reduction of defaults in lending programs as well as the effectiveness of government and central banking policy. the well - functioning of social preferences may assist society in paving the way to new developments through a decrease in the likelihood of market failures as well as a reduction in transaction costs. society may also utilize social preferences to increase the flow of information, transparency and accountability. = = formation = = biologists, social psychologists, and economists have proposed theories and documented evidence on the formation of social preferences over both the long run and the short run. the various theories explaining the formation and development of social preferences may be explained from a biological, cognitive and sociocultural perspective and are detailed as follows. = = = biological evolution = = = = = =
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over both the long run and the short run. the various theories explaining the formation and development of social preferences may be explained from a biological, cognitive and sociocultural perspective and are detailed as follows. = = = biological evolution = = = = = = = kin selection = = = = kin selection is an evolutionary strategy where some specific behavioral traits are favored to benefit close relatives'reproduction. hence, behavior that appears altruistic can align with the theory of the selfish gene. kin selection can explain altruistic behavior towards close relatives even at the cost of their own's survival, as long as one's sacrifice can help preserve a greater amount of the same genes in close relatives. for example, worker bees can die from attacking their predators in order to help preserve other bees'genes. = = = = reciprocity selection = = = = reciprocity selection suggests that one's altruistic act may evolve from the anticipation of future reciprocal altruistic behavior from others. an application of reciprocity selection in game theory is the tit - for - tat strategy in prisoner's dilemma, which is the strategy that the player cooperate at the initial encounter, and then follow the opponent's behavior on the previous encounter. robert axelrod and w. d. hamilton showed that tit - for - tat strategy can be an evolutionary stable strategy in a population where the probability of repeated encounters between two persons in a population is above a certain threshold. = = = social learning = = = psychologist albert bandura proposed that children learn about pro - social and moral behavior by imitating other pro - social models, including parents, other adults, and peers. there are also economic models proposing that parents transmit their social preferences to their children by demonstrating their own pro - social behavior. bandura conducted extensive psychological experimentation into the extent to which children will emulate aggressive behaviour by exposing them to models displaying behaviour before observing the child's behaviour once left alone. however, empirical support for parents'role in fostering pro - social behavior is mixed. for example, some researchers found a positive relation between the parent's use of induction and children's pro - social behavior, and others found no correlation between parent's adoption of punitive techniques and children's pro - social behavior. regarding other sources of social learning, recent field experiments have provided causal evidences for positive effects of school program and mentoring program on forming social preferences, and these research suggested that social interaction, prosocial role models as well as cultural transmission from
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behavior. regarding other sources of social learning, recent field experiments have provided causal evidences for positive effects of school program and mentoring program on forming social preferences, and these research suggested that social interaction, prosocial role models as well as cultural transmission from family and school are potential mechanisms. = = = cognitive factors = = = psychologist jean piaget was among the first to propose that cognitive development is a prerequisite in moral judgment and behavior. he argued for the importance of social interaction with others rather than learning in moral development, which requires the understanding of both rules and others'behavior. other important cognitive skills in fostering pro - social behavior include perspective taking and moral reasoning, which are supported by most empirical evidence. = = evidences of social preferences = = = = = experimental evidences = = = many initial evidences of social preferences came from lab experiments where subjects play economic games with others. however, many research found that subjects'behavior robustly and systematically deviated from the prediction from self - interest hypothesis, but could be explained by social preferences including altruism, inequity aversion and reciprocity. the ultimatum game, the dictator game, the trust game and the gift - exchange game are exercises that used to understand social preferences and their implications. = = = the ultimatum game = = = ultimatum game is one of the first experiments that shows self - interest hypothesis fails to predict people's behavior. in this game, the first mover proposes a split of a fixed amount, and the second mover decides to accept or reject the offer. if the second mover accepts the offer, the final payoff is exactly determined by the offer. however, if the second mover rejects the offer, both subjects will have zero payoff. contrary to the self - interest hypothesis's prediction that the first mover will propose zero amount and the second mover will accept the offer, experimenters found proposers will typically offer 25 % - 50 % of the fixed amount, and responders tend to reject the offer when the split is below 20 %. = = = the dictator game = = = a relevant game is dictator game, where one subject proposes the split of a fixed amount and the other subject is only allowed to accept the offer. the dictator game helps to isolate pure altruism from the strategic concern of the first mover ( i. e. the first mover proposes a larger share to second mover to avoid second mover's rejection ) in the ultimatum game. in this game,
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to isolate pure altruism from the strategic concern of the first mover ( i. e. the first mover proposes a larger share to second mover to avoid second mover's rejection ) in the ultimatum game. in this game, the average share decreases to 20 % of the fixed amount, however, more than 60 % of the subjects still propose a positive offer. = = = the trust and gift - exchange games = = = two other games, trust game ( also called investment game ) and gift - exchange game provide evidence for reciprocal behavior. in the trust game, the first mover is endowed with a fixed amount c, and decides the amount of money b to pass on to the second mover. this amount is multiplied by a factor of k when it reaches the second mover, and then the second mover decides how much of this amount ( kb ) is returned to the first mover. while self - interest model predicts no transfer and no return, experimenters found that first mover typically transfers roughly 50 % of endowment and responder's return increases with the transfer. in gift exchange game, the first mover proposes some offer to the second mover and asks for certain effort level from the second mover, and then the second mover decides his / her effort that is costly but can increase first mover's payoff. also contrary to the self - interest prediction, first mover's offer in experiments is usually greater than zero, and the second mover's effort level increases with offer. prisoner's dilemma and its generalized game, public goods game also provide indirect evidence for social preference, and there are many evidences of conditional cooperation among subjects. the prisoner's dilemma game illustrates the fact that the process of cooperation itself can create incentives to not cooperate. each player may make a contribution to a notional public good before all contributions are summed and distributed to players where the " selfish " players are given the opportunity to " free ride ". this game depicts the way in which consumers will tend to free ride without active intervention yet also the way consumers will change their behaviour with experience. = = = field evidences = = = many field evidences documented agent's fairness and reciprocal concern. for example, daniel kahneman, jack knetsch and richard thaler found that the concern for fairness constrains firm's profit seeking behavior ( e. g. raise price after an increase in demand ). many field experiments examine relative pay concerns and reciprocity in
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, jack knetsch and richard thaler found that the concern for fairness constrains firm's profit seeking behavior ( e. g. raise price after an increase in demand ). many field experiments examine relative pay concerns and reciprocity in work settings. for example, economists uri gneezy and john list conducted field experiments where subjects were hired for a typing job and for door - to - door fundraising and found subjects exerted larger effort level in group with a higher wage. however, this positive reciprocity was short lived. researchers have also found that positive reciprocity is smaller than negative reciprocity. in another study, job applicants were hired to catalog books for 6 hours with a pronounced wage, but applicants were later informed with either wage increase or wage cut. researchers found the decrease in effort in wage cut group was larger than the increase in effort in wage increase group. however, positive reciprocity did not extend to other activities ( volunteering to work for one more hour ). = = economic models = = existing models of social preferences can be divided into two types : distributive preferences and reciprocal preferences. distributive preferences are the preferences over the distribution and total magnitude of the payoff among the reference groups, including altruism and spitefulness, fairness and inequity aversion, and efficiency concern. reciprocal preferences reflect agent's concern over the intention of other's behavior. = = = pure altruism, warm glow, and spitefulness = = = pure altruism in economic models represents an agent's concern on other's well - being. a person exhibits altruistic preference if this person's utility increases with other's payoff. a related economic model is impure altruism, or warm - glow, where individuals feel good ( i. e. gain a " warm - glow " utility ) from doing something good without caring about other's payoff. spitefulness or envy preference is the opposite of pure altruism. in this instance, an agent's utility decreases with other's payoff. = = = fairness and inequity aversion = = = fairness and inequity aversion models capture the agent's concern on the fair distribution of payoffs across agents and especially the aversion to payoff differences. in the fehr - schmidt model, an agent compares his payoff to each other opponents in the group. however, the agent's utility decreases
