Pat'sBlog: April 2019

Saturday 6 April 2019

On This Day in Math - April 6

Abel Statue at Univ of Oslo, *Monuments on Mathematicians

Niels Henrik Abel
1802 - 1829
mathematician, famed due to
epoch-making works in
theory of equations, [theory of] infinite series
and elliptic functions

Science can amuse and fascinate us all, but it is engineering that changes the world.

~Isaac Asimov

The 96th day of the year; 96 is the smallest number that can be written as the difference of 2 squares in 4 ways. *What's So Special About This Number?
(students are encouraged to find them all...Is there a smaller number that can be so expressed in 3 ways?)

The sum of 96 consecutive squared integers is a square number ( $ x^2 + (x+1)^2 + (x+2)^2 +(x+3)^2 + \dotsm + (x+95)^2 = y^2 $ ) can be solved with eight sets of 96 consecutive year days. One solution is $ 13^2 + 14^2 + \dotsm + 108^2 = 652^2 $ *Ben Vitale

Ninety Six, South Carolina. There is much confusion about the mysterious name, "Ninety-Six," and the true origin may never be known. Speculation has led to the mistaken belief that it was 96 miles to the nearest Cherokee settlement of Keowee; to a counting of creeks crossing the main road leading from Lexington, SC, to Ninety-Six; to an interpretation of a Welsh expression, "nant-sych," meaning "dry gulch." Pitcher Bill Voiselle of the Boston Braves was from Ninety Six, South Carolina, and wore uniform number 96.

EVENTS

648 B.C. First Greek record of a total solar eclipse is made. See June 4, 780 B.C., and October 13, 2128 B.C. *VFR

"Zeus, the father of the Olympic Gods, turned mid-day into night,
hiding the light of the dazzling Sun; and sore fear came upon men."
"Nothing can be surprising any more or impossible or miraculous,
now that Zeus, father of the Olympians has made night out of noonday,
hiding the bright sunlight, and . . . fear has come upon mankind.
After this, men can believe anything, expect anything.
~ Archilochus, Greek poet

*Ted Pedas, Eclipse History web site.

1741 Euler's First paper on partitions is given. (This may be where generating functions are first used). On September 4, 1740, Philip Naudé the younger (1684-1747) wrote Euler from Berlin to ask “how many ways can the number 50 be written as a sum of seven different positive integers?” The problem seems to have captured Euler’s imagination. Euler gave his first answer on April 6, 1741, in a paper he read at the weekly meeting of the St. Petersburg Academy. That paper was published ten years later and is number 158 on Eneström’s index. (Ed Sandifer, "How Euler Did It") [E158 can be downloaded in English from Euler Project]

1841 William Thompson, the future Lord Kelvin, age 16, was formally entered at St. Peter's (or Peterhouse) Cambridge as a student of the college. His father, Professor James Thompson, may have urged him to enter Peterhouse because of the college's mathematical coach, Hopkins, whom Professor Kelvin admired. It would become the college of choice for many young Scots for a while. Tait went there, and Maxwell began there but later transferred to Trinity. *Silvanus Phillips Thompson, The life of Lord Kelvin

1852, Edward Sabine announced that the 11 year sunspot cycle was "absolutely identical" with the geomagnetic cycle. Later, using a larger dataset, Rudolf Wolf confirmed this fact. Since Newton's explanation of the effect of the sun's gravity on earth, this was the first new phenomenon of the sun interacting with the earth. Thus began continuing studies of the solar-terrestrial activity. Sabine was an Irish geophysicist, astronomer, and explorer, who made extensive pendulum measurements to determine the shape of the earth, and established magnetic observatories to relate sunspot activity with disturbances in terrestrial magnetism. Sabine was knighted in 1869.

In 1869, the American Museum of Natural History in New York City was officially created with the signing of a bill by the Governor of New York, John Thompson Hoffman. The museum began from the efforts of Albert Smith Bickmore, one-time student of Harvard zoologist Louis Agassiz, who was successful in his proposal to create a natural history museum in Central Park, New York City, with the support of William E. Dodge, Jr., Theodore Roosevelt, Sr., Joseph Choate, and J. Pierpont Morgan. It opened to the public 27 Apr 1871. With a series of exhibits, the Museum's collection went on view for the first time in the Central Park Arsenal, the Museum's original home, on the eastern side of Central Park. *TIS

1909 Ernst Zermelo (1871–1953) liked to argue that it is impossible for anyone ever to reach the North Pole, because the amount of whiskey needed to reach any latitude is proportional to the tangent of that latitude. Unaware of this argument, Robert E. Peary wrote in his diary on this date. “The Pole at last!!! The prize of 3 centuries, my dream & ambition for 23 years. Mine at last. I cannot bring myself to realize it. It all seems so simple ... .” Peary, his remarkable Black associate, Matthew Henson, and four Eskimos were the ﬁrst humans to reach the North Pole. See The National Geographic Society. 100 Years of Adventure and Discovery (1987), pp. 53 & 59. [Reid, Hilbert, p. 97.]*VFR

1922 Emmy Noether named “unofficial associate professor” at Gottingen. This purely honorary position reveals the strong prejudice of the day against women. [DSB 10, 138 and A. Dick, xiii.] Thony Christie sent me a note that says "In 1922 Emmy Noether was appointed 'außerordentliche Professur' which is not an 'unofficial associate professor' but is an official professorial post without a chair. " Thanks, Thony

1929 To celebrate the centenary of the Death of Neils Henrik Abel, Norway issued a set of stamps in his honor. This is the first set of stamps honoring a mathematician in Philatelic history.

1938 DuPont researcher Roy Plunkett and his assistant, Jack Rebok, discovered polytetraﬂuoroethylene, the slipperiest man-made substance. Teﬂon became a household word in 1960 when Teﬂon-coated frying pans were introduced. The Manhattan Project used it in producing Uranium-235, for it was the only gasket material that would contain the corrosive hexaﬂouride.

In 1955, a report that Jupiter emitted radio waves was the subject of a page-length column of the New York Times. Discovered by Bernard F. Burke and Kenneth L. Franklin, astronomers at the Carnegie Institution in Washington, the waves resembled short bursts of static, similar to the interference on home radios caused by lightning. This was the first time radio waves were detected from any planet in our solar system. The astronomers announced their find at the semi-annual meeting of the American Astronomical Society in Princeton, N.J. Discovered at first by chance, it took several weeks to pinpoint Jupiter as the origin, rather than any local source on Earth.*TIS

1956 The ﬁrst circular office building, the Capital Tower, at Hollywood and Vine in Los Angeles, was dedicated. The building has a diameter of 92 feet and a height of 150 feet. Above the 13 ﬂoors was a 90 foot spire from which a beacon ﬂashed the word “Hollywood” in Morse code. *FFF

1963 Watson-Watt is remembered as the inventor of Radar, for which he received a patent on April 2, 1935. Twenty-eight years later he read a poem in a science meeting in San Francisco about the strange twist of Technological Karma that led to his getting a speeding ticket in Canada in 1956. Reportedly he told the officer who stopped him, "If I knew what you were going to do with it, I would never have invented it." The poem reads:

Pity Sir Watson-Watt,
strange target of this radar plot
and thus, with others I can mention,
the victim of his own invention.
His magical all-seeing eye
enabled cloud-bound planes to fly
but now by some ironic twist
it spots the speeding motorist
and bites, no doubt with legal wit,
the hand that once created it.

Oh Frankenstein who lost control
of monster man created whole,
with fondest sympathy regard
one more hoist with his petard.
As for you courageous boffins
who may be nailing up your coffins,
particularly those whose mission
deals in the realm of nuclear fission,
pause and contemplate fate's counter plot
and learn with us what's Watson-Watt.

