The unreasonable efficiency of mathematics in science is a gift we neither understand nor deserve.
~Eugene Paul Wigner
The first day of the year 2015; 2013, 2014, and 2015 are each a product of three prime numbers.
2015 is 11111011111 a binary palindrome. In base four it is 113113, in base eight it is 3737 and in base twelve it is 11bb.
And 2015, 2015 raised to the power of 137 is a big number whose digit sum is 2015. *Futility Closet, w/ hat tip to David Brooks
The equation (69-4) x (38-7) = 2015 uses every digit only once. *Alex Bellos @alexbellos
4713 B.C. This was Julian day 1 and began at noon Greenwich or Universal Time (U.T.). It provides a convenient way to keep track of the number of days between events. Noon, January 1, 2014, begins Julian Day Number(JDN) 2,457,024 For the use of the Chinese remainder theorem in determining this date, see American Journal of Physics, 49(1981), 658–661.
JDN 2,400,000 was November 16, 1858. JD 2,500,000.0 will occur on August 31, 2132 at noon UT. *VFR
1675 Edmund Halley, along with Robert Hooke, Thomas Streete and Jonas Moore observe a lunar eclipse from Tower of London using Hooke's balance spring watch to time the event. *Lisa Jardine, Ingenious Pursuits, pg 158
1800 Cauchy’s father was elected Secretary of the Senate in France. The young Cauchy used a corner of his father’s office in Luxembourg Palace for his own desk. LaGrange and Laplace frequently stopped in on business and so took an interest in the boys mathematical talent. One day, in the presence of numerous dignitaries, Lagrange pointed to the young Cauchy and said “You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.”
[E. T. Bell, Men of Mathematics, p. 274] *VFR
1801 Giuseppe Piazzi (1746–1826) discovered the first asteroid, Ceres, but lost it in the sun 41 days later, after only a few observations. Using his new technique of least squares, Gauss correctly predicted where it could be found. This made Gauss a celebrity. [DSB 5, 300 and 10, 591–592;
Buhler, Gauss, pp. 40–45] *VFR
It was originally considered to be a new planet This was followed by the discovery of other similar bodies, which with the equipment of the time appeared to be points of light, like stars, showing little or no planetary disc, though readily distinguishable from stars due to their apparent motions. This prompted the astronomer Sir William Herschel to propose the term "asteroid", from Greek αστεροειδής, asteroeidēs 'star-like, star-shaped', from ancient Greek αστήρ, astēr 'star, planet'. *Wik
1806 France adopts the Gregorian calendar for the second time. France had used the Gregorian calendar prior to Oct 5, 1783 when the French revolutionaries, in their anticlerical zeal, adopted the “calendar of reason.” The year had twelve months, each with three weeks of ten days, plus five or six epagomenal days (days within a solar calendar that are outside any regular month). The day was divided into 10 hours of 100 minutes each. *VFR
1840 The metric system became the only legal system in France on 1 Jan 1840. Delambre and Méchain measured the meridian from Dunkirk to Barcelona, completing their work in 1799 and leading to the formal definition of the metre on 10 Dec 1799. In 1812, it was decreed that a hybrid 'systeme usuelle' could be used. *VFR
In 1872, the metric system was officially introduced throughout the German Empire. (Since 1795, the metric system had been the standard in France. A committeee from the French Academy used a decimal system and defined the meter to be one 10-millionths of the distance from the equator to the Earth's Pole. For the metric unit of mass, the gram was defined as the mass of one cubic centimeter of pure water at a given temperature.) By act Congress, the use of the metric system was legalized in the U.S. (1866), but was not made obligatory. On 20 May 1875, the Treaty of the Meter was signed by twenty countries, including the United States, at the International Metric Convention. *TIS
1876 The British Trade Marks Registration Act was passed which allowed formal registration of trade marks at the UK Patent Office for the first time. Registration of marks began on 1 January 1876. The first trade mark to be so registered was the red triangle of the Bass Brewery. *Wik1896, German scientist, Wilhelm Röntgen announced his discovery of x-rays. He sent copies of his manuscript and some of his x-ray photographs to several renowned physicists and friends, including Lord Kelvin in Glasgow and Henre Poincare in Paris. Four days later, on 5 Jan 1896, Die Presse published the news in a front-page article which described the discovery and suggested new methods of medical diagnoses might be made with this new kind of radiation. One day later, the London Standard cabled the news to other countries around the world about "a light which for the purpose of photography will penetrate wood, flesh, cloth, and most other organic substances." It printed the first English-language account the next day. *TIS
(sometime in the 1920's) The British Mathematician G. H. Hardy wrote a postcard to a friend listing the following six New Year's Wishes:
(1) Prove the Riemann hypothesis.
