A divisibility sequence is an integer sequence math a n n in N math such that math forall m,n in N m mid n Rightarrow a m mid a n math , i.e., whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring mathematics ring where the concept of divisibility is defined. Examples Any constant sequence is a divisibility sequence. Any sequence of the form a n kn , for some nonzero integer k , is a divisibility sequence. The Fibonacci numbers F 0, 1, 1, 2, 3, 5, 8,... form a divisibility sequence. Elliptic divisibility sequence s are another class of such sequences. References cite journal first1 Marshall last1 Hall title Divisibility sequences of third order journal Am. J. Math year 1936 pages 577&ndash 584 volume 58 jstor 2370976 cite journal first1 Morgan last1 Ward title A note on divisibility sequences journal Bull. Amer. Math. Soc volume 45 year 1939 pages 334&ndash 336 url http projecteuclid.org euclid.bams 1183501776 cite journal first1 V. E. last1 Hoggat, Jr. first2 C. T. last2 Long title Divisibility properties of generalized fibonacci polynomials year 1973 page 113 url http www.fq.math.ca Scanned 12 2 hoggatt1.pdf journal Fibonacci Quarterly cite journal first1 J. P. last1 Bé zivin first2 A. last2 Ethö first3 A. J. last3 van der Porten journal Am. J. Math. volume 112 issue 6 year 1990 pages 985&ndash 1001 title A full characterization of divisibility sequences jstor 2374733 External links Some http oeis.org search?q divisibility sequence divisibility sequences listed in the On line Encyclopedia of Integer Sequences . Category Sequences and series Category Integer sequences Category Arithmetic functions math stub ... more details
The concept of infinite divisibility arises in different ways in philosophy , physics , economics , order theory a branch of mathematics , and probability theory also a branch of mathematics . One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum theory continuum . In philosophy This theory is exposed in Plato s Timaeus dialogue dialogue Timaeus and was also supported by Aristotle . Andrew Pyle philosopher Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics . There he shows how infinite divisibility involves the idea that there is some extended item , such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort. Many professional philosophers claim that infinite divisibility involves either a collection of an infinite number of items since there are infinite divisions, there must be an infinite collection of objects , or more rarely , point sized items , or both. Pyle states that the mathematics of infinitely divisible extensions involve neither of these that there are infinite ... . Perhaps counter intuitively, atomism is compatible with infinite divisibility. For example ... conundrum of the divisibility of matter. The multiplicity of a material object &mdash the number of its ... divisible. Infinite divisibility does not imply gap less ness the rationals do not enjoy the supremum ... . Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify. In probability distributions Main infinite divisibility probability To say that a probability ... for any finite number of intervals . This concept of infinite divisibility of probability distributions ... Arteaga, A. 2007 On the Infinite Divisibility of some Skewed Symmetric Distributions . Statistics ... translation of Russian Wikipedia page DEFAULTSORT Infinite Divisibility Category Probability ... more details
refimprove date January 2011 lead too short date January 2011 A divisibility rule is a shorthand way ..., usually by examining its digits. Although there are divisibility tests for numbers in any base mathematics radix , and they are all different, we present rules only for decimal numbers. Divisibility ... number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious for others such as examining the last n digits ... with fewer digits. Note To test divisibility by any number that can be expressed as 2 sup n sup ... Divisibility condition Examples 1 number 1 Automatic. Any integer is divisible by 1. 2 number .... Examine the last three digits 34152 Examine divisibility of just 152 19 8 Add four times the hundreds .... Step by step examples Divisibility by 2 First, take any even number for this example it will be 376 ..., then the whole number is divisible by 2 Divisibility by 3 First, take any number for this example ... The original number 6 7 8 336 336 3 112 Divisibility by 4 The basic rule for divisibility by 4 is that if the number ... by 4 Divisibility by 5 Divisibility by 5 is easily determined by checking the last digit in the number ... number divided by 5 Divisibility by 6 Divisibility by 6 is determined by checking the original number to see if it is both an even number Divisibility by 2 divisible by 2 and Divisibility by 3 divisible by 3 . This is the best test to use. Alternatively, one can check for divisibility by six ... rightmost digit 2 br Sum 51 br 51 modulo 6 3 br Remainder 3 Divisibility by 7 Cleanup section date August 2010 Divisibility by 7 can be tested by a recursive method. A number of the form 10 x ... divisibility by 7 uses the fact that 10 sup 0 sup 1, 10 sup 1 sup 3, 10 ... E. title Divisibility by Seven Mudd Math Fun Facts url http www.math.hmc.edu funfacts ffiles 10005.5.shtml ... more details