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on the fair distribution of payoffs across agents and especially the aversion to payoff differences. in the fehr - schmidt model, an agent compares his payoff to each other opponents in the group. however, the agent's utility decreases with both positive and negative payoff differences between self and each other opponent in the reference group. moreover, the agent dislikes payoff disadvantage more than payoff advantage. hence, the agent presents altruistic behavior towards others when agent is better off than others, and displays spiteful behavior when agent is worse off than others. = = = efficiency concern and quasi - maximin preferences = = = economists gary charness and matthew rabin found that in some cases, agents prefer more efficient outcomes ( i. e. outcome with larger social welfare ) than more equal outcomes and they developed a model where agents'utility is a convex combination of own's material payoff and the social welfare. moreover, they assumed agents have quasi - maximin preferences, meaning that agents'care on social welfare includes the minimum payoff among agents as well as the total payoff for all agents in the group. however, the agent will care less about others'payoff if other is better off than self. = = = reciprocity = = = agent has the motivation to reciprocate towards both kind and unfair behavior. rabin ( 1993 )'s model is one of the earliest model that characterizes reciprocal behavior. in this model, the agent's payoff depends on the other opponent, and agent forms belief of the other opponent's kindness, which is based on the difference between the actual payoff that agent receives and the fair payoff. agents will reciprocate positively if he / she perceives the other individual's behavior as kind and fair and respond negatively if he / she perceives it as unfair. other researchers further generalize rabin ( 1993 )'s model by studying repeated interactions in n - person extensive form games, and also by including inequity aversion into agent's preference. charness and rabin also augmented their quasi - maximin preference with reciprocity concern. = = economic applications = = researchers have argued that the failure of recognizing social preference will lead to a biased understanding of much important economic behavior. three important ways in which social preferences are applied to real world economics are explained below. = = = understanding cooperation = = = research on social preferences showed that reciprocal and inequity averse individuals can cooperate if they
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of much important economic behavior. three important ways in which social preferences are applied to real world economics are explained below. = = = understanding cooperation = = = research on social preferences showed that reciprocal and inequity averse individuals can cooperate if they are sure that others will cooperate too and can punish the free riders. this has implications for designing proper social mechanisms to solve the free - riding problem. for example, fischbacher and gachter found that, through public goods experimentation, people contribute more to public goods than self - interest alone would suggest. this provides support for the notion of voluntary contribution. = = = design of economic incentive = = = accounting employee's reciprocity and fairness concerns can help design better contracts ( e. g. trust contract, bonus contract ) to enhance employee's effort and to solve firm's agency problems. moreover, the design of relative pay in the workplace can affect employee's job satisfaction and well - being. research on social preference has also facilitated the understanding of monetary incentives'crowding - out effect. = = = design of social policies = = = the distributive and reciprocal preferences mentioned previously are integral in good government and the upholding of ethical standards. without the existence of these preferences, it is unlikely that society would achieve desirable allocations of economic goods due to self - interest and the " free rider " problem. research and experimentation into social preferences assists in the design of optimal incentives used in public policy. accounting individual's fairness concerns can affect the design of the social policies, especially for redistributive policies. in addition, reciprocal preferences can affect people's evaluation of different policies towards the poor depending on the individual's belief that whether the poor are deserving or undeserving. = = see also = = altruism homo economicus homo reciprocans inequity aversion moral psychology neuroeconomics Β§ social decision making norm of reciprocity β repayment in kind ophelimity pro - social behavior reciprocity social value orientation warm - glow giving = = references = =
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in astrophysics and condensed matter physics, electron degeneracy pressure is a quantum mechanical effect critical to understanding the stability of white dwarf stars and metal solids. it is a manifestation of the more general phenomenon of quantum degeneracy pressure. the term " degenerate " here is not related to degenerate energy levels, but to fermi β dirac statistics close to the zero - temperature limit ( temperatures much smaller than the fermi temperature, which for metals is about 10000 k. ) in metals and in white dwarf stars, electrons can be modeled as a gas of non - interacting electrons confined to a finite volume. although there are strong electromagnetic forces between the negatively charged electrons, these forces are approximately balanced by the positive nuclei and so can be neglected in the simplest models. the pressure exerted by the electrons is related to their kinetic energy. the degeneracy pressure is most prominent at low temperatures : if electrons were classical particles, the movement of the electrons would cease at absolute zero and the pressure of the electron gas would vanish. however, since electrons are quantum mechanical particles that obey the pauli exclusion principle, no two electrons can occupy the same state, and it is not possible for all the electrons to have zero kinetic energy. instead, the confinement makes the allowed energy levels quantized, and the electrons fill them from the bottom upwards. if many electrons are confined to a small volume, on average the electrons have a large kinetic energy, and a large pressure is exerted. : 32 β 39 in white dwarf stars, the positive nuclei are completely ionized β disassociated from the electrons β and closely packed β a million times more dense than the sun. at this density gravity exerts immense force pulling the nuclei together. this force is balanced by the electron degeneracy pressure keeping the star stable. in metals, the positive nuclei are partly ionized and spaced by normal interatomic distances. gravity has negligible effect ; the positive ion cores are attracted to the negatively charged electron gas. this force is balanced by the electron degeneracy pressure. : 410 = = from the fermi gas theory = = electrons are members of a family of particles known as fermions. fermions, like the proton or the neutron, follow pauli's principle and fermi β dirac statistics. in general, for an ensemble of non - interacting fermions, also known as a fermi gas, each particle can be treated independently with a single - fermion energy
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follow pauli's principle and fermi β dirac statistics. in general, for an ensemble of non - interacting fermions, also known as a fermi gas, each particle can be treated independently with a single - fermion energy given by the purely kinetic term, e = p 2 2 m, { \ displaystyle e = { \ frac { p ^ { 2 } } { 2m } }, } where p is the momentum of one particle and m its mass. every possible momentum state of an electron within this volume up to the fermi momentum pf being occupied. the degeneracy pressure at zero temperature can be computed as p = 2 3 e tot v = 2 3 p f 5 10 Ο 2 m 3, { \ displaystyle p = { \ frac { 2 } { 3 } } { \ frac { e _ { \ text { tot } } } { v } } = { \ frac { 2 } { 3 } } { \ frac { p _ { \ text { f } } ^ { 5 } } { 10 \ pi ^ { 2 } m \ hbar ^ { 3 } } }, } where v is the total volume of the system and etot is the total energy of the ensemble. specifically for the electron degeneracy pressure, m is substituted by the electron mass me and the fermi momentum is obtained from the fermi energy, so the electron degeneracy pressure is given by p e = ( 3 Ο 2 ) 2 / 3 2 5 m e Ο e 5 / 3, { \ displaystyle p _ { \ text { e } } = { \ frac { ( 3 \ pi ^ { 2 } ) ^ { 2 / 3 } \ hbar ^ { 2 } } { 5m _ { \ text { e } } } } { \ rho _ { \ text { e } } } ^ { 5 / 3 }, } where Οe is the free electron density ( the number of free electrons per unit volume ). for the case of a metal, one can prove that this equation remains approximately true for temperatures lower than the fermi temperature, about 106 kelvins. when particle energies reach relativistic levels, a modified formula is required. the relativistic degeneracy pressure is proportional to Οe4 / 3. = = examples = = = = = metals = = = for the case of electrons in crystalline solid, several approximations
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, a modified formula is required. the relativistic degeneracy pressure is proportional to Οe4 / 3. = = examples = = = = = metals = = = for the case of electrons in crystalline solid, several approximations are carefully justified to treat the electrons as independent particles. usual models are the free electron model and the nearly free electron model. in the appropriate systems, the free electron pressure can be calculated ; it can be shown that this pressure is an important contributor to the compressibility or bulk modulus of metals. : 39 = = = white dwarfs = = = electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the chandrasekhar limit ( 1. 44 solar masses ). this is the pressure that prevents a white dwarf star from collapsing. a star exceeding this limit and without significant thermally generated pressure will continue to collapse to form either a neutron star or black hole, because the degeneracy pressure provided by the electrons is weaker than the inward pull of gravity. = = see also = = exchange interaction fermi level bose β einstein condensate nuclear density = = references = =
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an axial fan is a type of fan that causes gas to flow through it in an axial direction, parallel to the shaft about which the blades rotate. the flow is axial at entry and exit. the fan is designed to produce a pressure difference, and hence force, to cause a flow through the fan. factors which determine the performance of the fan include the number and shape of the blades. fans have many applications including in wind tunnels and cooling towers. design parameters include power, flow rate, pressure rise and efficiency. axial fans generally comprise fewer blades ( two to six ) than centrifugal fans. axial fans commonly have larger radius and lower speed ( Ο ) than ducted fans ( esp. at similar power. stress proportional to r ^ 2 ). = = calculation of parameters = = since the calculation cannot be done using the inlet and outlet velocity triangles, which is not the case in other turbomachines, calculation is done by considering a mean velocity triangle for flow only through an infinitesimal blade element. the blade is divided into many small elements and various parameters are determined separately for each element. there are two theories that solve the parameters for axial fans : slipstream theory blade element theory = = = slipstream theory = = = in the figure, the thickness of the propeller disc is assumed to be negligible. the boundary between the fluid in motion and fluid at rest is shown. therefore, the flow is assumed to be taking place in an imaginary converging duct where : d = diameter of the propeller disc. ds = diameter at the exit. in the figure, across the propeller disc, velocities ( c1 and c2 ) cannot change abruptly across the propeller disc as that will create a shockwave but the fan creates the pressure difference across the propeller disc. c 1 = c 2 = c { \ displaystyle c _ { \ rm { 1 } } = c _ { \ rm { 2 } } = c } and p 1 = p 2 { \ displaystyle p _ { \ rm { 1 } } \ neq p _ { \ rm { 2 } } } the area of the propeller disc of diameter d is : a = Ο d 2 4 { \ displaystyle a = { \ frac { \ pi d ^ { 2 } } { 4 } } } the mass flow rate across the propeller is : m = Ο a c { \ displaystyle { \ dot { m } } = { \ rho ac } } since thrust is change in mass multiplied by the velocity