*nndb.com

1967 Spain issued a stamp picturing Averroes (1126–1198) physician and philosopher. [Scott #1461] *VFR

1972 Cray Research is an American supercomputer manufacturer based in Seattle, Washington. The company's predecessor, Cray Research, Inc. (CRI), was founded in 1972 by computer designer Seymour Cray.

1992 Microsoft Releases Windows 3.1:
Microsoft Corporation releases Windows 3.1, an operating system that provided IBM and IBM-compatible PCs with a graphical user interface (though Windows was not the first such interface for PCs). Retail price was $149.00. In replacing the previous DOS command line interface with its Windows system, however, Microsoft created a program similar to the Macintosh operating system, and was sued by Apple for copyright infringement. (Microsoft later prevailed in this suit).
Windows 3.1 added multimedia extensions allowing support for sound cards, MIDI, and CD Audio, Super VGA (800 x 600) monitors, and increased the speed of modem it would support to 9600 bps. It also finally abandoned "Real Mode," a vestigial environment dating back to the 8086 CPU. It provided scalable fonts and trapped the "three finger salute" (CTRL-ALT-DEL), prompting the user to avoid inadvertent re-boots. It also refined its OLE (Object Linking and embedding) concept, allowing users to cut and paste between applications.*CHM

1995 Stephen Hawking, in response to a request for a "time travel equation" from the editors of THE FACE magazine, sent the following fax: "Thank you for your recent fax. I do not have any equations for time travel. If I had, I would win the National Lottery every week." *Letters of Note web site

BIRTHS

1749 Samuel Vince (6 April 1749; Fressingfield – 28 November 1821; Ramsgate) was an English clergyman, mathematician and astronomer at the University of Cambridge.
The son of a plasterer, Vince was admitted as a sizar to Caius College, Cambridge in 1771. In 1775 he was Senior Wrangler, and Winner of the Smith Prize at Cambridge. Migrating to Sidney Sussex College in 1777, he gained his M.A. in 1778 and was ordained a clergyman in 1779.
He was awarded the Copley Medal in 1780 and was Plumian Professor of Astronomy and Experimental Philosophy at Cambridge from 1796 until his death.
As a mathematician, Vince wrote on many aspects of his expertise, including logarithms and imaginary numbers. His Observations on the Theory of the Motion and Resistance of Fluids and Experiments upon the Resistance of Bodies Moving in Fluids had later importance to aviation history. He was also author of the influential A Complete System of Astronomy (3 vols. 1797-1808).
Vince also published the pamphlet The Credibility of Christianity Vindicated, In Answer to Mr. Hume's Objections; In Two Discourses Preached Before the University of Cambridge by the Rev. S. Vince. In this work, Vince made an apology of the Christian religion and, like Charles Babbage, sought to present rational arguments in favor of the belief in miracles, against David Hume's criticism. *Wik

1801 William Hallowes Miller (6 Apr 1801; 20 May 1880 at age 79) Welsh minerologist known for his Millerian indices built on his system of reference axes for crystals by which the different systems of crystal forms can be designated using a a set of three integers for each crystal face. When he published this scheme in A Treatise on Crystallography (1839), he provided an alternative to the existing confusion due to the many different descriptive systems previously in use. In his early career he published successful textbooks for hydrostatics and hydrodynamics (1831) and differential calculus (1833). Miller also prepared new standards in 1843 to replace the National Standards of weight and length that had been lost in the 1834 fire that destroyed the Parliament buildings. *TIS For an interesting story about how mathematics was related to the fire, read here.

1890 André-Louis Danjon (6 Apr 1890; 21 Apr 1967 at age 76) French astronomer who devised a now standard five-point scale for rating the darkness and colour of a total lunar eclipse, which is known as the Danjon Luminosity Scale. He studied Earth's rotation, and developed astronomical instruments, including a photometer to measure Earthshine - the brightness of a dark moon due to light reflected from Earth. It consisted of a telescope in which a prism split the Moon's image into two identical side-by-side images. By adjusting a diaphragm to dim one of the images until the sunlit portion had the same apparent brightness as the earthlit portion on the unadjusted image, he could quantify the diaphragm adjustment, and thus had a real measurement for the brightness of Earthshine.*TIS

1903 Harold Eugene Edgerton (6 Apr 1903; 4 Jan 1990 at age 86) was an American engineer and ultra-high-speed photographer who, as a graduate at the Massachusetts Institute of Technology (1926), used a strobe light in his studies, which,. by 1931, he applied the strobe to ultra-high-speed photography. He formed a company (1947) to specialize in electronic technology, which led to inventing the Rapatronic camera, capable of photographing US nuclear bomb test explosions from a distance of 7 miles. Throughout his career he applied high-speed photography as a tool in various scientific applications. He also developed sonar to study the ocean floor. Using side-scan sonar, in 1973, he helped locate the sunken Civil War battleship USS Monitor, lost since 1862, off Cape Hatteras, NC. *TIS

1947 Michael Worboys (born April 6, 1947, ) is a British mathematician and computer scientist. He is professor of spatial informatics and director of the School of Computing and Information Science at the University of Maine.
Worboys is mostly known for his research on the computational and mathematical foundations of Geographic Information Science (GIS). In 1993 he founded the GIS Research UK (GISRUK) conference series, which is still held annually. With Matt Duckham, he wrote the well-known text book GIS: a computing perspective.*Wik

DEATHS

1528 Albrecht Durer (21 May 1471, 6 April 1528)
German artist who published a book on geometric constructions (1535) using a straight-edge and compass. Although designed to enable artists better represent a natural three-dimensional scene on a canvas, Dürer included careful proofs to establish the validity of the constructions. In this respect, it could be regarded as the oldest surviving text on applied mathematics. He also wrote on the proportions of the human body.*TIS

1829 Niels Henrik Abel, age 26, died of tuberculosis. In 1929 Norway issued four stamps for the centenary of his death. [Scott #145–148] Neils Henrik Abel was born at Fomm¨oy, a small island near Stavanger in Norway. Before going to the university in 1821 he attacked, with the vigor and immodesty of youth, the problem of the solution of the quintic equation. He submitted a solution for publication but found an error before it was published. In 1823 he proved the impossibility of a solution involving radicals that solves ﬁfth or higher degree equations. *VFR
He developed the concept of elliptic functions independently of Carl Gustav Jacobi, and the theory of Abelian integrals and functions became a central theme of later 19th-century analysis. He had difficulty finding an academic position, was troubled by poverty, and died in poverty in his late twenties.*TIS
I love Abel's commet on Gauss' writing style, "He is like the fox, who effaces his tracks in the sand with his tail."
The early death of this talented mathematician, of whom Adrien-Marie Legendre said "quelle tête celle du jeune Norvégien!" ("what a head the young Norwegian has"), cut short a career of extraordinary brilliance and promise. Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. His works, the greater part of which originally appeared in Crelle's Journal, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881. The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety). (Wikipedia)

1963 Otto Struve (12 Aug 1897, 6 Apr 1963 at age 65) Russian-American astronomer who was a fourth generation astronomer, the great-grandson of Friedrich Struve. He made detailed spectroscopic investigations of stars, especially close binaries and peculiar stars, the interstellar medium (where he discovered H II regions), and gaseous nebulae. He contributed to the understanding of the broadening of spectral lines due to stellar rotation, electric fields, and turbulence and worked to separate these effects from each other and from chemical abundances. He was a pioneer in the study of mass transfer in closely interacting binary stars. Struve emigrated to the USA (1921) and joined the Yerkes Observatory, Wisconsin, becoming its director in 1932. *TIS