Unreferenced date November 2007 The base conversion divisibility test is a process that can be used to determine whether or not a certain positive natural number a can be divided evenly into a larger natural number b . It is the general case for the well known test for 9 number divisibility by nine . For other divisor s, applying this test is generally harder than figuring it out by normal division. Example Is 312 evenly divisible by 13? a 13 b 312 x a 1 14 y b base 14 184 312 in base x z 1 8 4 13 z a 13 13 1, a natural number 312 is evenly divisible by 13. Dividing by nine The trick for determining if a number is divisible by nine is well known If the sum of the digits of a number is divisible by nine, then the number itself is as well. This is a special case of the general rule, made easy because no base conversion is necessary since 9 1 10, and we already use base 10. Example Is 2,340 evenly divisible by 9? a 9 b 2,340 x a 1 10 y b base 10 2,340 z 2 3 4 0 9 z a 9 9 1, a natural number 2,340 is evenly divisible by 9. Proof Any number can be expressed as math number base sum i 0 n digits i times base i math We know that under Modulo Arithmetic , math base equiv base 1 1 math Thus math number equiv base 1 sum i 0 n digits i times 1 math Category Arithmetic ... more details
In mathematics, an elliptic divisibility sequence EDS is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomial s on elliptic curve s. EDS were first defined, and their arithmetic properties studied, by Morgan Ward ref name Ward Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 1948 , 31&ndash 74. ref in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography . Definition A nondegenerate elliptic divisibility sequence EDS is a sequence of integers math var W sub n sub var sub var n var &ge 1 sub defined recursively by four initial values math var W var sub 1 sub , math var W var sub 2 sub , math var W var sub 3 sub , math var W var sub 4 sub ... , then every term math var W sub n sub var in the sequence is an integer. Divisibility property An EDS is a divisibility sequence in the sense that math m mid n Longrightarrow W m mid W n. math In particular ... term in the sequence is an integer. General recursion A fundamental property of elliptic divisibility ... to Ward. See the appendix to J. H. Silverman and N. Stephens. The sign of an elliptic divisibility ... math var D sub n sub var is also called an elliptic divisibility sequence . It is a divisibility ... ref name Einsiedler M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences ... rachthesis.ps.gz Elliptic divisibility sequences . PhD thesis, Goldsmith s College University ... Everest s EDS web page. http www.mth.uea.ac.uk h090 primeEDS.html Prime Values of Elliptic Divisibility ... of Elliptic Divisibility Sequences. Category Number theory Category Integer sequences ... more details
The concepts of infinite divisibility and the Decomposable distributions decomposition of distributions arise in probability and statistics in relation to seeking families of probability distributions that might be a natural choice in certain applications, in the same way that the normal distribution is. The distributions sought correspond to random variables which are equivalent to the sums of a number of independent and identically distributed random variables , where the number of such variables can be set to any pre specified number. The term infinitely divisible characteristic function is used for the characteristic function of any infinitely divisible distribution. ref name Lukacs Lukacs, E. 1970 Characteristic Functions , Griffin , London. p. 107 ref The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti . These distributions play a very important role in probability theory in the context of limit theorems. ref name Lukacs Lukacs, E. 1970 Characteristic Functions , Griffin , London. p. 107 ref Definition In probability theory , to say that a probability distribution F on the real line is infinitely divisible means that, for every positive integer n , there exist n statistical independence independent identically distributed random variables X sub n 1 sub , ..., X sub nn sub whose sum S sub n sub X sub n 1 sub &hellip X sub nn sub has the distribution F . Examples The Poisson distribution , the negative binomial distribution , the exponential distribution , the geometric distribution , the Gamma distribution ... References references Dom nguez Molina, J.A. Rocha Arteaga, A. 2007 On the Infinite Divisibility of some ... doi 10.1016 j.spl.2006.09.014 Steutel, F. W. 1979 , Infinite Divisibility in Theory and Practice with discussion ... , Infinite Divisibility of Probability Distributions on the Real Line Marcel Dekker . ProbDistributions Infinite divisibility Infinite divisibility in probability distributions Category Theory of probability ... more details