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} { 4 } } } the mass flow rate across the propeller is : m = Ο a c { \ displaystyle { \ dot { m } } = { \ rho ac } } since thrust is change in mass multiplied by the velocity of the mass flow i. e., change in momentum, the axial thrust on the propeller disc due to change in momentum of air, which is : f x = m ( c s β c u ) = Ο a c ( c s β c u ) { \ displaystyle f _ { \ rm { x } } = { \ dot { m } } { ( c _ { \ rm { s } } - c _ { \ rm { u } } ) } = { \ rho ac } { ( c _ { \ rm { s } } - c _ { \ rm { u } } ) } } applying bernoulli's principle upstream and downstream : p a + 1 2 Ο c u 2 = p 1 + 1 2 Ο c 2 p a + 1 2 Ο c s 2 = p 2 + 1 2 Ο c 2 { \ displaystyle { \ begin { aligned } p _ { a } + { \ frac { 1 } { 2 } } { \ rho c _ { u } ^ { 2 } } & = p _ { 1 } + { \ frac { 1 } { 2 } } { \ rho c ^ { 2 } } \ \ p _ { a } + { \ frac { 1 } { 2 } } { \ rho c _ { s } ^ { 2 } } & = p _ { 2 } + { \ frac { 1 } { 2 } } { \ rho c ^ { 2 } } \ end { aligned } } } on subtracting the above equations : p 2 β p 1 = 1 2 Ο ( c s 2 β c u 2 ) { \ displaystyle p _ { 2 } - p _ { 1 } = { \ frac { 1 } { 2 } } \ rho ( c _ { s } ^ { 2 } - c _ { u } ^ { 2 } ) } thrust difference due to pressure difference is projected area multiplied by the pressure difference. axial thrust due to pressure difference comes out to be : f x = a ( p 2 β p 1 ) = 1 2 Ο a ( c s 2 β c u 2 ) { \ displaystyle f _ { x } = a ( p _ { 2 } -
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pressure difference comes out to be : f x = a ( p 2 β p 1 ) = 1 2 Ο a ( c s 2 β c u 2 ) { \ displaystyle f _ { x } = a ( p _ { 2 } - p _ { 1 } ) = { \ frac { 1 } { 2 } } \ rho a \ left ( c _ { s } ^ { 2 } - c _ { u } ^ { 2 } \ right ) } comparing this thrust with the axial thrust due to change in momentum of air flow, it is found that : c = c s + c u 2 { \ displaystyle c = { \ frac { c _ { s } + c _ { u } } { 2 } } } a parameter'a'is defined such that - c = ( 1 + a ) c u { \ displaystyle c = ( 1 + a ) c _ { u } } where a = c c u β 1 { \ displaystyle a = { \ frac { c } { c _ { u } } } - 1 } using the previous equation and " a ", an expression for cs comes out to be : c s = ( 1 + 2 a ) c u { \ displaystyle c _ { s } = ( 1 + 2a ) c _ { u } } calculating the change in specific stagnation enthalpy across disc : Ξ΄ h o = h o d β h o u = ( h d + 1 2 c s 2 ) β ( h u + 1 2 c u 2 ) = 1 2 ( c s 2 β c u 2 ) { \ displaystyle \ delta h _ { o } = h _ { od } - h _ { ou } = \ left ( h _ { d } + { \ frac { 1 } { 2 } } c _ { s } ^ { 2 } \ right ) - \ left ( h _ { u } + { \ frac { 1 } { 2 } } c _ { u } ^ { 2 } \ right ) = { \ frac { 1 } { 2 } } \ left ( c _ { s } ^ { 2 } - c _ { u } ^ { 2 } \ right ) } now, ideal value of power supplied to the propeller = mass flow rate * change in stagnation enthalpy ; p i = m Ξ΄ h o { \ displaystyle p _ { i } = { \ dot { m } } { \ delta
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ideal value of power supplied to the propeller = mass flow rate * change in stagnation enthalpy ; p i = m Ξ΄ h o { \ displaystyle p _ { i } = { \ dot { m } } { \ delta h _ { o } } } where m = Ο a c { \ displaystyle { \ dot { m } } = \ rho ac } if propeller was employed to propel an aircraft at speed = cu ; then useful power = axial thrust * speed of aircraft ; p = f x c u { \ displaystyle p = f _ { x } c _ { u } } hence the expression for efficiency comes out to be : Ξ· p = actual power ( p ) ideal power ( p i ) = f x c u 1 2 Ο a c ( c s 2 β c u 2 ) = c u c = 1 1 + a { \ displaystyle \ eta _ { p } = { \ frac { { \ text { actual power } } ( p ) } { { \ text { ideal power } } ( p _ { i } ) } } = { \ frac { f _ { x } c _ { u } } { { \ frac { 1 } { 2 } } \ rho ac \ left ( c _ { s } ^ { 2 } - c _ { u } ^ { 2 } \ right ) } } = { \ frac { c _ { u } } { c } } = { \ frac { 1 } { 1 + a } } } let ds be the diameter of the imaginary outlet cylinder. by continuity equation ; c Ο d 2 4 = c s Ο d s 2 4 β d s 2 = c c s d 2 { \ displaystyle { \ begin { aligned } c { \ frac { \ pi d ^ { 2 } } { 4 } } & = c _ { s } { \ frac { \ pi d _ { s } ^ { 2 } } { 4 } } \ \ \ rightarrow d _ { s } ^ { 2 } & = { \ frac { c } { c _ { s } } } d ^ { 2 } \ end { aligned } } } from the above equations it is known that - c s = 1 + 2 a 1 + a c { \ displaystyle c _ { s } = { \ frac { 1 + 2a } { 1 + a } } c } therefore ; d s 2 = ( 1 +
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known that - c s = 1 + 2 a 1 + a c { \ displaystyle c _ { s } = { \ frac { 1 + 2a } { 1 + a } } c } therefore ; d s 2 = ( 1 + a 1 + 2 a ) d 2 { \ displaystyle d _ { s } ^ { 2 } = \ left ( { \ frac { 1 + a } { 1 + 2a } } \ right ) d ^ { 2 } } hence the flow can be modeled where the air flows through an imaginary diverging duct, where diameter of propeller disc and diameter of the outlet are related. = = = blade element theory = = = in this theory, a small element ( dr ) is taken at a distance r from the root of the blade and all the forces acting on the element are analysed to get a solution. it is assumed that the flow through each section of small radial thickness dr is assumed to be independent of the flow through other elements. resolving forces in the figure - Ξ΄ f x = Ξ΄ l sin ( Ξ² ) β Ξ΄ d cos ( Ξ² ) { \ displaystyle \ delta f _ { x } = \ delta l \ sin ( \ beta ) - \ delta d \ cos ( \ beta ) } Ξ΄ f y = Ξ΄ l cos ( Ξ² ) + Ξ΄ d sin ( Ξ² ) { \ displaystyle \ delta f _ { y } = \ delta l \ cos ( \ beta ) + \ delta d \ sin ( \ beta ) } lift coefficient ( cl ) and drag coefficient ( cd ) are given as - l i f t ( Ξ΄ l ) = 1 2 c l Ο w 2 ( l d r ) { \ displaystyle \ mathrm { lift } ( \ delta l ) = { \ frac { 1 } { 2 } } c _ { l } \ rho w ^ { 2 } ( ldr ) } d r a g ( Ξ΄ d ) = 1 2 c d Ο w 2 ( l d r ) { \ displaystyle \ mathrm { drag } ( \ delta d ) = { \ frac { 1 } { 2 } } c _ { d } \ rho w ^ { 2 } ( ldr ) } also from the figure - tan ( ) = Ξ΄ d Ξ΄ l = c d c l { \ displaystyle \ tan ( \ phi ) = { \ frac { \ delta d } { \ delta l } } = { \ frac { c _ {
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- tan ( ) = Ξ΄ d Ξ΄ l = c d c l { \ displaystyle \ tan ( \ phi ) = { \ frac { \ delta d } { \ delta l } } = { \ frac { c _ { d } } { c _ { l } } } } now, Ξ΄ f x = Ξ΄ l ( sin Ξ² β Ξ΄ d Ξ΄ l cos Ξ² ) = Ξ΄ l ( sin Ξ² β tan cos Ξ² ) = 1 2 c l Ο w 2 l d r sin ( Ξ² β ) cos { \ displaystyle \ delta f _ { x } = \ delta l ( \ sin \ beta - { \ frac { \ delta d } { \ delta l } } \ cos \ beta ) = \ delta l ( \ sin \ beta - \ tan \ phi \ cos \ beta ) = { \ frac { 1 } { 2 } } c _ { l } \ rho w ^ { 2 } ldr { \ frac { \ sin ( \ beta - \ phi ) } { \ cos \ phi } } } no. of blades ( z ) and spacing ( s ) are related as, s = 2 Ο r z { \ displaystyle s = { \ frac { 2 \ pi r } { z } } } and the total thrust for the elemental section of the propeller is zΞ΄fx. therefore, Ξ΄ p ( 2 Ο r d r ) = z Ξ΄ f x { \ displaystyle \ delta p ( 2 \ pi rdr ) = z \ delta f _ { x } } β Ξ΄ p = 1 2 c l Ο w 2 ( l s ) sin ( Ξ² β ) cos = 1 2 c d Ο w 2 ( l s ) sin ( Ξ² β ) sin { \ displaystyle \ rightarrow \ delta p = { \ frac { 1 } { 2 } } c _ { l } \ rho w ^ { 2 } ( { \ frac { l } { s } } ) { \ frac { \ sin ( \ beta - \ phi ) } { \ cos \ phi } } = { \ frac { 1 } { 2 } } c _ { d } \ rho w ^ { 2 } ( { \ frac { l } { s } } ) { \ frac { \ sin ( \ beta - \ phi ) } { \ sin \ phi } } } similarly, solving for Ξ΄fy
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\ rho w ^ { 2 } ( { \ frac { l } { s } } ) { \ frac { \ sin ( \ beta - \ phi ) } { \ sin \ phi } } } similarly, solving for Ξ΄fy, Ξ΄fy is found out to be - Ξ΄ f y = 1 2 c l Ο w 2 l d r cos ( Ξ² β ) cos { \ displaystyle \ delta f _ { y } = { \ frac { 1 } { 2 } } c _ { l } \ rho w ^ { 2 } ldr { \ frac { \ cos ( \ beta - \ phi ) } { \ cos \ phi } } } and ( t o r q u e ) Ξ΄ q = r Ξ΄ f y { \ displaystyle ( \ mathrm { torque } ) \ delta q = r \ delta f _ { y } } finally, thrust and torque can be found out for an elemental section as they are proportional to fx and fy respectively. = = performance characteristics = = the relationship between the pressure variation and the volume flow rate are important characteristics of fans. the typical characteristics of axial fans can be studied from the performance curves. the performance curve for the axial fan is shown in the figure. ( the vertical line joining the maximum efficiency point is drawn which meets the pressure curve at point " s " ) the following can be inferred from the curve - as the flow rate increases from zero the efficiency increases to a particular point reaches maximum value and then decreases. the power output of the fans increases with almost constant positive slope. the pressure fluctuations are observed at low discharges and at flow rates ( as indicated by the point " s " ) the pressure deceases. the pressure variations to the left of the point " s " causes for unsteady flow which are due to the two effects of stalling and surging. = = causes of unstable flow = = stalling and surging affects the fan performance, blades, as well as output and are thus undesirable. they occur because of the improper design, fan physical properties and are generally accompanied by noise generation. = = = stalling effect / stall = = = the cause for this is the separation of the flow from the blade surfaces. this effect can be explained by the flow over an air foil. when the angle of incidence increases ( during the low velocity flow ) at the entrance of the air foil, flow pattern changes and separation occurs. this is the first stage
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the blade surfaces. this effect can be explained by the flow over an air foil. when the angle of incidence increases ( during the low velocity flow ) at the entrance of the air foil, flow pattern changes and separation occurs. this is the first stage of stalling and through this separation point the flow separates leading to the formation of vortices, back flow in the separated region. for a further the explanation of stall and rotating stall, refer to compressor surge. the stall zone for the single axial fan and axial fans operated in parallel are shown in the figure. the following can be inferred from the graph : for the fans operated in parallel, the performance is less when compared to the individual fans. the fans should be operated in safe operation zone to avoid the stalling effects. = = = = vfds are not practical for some axial fans = = = = many axial fan failures have happened after controlled blade axial fans were locked in a fixed position and variable frequency drives ( vfds ) were installed. the vfds are not practical for some axial fans. axial fans with severe instability regions should not be operated at blades angles, rotational speeds, mass flow rates, and pressures that expose the fan to stall conditions. = = = surging effect / surge = = = surging should not be confused with stalling. stalling occurs only if there is insufficient air entering into the fan blades causing separation of flow on the blade surface. surging or the unstable flow causing complete breakdown in fans is mainly contributed by the three factors system surge fan surge paralleling = = = = system surge = = = = this situation occurs when the system resistance curve and static pressure curve of the fan intersect have similar slope or parallel to each other. rather than intersecting at a definite point the curves intersect over certain region reporting system surge. these characteristics are not observed in axial fans. = = = = fan surge = = = = this unstable operation results from the development of pressure gradients in the opposite direction of the flow. maximum pressure is observed at the discharge of the impeller blade and minimum pressure on the side opposite to the discharge side. when the impeller blades are not rotating these adverse pressure gradients pump the flow in the direction opposite to the direction of the fan. the result is the oscillation of the fan blades creating vibrations and hence noise. = = = paralleling = = = this effect is seen only in case of multiple fans. the air flow capacities of the fans are compared and connected in same outlet or same inlet conditions.