(2) make 211 not out in the fourth innings of the last test match at the Oval.
(3) find an argument for the nonexistence of God which shall convince the general public.
(4) be the first man to the top of Mt. Everst.
(5) be proclaimed the first president of the U.S.S.R. of Great Britain and Germany.
(6) murder Mussolini.
It seems that, like most New Year's Resolutions, they went unfulfilled.
*Clifford A. Pickover, A Passion for Mathematics, pg 23
1925 Hubble Paper changes the size of the universe: Hubble's observations, made in 1922–1923, proved conclusively that these nebulae were much too distant to be part of the Milky Way and were, in fact, entire galaxies outside our own. This idea had been opposed by many in the astronomy establishment of the time, in particular by the Harvard University-based Harlow Shapley. (Shapley wrote sarcastically that Hubble's letter informing him of his results was “the most entertaining piece of literature I have seen for a long time.” ) Despite the opposition, Hubble, then a thirty-five year old scientist, had his findings first published in The New York Times on November 23, 1924, and then more formally presented in the form of a paper at the January 1, 1925 meeting of the American Astronomical Society. Hubble's findings fundamentally changed the scientific view of the universe.*Wik
1938 Hungary issued a stamp portraying Pope Sylvester II, Archbishop of Astrik.(Often known in mathematics as Gerbert of Aurillac) [Scott #511] *VFR
1945 Counterfeit coins must have been around as long as coins, but the first written example of a counterfeit coin problem was printed in the American Mathematical Monthly in the January issue of 1945. The problem was submitted by E. D. Schell and asked, "You have eight similar coins and a beam balance. At most one coin is counterfeit and hence underweight. How can you detect whether there is an underweight coin, and if so, which one, using the balance only twice." *Counterfeit Coin Problems, Bennet Manvel, Mathematics Magazine, Vol. 50, No. 2 (Mar., 1977), pp. 90-92
In 1946, ENIAC, the first U.S. computer was finished by John Mauchly and J. Presper Eckert. It was built at the Moore School of Engineering, University of Pennsylvania, Philadelphia, based on ideas developed by John Atanasoff of Iowa State College. Though not the first ever computer, ENIAC is regarded as the first successful, general digital computer. It weighed over 27,000 kg (60,000 lb), and contained more than 18,000 vacuum tubes. A staff of six technicians replaced about 2000 of the tubes each month. Many of ENIAC's first tasks were for military purposes, such as calculating ballistic firing tables and designing atomic weapons. Since ENIAC was initially not a stored program machine, it had to be reprogrammed for each task. *TIS
1962 The standard meter is re-defined as “the length of 1,656,763.83 wave lengths of a certain type of orange colored radiation given off in a vacuum by the atom of krypton 86.” *Gardner, Relativity for the Million, pg 5
In 1972, Coordinated Universal Time (UTC) was adopted worldwide. UTC is determined from six primary atomic clocks that are coordinated by the International Bureau of Weights and Measures located in France. The abbreviation - UTC - was chosen as an international compromise between the initials of the English language form "coordinated universal time" and the French "temps universel coordonné." Time zone boundaries as used by nations are drawn according to political considerations. Leap seconds are added to UTC periodically, about once each 18 months, so the highly accurate atomic clock time matches the time measured by Earth's rotation, which is very slightly variable due to tidal forces with the Moon.*TIS
In 2000, Greenwich Electronic Time - known as GeT - was initiated in Britain to act as an international standard for all electronic commerce. All e-mail messages and e-commerce transactions already carry a "time stamp" based on Co-ordinated Universal Time - the modern equivalent of GMT. Most computer clocks have software which converts e-mail and message dates into local time. The move will provide a single time standard for worldwide Internet traders and users around the world in the same way that Greenwich Mean Time has helped travellers to keep time since 1884.*TIS
1614 John Wilkins FRS (1 January 1614 – 19 November 1672) was an English clergyman, natural philosopher and author, as well as a founder of the Invisible College and one of the founders of the Royal Society, and Bishop of Chester from 1668 until his death.