1992 Isaac Asimov (2 Jan 1920; 6 Apr 1992) American author and biochemist, who was a prolific writer of science fiction and of science books for the layperson. Born in Petrovichi, Russia, he emigrated with his family to New York City at age three. He entered Columbia University at the age of 15 and at 18 sold his first story to Amazing Stories. After earning a Ph.D., he taught biochemistry at Boston University School of Medicine after 1949. By 18 Mar 1941, Asimov had already written 31 stories, sold 17, and 14 had been published. As an author, lecturer, and broadcaster of astonishing range, he is most admired as a popularizer of science (The Collapsing Universe; 1977) and a science fiction writer (I, Robot;1950). He coined the term "robotics." He published about 500 volumes.*TIS

1993 John Charles Burkill FRS (1 February 1900 Holt, Norfolk, England – 6 April 1993 Sheffield, England) was an English mathematician who worked on analysis and introduced the Burkill integral. He was elected a fellow of the Royal Society in 1953*Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*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

Friday 5 April 2019

On This Day in Math - April 5

How dare we speak of the laws of chance? Is not chance the antithesis of all law?
~Joseph Bertrand

The 95th day of the year; 95⁰ + 95¹ + 95² + 95³ + 95⁴ + 95⁵ + 95⁶ is prime. *Prime Curios

95and its reversal (59) begin fewer four-digit prime numbers (seven) than any other two-digit number.

95 is the number of planar partitions of 10. (A plane partition is a two-dimensional array of integers n_(i,j) that are nonincreasing both from left to right and top to bottom and that add up to a given number n. Here's one plane partition of 22

5 4 2 1 1
3 2
2 2

95 is the sum of 7 consecutive primes = 5 + 7 + 11 + 13 + 17 + 19 + 23

EVENTS

1610 Writing to Galileo, Kepler was impressed by the observation that stars seen through the telescope still sparkled, in contrast to the circular appearance of planets. He asked:

"What other conclusion shall we draw from this difference, Galileo, than that the fixed stars generate their light from within, whereas the planets, being opaque, are illuminated from without; that is, to use Bruno’s terms, the former are suns, the latter, moons, or earths?"

*Steven Soter, Ciclops.org

1752 Taxes are due in England. Previously they were due on March 25, the ﬁrst day of the year, but because the adoption of the Gregorian calendar reform necessitated the dropping of eleven days, the tax date was changed also. Apparently the tax collectors couldn’t do fractions. *VFR

In 1753, the British Museum was founded by an Act of Parliament granting £20,000 to purchase the 50,000 volume library of Sir Hans Sloane and his vast collection of 69,352 items of nature and art. Sloane was a prominent London physician who made the collection available in his will at much below its intrinsic value. Montagu House, Bloomsbury, was purchased in 1754 by the government to house this and other collections. Since it opened, on 15 Jan 1759, the Museum has been collecting, conserving and studying millions of artefacts. The British Museum established its Research Laboratory in 1920 with the appointment of Dr Alexander Scott as its first scientist.*TIS The myth that Sloan had invented the process of Hot Chocolate, which is still strongly promoted in the shops in Chelsea that feature this product, is a myth. See James Delbourgo's Article on Sloan and Cocoa here. Skipping to page 78 for details of the history of Chocolate in use around Europe in the 17th century. It had been used for much longer by the natives of South America with some apparent religious or spiritual relationship.

1792 George Washington cast the ﬁrst presidential veto in the USA. Amazingly, mathematics was involved. It seemed so easy. The 1787 US Constitution laid out simple rules for deciding how many representatives each state shall receive:
"Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers, ... The number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative ...". It may have seemed easy, but for the 200+ years of US government, the question of "Who gets how many?" continues to perplex and promote controversy. When congress discussed mathematical methods of applying this constitutional directive there were two methods of prime consideration, Jefferson's method, and Hamilton's method. Congress selected Hamilton's method and in the first use of the Presidential veto . President Washington rejected the bill. Congress submitted and passed another bill using Jefferson's method. The method used has changed frequently over the years with a method by Daniel Webster adopted in 1842, (the original 65 Representatives had grown to 223) and then replaced with Hamilton's method in 1852 (234 Representatives). In a strange "Only in America" moment in 1872, the congress reapportioned without actually adopting an official method and some analysis suggest that the difference caused Rutherford Hayes to win instead of Samuel Tilden who would have won had Hamilton's method been used. Since 1931 the US House has had 435 Representatives with the brief exception of when Alaska and Hawaii became states. Then there was a temporary addition of one seat for each until the new apportionment after the 1960 Census. In 1941 the Huntington-Hill Method was adopted and has remained in continuous (and contentious) use ever since.(Pat B)

Chaotic Elections! A Mathematician Looks at Voting

1800 A UFO sighting near Baton Rouge, Louisiana will be reported to the American Philosophical Society by Thomas Jefferson, President of the society, and (at that time) Vice-President of the United States. The report of a UFO by a Vice-President is still the highest government official to report a UFO. The report itself was written by the naturalist William Dunbar: "A phenomenon was seen to pass Baton Rouge on the night of the 5th April 1800, of which the following is the best description I have been able to obtain. It was first seen in the South West, and moved so rapidly, passing over the heads of the spectators, as to disappear in the North East in about a quarter of a minute. It appeared to be of the size of a large house, 70 or 80 feet long"

In 1881, Hermann von Helmholtz presented The Faraday Lecture before the Fellows of the Chemical Society in London. His topic was The Modern Development of Faraday's Conception of Electricity. Helmholtz recognized Michael Faraday as being the person who most advanced the general scientific method, saying “His principal aim was to express in his new conceptions only facts, with the least possible use of hypothetical substances and forces.” *TIS

1893, Thomas Corwin Mendenhall, then Superintendent of Weights and Measures, with the approval of the Secretary of the Treasury, decided that the International Meter and Kilogram would in the future be regarded as the fundamental standards of length and mass in the United States, both for metric and customary weights and measures. This decision, which has come to be known as "The Mendenhall Order," was first published as Bulletin No. 26 of the Coast and Geodetic Survey under the title Fundamental Standards of Length and Mass. The Mendenhall Order initiated a departure from the previous policy of attempting to maintain our standards of length and mass to be identical with those of Great Britain.*TIS (And after all this time we have completely converted to metric ;-] )

1955 On the 5th of April, 1955, Nobel laureate Bertrand Russell sent a following letter to Albert Einstein along with a rough draft of what would soon be known as the Russell-Einstein Manifesto - a written warning to the world's population on the dangers of nuclear weapons, and a plea for all leaders to avoid war when faced with conflict - and asked him to be both a signatory and supporter. Einstein's short reply, and in fact the last letter he ever wrote, arrived a week later:

Dear Bertrand Russell,
Thank you for your letter of April 5. I am gladly willing to sign your excellent statement. I also agree with your choice of the prospective signers.