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##cillation of the fan blades creating vibrations and hence noise. = = = paralleling = = = this effect is seen only in case of multiple fans. the air flow capacities of the fans are compared and connected in same outlet or same inlet conditions. this causes noise, specifically referred to as beating in case of fans in parallel. to avoid beating use is made of differing inlet conditions, differences in rotational speeds of the fans, etc. = = methods to avoid unsteady flow = = by designing the fan blades with proper hub - to - tip ratio and analyzing performance on the number of blades so that the flow doesn't separate on the blade surface these effects can be reduced. some of the methods to overcome these effects are re - circulation of excess air through the fan, axial fans are high specific speed devices operating them at high efficiency and to minimize the effects they have to be operated at low speeds. for controlling and directing the flow use of guide vanes is suggested. turbulent flows at the inlet and outlet of the fans cause stalling so the flow should be made laminar by the introduction of a stator to prevent the effect. = = see also = = mechanical fan propeller ( marine ) propeller ( aircraft ) industrial fan ceiling fan turbofan ducted propeller window fan compressor surge compressor stall propeller walk cavitation azimuth thruster kitchen rudder paddle steamer propulsor cleaver folding propeller modular propeller supercavitating propeller = = notes = = = = references = = = = external links = = plastic blades for gas turbine engines - orenda engines ltd fibre blade attatchment - us department of navy turbine blade - ex - cell - o corp composite fan blade - williams international corp method for manufacturing a fiber reinforcement body for a metal matrix composite - toyota industries corp
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mont saint - hilaire ( french pronunciation : [ mΙ sΙt βΏ ilΙΚ ] ; english : mount saint - hilaire ; western abnaki : wigwomadenek, wigwomadensizek ;, western abnaki : wigwomaden ; see Β§ names of mont saint - hilaire for other names ) is a mountain, 414 m ( 1, 358 ft ) high, in the monteregie region of southern quebec. it is about thirty kilometres east of montreal, and immediately east of the richelieu river. it is one of the monteregian hills. around the mountains are the towns of mont - saint - hilaire and saint - jean - baptiste. other nearby towns include otterburn park, beloeil and mcmasterville. the area surrounding the mountain is a biosphere reserve, as one of the last remnants of the primeval forests of the saint - lawrence valley. most of the mountain is currently the property of mcgill university, as the gault nature reserve, which is considered the third mcgill campus. the university has opened the western half of the mountain to visitors ( at a fee ) for hiking and cross - country skiing, as the milieu naturel ( natural area ). the eastern half, or milieu de conservation ( preservation area ), is not accessible to the general public. until the late nineteenth century, the lack of information on more remote summits of quebec, as well as the relatively high prominence ( about 400 m ( 1, 312 ft ) ) of mont saint - hilaire, led to it being mistaken as the highest summit in quebec. in actuality, saint - hilaire's 414 metres falls far short of making it the highest mountain in quebec. mont saint - hilaire is home to a wide variety of fauna and flora, as well as a number of rare minerals, including some which were discovered on the mountain and some which are unique to the region. these minerals are exploited by a quarry on the north - eastern side of the mountain. in addition, the soil is ideal for the growth of apple trees, and the mountain's apple orchards draws tens of thousands of visitors each year. = = geography = = the mountain stands 414 m ( 1, 358 ft ) above the sea level, or 400 metres above the surrounding plains. it has several summits, surrounding a central lake, lac hertel ( lake hertel ). most of the well - known summit
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414 m ( 1, 358 ft ) above the sea level, or 400 metres above the surrounding plains. it has several summits, surrounding a central lake, lac hertel ( lake hertel ). most of the well - known summits of the mountain are in the western ( open to the public ) part. they are the pain de sucre ( sugar loaf ), 414 m ( 1, 358 ft ) high ; the sunrise, 405 m ( 1, 329 ft ) high ; the rocky, 403 m ( 1, 322 ft ) high ; and the sommet dieppe ( dieppe summit ), 371 m ( 1, 217 ft ) high as well as burnt hill, 320 m ( 1, 050 ft ) high. the summits of the eastern half of the mountain, which is closed to the public, are generally little known, to the point that most official maps fail to identify the distinct summits at all. only a few names are known, such as east hill and lake hill. these summits range from 277 to 392 metres ( 909 to 1, 286 ft ) in elevation. = = = cliffs = = = the best - known feature of the mountain is the cliffs. collectively known as the falaise dieppe, or falaise de dieppe ( dieppe cliffs ) the cliffs are part of the dieppe summit, and nearly 175 m ( 574 ft ) high. some of the best known features of the cliffs include the 60 - metre high tour rouge ( red tower ), as well as two slabs, the dalle noire ( black slab ) and dalle verte ( green slab ), which rise at a 75 - degree angle. the cliff's unique ecosystem hosts lichens, as well as cedar trees, some of which may be as much as five hundred years old. it also hosts the mountain's population of peregrine falcons. however, the action of rock climbers has proven destructive to the ecosystem, in addition to being dangerous to the climbers themselves. a white cross on the cliff commemorates the death of a boy scout in 1941. = = = lac hertel = = = at the centre of the mountain is lac hertel, a lake in a glacially - formed depression in the middle of the various summits. it covers an area of 0. 3 square kilometres ( 0. 12 sq mi ), and has a maximum depth of 9 m ( 30 ft ). it is fed by three permanent streams ; a fourth
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in the middle of the various summits. it covers an area of 0. 3 square kilometres ( 0. 12 sq mi ), and has a maximum depth of 9 m ( 30 ft ). it is fed by three permanent streams ; a fourth flows from the lake toward the richelieu river. the lake serves as a secondary reservoir of drinking water to the region, and, as such, swimming, fishing and boating are forbidden. the central position of the lake on the mountain has led to claims that mont saint - hilaire is a volcanic caldera. however, the lake is actually the result of glacial erosion, and in no way an ancient volcanic crater. = = geology = = mont saint - hilaire is one of the monteregian hills, a group of erosional remnants of intrusive mountains spread across southern quebec. it is composed of three distinct plutonic intrusions that formed during the cretaceous period between 133 and 120 million years ago. like the other monteregian hills, mont saint - hilaire forms part of the great meteor hotspot track, which was created when the north american plate slid over the new england hotspot. during this time, melting occurred, creating subsurface magma intrusions. erosion of the surrounding softer sedimentary rocks revealed the more resistant rocks of mont saint - hilaire. = = = mineral wealth = = = mont saint - hilaire is a famous mineral locality because of its great number of rare and exotic mineral species. annite ( iron rich biotite ) from mont saint - hilaire is among the most iron - rich found in nature. in the gabbro, biotite is less iron - rich, has lower manganese content, but is titanium - rich. phlogopite is found as small metamorphic crystals in marble xenoliths within the syenite. siderophyllite, a relatively rare mineral, occurs as large crystals in a metasomatised albite - rich albitite dike. in addition to gabbro, the second intrusive suite included nepheline syenite, diorite and monzonite. the third intrusive occupies the eastern side and is mainly peralkaline nepheline syenites and porphyrites. the most mineralogically interesting are the associated agpaitic ( alkali rich, low aluminium and silicon ) pegmatites, the intrusive breccias, and the hornfels derived from the
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##enites and porphyrites. the most mineralogically interesting are the associated agpaitic ( alkali rich, low aluminium and silicon ) pegmatites, the intrusive breccias, and the hornfels derived from the metasomatised sedimentary wall rocks. there have been over 366 distinct species of minerals collected at mont saint - hilaire, 50 of which have this site as type locality. = = ecology = = as the last remnant in quebec of the ancient gulf of st. lawrence lowland forests, the area has been a provincial biosphere reserve since 1978 and a federal migratory bird sanctuary since 1960. the area hosts 21 at risk and two endangered species of plant under current statutes. = = history = = the mountain, particularly the pain - de - sucre summit, was well known by the algonquin first nations, who used it as a vantage point to survey the valley of the richelieu river below. the mountain is located in abenaki traditional territory and its name, wigwomadenizibo, which means little house - shaped mountain. the first european to visit the region was samuel de champlain, who explored the richelieu river in two expeditions in 1603 and 1609. it was during the later expedition that he discovered the mountain. = = = development of the region = = = settlement around the mountain began in 1694 when a seigneury was granted to jean - baptiste hertel de rouville. a village slowly grew on the slopes of the mountain, near the streams emptying out of lake hertel. the combination of sugar bushes, the orchards alongside the mountain slopes, and the stream flowing from lake hertel which facilitated the construction of watermills provided for growth of the village in the eighteenth and early nineteenth centuries. in the nineteenth century, the mountain became a tourist destination, particularly after tomas edmond campbell bought the seigneury from the hertel de rouville family in 1844. a cafe, the campbell cafe, was established in 1851, and in 1874, a 150 - room hotel, the iroquois hotel. both burned down, the cafe in 1861 and the hotel in 1895. the tourist value of the mountain dwindled as development of the eastern townships made the northern appalachians more accessible. = = = the gault reserve = = = campbell sold the mountain in 1913 to andrew hamilton gault. while he saw to the development of the region, gault also insisted on protecting the wild nature of mont saint - hilaire, where