Wilkins is one of the few persons to have headed a college at both the University of Oxford and the University of Cambridge. He was a polymath, although not one of the most important scientific innovators of the period. His personal qualities were brought out, and obvious to his contemporaries, in reducing political tension in Interregnum Oxford, in founding the Royal Society on non-partisan lines, and in efforts to reach out to religious nonconformists. He was one of the founders of the new natural theology compatible with the science of the time.
He is particularly known for An Essay towards a Real Character and a Philosophical Language in which, amongst other things, he proposed a universal language and a decimal system of measure not unlike the modern metric system.*wik
1803 Count Guglielmo Libri Carucci dalla Sommaja (1 Jan 1803, 28 Sept 1869) Libri's early work was on mathematical physics, particularly the theory of heat. However he made many contributions to number theory and to the theory of equations. His best work during the 1830s and 1840s was undoubtedly his work on the history of mathematics. From 1838 to 1841 he published four volumes of Histoire des sciences mathématiques en Italie, depuis la rénaissanace des lettres jusqu'à la fin du dix-septième siècle. He intended to write a further two volumes, but never finished the task. It is an important work but suffers from over-praise of Italians at the expense of others. *SAU
1806 Horatio Nelson Robinson, (Jan 1, 1806; Hartwick, Otsego County, New York - 19 Jan, 1867; Elbridge, New York) received only a common-school education, but early evinced a genius for mathematics, making the calculations for an almanac at the age of sixteen. A wealthy neighbor gave him the means to study at Princeton, and at the age of nineteen he was appointed an instructor of mathematics in the navy, which post he retained for ten years. He then taught an academy at Canandaigua, and afterward one at Genesee, New York, until in 1844 he gave up teaching because his health was impaired, and removed to Cincinnati, Ohio. There he prepared the first of a series of elementary mathematical text-books, which have been adopted in many of the academies and colleges of the United States. In revising and completing the series he had the assistance of other mathematicians and educators. He removed to Syracuse, New York, in 1850, and to Elbridge in 1854. His publications include "University Algebra" (Cincinnati, 1847), with a "Key" (1847) ; "Astronomy, University Edition" (1849) ; " Geometry and Trigonometry" (1850) ; "Treatise on Astronomy" (Albany, 1850) ; "Mathematical Recreations" (Albany, 1851); "Concise Mathematical Operations" (Cincinnati, 1854); "Treatise on Surveying and Navigation" (1857), which, in its revised form, was edited by Oren Root (New York, 1863); "Analytical Geometry and Conic Sections" (New York, 1864) ; "Differential and Integral Calculus" (1861), edited by Isaac F. Quinby (l868). *famousamericans.net
1878 Agner Krarup Erlang (January 1, 1878 – February 3, 1929) was a Danish mathematician, statistician and engineer, who invented the fields of traffic engineering and queueing theory.*Wik
1879 Albert Hoyt Taylor (1 Jan 1879; 11 Dec 1961) American physicist and radio engineer, known as the "father of navy radar" whose work laid the foundation for U.S. radar development. In Sep 1922, with Leo C. Young, he proposed the detection of intruding ships by transmitting a curtain of high-frequency radio waves across harbour entrances, or between ships, with a receiver to detect disturbances caused by ships moving in the electromagnetic field. Taylor became superintendent of the Radio Division at the newly-established Naval Research Laboratory (1923-45). In 1934, he directed Robert Page to experiment with pulsed high-frequency radio signals for aircraft detection. In 1937, the first 200-MHz shipboard radar was installed. He also investigated ionospheric effects. *TIS
1894 Satyendra Nath Bose (1 Jan 1894; 4 Feb 1974) Indian physicist and mathematician who collaborated with Albert Einstein to develop a theory of statistical quantum mechanics, now called Bose-Einstein statistics. In his early work in quantum theory (1924), Bose wrote about the Planck black-body radiation law using a quantum statistics of photons, Plank's Law and the Light Quantum Hypothesis. Bose sent his ideas to Einstein, who extended this technique to integral spin particles. Dirac coined the name boson for particles obeying these statistics. Among other things, Bose-Einstein statistics explain how an electric current can flow in superconductors forever, with no loss. Bose also worked on X-ray diffraction, electrical properties of the ionosphere and thermoluminescence. *TIS
1905 Stanisław Mazur (1 January 1905, Lviv - 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences. He made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. Mazur was also interested in summability theory, infinite games and computable functions.