With kind regards,

A. Einstein.

Einstein passed away on the 18th of that month, and the manifesto was released to the public on July 9th.
* Shaun Usher, Letters of Note Web site

In 1963, the U.S. Atomic Energy Commission gave the Fermi Award to J. Robert Oppenheimer for research in nuclear energy. Oppenheimer was the chief scientist of the Manhattan Project during WWII that created the atomic bomb. Later, he opposed the more destructive hydrogen bomb development and his security clearance was revoked (1954). Nine years later, a wiser U.S. government awarded Oppenheimer the prestigious Fermi Award, "For contributions to theoretical physics as a teacher and originator of ideas, and for leadership of the Los Alamos Laboratory and the atomic energy program during critical years." The actual presentation of the medal and $50,000 was made 2 Dec 1963 by President Lyndon B. Johnson. *TIS
American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer

BIRTHS

1588- Thomas Hobbes(5 April 1588 – 4 December 1679) was an English scholar and amateur mathematician who wrote on optics and on geometry. He attacked the 'new' methods of mathematical analysis. Hobbes was caught in a series of conflicts from the time of publishing his De Corpore in 1655. In Leviathan he had assailed the system of the original universities. Because Hobbes was so evidently opposed to the existing academic arrangements, and because De Corpore contained not only tendentious views on mathematics, but an unacceptable proof of the squaring of the circle (which was apparently an afterthought), mathematicians took him to be a target for polemics. John Wallis was not the first such opponent, but he tenaciously pursued Hobbes. The resulting controversy continued well into the 1670s. *Wik

1607 Honor´e Fabri, or Honoratus Fabrius,(1607 in Ain, France; 8 March 1688 at Rome,) He developed the inﬁnitesimal methods of Cavalieri and Torricelli and his quadrature of the cycloid inspired Leibniz. Some of his geometrical work boils down to special cases of xn sin x dx, sinn x dx and arcsin x dx dy. [DSB 4, 506] *VFR In his treatise on man he claims to have discovered the circulation of the blood, prior to William Harvey, but after having investigated this question, Father Auguste Bellynck arrives at the conclusion that, at best, Father Fabri may have made the discovery independently of Harvey. *Wik

1622 – Vincenzo Viviani,(April 5, 1622 – September 22, 1703) Italian mathematician In 1639, at the age of 17, he was an assistant of Galileo Galilei in Arcetri. He remained a disciple until Galileo's death in 1642. From 1655 to
1656, Viviani edited the first edition of Galileo's collected works. He was a leader in his field and founded the Accademia del Cimento. As one of the first important scientific societies, this organization came before England's Royal Society. In 1639, at age 17, he became the student, secretary and assistant of Galileo (now blind) in Arcetri, until Galileo died in 1642. A note from Thony Christie informed me that after Galileo's death, his papers were being used by the local butcher to wrap his meat and sausages until Viviani rescued what was left of them.

During his long career, Viviani published a number of books on mathematical and scientific subjects. He edited the first edition of Galileo's collected works (1655-1656), and worked tirelessly to have his master's memory rehabilitated. In 1660, together with Borelli, he measured the velocity of sound by timing the difference between the flash and the sound of a cannon. They obtained the value of 350 metres per second.*TIS
Viviani's Theorem is named for him. The theorem states that in an equilateral triangle, the sum of the perpendicular distances to the sides is equal to the altitude of the triangle. In the figure h=PE+PF+PG. If the point is outside the triangle, the relationship will still hold if one or more of the perpendiculars is treated as a negative value. The theorem can be generalized to a regular n-gon to state, for any point P interior to a regular n-gon, the sum of the perpendicular distances to the n sides is n times the apothem of the figure.

1877 Georg Faber (5 April 1877 in Kaiserslautern, Germany - 7 March 1966 in Munich, Germany) Faber's most important work was on the polynomial expansion of functions. This is the problem of expanding an analytical function in an area bounded by a smooth curve as a sum of polynomials, where the polynomials are determined by the area. These polynomials are now known as 'Faber polynomials' and first appear in Faber's 1903 paper Über polynomische Entwickelungen published in Mathematische Annalen. Another important paper which he also published in Mathematische Annalen, this time in 1909, was Über stetige Funktionen. In this paper he introduced the 'hierarchical basis' and explicitly used it for the representation of functions. In fact Faber was building on the idea of Archimedes who computed approximately using a hierarchy of polygonal approximations of a circle. Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations. One further achievement of Faber is worthy of mention. In 1894 Lord Rayleigh made the following claim:" ... given a fixed area of ox-hide to make a drum, the ground tone is lowest if you make your drum circular. " Two mathematicians independently verified Rayleigh's conjecture, Faber and Edgar Krahn. *SAU

1901 Subbayya Sivasankaranarayana Pillai (April 5, 1901 Nagercoil, Tamil Nadu - 31 August 1950, Cairo, Egypt) was an Nagercoil native Indian mathematician specializing in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan". In number theory, a Pillai prime, named for him, is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, $n! \equiv -1 \mod p but p \not\equiv 1 \mod n $. The first few Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... (sequence A063980 in OEIS). *Wik

1911 Computer Pioneer Cuthbert Hurd Is Born:
The mathematician son of an itinerant preacher, IBM President Thomas Watson Sr. hired Hurd in early 1949 as IBM's second Ph.D. A figure generally unknown to history, Hurd quietly encouraged IBM upper management to enter into the computer field, convincing them in the early 1950s that a market for scientific computers existed after a cross-country sales trip revealed pent-up demand. At the time, IBM enjoyed large profits from its traditional punch card accounting business so the change was difficult for IBM to make internally. Hurd's first great success was in selling 10 of IBM's 701 computers, its first commercial scientific machine which rented for $18,000 a month. Shortly thereafter, he became manager of the IBM team that invented and developed the FORTRAN programming language under John Backus. Hurd died on May 22, 1996 in Portola Valley, California.*CHM

1911 Walter Warwick Sawyer (or W. W. Sawyer) (April 5,1911–February 15, 2008) was a mathematician, mathematics educator and author, who taught on several continents.
Born in London, England , he attended Highgate School and was an undergraduate at St. John's College, Cambridge, obtaining a BA in 1933. He was an assistant lecturer in mathematics from 1933 to 1937 at University College, Dundee and from 1937 to 1944 at Manchester University. From 1945 to 1947, he was the head of mathematics at Leicester College of Technology.
In 1948 W. W. Sawyer became the first head of the mathematics department of what is now the University of Ghana. From 1951 to 1956, he was at Canterbury College (now the University of Canterbury in New Zealand). He left Canterbury College to become an associate professor at the University of Illinois, where he worked from winter 1957 through June 1958. While there, he criticized the New Math movement, which included the people who had hired him. From 1958 to 1965, he was a professor of mathematics at Wesleyan University. In the fall of 1965 he became a professor at the University of Toronto, appointed to both the College of Education and the Department of Mathematics. He retired in 1976.
W. W. Sawyer was the author of some 11 books. He is probably best known for his semi-popular works Mathematicians Delight and Prelude to Mathematics. Both of these have been translated into many languages. Mathematician's Delight was still in print 65 years after it was written. Some mathematicians have credited these books with helping to inspire their choice of a career.
W.W. Sawyer died on February 15, 2008, at the age of 96. He was survived by a daughter, Anne. *Wik
His first book "Mathematician’s Delight" (1943), was written with the aim "to dispel the fear of mathematics." It is one of the most successful math book ever written, going through numerous editions, translations into 10 languages, and selling more than 500,000 copies.

My favorite Sawyer quote:

Complete success would mean that every individual felt,
"I enjoyed the mathematics that I had time to learn.
If I ever need or want to learn some more,
I shall not be afraid to do so."

DEATHS

1678 - Claude Hardy (1598 in Le Mans, France- 5 April 1678 in Paris, France) was a French lawyer and amateur mathematician who made Latin translations of some of Euclid's work. A translation into French of Viète's book on algebra, originally written in Latin, appeared around 1630 with Antoine Vasset as the translator. It is believed that "Antoine Vasset" was a pseudonym for Claude Hardy. In 1630, under his own name, Hardy published Examen and in 1638 he published Refutation. These works dealt with the problem of the duplication of the cube*SAU

1684 William Brouncker (1620 – 5 April 1684) He was the King’s nominee and ﬁrst president of the Royal Society of London (1666–1677). His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series. He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi *Wik
In 1656 he gave the continued fraction expansion

. . . and used it to calculate π correct to ten decimal places. *VFR

1861 Ferdinand Joachimsthal (9 March 1818 in Goldberg, Prussian Silesia (now Złotoryja, Poland) - 5 April 1861 in Breslau, Germany (now Wrocław, Poland)) Influenced by the work of Jacobi, Dirichlet and Steiner, Joachimsthal wrote on the theory of surfaces where he made substantial contributions, particularly to the problem of normals to conic sections and second degree surfaces. *SAU

I've told you once and I'll tell you again
There's always a prime between n and 2n.