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= = = the gault reserve = = = campbell sold the mountain in 1913 to andrew hamilton gault. while he saw to the development of the region, gault also insisted on protecting the wild nature of mont saint - hilaire, where he planned to build a mansion home for his retirement. construction of the mansion began in 1957, but gault lived there a mere three weeks before he died. he bequeathed the property to mcgill university, where he had received his education, which made it into the " gault reserve ". the natural value of the mountain led to its being proclaimed a migratory bird sanctuary in 1960. in 1970, the mountain was divided into a preservation area, closed to the public, and an area open to the public, which became the responsibility of the centre de conservation de la nature ( nature conservation centre ) in 1972. in 1978, the mountain was made the first world biosphere reserve of canada as a result of it being the last remnants of the primeval forests of the saint - lawrence valley. although initial documents indicate the biosphere reserve covered large areas surrounding the mountains, it appears that today, the effective definition of the biosphere reserve corresponds to that of the gault reserve. = = names = = in the modern abenaki language, mont saint - hilaire is known as wigwomadenizibo, meaning the small house - shaped mountain. the neighbouring mount yamaska is similarly referred to in abenaki as wigwomadenek ( wigwam - shaped mountain ), without the diminutive suffix. the mountain was formerly rendered by europeans as wigwomadensis ( wigwam - shaped mountain ). when samuel de champlain visited the mountain, he named it mont fort ( which can be interpreted as " fort mountain " or " mount strong " ). the establishment of the town and parish of chambly to the south led to the mountain temporarily becoming mont chambly in the later seventeenth century ( although the name persisted in english until at least 1830 ). after 1697, the mountain became known as mont rouville, after the newly established seigneury of the hertel de rouville family. when the campbell family replaced the hertel de rouville family, the mountain took up the name mont beloeil, after the nearby municipality of beloeil, on the other side of the richelieu river. however, the name mont saint - hilaire, after the parish of mont - saint - hilaire established at the
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mont beloeil, after the nearby municipality of beloeil, on the other side of the richelieu river. however, the name mont saint - hilaire, after the parish of mont - saint - hilaire established at the foot of the mountain, became prevalent by the early twentieth century. even so, quarrels between inhabitants of beloeil and mont - saint - hilaire, both near the mountain, as to which the mountain should be named after lasted well into the twentieth century, with the town newspapers of beloeil attempting to resurrect the debate as late as 1986. = = in culture = = mont saint - hilaire, owing to its stark form, has always played a significant part in the culture of the nearby region. it is an important regional icon, its silhouette appearing in the symbols of many towns and cities of the region, such as beloeil, mcmasterville, mont - saint - hilaire and otterburn park. = = = in art = = = several artists have painted mont saint - hilaire over the years. the first was the english painter john bainbrigge, who made three separate watercolour paintings of the mountains around 1838, while garrisoned in the region. a few decades later, the mountain also appeared in a painting by cornelius krieghoff. however, the famous artists most closely linked with the mountains are the three who were born or lived in mont - saint - hilaire itself over the course of the late nineteenth and twentieth century : ozias leduc, born in 1864 in mont - saint - hilaire, his student paul - emile borduas, also born in mont - saint - hilaire in 1905, and finally, jordi bonet, who, after emigrating to quebec, settled down at mont - saint - hilaire in 1969. leduc especially is known for his numerous representations of the mountain in several paintings such as l'heure mauve and neige doree, but the mountain also inspired the work of borduas, such as his le trou des fees and synthese d'un paysage de mont - saint - hilaire, and bonnet, who included the silhouette of the mountain in several of his works. = = = in religion = = = some evidence suggests that mont saint - hilaire, particularly the pain de sucre summit, was a sacred site of the algonquin natives, who conducted rituals there. despite a
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several of his works. = = = in religion = = = some evidence suggests that mont saint - hilaire, particularly the pain de sucre summit, was a sacred site of the algonquin natives, who conducted rituals there. despite a slow establishment in the region ( the first two parishes at the foot of the mountain, saint - jean - baptiste and saint - hilaire, were only established in 1796 and 1798 ), the catholic church soon established itself on the mountain. in 1841, a great wooden cross was built atop the mountain, more than 30 metres ( 98 ft ) high and 9 metres ( 30 ft ) across, with a chapel at its base. the cross was hollow, allowing visitors to climb to the top. a winding trail was established leading up to the mountain, with stations of the cross along the way. the cross was destroyed in a storm in 1846. it was replaced in 1871 by a stone chapel, which burned in 1876. = = = in legends = = = oral tradition in the area surrounding mont saint - hilaire has recorded several legends concerning the mountains. local figures such as the painter ozias leduc and local historians armand cardinal and pierre lambert have set down in writing several of those legends ; even so much of the oral tradition is now lost. legends of the mountain generally centre on the figures of either the mountain's three fairies, who are said to have lived in a cave in the cliff - face for several centuries before abandoning the mountain and their immortality to marry their mortal lovers, or the devil, and the existence of two passage to hell, including the location known as les portes de fer ( the iron gates ) ( two great stone slabs found next to one another on the slopes of the mountain ). lac hertel is also at the centre of several legends. according to different legends, it is either bottomless, connected by underground passages to lake champlain, or was formed as a direct result of the devil successfully convincing several of the early settlers to abandon their promise to attend mass at the newly established chapel every sunday. another legend, based on the unusual ice patterns found on the northern cliff - face of the mountain, where a vaguely horse - shaped ice formation does not melt until late in the spring, state that the spirit of a local farmer's horse warns farmers against planting their crop so long as the ice formation remains on the mountain. = = = in esotericism = = = since the late twentieth century, the mountain has become a major hub of alleged supernatural activity
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a local farmer's horse warns farmers against planting their crop so long as the ice formation remains on the mountain. = = = in esotericism = = = since the late twentieth century, the mountain has become a major hub of alleged supernatural activity. the towns surrounding the mountain have shown the highest number of ufo sightings in quebec. the ufologist organisation ufo - quebec has claimed mont saint - hilaire to be the central hub of all ufo observations in southern quebec. on the night of may 22 β 23, 1981, several hundred people came to the mountain after richard glenn claimed to have been warned of ufo visits to the mountain that night. the observations were severely limited by a rainstorm, although some alleged sightings still took place. glenn also introduced the concept of the hollow earth to the mountain, claiming mont saint - hilaire to be an entrance leading to the domain of the inhabitants of the hollow earth. the abundance of unusual minerals and crystals at mont saint - hilaire, as well as the effect of its mass upon local magnetism, have also been of great interest to followers of the new age movement and conspiracy theories. = = see also = = mont - saint - hilaire, quebec = = references = = = = external links = = alkali nuts unesco mab page mont saint - hilaire at mcgill university