Summary From magma to group via two alternative paths. Key M Magma, i Invertibility, Q Quasigroup, S Semigroup, d Divisibility, a Associativity, N moNoid, G Group, L Loop, e idEntity. Licensing GFDL migration redundant cc by sa 3.0 ... more details
s writing. Infinite divisibility Infinite divisibility refers to the idea that extension, or quantity ... divisibility of extension. Actual divisibility may be limited due to unavailability of cutting instruments ... more details
Unreferenced date November 2008 Continuous modelling is the mathematical practice of applying a mathematical model model to continuous function continuous data data which has a potentially infinite number, and divisibility, of attributes . They often use differential equation s and are converse to discrete modelling . Modelling is generally broken down into several steps Making assumptions about the data The modeller decides what is influencing the data and what can be safely ignored. Making equations to fit the assumptions. Solving the equations. Verifying the results Various statistical tests are applied to the data and the model and compared. If the model passes the verification progress it is put into practice. External links http www.npl.co.uk scientific software research math modelling Definition by the UK National Physical Laboratory Category Applied mathematics Mathapplied stub ... more details
Summary Non free image data Description Indefinite Divisibility by Yves Tanguy , 1942 Oil on canvas. Source http www.ibiblio.org wm paint auth tanguy WebMuseum, Paris Albright Knox Art Gallery , New York http www.msubillings.edu art Surrealism.htm Portion Entire image. Low resolution It is a low resolution image. Non free image rationale Article Yves Tanguy Purpose It illustrates an educational article about the artist who created the painting that is represented in this image. br Its inclusion in the article adds significantly to the article because it is one of the most famous works by the artist who is the main subject of the article. Replaceability It is not replaceable with an uncopyrighted or freely copyrighted image of comparable educational value. Non free image rationale Article Surrealism Purpose It illustrates an educational article about the surrealist art movement, to which the artist who created this painting has made a significant contribution. Replaceability It is not replaceable with an uncopyrighted or freely copyrighted image of comparable educational value. Licensing Non free 2D art Category Images of paintings ... more details
In mathematics , Higman s lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is well quasi ordering well quasi ordered . That is, if math w 1, w 2, ldots math is an infinite sequence of words over some fixed finite alphabet, then there exist indices math i j math such that math w i math can be obtained from math w j math by deleting some possibly none symbols. More generally this remains true when the alphabet is not necessarily finite, but is itself well quasi ordered, and the subsequence relation allows the replacement of symbols by earlier symbols in the well quasi ordering of labels. This is a special case of the later Kruskal s tree theorem . References citation first Graham last Higman authorlink Graham Higman title Ordering by divisibility in abstract algebras journal Proceedings of the London Mathematical Society series 3 volume 2 issue 7 pages 326 336 year 1952 doi 10.1112 plms s3 2.1.326 Category Wellfoundedness Category Order theory Category Lemmas combin stub ... more details
All infinite divisibility probability infinitely divisible distributions are a fortiori decomposable ... At the other extreme from indecomposability is Infinite divisibility probability infinite divisibility ... divisibility probability References Lukacs, Eugene, Characteristic Functions , New York, Hafner Publishing ... more details