Mazur was a student of Stefan Banach at University of Lwów. His doctorate, under Banach's supervision, was awarded in 1935.
Mazur was a close collaborator with Banach at Lwów and was a member of the Lwów School of Mathematics, where he participated in the mathematical activities at the Scottish Café. On 6 November 1936, Mazur posed the "basis problem" of determining whether every Banach space has a Schauder basis, with Mazur promising a "live goose" as a reward: Thirty-seven years later, a live goose was awarded by Mazur to Per Enflo in a ceremony that was broadcast throughout Poland.*Wik
1917 Jule Gregory Charney (1 Jan 1917; 16 Jun 1981) American meteorologist who, working with John von Neumann, first introduced the electronic computer into weather prediction (1950) and improved understanding of the large-scale circulation of the atmosphere. The entire Oct 1947 issue of the Journal of Meteorology published his Ph.D. dissertation, (UCLA, 1936) Dynamics of long waves in a baroclinic westerly current. It emphasized the influence of "long waves" in the upper atmosphere rather than the existing practice of emphasis on the polar front. It also simplified analysis of perturbations of these waves using mathematically rigorous methods that yielded useful physical interpretation. He helped the U.S. Weather Bureau set up (1954) a numerical weather prediction unit. *TIS
1920 Heinz Zemanek, one of the European pioneers in computer technology, is born in Vienna, Austria. He studied at the University of Technology Vienna and received a doctorate degree in 1951. During 1954 - 1959 Zemanek gathered a group of students to develop the Mailuefterl, one of the earliest fully transistorized computers in Europe. From 1961 to 1975 Zemanek was Director of the IBM Laboratory Vienna which was established for his team. During this period Zemanek designed the PL/I programming language. The formal definition was written in the Vienna Definition Language, which was later extended to the Vienna Definition Method. (VDL and VDM). From 1968 to 1971 he founded the Austrian Computer Society and was President of the International Federation for Information Processing (IFIP). In 1976 Professor Zemanek was appointed IBM Fellow.*CHM
1923 Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings. He worked on commutative algebra, and then was a major influence on the classification of finite simple groups.*wik
1942 Edward Joseph Hoffman (1 Jan 1942; 1 Jul 2004) American biomedical physicist who achieved international recognition in the science field of medical imaging. In 1974, working with Michael E Phelps and others, he co-invented the PET Scanner (Positron Emission Tomography) which is used to detect cancers and other diseases. Hoffman further developed its use for quantitative measurements. A patient is prepared for a PET scan with an injection of slightly radioactive material such as molecules designed to mimic glucose as they travel through the body. Since cancerous tissues consume glucose, the scanner can then detect their location. PET technology can also be employed in the diagnosis of cardiovascular disease and Alzheimer's disease. Today, some 1,500 scanners are in use.*TIS
1748 Johann Bernoulli (aka Jean Bernouilli) (6 Aug 1667, 1 Jan 1748)Swiss mathematician who is noted for his discovery of the exponential calculus (1691) and the equation of the catenary (1690). His first publication was on the process of fermentation (1690), but thereafter, he studied and taught mathematics for the rest of his life. He followed his his brother Jacques as professor of mathematics at Basle. He was the first to use g to represent the acceleration due to gravity. He applied the then new calculus to the measurement of curves, to differential equations, and to mechanical problems. He introduced the famous brachistochrome problem. “Archimedes of his age” was inscribed on his tombstone. The mathematician Daniel Bernoulli was his son.*TIS
1779 Jeremiah Fenwicke Dixon (27 July 1733 – 22 January 1779) was an English surveyor and astronomer who is best known for his work with Charles Mason, from 1763 to 1767, in determining what was later called the Mason-Dixon line.