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the lenstra elliptic - curve factorization or the elliptic - curve factorization method ( ecm ) is a fast, sub - exponential running time, algorithm for integer factorization, which employs elliptic curves. for general - purpose factoring, ecm is the third - fastest known factoring method. the second - fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. the lenstra elliptic - curve factorization is named after hendrik lenstra. practically speaking, ecm is considered a special - purpose factoring algorithm, as it is most suitable for finding small factors. currently, it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor p rather than by the size of the number n to be factored. frequently, ecm is used to remove small factors from a very large integer with many factors ; if the remaining integer is still composite, then it has only large factors and is factored using general - purpose techniques. the largest factor found using ecm so far has 83 decimal digits and was discovered on 7 september 2013 by r. propper. increasing the number of curves tested improves the chances of finding a factor, but they are not linear with the increase in the number of digits. = = algorithm = = the lenstra elliptic - curve factorization method to find a factor of a given natural number n { \ displaystyle n } works as follows : pick a random elliptic curve over z / n z { \ displaystyle \ mathbb { z } / n \ mathbb { z } } ( the integers modulo n { \ displaystyle n } ), with equation of the form y 2 = x 3 + a x + b ( mod n ) { \ displaystyle y ^ { 2 } = x ^ { 3 } + ax + b { \ pmod { n } } } together with a non - trivial point p ( x 0, y 0 ) { \ displaystyle p ( x _ { 0 }, y _ { 0 } ) } on it. this can be done by first picking random x 0, y 0, a β z / n z { \ displaystyle x _ { 0 }, y _ { 0 }, a \ in \ mathbb { z } / n \ mathbb { z } }, and then setting b = y 0 2 β x 0 3 β a x 0 ( mod n ) { \ displaystyle b =
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_ { 0 }, a \ in \ mathbb { z } / n \ mathbb { z } }, and then setting b = y 0 2 β x 0 3 β a x 0 ( mod n ) { \ displaystyle b = y _ { 0 } ^ { 2 } - x _ { 0 } ^ { 3 } - ax _ { 0 } { \ pmod { n } } } to assure the point is on the curve. one can define addition of two points on the curve, to define a group. the addition laws are given in the article on elliptic curves. we can form repeated multiples of a point p { \ displaystyle p } : [ k ] p = p + β¦ + p ( k times ) { \ displaystyle [ k ] p = p + \ ldots + p { \ text { ( k times ) } } }. the addition formulae involve taking the modular slope of a chord joining p { \ displaystyle p } and q { \ displaystyle q }, and thus division between residue classes modulo n { \ displaystyle n }, performed using the extended euclidean algorithm. in particular, division by some v mod n { \ displaystyle v { \ bmod { n } } } includes calculation of the gcd ( v, n ) { \ displaystyle \ gcd ( v, n ) }. assuming we calculate a slope of the form u / v { \ displaystyle u / v } with gcd ( u, v ) = 1 { \ displaystyle \ gcd ( u, v ) = 1 }, then if v = 0 mod n { \ displaystyle v = 0 { \ bmod { n } } }, the result of the point addition will be β { \ displaystyle \ infty }, the point " at infinity " corresponding to the intersection of the " vertical " line joining p ( x, y ), p β² ( x, β y ) { \ displaystyle p ( x, y ), p'( x, - y ) } and the curve. however, if gcd ( v, n ) = 1, n { \ displaystyle \ gcd ( v, n ) \ neq 1, n }, then the point addition will not produce a meaningful point on the curve ; but, more importantly, gcd ( v, n ) { \ displaystyle \ gcd ( v, n ) } is a non - trivial factor of n { \ displaystyle n
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point addition will not produce a meaningful point on the curve ; but, more importantly, gcd ( v, n ) { \ displaystyle \ gcd ( v, n ) } is a non - trivial factor of n { \ displaystyle n }. compute [ k ] p { \ displaystyle [ k ] p } on the elliptic curve ( mod n { \ displaystyle { \ bmod { n } } } ), where k { \ displaystyle k } is a product of many small numbers : say, a product of small primes raised to small powers, as in the p - 1 algorithm, or the factorial b! { \ displaystyle b! } for some not too large b { \ displaystyle b }. this can be done efficiently, one small factor at a time. say, to get [ b! ] p { \ displaystyle [ b! ] p }, first compute [ 2 ] p { \ displaystyle [ 2 ] p }, then [ 3 ] ( [ 2 ] p ) { \ displaystyle [ 3 ] ( [ 2 ] p ) }, then [ 4 ] ( [ 3! ] p ) { \ displaystyle [ 4 ] ( [ 3! ] p ) }, and so on. b { \ displaystyle b } is picked to be small enough so that b { \ displaystyle b } - wise point addition can be performed in reasonable time. if we finish all the calculations above without encountering non - invertible elements ( mod n { \ displaystyle { \ bmod { n } } } ), it means that the elliptic curves'( modulo primes ) order is not smooth enough, so we need to try again with a different curve and starting point. if we encounter a gcd ( v, n ) = 1, n { \ displaystyle \ gcd ( v, n ) \ neq 1, n } we are done : it is a non - trivial factor of n { \ displaystyle n }. the time complexity depends on the size of the number's smallest prime factor and can be represented by exp [ ( β2 + o ( 1 ) ) βln p ln ln p ], where p is the smallest factor of n, or l p [ 1 2, 2 ] { \ displaystyle l _ { p } \ left [ { \ frac { 1 } { 2 } }, { \ sqrt { 2 } } \ right ] }, in l -
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, or l p [ 1 2, 2 ] { \ displaystyle l _ { p } \ left [ { \ frac { 1 } { 2 } }, { \ sqrt { 2 } } \ right ] }, in l - notation. = = explanation = = if p and q are two prime divisors of n, then y2 = x3 + ax + b ( mod n ) implies the same equation also modulo p and modulo q. these two smaller elliptic curves with the { \ displaystyle \ boxplus } - addition are now genuine groups. if these groups have np and nq elements, respectively, then for any point p on the original curve, by lagrange's theorem, k > 0 is minimal such that k p = β { \ displaystyle kp = \ infty } on the curve modulo p implies that k divides np ; moreover, n p p = β { \ displaystyle n _ { p } p = \ infty }. the analogous statement holds for the curve modulo q. when the elliptic curve is chosen randomly, then np and nq are random numbers close to p + 1 and q + 1, respectively ( see below ). hence it is unlikely that most of the prime factors of np and nq are the same, and it is quite likely that while computing ep, we will encounter some kp that is β modulo p but not modulo q, or vice versa. when this is the case, kp does not exist on the original curve, and in the computations we found some v with either gcd ( v, p ) = p or gcd ( v, q ) = q, but not both. that is, gcd ( v, n ) gave a non - trivial factor of n. ecm is at its core an improvement of the older p β 1 algorithm. the p β 1 algorithm finds prime factors p such that p β 1 is b - powersmooth for small values of b. for any e, a multiple of p β 1, and any a relatively prime to p, by fermat's little theorem we have ae β‘ 1 ( mod p ). then gcd ( ae β 1, n ) is likely to produce a factor of n. however, the algorithm fails when p β 1 has large prime factors, as is the case for numbers containing strong primes, for example. ecm gets around this obstacle by considering the group of a
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) is likely to produce a factor of n. however, the algorithm fails when p β 1 has large prime factors, as is the case for numbers containing strong primes, for example. ecm gets around this obstacle by considering the group of a random elliptic curve over the finite field zp, rather than considering the multiplicative group of zp which always has order p β 1. the order of the group of an elliptic curve over zp varies ( quite randomly ) between p + 1 β 2βp and p + 1 + 2βp by hasse's theorem, and is likely to be smooth for some elliptic curves. although there is no proof that a smooth group order will be found in the hasse - interval, by using heuristic probabilistic methods, the canfield β erdos β pomerance theorem with suitably optimized parameter choices, and the l - notation, we can expect to try l [ β2 / 2, β2 ] curves before getting a smooth group order. this heuristic estimate is very reliable in practice. = = example usage = = the following example is from trappe & washington ( 2006 ), with some details added. we want to factor n = 455839. let's choose the elliptic curve y2 = x3 + 5x β 5, with the point p = ( 1, 1 ) on it, and let's try to compute ( 10! ) p. the slope of the tangent line at some point a = ( x, y ) is s = ( 3x2 + 5 ) / ( 2y ) ( mod n ). using s we can compute 2a. if the value of s is of the form a / b where b > 1 and gcd ( a, b ) = 1, we have to find the modular inverse of b. if it does not exist, gcd ( n, b ) is a non - trivial factor of n. first we compute 2p. we have s ( p ) = s ( 1, 1 ) = 4, so the coordinates of 2p = ( x β², y β² ) are x β² = s2 β 2x = 14 and y β² = s ( x β x β² ) β y = 4 ( 1 β 14 ) β 1 = β53, all numbers understood ( mod n ). just to check that this 2p is indeed on the curve : ( β53 ) 2 = 2809 = 143 + 5 Β· 14 β 5.
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4 ( 1 β 14 ) β 1 = β53, all numbers understood ( mod n ). just to check that this 2p is indeed on the curve : ( β53 ) 2 = 2809 = 143 + 5 Β· 14 β 5. then we compute 3 ( 2p ). we have s ( 2p ) = s ( 14, β53 ) = β593 / 106 ( mod n ). using the euclidean algorithm : 455839 = 4300 Β· 106 + 39, then 106 = 2 Β· 39 + 28, then 39 = 28 + 11, then 28 = 2 Β· 11 + 6, then 11 = 6 + 5, then 6 = 5 + 1. hence gcd ( 455839, 106 ) = 1, and working backwards ( a version of the extended euclidean algorithm ) : 1 = 6 β 5 = 2 Β· 6 β 11 = 2 Β· 28 β 5 Β· 11 = 7 Β· 28 β 5 Β· 39 = 7 Β· 106 β 19 Β· 39 = 81707 Β· 106 β 19 Β· 455839. hence 106β1 = 81707 ( mod 455839 ), and β593 / 106 = β133317 ( mod 455839 ). given this s, we can compute the coordinates of 2 ( 2p ), just as we did above : 4p = ( 259851, 116255 ). just to check that this is indeed a point on the curve : y2 = 54514 = x3 + 5x β 5 ( mod 455839 ). after this, we can compute 3 ( 2 p ) = 4 p 2 p { \ displaystyle 3 ( 2p ) = 4p \ boxplus 2p }. we can similarly compute 4! p, and so on, but 8! p requires inverting 599 ( mod 455839 ). the euclidean algorithm gives that 455839 is divisible by 599, and we have found a factorization 455839 = 599 Β· 761. the reason that this worked is that the curve ( mod 599 ) has 640 = 27 Β· 5 points, while ( mod 761 ) it has 777 = 3 Β· 7 Β· 37 points. moreover, 640 and 777 are the smallest positive integers k such that kp = β on the curve ( mod 599 ) and ( mod 761 ), respectively. since 8! is a multiple of 640 but not a multiple of 777, we have 8! p = β