Unreferenced date June 2007 In educational testing , the process of elimination is a test taking tactic for increasing the chances of answering multiple choice question multiple choice questions correctly. A test taker is presented with several possibilities, of which only one answers the question. Even if only one is eliminated and the test taker guesses among the rest, it is rather more probability probable he will hit it when there are only five or four the gain in luck is substantial. Method The method of elimination is iterative algorithm iterative . One looks at the answers, determines that several answers are unfit, eliminates these, and repeats, until one cannot eliminate any more. This iteration is most effectively applied when there is logic logical structure between the answers that is to say, when by eliminating an answer one can eliminate several others. In this case one can find the answers which one cannot eliminate by eliminating any other answers and test them alone the others are eliminated as a logical consequence. This is the idea behind optimizations for computerized searches when the input is sorted as, for instance, in binary search algorithm binary search . Application Howto date January 2011 Here are two questions of one sort, to illustrate how this tactic is applied. In the first, elimination produces an answer almost at once if you know how to go at it in the other, there is no way around it you must try every answer. By which of the following is the number 2135 divisibility divisible 2, 3, 4, 15, 7? Since see divisibility rule for a refresher 2135 is not divisible by 2, it is not divisible by 4 since 2 1 3 5 11 and it is not divisible by 3, it is not divisible by 15. Then only 7 is left and, indeed 305 times 7 is 2135. Note that, if we had a number divisible by 2 but not by 4 and not divisible by 7 , then testing 2 would give us the answer at once. It is always worth testing answers whose exclusion eliminates possibilities, for then, as l ... more details
and digital roots can be used for quick divisibility rule divisibility tests a natural number is divisible ... chess . Harshad number s are defined in terms of divisibility by their digit sums, and Smith ... more details
EDS may refer to Education Educational Specialist Ed.S. , a terminal academic degree in the U.S. Episcopal Divinity School , an Episcopal Seminary in Cambridge, Massachusetts Evansville Day School , an independent college prep school in Evansville, Indiana Politics Environmental Defence Society , a New Zealand environmental organisation European Democrat Students , a centre right political students union Evropsk demokratick strana , a Czech political party Science and mathematics Electrodynamic suspension , a type of magnetic levitation Elliptic divisibility sequence , a class of integer sequences in mathematics Energy dispersive X ray spectroscopy , a method used to determine the energy spectrum of X ray radiation Medicine Ehlers Danlos syndrome , a group of heritable connective tissue disorders Episodic Dyscontrol Syndrome , a pattern of episodic, abnormal, and often violent and uncontrollable social behavior Excessive Daytime Sleepiness , a sleep disorder symptom, especially common in Sleep Apnea and Narcolepsy Technology Earth Departure Stage , a rocket stage forming part of NASA s project Constellation Electronic Document System , an early graphical hypertext system Explosive detection Explosive Detection System, a mechanism for detecting explosive material Extended Data Services now XDS , a standard for the delivery of metadata on NTSC video signals Other Eau de Solide , Eau de Sport EdS , since 1993 94 a new class of perfumes with very low allergic irritating potential. EdS is since 1997 a registered trademark for natural perfumes and cosmetics. Electronic Data Systems , now HP Enterprise Services Encyclopedia Dramatica , a wiki concerning internet humor disambig cs EDS de EDS fr EDS it EDS nl EDS ja EDS sk EDS sl EDS zh EDS ... more details
101 one hundred and one is the natural number following 100 number 100 and preceding 102 number 102 . br It is variously pronounced one hundred and one a hundred and one , one hundred one a hundred one , and one oh one . As an Ordinal number linguistics ordinal number , 101st rather than 101th is the correct form. Number number 101 range 100s cardinal one hundred and one ordinal st ordinal text one hundred and first numeral 101 factorization prime number prime prime 26th divisor 1, 101 roman CI unicode greek prefix latin prefix bin 1100101 oct duo hex 65 misc Wiktionary one hundred and one one hundred one In mathematics 101 is the 26th prime number and a palindromic number and so a palindromic prime . The next prime is 103 number 103 , with which it makes a twin prime pair, making 101 a Chen prime . Because the period length of its reciprocal is unique among primes, 101 is a unique prime . 101 is an Eisenstein prime with no imaginary part and real part of the form math 3n 1 math . 101 is the sum of five consecutive primes 13 number 13 17 number 17 19 number 19 23 number 23 29 number 29 . Given 101, the Mertens function returns 0 number 0 . 101 is the fifth alternating factorial . 101 is a centered decagonal number . The smallest prime for which 2 p 1 has not been completely factored. For a 3 digit number in base 10, this number has a relatively simple divisibility rule divisibility test . The candidate number is split into groups of four, starting with the rightmost four, and added up to produce a 4 digit number. If this 4 digit number is of the form 1000 a 100 b 10 a b where a and b are integers from 0 to 9 , such as 3232 or 9797, or of the form 100 b b , such as 707 and 808, then the number is divisible by 101. This might not be as simple as the divisibility tests for numbers like 3 or 5, and it might not be terribly practical, but it is simpler than the divisibility tests for other 3 digit numbers. citation needed On the seven segment display of a calculator, 101 ... more details