Dixon was born in Cockfield, near Bishop Auckland, County Durham, the fifth of seven children, to Sir George Fenwick Dixon 5th Bt. and Lady Mary Hunter. His father was a wealthy Quaker coal mine owner and aristocrat. His mother came from Newcastle, and was said to have been "the cleverest woman" to ever marry into the Dixon family. Dixon became interested in astronomy and mathematics during his education at Barnard Castle. Early in life he made acquaintances with the eminent intellectuals of Southern Durham: mathematician William Emerson, and astronomers John Bird and Thomas Wright. In all probability it was John Bird, who was an active Fellow of the Royal Society, who recommended Dixon as a suitable companion to accompany Mason.
Jeremiah Dixon served as assistant to Charles Mason in 1761 when the Royal Society selected Mason to observe the transit of Venus from Sumatra. However, their passage to Sumatra was delayed, and they landed instead at the Cape of Good Hope where the transit was observed on June 6, 1761. Dixon returned to the Cape once again with Nevil Maskelyne's clock to work on experiments with gravity.
Dixon and Mason signed an agreement in 1763 with the proprietors of Pennsylvania and Maryland, Thomas Penn and Frederick Calvert, sixth Baron Baltimore, to assist with resolving a boundary dispute between the two provinces. They arrived in Philadelphia in November 1763 and began work towards the end of the year. The survey was not complete until late 1766, following which they stayed on to measure a degree of Earth's meridian on the Delmarva Peninsula in Maryland, on behalf of the Royal Society. They also made a number of gravity measurements with the same instrument that Dixon had used with Maskelyne in 1761. Before returning to England in 1768, they were both admitted to the American Society for Promoting Useful Knowledge, in Philadelphia.
Dixon sailed to Norway in 1769 with William Bayly to observe another transit of Venus. The two split up, with Dixon at Hammerfest Island and Bayly at North Cape, in order to minimize the possibility of inclement weather obstructing their measurements. Following their return to England in July, Dixon resumed his work as a surveyor in Durham. He died unmarried in Cockfield on 22 January 1779, and was buried in an unmarked grave in the Quaker cemetery in Staindrop.
Although he was recognized as a Quaker, he was not a very good one, dressing in a long red coat and occasionally drinking to excess. *Wik
1787 José Anastácio da Cunha (1744 in Lisbon, Portugal - 1 Jan 1787 in Lisbon, Portugal) was a Portuguese mathematician who wrote an encyclopaedia of mathematics.*SAU
1894 Heinrich Rudolf Hertz (22 Feb 1857, 1 Jan 1894) was a German physicist who was the first to broadcast and receive radio waves. He studied under Kirchhoff and Helmholtz in Berlin, and became professor at Bonn in 1889. His main work was on electromagnetic waves (1887). Hertz generated electric waves by means of the oscillatory discharge of a condenser through a loop provided with a spark gap, and then detecting them with a similar type of circuit. Hertz's condenser was a pair of metal rods, placed end to end with a small gap for a spark between them. Hertz was also the first to discover the photoelectric effect. The unit of frequency - one cycle per second - is named after him. Hertz died of blood poisoning in 1894 at the age of 37. *TIS
1796 Alexandre-Theophile Vandermonde (28 Feb 1735 in Paris, France - 1 Jan 1796 in Paris, France) was a French mathematician best known for his work on determinants. *SAU
1896 Alfred Ely Beach (1 Sep 1826, 1 Jan 1896) was an American inventor and publisher of Scientific American magazine which reported on technology developments and patents in the 19th-century. It is still published today, one of the world's leading science magazines. Beach himself invented a tunneling shield and built the pneumatic tube subway which propelled a carriage by means of air pressure generated by huge fans. The tunnel was short - one block - so it operated as a demonstration (1870-73), with one station and train car. In 1856 he won First Prize and a gold medal at New York's Crystal Palace Exhibition. Beach had invented a typewriter for the blind, resembling the modern typewriter in the arrangement of its keys and typebars, but embossed its letters on a narrow paper strip instead of a sheet. *TIS
1922 Wooster Woodruff Beman (May 28, 1850 - January 1, 1922). He attended school in Valparaiso, Ind., and entered the University of Michigan in 1866, receiving his B.A. degree in 1870. After teaching for a year at Kalamazoo College as instructor in Greek and mathematics, he returned to the University of Michigan as an instructor while also working for his master's degree, which he received in 1873. In 1874, he became assistant professor, in 1882 associate professor, and in 1887 full professor.