In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the reduction of data from observations. *TIS At age 11 he started to attend classes at the Ecole Polytechnique, where his Uncle Duhamel was a well-known professor of mathematics. At 17 he received his doctor of science degree. *VFR
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count?" which Bertrand asked, and proved in 1887 in Comptes Rendus de l'Académie des Sciences.
The answer is $\frac{p-q}{p+q}.$

2004 – Heiner Zieschang (12 November 1936 in Kiel – 5 April 2004) was a German mathematician. He was a professor at Ruhr University in Bochum from 1968 till 2002. He was a topologist. In 1996 he was an honorary doctor of University of Toulouse and in 1997 he was an honorary professor of Moscow State University.

2009 Irving John ("I.J."; "Jack") Good (9 December 1916 – 5 April 2009) was a British mathematician who worked as a cryptologist at Bletchley Park with Alan Turing. After World War II, Good continued to work with Turing on the design of computers and Bayesian statistics at the University of Manchester. Good moved to the United States where he was professor at Virginia Tech.
He was born Isadore Jacob Gudak to a Polish-Jewish family in London. He later anglicized his name to Irving John Good and signed his publications "I. J. Good."
An originator of the concept now known as "technological singularity," Good served as consultant on supercomputers to Stanley Kubrick, director of the 1968 film 2001: A Space Odyssey. Good's published work ran to over three million words. He was known for his work on Bayesian statistics. He published a number of books on probability theory. In 1958 he published an early version of what later became known as the Fast Fourier Transform but in a journal so obscure that it never became widely known.*Wik

Thursday 4 April 2019

On This Day in Math - April 4

Cover to Towers of Hanoi, *Wik

If the Lord Almighty had consulted me before embarking upon his creation, I should have recommended something simpler.
Remarking on the complexity of Ptolemaic model of the universe after it was explained to him.
~Alfonso X of Castile

The 94th day of the year; 94!-1 is prime. The number 94!-1 ends in 21 consecutive nines. Students might inquire how they could have known this without being told.

94 begins the smallest string of three consecutive numbers none of which is a palindrome in any base, b $ 2 \leq b \leq 10 $

Add the prime factors of 94 and the result is 49, 94 reversed.

94 is the smallest even number greater than four which cannot be written as a sum of two twin primes.

EVENTS

1597 Galileo writes to Kepler, "....Like you, I accepted the Copernican position several years ago and discovered from thence the causes of many natural effects which are doubtless inexplicable by the current theories. I have written up many of my reasons and refutations on the subject, but I have not dared until now to bring them into the open, being warned by the fortunes of Copernicus himself, our master, who procured immortal fame among a few but stepped down among the great crowd (for the foolish are numerous), only to be derided and dishonored. I would dare publish my thoughts if there were many like you; but, since there are not, I shall forebear.... " *Giorgio de Santillana, The Crime of Galileo (1955).

1615 In response to Galileo's assertion that Copernican cosmology was so scientifically confirmed that the Scriptures must be conformed to it, Cardinal Robert Bellarmine writes that, "I do not think there is any such proof, since none has been shown to me." He then cautions Galileo that without such evidence, teaching Copernicanism would be, "a very dangerous attitude, ...to injure our holy faith by contradicting the scriptures." *Timothy Ferris, Coming of Age in the Milky Way (Thony Christie has informed me that Bellermine's comments were in a letter to Paolo Antonio Foscarini, and not directly to Galileo.)

1687 Edmond Halley received Book Three of Newton's Masterpiece, the Principia. He would spend months pushing the publication to print, with a run of 250+ copies completed on July 5th of that year. The first edition sold out almost immediately. He writes to John Wallis that Newton "now falls in with Mr. Hooke." Newton had added Proposition XIX that the earth's diameter was greater at the equator than between the poles. Newton had previously argue for a spherical earth, and now agreed it was more oblate. *Stephen Inwood, Forgotten Genius

1692 Acta eruditorum contained, under a pseudonym, Vincenzo Viviani’s problem of constructing in a hemispherical cupola four equal-sized windows such that the remaining area of the cupola is quadrable. The problem was solved by Leibniz (date?), Guido Grandi (1699), and Viviani himself (1692). *VFR

1870 Benjamin Peirce wrote in the introduction of his “Linear Associative Algebra,” doubtless his most original mathematical work: “This work has been the pleasantest mathematical eﬀort of my life. In no other have I seemed to myself to have received so full a reward for my mental labor in the novelty and breadth of the results.” [MAA 32(1925); p. 15 of Benjamin Peirce, MAA oﬀprint of 1925] *VFR

In 1930, the American Interplanetary Society was founded by G. Edward Pendray, David Lasser, Laurence Manning and others. Its was known as the American Rocket Society from 6 Apr 1934. Through the 1930s, the group designed an experimental test stand and tested liquid-fuelled rockets. Their pioneering work led the way to the United States space program. Their ARS-4 rocket, was the first launched in America to break the sound barrier (9 Sep 1934). It was fired from from Marine Park, Staten Island, N.Y., reached a top speed of 700 mph, travelled to a maxium height of 400-ft and a horizontal range of 1,600-ft. In early 1963, it merged with the American Institute of Aeronautics and Astronautics *TIS

1994 Marc Andreesen Founds Netscape with Jim Clark:
Marc Andreessen and Jim Clark found Mosaic Communications Corp, later renamed Netscape Communications Corp. Andreessen developed the software used for browsing the World Wide Web while working at the National Center for Supercomputing Applications (NCSA) at the University of Illinois. Clark co-founded high-performance computer maker Silicon Graphics Inc.*CHM

BIRTHS

1688 Joseph-Nicolas Delisle (4 Apr 1688; 11 Sep 1768 at age 80) French astronomer who proposed that the series of coloured rings sometimes observed around the Sun is caused by diffraction of sunlight through water droplets in a cloud. He also worked to find the distance of the Sun from the Earth by observing transits of Venus and Mercury across the face of the Sun. *TIS

1809 Benjamin Peirce (4 Apr 1809, 6 Oct 1880) American astronomer, mathematician and educator who computed the general perturbations of the planets Uranus and Neptune. He was Harvard's Perkins Professor of Astronomy and Mathematics for nearly 40 years, and was largely responsible for introducing mathematics as a subject for research in American institutions. He is known especially for his contributions to analytic mechanics and linear associative algebra, but he is also remembered for his early work in astronomy and for playing a role in the discovery of Neptune. *TIS In number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras. He first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. *Wik He taught at Harvard for 49 years. Early on, he and the other young mathematics tutor, Charles W. Eliot, secured the innovation of written ﬁnal examinations. Previously all exams were oral. Eliot later became president of Harvard. *VFR

1842 François Edouard Anatole Lucas (4 April 1842, 3 Oct 1891) Lucas is best known (to formal mathematicaticians) for his results in number theory, in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him. He gave the well-known formula for the Fibonacci numbers

√5 f_n = ((1 + √5)/2)ⁿ - ((1 - √5)/2)ⁿ.