William Heytesbury ref Known as Gugliemus Hentisberus or Tisberus. ref ca. 1313 1372 1373 , philosopher and logician, is best known as one of the Oxford Calculators of Merton College , where he was a fellow by 1330 . In his work he applied logical techniques to the problems of Infinitely divisible divisibility , the Continuum set theory continuum , and kinematics . His masterpiece magnum opus was the Regulae solvendi sophismata Rules for Solving Sophism s , written c. 1335 . He was Chancellor education Chancellor of the University of Oxford from 1371 until 1372. Works 1335 Regulae solvendi sophismata Rules for Solving Sophisms 1. On insoluble sentences 2. On knowing and doubting 3. On relative terms 4. On beginning and ceasing 5. On maxima and minima 6. On the three categories 1483 De probationibus conclusionum tractatus regularum solvendi sophismata , Pavia 1483 De tribus praedicamentis De probationibus conclusionum tractatus regularum solvendi sophismata On the Proofs of Conclusions from the Treatise of Rules for Resolving Syllogisms Liber Calculationum Further reading Sylla, Edith 1982 The Oxford Calculators , in Norman Kretzmann , Anthony Kenny & Pinborg edd. , The Cambridge History of Later Medieval Philosophy Murdoch, John 1982 Infinity and Continuity , in Kretzmann, Kenny & Pinborg edd. , The Cambridge History of Later Medieval Philosophy References sep entry heytesbury William Heytesbury John Longeway Notes references DEFAULTSORT Heytesbury, William Category 14th century mathematicians Category 14th century philosophers Category 14th century English people Category 14th century Latin writers Category Scholastic philosophers Category Fellows of Merton College, Oxford Category Medieval European mathematics philosopher stub de William Heytesbury fr William Heytesbury nl William van Heytesbury pt William de Heytesbury ru , ... more details
In mathematics , a Wieferich pair is a pair of prime number s p and q that satisfy p sup q &minus 1 sup 1 Modular arithmetic mod q sup 2 sup and q sup p &minus 1 sup 1 mod p sup 2 sup Wieferich pairs are named after Germany German mathematician Arthur Wieferich . There are only six Wieferich pairs known ref MathWorld title Double Wieferich Prime Pair urlname DoubleWieferichPrimePair ref 2, 1093 , 3, 1006003 , 5, 1645333507 , 83, 4871 , 911, 318917 , and 2903, 18787 sequence OEIS2C id A124121 and OEIS2C id A124122 in On Line Encyclopedia of Integer Sequences OEIS Wieferich pairs play an important role in Preda Mih ilescu s 2002 proof ref cite journal author Preda Mih ilescu authorlink Preda Mih ilescu title Primary Cyclotomic Units and a Proof of Catalan s Conjecture journal J. Reine Angew. Math. volume 572 year 2004 pages 167 195 id MR 2076124 ref of Mih ilescu s theorem formerly known as Catalan s conjecture . ref Jeanine Daems http www.math.leidenuniv.nl jdaems scriptie Catalan.pdf A Cyclotomic Proof of Catalan s Conjecture . ref See also Wieferich prime Fermat quotient References Reflist Further reading cite journal author Yuri Bilu title Catalan s conjecture after Mih ilescu journal Ast risque volume 294 year 2004 pages vii, 1 &ndash 26 cite journal author R. Ernvall coauthors T. Mets nkyl title On the p divisibility of Fermat quotients journal Math. Comp. volume 66 issue 219 year 1997 pages 1353 1365 url http www.ams.org mcom 1997 66 219 S0025 5718 97 00843 0 home.html doi 10.1090 S0025 5718 97 00843 0 cite journal author Ray Steiner title Class number bounds and Catalan s equation journal Math. Comp. volume 67 issue 213 year 1998 pages 1317 1322 url http www.ams.org mcom 1998 67 223 S0025 5718 98 00966 1 home.html doi 10.1090 S0025 5718 98 00966 1 Category Prime numbers ... more details
A sub 3 sub , another divisibility condition giving type A sub 4 sub , and a final non divisibility condition giving type exactly A sub 4 sub . To see where these extra divisibility conditions come ... four in x sub 1 sub and y sub 1 sub . The divisibility condition for type A sub 4 sub is that x sub ... more details