In addition to his teaching, Beman wrote books and articles on the history and teaching of elementary mathematics. Among his works are "Nature and Meaning of Numbers" (from the German), and "Continuity and Irrational Numbers." He was the joint author, with D. E. Smith, of "Plane and Solid Geometry," "Higher Arithmetic," "New Plane and Solid Geometry," "Elements of Algebra," "Academic Algebra," translations of "Famous Problems of Elementary Geometry," and "A Brief History of Mathematics." *Michigan Historical Collections
1992 Grace Hooper dies: The US Navy recalled Captain Grace Murray Hopper to active duty to help develop the programming language COBOL. With a team drawn from several computer manufacturers and the Pentagon, Hopper -- who had worked on the Mark I and II computers at Harvard in the 1940s -- created the specifications for COBOL (COmmon Business Oriented Language) with business uses in mind. These early COBOL efforts aimed at creating easily-readable computer programs with as much machine independence as possible. Designers hoped a COBOL program would run on any computer for which a compiler existed with only minimal modifications.
Hopper made many major contributions to computer science throughout her very long career, including what is likely the first compiler ever written, "A-0." She appears to have also been the first to coin the word "bug" in the context of computer science, taping into her logbook a moth which had fallen into a relay of the Harvard Mark II computer. She died on January 1, 1992. (the term "bug" in the meaning of technical error dates back at least to 1878 and Thomas Edison , and "debugging" seems to have been used as a term in aeronautics before entering the world of computers. Indeed, in an interview Grace Hopper remarked that she was not coining the term. The moth fit the already existing terminology, so it was saved.)
The U.S. Navy commissioned their most advanced ship, the U.S.S. Hopper (DDG 70), on September 6, 1997 named in honor of Grace Hopper. *CMH
1995 Eugene Paul Wigner
Thumbnail - Eugene Paul Wigner (17 Nov 1902, 1 Jan 1995). Hungarian-American physicist who shared the 1963 Nobel Prize for Physics (with Maria Goeppert Mayer and Johannes Hans Jensen) for his insight into quantum mechanics, for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles. He made many contributions to nuclear physics and played a prominent role in the development of the atomic bomb and nuclear energy. *TIS
1996 Gertrude Blanch (2 Feb, 1897 (sometimes 1898) - 1 Jan,1996) was an American mathematician who did pioneering work in numerical analysis and computation. After the war, Blanch's career was hampered by FBI suspicions that she was secretly a communist. Their evidence for this seems scarce, and included, for example, the observation that she had never married or had children. In what must have been a remarkable showdown, the diminutive fifty-year-old mathematician demanded, and won, a hearing to clear her name.
Subsequently, she worked for the Institute for Numerical Analysis at UCLA and the Aerospace Research Laboratory at Wright-Patterson Air Force Base in Dayton, Ohio. She was one of the founders of the ACM.
She published over thirty papers on functional approximation, numerical analysis and Mathieu functions. In 1962, she was elected a Fellow in the American Association for the Advancement of Science.
Blanch retired in 1967 at the age of 69, but continued working under a consulting contract for the Air Force for another year. Thereafter she moved to San Diego and continued to work on numerical solutions of Mathieu functions until her death in 1996, concentrating on the use of continued fractions to achieve highly accurate results in a small number of computational steps. This work has not been published. The Gertrude Blanch Papers, 1932-1996 are stored at the Charles Babbage Institute, University of Minnesota, Minneapolis. *Wik
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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