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smallest positive integers k such that kp = β on the curve ( mod 599 ) and ( mod 761 ), respectively. since 8! is a multiple of 640 but not a multiple of 777, we have 8! p = β on the curve ( mod 599 ), but not on the curve ( mod 761 ), hence the repeated addition broke down here, yielding the factorization. = = the algorithm with projective coordinates = = before considering the projective plane over ( z / n z ) /, { \ displaystyle ( \ mathbb { z } / n \ mathbb { z } ) / \ sim, } first consider a'normal'projective space over r { \ displaystyle \ mathbb { r } } : instead of points, lines through the origin are studied. a line may be represented as a non - zero point ( x, y, z ) { \ displaystyle ( x, y, z ) }, under an equivalence relation ~ given by : ( x, y, z ) ( x β², y β², z β² ) { \ displaystyle ( x, y, z ) \ sim ( x ', y ', z') } c = 0 such that x'= cx, y'= cy and z'= cz. under this equivalence relation, the space is called the projective plane p 2 { \ displaystyle \ mathbb { p } ^ { 2 } } ; points, denoted by ( x : y : z ) { \ displaystyle ( x : y : z ) }, correspond to lines in a three - dimensional space that pass through the origin. note that the point ( 0 : 0 : 0 ) { \ displaystyle ( 0 : 0 : 0 ) } does not exist in this space since to draw a line in any possible direction requires at least one of x ', y'or z'= 0. now observe that almost all lines go through any given reference plane - such as the ( x, y, 1 ) - plane, whilst the lines precisely parallel to this plane, having coordinates ( x, y, 0 ), specify directions uniquely, as'points at infinity'that are used in the affine ( x, y ) - plane it lies above. in the algorithm, only the group structure of an elliptic curve over the field r { \ displaystyle \ mathbb { r } } is used. since we do not necessarily need the field r { \ displaystyle \ mathbb
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it lies above. in the algorithm, only the group structure of an elliptic curve over the field r { \ displaystyle \ mathbb { r } } is used. since we do not necessarily need the field r { \ displaystyle \ mathbb { r } }, a finite field will also provide a group structure on an elliptic curve. however, considering the same curve and operation over ( z / n z ) / { \ displaystyle ( \ mathbb { z } / n \ mathbb { z } ) / \ sim } with n not a prime does not give a group. the elliptic curve method makes use of the failure cases of the addition law. we now state the algorithm in projective coordinates. the neutral element is then given by the point at infinity ( 0 : 1 : 0 ) { \ displaystyle ( 0 : 1 : 0 ) }. let n be a ( positive ) integer and consider the elliptic curve ( a set of points with some structure on it ) e ( z / n z ) = { ( x : y : z ) β p 2 | y 2 z = x 3 + a x z 2 + b z 3 } { \ displaystyle e ( \ mathbb { z } / n \ mathbb { z } ) = \ { ( x : y : z ) \ in \ mathbb { p } ^ { 2 } \ | \ y ^ { 2 } z = x ^ { 3 } + axz ^ { 2 } + bz ^ { 3 } \ } }. pick x p, y p, a β z / n z { \ displaystyle x _ { p }, y _ { p }, a \ in \ mathbb { z } / n \ mathbb { z } } with a = 0. calculate b = y p 2 β x p 3 β a x p { \ displaystyle b = y _ { p } ^ { 2 } - x _ { p } ^ { 3 } - ax _ { p } }. the elliptic curve e is then in weierstrass form given by y 2 = x 3 + a x + b { \ displaystyle y ^ { 2 } = x ^ { 3 } + ax + b } and by using projective coordinates the elliptic curve is given by the homogeneous equation z y 2 = x 3 + a z 2 x + b z 3 { \ displaystyle zy ^ { 2 } = x ^ { 3 } + az ^ { 2 } x + bz ^ {
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curve is given by the homogeneous equation z y 2 = x 3 + a z 2 x + b z 3 { \ displaystyle zy ^ { 2 } = x ^ { 3 } + az ^ { 2 } x + bz ^ { 3 } }. it has the point p = ( x p : y p : 1 ) { \ displaystyle p = ( x _ { p } : y _ { p } : 1 ) }. choose an upperbound b β z { \ displaystyle b \ in \ mathbb { z } } for this elliptic curve. remark : you will only find factors p if the group order of the elliptic curve e over z / p z { \ displaystyle \ mathbb { z } / p \ mathbb { z } } ( denoted by # e ( z / p z ) { \ displaystyle \ # e ( \ mathbb { z } / p \ mathbb { z } ) } ) is b - smooth, which means that all prime factors of # e ( z / p z ) { \ displaystyle \ # e ( \ mathbb { z } / p \ mathbb { z } ) } have to be less or equal to b. calculate k = l c m ( 1, β¦, b ) { \ displaystyle k = { \ rm { lcm } } ( 1, \ dots, b ) }. calculate k p : = p + p + + p { \ displaystyle kp : = p + p + \ cdots + p } ( k times ) in the ring e ( z / n z ) { \ displaystyle e ( \ mathbb { z } / n \ mathbb { z } ) }. note that if # e ( z / n z ) { \ displaystyle \ # e ( \ mathbb { z } / n \ mathbb { z } ) } is b - smooth and n is prime ( and therefore z / n z { \ displaystyle \ mathbb { z } / n \ mathbb { z } } is a field ) that k p = ( 0 : 1 : 0 ) { \ displaystyle kp = ( 0 : 1 : 0 ) }. however, if only # e ( z / p z ) { \ displaystyle \ # e ( \ mathbb { z } / p \ mathbb { z } ) } is b - smooth for some divisor p of n, the product might not be ( 0 : 1 : 0
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z ) { \ displaystyle \ # e ( \ mathbb { z } / p \ mathbb { z } ) } is b - smooth for some divisor p of n, the product might not be ( 0 : 1 : 0 ) because addition and multiplication are not well - defined if n is not prime. in this case, a non - trivial divisor can be found. if not, then go back to step 2. if this does occur, then you will notice this when simplifying the product k p. { \ displaystyle kp. } in point 5 it is said that under the right circumstances a non - trivial divisor can be found. as pointed out in lenstra's article ( factoring integers with elliptic curves ) the addition needs the assumption gcd ( x 1 β x 2, n ) = 1 { \ displaystyle \ gcd ( x _ { 1 } - x _ { 2 }, n ) = 1 }. if p, q { \ displaystyle p, q } are not ( 0 : 1 : 0 ) { \ displaystyle ( 0 : 1 : 0 ) } and distinct ( otherwise addition works similarly, but is a little different ), then addition works as follows : to calculate : r = p + q ; { \ displaystyle r = p + q ; } p = ( x 1 : y 1 : 1 ), q = ( x 2 : y 2 : 1 ) { \ displaystyle p = ( x _ { 1 } : y _ { 1 } : 1 ), q = ( x _ { 2 } : y _ { 2 } : 1 ) }, Ξ» = ( y 1 β y 2 ) ( x 1 β x 2 ) β 1 { \ displaystyle \ lambda = ( y _ { 1 } - y _ { 2 } ) ( x _ { 1 } - x _ { 2 } ) ^ { - 1 } }, x 3 = Ξ» 2 β x 1 β x 2 { \ displaystyle x _ { 3 } = \ lambda ^ { 2 } - x _ { 1 } - x _ { 2 } }, y 3 = Ξ» ( x 1 β x 3 ) β y 1 { \ displaystyle y _ { 3 } = \ lambda ( x _ { 1 } - x _ { 3 } ) - y _ { 1 } }, r = p + q = ( x 3 : y 3 : 1 ) { \ displaystyle r = p + q = ( x
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\ lambda ( x _ { 1 } - x _ { 3 } ) - y _ { 1 } }, r = p + q = ( x 3 : y 3 : 1 ) { \ displaystyle r = p + q = ( x _ { 3 } : y _ { 3 } : 1 ) }. if addition fails, this will be due to a failure calculating Ξ». { \ displaystyle \ lambda. } in particular, because ( x 1 β x 2 ) β 1 { \ displaystyle ( x _ { 1 } - x _ { 2 } ) ^ { - 1 } } can not always be calculated if n is not prime ( and therefore z / n z { \ displaystyle \ mathbb { z } / n \ mathbb { z } } is not a field ). without making use of z / n z { \ displaystyle \ mathbb { z } / n \ mathbb { z } } being a field, one could calculate : Ξ» β² = y 1 β y 2 { \ displaystyle \ lambda'= y _ { 1 } - y _ { 2 } }, x 3 β² = Ξ» β² 2 β x 1 ( x 1 β x 2 ) 2 β x 2 ( x 1 β x 2 ) 2 { \ displaystyle x _ { 3 }'= { \ lambda'} ^ { 2 } - x _ { 1 } ( x _ { 1 } - x _ { 2 } ) ^ { 2 } - x _ { 2 } ( x _ { 1 } - x _ { 2 } ) ^ { 2 } }, y 3 β² = Ξ» β² ( x 1 ( x 1 β x 2 ) 2 β x 3 β² ) β y 1 ( x 1 β x 2 ) 3 { \ displaystyle y _ { 3 }'= \ lambda'( x _ { 1 } ( x _ { 1 } - x _ { 2 } ) ^ { 2 } - x _ { 3 }') - y _ { 1 } ( x _ { 1 } - x _ { 2 } ) ^ { 3 } }, r = p + q = ( x 3 β² ( x 1 β x 2 ) : y 3 β² : ( x 1 β x 2 ) 3 ) { \ displaystyle r = p + q = ( x _ { 3 }'( x _ { 1 } - x _ { 2 } ) : y _ { 3 }': ( x _ { 1 } - x _ { 2 } ) ^
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displaystyle r = p + q = ( x _ { 3 }'( x _ { 1 } - x _ { 2 } ) : y _ { 3 }': ( x _ { 1 } - x _ { 2 } ) ^ { 3 } ) }, and simplify if possible. this calculation is always legal and if the gcd of the z - coordinate with n = ( 1 or n ), so when simplifying fails, a non - trivial divisor of n is found. = = twisted edwards curves = = the use of edwards curves needs fewer modular multiplications and less time than the use of montgomery curves or weierstrass curves ( other used methods ). using edwards curves you can also find more primes. definition. let k { \ displaystyle k } be a field in which 2 = 0 { \ displaystyle 2 \ neq 0 }, and let a, d β k { 0 } { \ displaystyle a, d \ in k \ setminus \ { 0 \ } } with a = d { \ displaystyle a \ neq d }. then the twisted edwards curve e e, a, d { \ displaystyle e _ { e, a, d } } is given by a x 2 + y 2 = 1 + d x 2 y 2. { \ displaystyle ax ^ { 2 } + y ^ { 2 } = 1 + dx ^ { 2 } y ^ { 2 }. } an edwards curve is a twisted edwards curve in which a = 1 { \ displaystyle a = 1 }. there are five known ways to build a set of points on an edwards curve : the set of affine points, the set of projective points, the set of inverted points, the set of extended points and the set of completed points. the set of affine points is given by : { ( x, y ) β a 2 : a x 2 + y 2 = 1 + d x 2 y 2 } { \ displaystyle \ { ( x, y ) \ in \ mathbb { a } ^ { 2 } : ax ^ { 2 } + y ^ { 2 } = 1 + dx ^ { 2 } y ^ { 2 } \ } }. the addition law is given by ( e, f ), ( g, h ) β¦ ( e h + f g 1 + d e g f h, f h β a e g 1 β d e g f h ). { \ displaystyle (