Lucas also devised methods of testing primality, essentially those used today. In 1876 he used his methods to prove that the Mersenne number 2127 - 1 is prime. This remains the largest prime number discovered without the aid of a computer. (For recreational mathematicians), Lucas is also well known for his invention of the Tower of Hanoi puzzle and other mathematical recreations. The Tower of Hanoi puzzle appeared in 1883 under the name of M. Claus. Notice that Claus is an anagram of Lucas! His four volume work on recreational mathematics Récréations mathématiques (1882-94) has become a classic.*SAU Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflamation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire". *Pballew.net

1868 Philippa Garrett Fawcett (4 April 1868 - 10 June 1948) the first woman at Cambridge to come top in the Mathematical Tripos Examinations. A description of the event is recorded in the North Hall Diary of Newnham College:-

The great event of the year was Philippa Garrett Fawcet's achievement in the Mathematical Tripos. For the first time a woman has been placed above the Senior Wrangler. The excitement in the Senate House when the lists were read was unparalleled. The deafening cheers of the throng of undergraduates redoubled as Miss Fawcett left the Senate House by the side of the Principal. On her arrival at the College she was enthusiastically greeted by a crowd of fellow-students, and carried in triumph into Clough Hall. Flowers, letters, and telegrams poured in upon her throughout the day. The College was profusely decorated with flags. In the evening the whole College dined in Clough Hall. After dinner toasts were proposed: the healths drunk were those of the Principal, Miss Fawcett, her Coach (Mr Hobson) and Senior and Junior Optimes. At 9.30 p.m. the College gardens were illuminated, and a bonfire was lighted on the hockey-ground, round which Miss Fawcett was three times carried amid shouts of triumph and strains of "For she's a jolly good fellow."

This blog from the Smithsonian has some nice detail about her.

1884 Thomas Murray MacRobert FRSE (4 April 1884, Dreghorn, Ayrshire – 1 November 1962, Glasgow) was a Scottish mathematician. He became professor of mathematics at the University of Glasgow and introduced the MacRobert E function, a generalization of the generalized hypergeometric series.*Wik

1902 Eberhard Frederich Ferdinand Hopf (April 4, 1902, Salzburg, Austria-Hungary – July 24, 1983, Bloomington, Indiana) was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry. The Hopf maximum principle is an early result of his (1927) which is one of the most important techniques in the theory of elliptic partial differential equations.*Wik

1949 Shing-Tung Yau (4 Apr 1949, )Chinese-born mathematician who was awarded the Fields Medal in 1982 for his work in partial differential equations and differential geometry. His work also has applications in topology, algebraic geometry, representation theory and general relativity. Working collaboratively with Richard M. Schoen, Yau solved a long-standing open problem in relativity theory, by showing the positivity of mass for space-time. As a consequence, Schoen and Yau were able to give the first rigorous demonstration of how black holes can be formed because of the condensation of matter. A black hole possesses a gravitational field so intense that no matter or radiation can escape from it. Yau was the 1997 National Medal of Science winner. *TIS

DEATHS

1284 Alfonso X of Castile (23 Nov 1221; 4 Apr 1284) Spanish monarch and astronomer who encouraged the preparation of revised planetary tables (1252), published on the day of his accession to the throne as king of Castile and León. These "Alfonsine Tables," a revision and improvement of the Ptolemaic tables, were the best available during the Middle Ages; they were not replaced by better ones for over three centuries. The astronomical data tabulating the positions and movements of the planets was compiled by about 50 astronomers he had assembled for this purpose. He questioned the complexity of the Ptolemaic model centuries before Copernicus. "If the Lord Almighty had consulted me before embarking on the Creation, I would have recommended something simpler." He also wrote a commentary on alchemy. *TIS

1617 John Napier of Merchiston (1550 – 4 April 1617) – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer of the logarithm. Napier is the inventor of the so-called "Napier's bones". Napier also made common the use of the decimal point in arithmetic and mathematics. Napier's birthplace, the Merchiston Tower in Edinburgh, Scotland, is now part of the facilities of Edinburgh Napier University. After his death from the effects of gout, Napier's remains were buried in St Cuthbert's Church, Edinburgh.
His work, Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms. The book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts. Modern English translations of both Napier's books on logarithms, and their description can be found on the web, as well as a discussion of Napier's Bones (see below) and Promptuary (another early calculating device). His invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters they discussed was a re-scaling of Napier's logarithms, in which the presence of the mathematical constant e (more accurately, the integer part of e times a large power of 10) was a practical difficulty. Napier delegated to Briggs the computation of a revised table. The computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics, physics; and also in astrology.
Napier made further contributions. He improved Simon Stevin's decimal notation.

Arab lattice multiplication, used by Fibonacci, was made more convenient by his introduction of Napier's bones, a multiplication tool using a set of numbered rods. He may have worked largely in isolation, but he had contact with Tycho Brahe who corresponded with his friend John Craig. Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from Anthony à Wood, perhaps not well substantiated, that Napier had a hint from Craig that Longomontanus, a follower of Brahe, was working in a similar direction.
In addition to his mathematical and religious interests, Napier was often perceived as a magician, and is thought to have dabbled in alchemy and necromancy. It was said that he would travel about with a black spider in a small box, and that his black rooster was his familiar spirit.
Napier used his rooster to determine which of his servants had been stealing from his home. He would shut the suspects one at a time in a room with the bird, telling them to stroke it. The rooster would then tell Napier which of them was guilty. Actually, what would happen is that he would secretly coat the rooster with soot. Servants who were innocent would have no qualms about stroking it but the guilty one would only pretend he had, and when Napier examined their hands, the one with the clean hands was guilty.
Another occasion which may have contributed to his reputation as a sorcerer involved a neighbor whose pigeons were found to be eating Napier's grain. Napier warned him that he intended to keep any pigeons found on his property. The next day, it is said, Napier was witnessed surrounded by unusually passive pigeons which he was scooping up and putting in a sack. The previous night he had soaked some peas in brandy, and then sown them. Come morning, the pigeons had gobbled them up, rendering themselves incapable of flight.
*Wik

1807 Joseph Jérôme Le Français de Lalande, (11 Jul 1732, 4 Apr 1807 at age 74)
He determined the Moon's parallax from Berlin for the French Academy (1751). He was appointed professor of Astronomy, Collège de France (1762), and subsequently, director of the Paris Observatory. He published his Traité d'astronomie in 1764 - tables of the planetary positions that were considered the best available for the rest of the century. In 1801 he also published a comprehensive star catalogue. He died in 1807, apparently of tuberculosis. *TIS

1919 Sir William Crookes, OM, FRS (17 June 1832 – 4 April 1919) was a British chemist and physicist who attended the Royal College of Chemistry, London, and worked on spectroscopy. He was a pioneer of vacuum tubes, inventing the Crookes tube. *Wik @LunarHeritage pointed out that Crookes also spectroscopically 1st discovered the element Thallium (1861) Tl atomic number 81 has two stable isotopes, one of which Tl-203, produces one of the workhorses of nuclear medicine

1923 John Venn FRS (4 August 1834 – 4 April 1923), was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science. Leibniz was the ﬁrst to systematically use geometric diagrams to represent syllogisms, and Euler developed the ideas, but Venn gets the credit for his book popularized them. *VFR He was a fellow of Gonville and Caius and there is a stained glass window memorial there in the dining hall, which I had the pleasure of visiting with Professor Anthony Edwards.

Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a mathematician specializing in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.
Among his teachers were Max Planck and Ferdinand Georg Frobenius, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best student was Jürgen Moser, one of the founders of KAM theory (Kolmogorov-Arnold-Moser), which lies at the foundations of chaos theory.
Siegel's work on number theory, diophantine equations, and celestial mechanics in particular won him numerous honours. In 1978, he was awarded the Wolf Prize in Mathematics, one of the most prestigious in the field.
Siegel's work spans analytic number theory; and his theorem on the finiteness of the integer points of curves, for genus greater than 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped. He worked on L-functions, discovering the (presumed illusory) Siegel zero phenomenon. His work derived from the Hardy-Littlewood circle method on quadratic forms proved very influential on the later, adele group theories encompassing the use of theta-functions. The Siegel modular forms are recognised as part of the moduli theory of abelian varieties. In all this work the structural implications of analytic methods show through.
André Weil, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. In the early 1970s Weil gave a series of seminars on the history of number theory prior to the 20th century and he remarked that Siegel once told him that when the first person discovered the simplest case of Faulhaber's formula then, in Siegel's words, "Es gefiel dem lieben Gott." (It pleased the dear Lord.) Siegel was a profound student of the history of mathematics and put his studies to good use in such works as the Riemann-Siegel formula.*Wik

Wednesday 3 April 2019

On This Day in Math - April 3

Alberti's Statue in the courtyard of the Uffizi Gallery, Florence; *Wik

Knowing what is big and what is small is more important than being able to solve partial differential equations.
~Stan Ulam

The 93rd day of the year; The first 93 digits of 93! form a prime number. *Prime Curios ( Can students find a smaller number n for which the first n digits of n! form a prime? Send results to me.)

93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467

315215203289516844845303838996289 ...

93 is the sum of three distinct squares, 93 = 2² + 5²+ 8² )
and six consecutive integers 93= 13 + 14 + 15 + 16 + 17 + 18

There are 93 five-digit prime palindromes. The smallest (I think) is 10301

A potato can be cut into 93 pieces with just nine straight cuts.

and 93 in base 10 is 333 in base 5

EVENTS

1736 Euler replied to Ehler on Konigsberg Bridge Challenge, "Not a Mathematical Problem..."

... Thus you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others. In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way ...

(See March 9th for Ehler's challenge to Euler) *Brian Hopkins, Robin Wilson; The Truth About Konigsberg

- See more at: http://www.maa.org/programs/maa-awards/writing-awards/the-truth-about-konigsberg#sthash.c6jO9L76.dpuf

1753 Goldbach wrote Euler with a conjecture that every odd number greater than 3 is the sum of an odd number and twice a square (he allowed 0²). Euler would reply on Dec 16 that it was true for the first 1000 odd numbers, and then again on April 3, 1753, to confirm it for the first 2500. A hundred years later, German mathematician Moritz Stern found two contradictions, 5777 and 5993. The story appears in Alfred S. Posamentier's Magnificent Mistakes in Mathematics, (but gloriously, has a mistake for the date, using 1852, but such a wonderful book can forgive a print error.)

1763 Easter Saturday 1763, Lalande recorded that the Astronomer Royal Nathaniel Bliss said ‘that Mr Lemonnier attached the wire to his quadrant with wax from his ears, that he went to Oxford with his sword broken, and that his observations agree less well with those of Mr Bevis than those of Caille.’ Pierre Charles Lemonnier or Le Monnier was a talented French astronomer 17 years Lalande’s senior who had a penchant for British instruments and astronomical methods and was a member of the Royal Society and the French Academy of Sciences. *Royal Museum Greenwich blog "MEOW!"

1769 A letter from Mr. Richard Price, F. R. S. to Benjamin Franklin, Esq; LL.D. discusses De Moivre's work on Population and survival rates. The paper runs to 38 pages. "Observations on the Expectations of Lives, the Increase of Mankind, the Influence of Great Towns on Population, and Particularly the State of London with Respect to Healthfulness and Number of Inhabitants." *Phil. Trans. January 1, 1769 59:89-125;

In 1934, a British patent application for the first catseye road marker was recorded for inventor Percy Shaw (1889-1975), described as "Improvements relating to Blocks for Road Surface." These are the familiar reflectors which mark the lines that are lit up at night by the lights of passing vehicles. The raised surface in which the reflectors are mounted have a construction that "will yield when traveled over by a vehicle wheel and sink to the level of the road surface" such as a resilient white rubber cushion mounted in a metal holder sunk below the road surface. The patent No. 436,290 was accepted 3 Oct 1935. A revised design was patented the following year as No. 457,536. Shaw started Reflecting Roadstuds Ltd. to manufacture them. *TIS (While this is of little mathematical interest, before I learned what the term meant, I regularly wondered about a sign I passed on the way from Stoke Ferry to King's Lynn in Norfolk Shire which said, "Cat's Eyes Removed Ahead."

1964 It is reported in the New York Times that the casinos in Las Vegas have changed their rules in blackjack so as to defeat the winning strategy devised by Edward O. Thorp. See 27 March 1964. *VFR
Thorp is described as an American mathematics professor, author, hedge fund manager, and blackjack player. He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.*Wik

1983 The Republic of China (Taiwan) marks the 400th anniversary of the arrival of Father Matteo Ricci (1552–1610) in China with a pair of stamps. [Scott #2359–2360] *VFR

2016 First day of the major league baseball season. The exact width of home plate is irrational: 12 times the square root of two.
History: The plate was originally a circle of diameter one, then a square of the same size(!), which, by mistake was a one-by-one square. Then the corners were filled in to make the current pentagonal plate. *VFR Home plate is an irregular pentagon. The front is 17 inches wide, faces the pitcher, and defines the width of the strike zone. Then parallel sides 8.5 inches long connect to the foul lines. Finally 12 inch sides run down the foul lines, connecting where the foul lines meet.
It can be thought of as a 17 inch square with the parts that would be in foul territory removed. The figure described in the official rules of MLB, as well as above, is technically impossible. One of two things must be true to make it possible:

the parallel sides of 8.5" are in reality approximately 8.5295" (the square root of 71.75)
or
the 12" sides that run along the foul lines are approximately 12.0208" (square root of 144.5)

The latter is more likely the case, as it would produce the angle measurements of 90º at the base and rear point and 135º at the sides.

Rick Pearce ‏@MrPearceMath has written to object: "The width is 17", not irrational. Other 4 sides are. But you're right about impossibility of defn." I'm not sure the difference of 17 and 12 square root (2) is a miniscule .0295" (Will wander down to nearest ball park with steel tape in hand, this challenge will not stand!)

BIRTHS

1529 – Michael Neander (April 3, 1529 – October 23, 1581) German mathematician and astronomer was born in Joachimsthal, Bohemia, and was educated at the University of Wittenberg, receiving his B.A. in 1549 and M.A. in 1550.
From 1551 until 1561 he taught mathematics and astronomy in Jena, Germany. He became a professor in 1558 when the school where he taught became a university. From 1560 until his death he was a professor of medicine at the University of Jena. He died in Jena, Germany. The crater Neander on the Moon is named after him. *Wik

1835 John Howard Van Amringe (3 April 1835 in Philadelphia, Pennsylvania, USA - 10 Sept 1915 in Morristown, New Jersey, USA) was a U.S. educator and mathematician. He was born in Philadelphia, and graduated from Columbia in 1860. Thereafter, he taught mathematics at Columbia, holding a professorship from 1865 to 1910 when he retired. Van Amringe was also the first Dean of Columbia College, the university's undergraduate school of arts and sciences, which he defended from dismemberment and incorporation into the larger university. During his long presence at the school, he made many addresses and enjoyed unrivaled popularity. He is memorialized with a bust enshrined in a column-supported cupola on "Van Am Quad" in the southeastern portion of the campus, surrounded by three College dormitories (John Jay Hall, Hartley Hall, and Wallach Hall) and by the main College academic building, Hamilton Hall. He is buried in Greenwood Cemetery in Brooklyn.
Van Amringe served as the first president of the American Mathematical Society between 1888 and 1890.
In honor of Van Amringe, Columbia University's Department of Mathematics has presented a "Van Amringe Mathematical Prize" each year (since 1911) to the best freshman or sophomore mathematics student, based on a very challenging examination. *Wik