In mathematics, Atkin Lehner theory is part of the theory of modular form s, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level. A newform is a cusp form new at a given level N , where the levels are the nested subgroups &Gamma sub 0 sub N of the modular group , with N ordered by divisibility . That is, if M divides N , sub 0 sub N is a subgroup of sub 0 sub M . The oldforms for sub 0 sub N are those modular forms f &tau of level N of the form g d &tau for modular forms g of level M with M a proper divisor of N , where d divides N M . The newforms are defined as a vector subspace of the modular forms of level N , complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product . The Hecke operator s, which act on the space of all cusp forms, preserve the subspace of newforms and are self adjoint and commuting operators with respect to the Petersson inner product when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite dimensional C algebra that is commutative and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra . References Citation authorlink A. O. L. Atkin last1 Atkin first1 A. O. L. authorlink2 Joseph Lehner last2 Lehner first2 J. title Hecke operators on sub 0 sub m doi 10.1007 BF01359701 id MathSciNet id 0268123 year 1970 journal Mathematische Annalen issn 0025 5831 volume 185 pages 134 160 Category Modular forms ... more details
The rule of nines , in mathematics, is a divisibility rule for the divisor 9. It is notable because it illustrates some interesting properties of modular arithmetic , and its proof is derived from that basis. The rule is that any positive integer is divisible by 9 if and only if the sum of its digits is also divisible by 9, when expressed in decimal decimal notation . Proof This proof, although not directly taken from that source, is based on the one by Flannery 2001 . Let the positive integer n be represented by the decimal digits a sub k sub a sub k 1 sub ... a sub 2 sub a sub 1 sub a sub 0 sub . Because math begin align 10 0 & equiv 1 pmod 9 10 1 & equiv 1 pmod 9 10 2 & equiv 1 pmod 9 &... end align math and multiplication functions the same way in modular arithmetic as it does in elementary algebra , ref with the caveat that the modulus must remain the same ref math begin align a 0 times 10 0 & equiv a 0 pmod 9 a 1 times 10 1 & equiv a 1 pmod 9 a 2 times 10 2 & equiv a 2 pmod 9 &... a k times 10 k & equiv a k pmod 9 . end align math Summing these equivalences, we get math a k times 10 k ... a 0 times 10 0 equiv a k ... a 0 pmod 9 . math Notice that the left term of this equivalence is equal to n , according to our definition. Therefore, the sum of the digits of n is equivalent to n itself modulo 9 and so this sum is divisible by 9 if and only if n is also divisible by 9. The proof is complete. Notes Not for actual references, it s for the note in the ref tag above references References cite book last Flannery title In Code A Mathematical Journey publisher Workman Publishing year 2001 isbn 0761123849 Category Elementary arithmetic Category Articles containing proofs ... more details
An instant is a infinitesimal moment in time , a moment whose passage is instantaneous. The continuous nature of time and its infinite divisibility was addressed by Aristotle in his Physics Aristotle Physics where he wrote on Zeno s paradoxes . The philosopher and mathematician Bertrand Russell was still seeking to define the exact nature of an instant thousands of years later. ref citation url http books.google.co.uk books?id 29E9AAAAIAAJ&pg PA129 title The structure of time author W. Newton Smith chapter The Russellian construction of instants page 129 publisher Routledge year 1984 isbn 9780710203892 ref In physics , a theoretical lower bound unit of time called the Planck time has been proposed, that being the time required for light to travel a distance of 1 Planck length . ref name gsu hbase cite web url http hyperphysics.phy astr.gsu.edu hbase astro planck.html title Big Bang models back to Planck time publisher Georgia State University date 19 June 2005 ref The Planck length is theorized to be the smallest time measurement that will ever be possible, ref cite encyclopedia url http astronomy.swin.edu.au cosmos P Planck Time title Planck Time encyclopedia COSMOS The SAO Encyclopedia of Astronomy publisher Swinburne University ref roughly 10 sup 43 sup seconds. Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change. As of May 2010, the smallest time interval that was directly measured was on the order of 12 attoseconds 12 10 sup 18 sup seconds , ref cite web url http www.physorg.com news192909576.html title 12 attoseconds is the world record for shortest controllable time ref about 10 sup 24 sup times larger than the Planck time. It is therefore physically impossible, with current technology, to determine if any action exists that causes a reaction in an instant , rather than a reaction occurring after an interval of time too short to observe or measure. See ... more details
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