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law is given by ( e, f ), ( g, h ) β¦ ( e h + f g 1 + d e g f h, f h β a e g 1 β d e g f h ). { \ displaystyle ( e, f ), ( g, h ) \ mapsto \ left ( { \ frac { eh + fg } { 1 + degfh } }, { \ frac { fh - aeg } { 1 - degfh } } \ right ). } the point ( 0, 1 ) is its neutral element and the inverse of ( e, f ) { \ displaystyle ( e, f ) } is ( β e, f ) { \ displaystyle ( - e, f ) }. the other representations are defined similar to how the projective weierstrass curve follows from the affine. any elliptic curve in edwards form has a point of order 4. so the torsion group of an edwards curve over q { \ displaystyle \ mathbb { q } } is isomorphic to either z / 4 z, z / 8 z, z / 12 z, z / 2 z Γ z / 4 z { \ displaystyle \ mathbb { z } / 4 \ mathbb { z }, \ mathbb { z } / 8 \ mathbb { z }, \ mathbb { z } / 12 \ mathbb { z }, \ mathbb { z } / 2 \ mathbb { z } \ times \ mathbb { z } / 4 \ mathbb { z } } or z / 2 z Γ z / 8 z { \ displaystyle \ mathbb { z } / 2 \ mathbb { z } \ times \ mathbb { z } / 8 \ mathbb { z } }. the most interesting cases for ecm are z / 12 z { \ displaystyle \ mathbb { z } / 12 \ mathbb { z } } and z / 2 z Γ z / 8 z { \ displaystyle \ mathbb { z } / 2 \ mathbb { z } \ times \ mathbb { z } / 8 \ mathbb { z } }, since they force the group orders of the curve modulo primes to be divisible by 12 and 16 respectively. the following curves have a torsion group isomorphic to z / 12 z { \ displaystyle \ mathbb { z } / 12 \ mathbb { z } } : x 2 + y 2 = 1
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##visible by 12 and 16 respectively. the following curves have a torsion group isomorphic to z / 12 z { \ displaystyle \ mathbb { z } / 12 \ mathbb { z } } : x 2 + y 2 = 1 + d x 2 y 2 { \ displaystyle x ^ { 2 } + y ^ { 2 } = 1 + dx ^ { 2 } y ^ { 2 } } with point ( a, b ) { \ displaystyle ( a, b ) } where b β { β 2, β 1 / 2, 0, Β± 1 }, a 2 = β ( b 2 + 2 b ) { \ displaystyle b \ notin \ { - 2, - 1 / 2, 0, \ pm 1 \ }, a ^ { 2 } = - ( b ^ { 2 } + 2b ) } and d = β ( 2 b + 1 ) / ( a 2 b 2 ) { \ displaystyle d = - ( 2b + 1 ) / ( a ^ { 2 } b ^ { 2 } ) } x 2 + y 2 = 1 + d x 2 y 2 { \ displaystyle x ^ { 2 } + y ^ { 2 } = 1 + dx ^ { 2 } y ^ { 2 } } with point ( a, b ) { \ displaystyle ( a, b ) } where a = u 2 β 1 u 2 + 1, b = β ( u β 1 ) 2 u 2 + 1 { \ displaystyle a = { \ frac { u ^ { 2 } - 1 } { u ^ { 2 } + 1 } }, b = - { \ frac { ( u - 1 ) ^ { 2 } } { u ^ { 2 } + 1 } } } and d = ( u 2 + 1 ) 3 ( u 2 β 4 u + 1 ) ( u β 1 ) 6 ( u + 1 ) 2, u β { 0, Β± 1 }. { \ displaystyle d = { \ frac { ( u ^ { 2 } + 1 ) ^ { 3 } ( u ^ { 2 } - 4u + 1 ) } { ( u - 1 ) ^ { 6 } ( u + 1 ) ^ { 2 } } }, u \ notin \ { 0, \ pm 1 \ }. } every edwards curve with a point of order 3 can be written in the ways shown above. curves with torsion group isomorphic to z / 2 z Γ z /
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}, u \ notin \ { 0, \ pm 1 \ }. } every edwards curve with a point of order 3 can be written in the ways shown above. curves with torsion group isomorphic to z / 2 z Γ z / 8 z { \ displaystyle \ mathbb { z } / 2 \ mathbb { z } \ times \ mathbb { z } / 8 \ mathbb { z } } and z / 2 z Γ z / 4 z { \ displaystyle \ mathbb { z } / 2 \ mathbb { z } \ times \ mathbb { z } / 4 \ mathbb { z } } may be more efficient at finding primes. = = stage 2 = = the above text is about the first stage of elliptic curve factorisation. there one hopes to find a prime divisor p such that s p { \ displaystyle sp } is the neutral element of e ( z / p z ) { \ displaystyle e ( \ mathbb { z } / p \ mathbb { z } ) }. in the second stage one hopes to have found a prime divisor q such that s p { \ displaystyle sp } has small prime order in e ( z / q z ) { \ displaystyle e ( \ mathbb { z } / q \ mathbb { z } ) }. we hope the order to be between b 1 { \ displaystyle b _ { 1 } } and b 2 { \ displaystyle b _ { 2 } }, where b 1 { \ displaystyle b _ { 1 } } is determined in stage 1 and b 2 { \ displaystyle b _ { 2 } } is new stage 2 parameter. checking for a small order of s p { \ displaystyle sp }, can be done by computing ( l s ) p { \ displaystyle ( ls ) p } modulo n for each prime l. = = gmp - ecm and eecm - mpfq = = the use of twisted edwards elliptic curves, as well as other techniques were used by bernstein et al to provide an optimized implementation of ecm. its only drawback is that it works on smaller composite numbers than the more general purpose implementation, gmp - ecm of zimmermann. = = hyperelliptic - curve method ( hecm ) = = there are recent developments in using hyperelliptic curves to factor integers. cosset shows in his article ( of 2010 ) that one can build a hyperelliptic curve
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##n. = = hyperelliptic - curve method ( hecm ) = = there are recent developments in using hyperelliptic curves to factor integers. cosset shows in his article ( of 2010 ) that one can build a hyperelliptic curve with genus two ( so a curve y 2 = f ( x ) { \ displaystyle y ^ { 2 } = f ( x ) } with f of degree 5 ), which gives the same result as using two " normal " elliptic curves at the same time. by making use of the kummer surface, calculation is more efficient. the disadvantages of the hyperelliptic curve ( versus an elliptic curve ) are compensated by this alternative way of calculating. therefore, cosset roughly claims that using hyperelliptic curves for factorization is no worse than using elliptic curves. = = quantum version ( geecm ) = = bernstein, heninger, lou, and valenta suggest geecm, a quantum version of ecm with edwards curves. it uses grover's algorithm to roughly double the length of the primes found compared to standard eecm, assuming a quantum computer with sufficiently many qubits and of comparable speed to the classical computer running eecm. = = references = = bernstein, daniel j. ; birkner, peter ; lange, tanja ; peters, christiane ( 2013 ). " ecm using edwards curves ". mathematics of computation. 82 ( 282 ) : 1139 β 1179. doi : 10. 1090 / s0025 - 5718 - 2012 - 02633 - 0. mr 3008853. bosma, w. ; hulst, m. p. m. van der ( 1990 ). primality proving with cyclotomy. ph. d. thesis, universiteit van amsterdam. oclc 256778332. brent, richard p. ( 1999 ). " factorization of the tenth fermat number ". mathematics of computation. 68 ( 225 ) : 429 β 451. bibcode : 1999macom.. 68.. 429b. doi : 10. 1090 / s0025 - 5718 - 99 - 00992 - 8. mr 1489968. cohen, henri ( 1993 ). a course in computational algebraic number theory. graduate texts in mathematics. vol. 138. berlin : springer - verlag. doi : 10. 1007 / 978 - 3 - 662 - 02945 - 9. isbn 978 - 0 - 38
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. a course in computational algebraic number theory. graduate texts in mathematics. vol. 138. berlin : springer - verlag. doi : 10. 1007 / 978 - 3 - 662 - 02945 - 9. isbn 978 - 0 - 387 - 55640 - 6. mr 1228206. s2cid 118037646. cosset, r. ( 2010 ). " factorization with genus 2 curves ". mathematics of computation. 79 ( 270 ) : 1191 β 1208. arxiv : 0905. 2325. bibcode : 2010macom.. 79. 1191c. doi : 10. 1090 / s0025 - 5718 - 09 - 02295 - 9. mr 2600562. s2cid 914296. lenstra, a. k. ; lenstra jr., h. w., eds. ( 1993 ). the development of the number field sieve. lecture notes in mathematics. vol. 1554. berlin : springer - verlag. pp. 11 β 42. doi : 10. 1007 / bfb0091534. isbn 978 - 3 - 540 - 57013 - 4. mr 1321216. lenstra jr., h. w. ( 1987 ). " factoring integers with elliptic curves " ( pdf ). annals of mathematics. 126 ( 3 ) : 649 β 673. doi : 10. 2307 / 1971363. hdl : 1887 / 2140. jstor 1971363. mr 0916721. pomerance, carl ; crandall, richard ( 2005 ). prime numbers : a computational perspective ( second ed. ). new york : springer. isbn 978 - 0 - 387 - 25282 - 7. mr 2156291. pomerance, carl ( 1985 ). " the quadratic sieve factoring algorithm ". advances in cryptology, proc. eurocrypt'84. lecture notes in computer science. vol. 209. berlin : springer - verlag. pp. 169 β 182. doi : 10. 1007 / 3 - 540 - 39757 - 4 _ 17. isbn 978 - 3 - 540 - 16076 - 2. mr 0825590. pomerance, carl ( 1996 ). " a tale of two sieves " ( pdf ). notices of the american mathematical society. 43 ( 12 ) : 1473 β 148
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Lenstra elliptic-curve factorization
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- 16076 - 2. mr 0825590. pomerance, carl ( 1996 ). " a tale of two sieves " ( pdf ). notices of the american mathematical society. 43 ( 12 ) : 1473 β 1485. mr 1416721. silverman, robert d. ( 1987 ). " the multiple polynomial quadratic sieve ". mathematics of computation. 48 ( 177 ) : 329 β 339. doi : 10. 1090 / s0025 - 5718 - 1987 - 0866119 - 8. mr 0866119. trappe, w. ; washington, l. c. ( 2006 ). introduction to cryptography with coding theory ( second ed. ). saddle river, nj : pearson prentice hall. isbn 978 - 0 - 13 - 186239 - 5. mr 2372272. samuel s. wagstaff, jr. ( 2013 ). the joy of factoring. providence, ri : american mathematical society. pp. 173 β 190. isbn 978 - 1 - 4704 - 1048 - 3. watras, marcin ( 2008 ). cryptography, number analysis, and very large numbers. bydgoszcz : wojciechowski - steinhagen. pl : 5324564. = = external links = = factorization using the elliptic curve method, a webassembly application which uses ecm and switches to the self - initializing quadratic sieve when it is faster. gmp - ecm archived 2009 - 09 - 12 at the wayback machine, an efficient implementation of ecm. ecmnet, an easy client - server implementation that works with several factorization projects. pyecm, a python implementation of ecm. distributed computing project yoyo @ home subproject ecm is a program for elliptic curve factorization which is used to find factors for different kinds of numbers. lenstra elliptic curve factorization algorithm source code simple c and gmp elliptic curve factorization algorithm source code. eecm - mpfq an implementation of ecm using edwards curves written with the mpfq finite field library.
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Lenstra elliptic-curve factorization
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