1842 Hermann Karl Vogel (3 Apr 1842; died 13 Aug 1907 at age 65) German astronomer who discovered spectroscopic binaries (double-star systems that are too close for the individual stars to be discerned by any telescope but, through the analysis of their light, have been found to be two individual stars rapidly revolving around one another). He pioneered the study of light from distant stars, and introduced the use of photography in this field.*TIS

1859 Karl Heun (3 April 1859 in Wiesbaden, Germany - 10 Jan 1929 in Karlsruhe, Germany) was a German mathematician best known for the Heun differential equation which generalizes the hypergeometric differential equation. *SAU

1892 Hans Rademacher (3 April 1892 in Wandsbeck (part of Hamburg), Schleswig-Holstein, Germany - 7 Feb 1969 in Haverford, Pennsylvania, USA) It was philosophy that he intended to take as his main university subject when he entered the university of Göttingen in 1911, but he was persuaded to study mathematics by Courant after having enjoyed the excellent mathematics teaching of Hecke and Weyl. He is remembered for the system of orthogonal functions (now known as Rademacher functions) which he introduced in a paper published in 1922. Berndt writes "Since its discovery, Rademacher's orthonormal system has been utilised in many instances in several areas of analysis." Rademacher's early arithmetical work dealt with applications of Brun's sieve method and with the Goldbach problem in algebraic number fields. About 1928 he began research on the topics for which he is best known among mathematicians today, namely his work in connection with questions concerning modular forms and analytic number theory. Perhaps his most famous result, obtained in 1936 when he was in the United States, is his proof of the asymptotic formula for the growth of the partition function (the number of representations of a number as a sum of natural numbers). This answered questions of Leibniz and Euler and followed results obtained by Hardy and Ramanujan. Rademacher also wrote important papers on Dedekind sums and investigated many problems relating to algebraic number fields. *SAU

1900 Albert Edward Ingham (3 April 1900–6 September 1967) was an English mathematician. His book On the distribution of prime numbers published in 1932 was his only book and it is a classic. Many of the ideas here, as in other work of Ingham's, came from the joint work undertaken by Harald Bohr and Littlewood.

1907 Mark Grigorievich Krein (3 April 1907 – 17 October 1989) was a Soviet Jewish mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory.
He was born in Kiev, leaving home at age 17 to go to Odessa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov.
He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony.
He died in Odessa.
On 14 January 2008, the memorial plaque of Mark Krein was unveiled on the main administration building of I.I. Mechnikov Odessa National University. *Wik

DEATHS

1472 Leone Battista Alberti (18 Feb 1404 in Genoa, French Empire (now Italy)- 3 April 1472 in Rome, Papal States (now Italy) The date of his death is given by Wikipedia as April 20, 1472) Italian mathematician who wrote the first general treatise on the laws of perspective and also wrote a book on cryptography containing the first example of a frequency table. Alberti died in Rome, but his ashes were brought from Rome and put in the family vault in the Santa Croce Cathedral (where Galileo is buried).

1717 Jacques Ozanam (16 June 1640, Sainte-Olive, Ain - 3 April 1718, Paris)In 1670, he published trigonometric and logarithmic tables more accurate than the then existing ones of Ulacq, Pitiscus, and Briggs. An act of kindness in lending money to two strangers secured for him the notice of M. d'Aguesseau, father of the chancellor, and an invitation to settle in Paris. There he enjoyed prosperity and contentment for many years. He married, had a large family, and derived an ample income from teaching mathematics to private pupils, chiefly foreigners. *Wik
He is remembered for his book on mathematical recreations. “He was wont to say that it was the business of the Sorbonne doctors to discuss, of the pope to decide, and of a mathematician to go straight to heaven in a perpendicular line.” [DSB 10, 264]. *VFR
On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait labelled Ozanam -- the only one I know of. This copy is at the Institut für Geschichte der Naturwissenschaft in Munich. *David Singmaster

1817 Friedrich Ludwig Wachter (1792–1817), a student of Gauss, called the geometry obtained by denying Euclid’s parallel postulate “anti-Euclidean geometry”. Had he returned from his customary evening walk on this date he might now be known as one of the founders of non-Euclidean geometry. [G. E. Martin, Foundations of Geometry and the Non-Euclidean Plane, p. 306] *VFR
(… In a letter to Gauss in 1816 he states that as a circle’s radius increases toward infinity as being identical to a plane, “even in the case of the fifth postulate being false, there would be geometry on this surface identical with that of the ordinary plane.)

1827 Ernst Florenz Friedrich Chladni, physicist and amateur musician, died. He is best remembered for the spectacular symmetrical patterns formed when a sand covered plate is vibrated with a violin bow. He was a Professor of Physics in Breslau when he developed Chladni figures c1800. He came to Paris in 1808 to present his work at the Institut and Laplace had him give a two hour demonstration to Napoleon, who gave him 6000 francs.
German physicist who is known as the “father of acoustics” for his mathematical investigations of sound waves. Chladni figures, seen when thin plates covered in sand at set in vibration, are complex patterns of vibration with nodal lines that remain stationary and retain sand. He demonstrated these to an audience of scientists in Paris in 1809. He measured the speed of sound in various gases by determining the pitch of the note of an organ pipe filled with different gases. To determine the speed of sound in solids, Chladni used analysis of the nodal pattern in standing-wave vibrations in long rods. He performed on the euphonium, an instrument he invented, made of glass and steel bars vibrated by rubbing with a moistened finger. He also investigated meteorites. *TIS

1900 Joseph-Louis-François Bertrand (11 Mar 1822; 5 Apr 1900 at age 78) was a French mathematician and educator and educator remembered for his elegant applications of differential equations to analytical mechanics, particularly in thermodynamics, and for his work on statistical probability and the theory of curves and surfaces. In 1845 Bertrand conjectured that there is at least one prime between n and (2n-2) for every n greater than 3, as proved five years later by Chebyshev. In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the reduction of data from observations. *TIS At age 11 he started to attend classes at the Ecole Polytechnique, where his Uncle Duhamel was a well-known professor of mathematics. At 17 he received his doctor of science degree. *VFR

1998 – Mary Cartwright, English mathematician (b. 1900) G.H. Hardy was her Doctorial advisor, She did early investigations with Littlewood of fine structure of solutions to some types of differential equations, today seen to be a typical instance of the butterfly effect. She was the first woman:

to receive the Sylvester Medal
to serve on the Council of the Royal Society
to be President of the London Mathematical Society (in 1961–62)

She also received the De Morgan Medal of the Society in 1968. In 1969 she received the distinction of being honoured by the Queen, becoming Dame Mary Cartwright, Dame Commander of the Order of the British Empire.

Tuesday 2 April 2019

On This Day in Math - April 2

12th century copy of Gerber's De geometria. *Wik

The notion of a set is too vague for the continuum hypothesis to have a positive or negative answer.
~Paul Cohen

The 92nd day of the year; 92 is the smallest composite number for which the reverse of its digits in hexadecimal, decimal, octal, and binary are all prime. *Prime Curios (Is there a smaller Prime that could also be prime when reversed in all these bases?)

And... There are exactly 92 Johnson Solids: The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely regular Platonic solids, the "semiregular" Archimedean solids, and the two infinite families of prisms and antiprisms). *Geometry Fact ‏@GeometryFact
and a related point, The snub dodecahedron has 92 faces (80 triangular, 12 pentagonal), the most an Archimedean solid can have.

92 is the number of different arrangements of 8 non-attacking Queens on an 8 by 8 chessboard (i.e. no two Queens should share the same row, column, or diagonal)

92= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 the sum of eight consecutive integers

92 is a palindrome in bases 6 (232₆), and 7 (161₇)

EVENTS

999 Gerbert was elected Pope Sylvester II. He introduced into the West the practice of making calculations by using marked discs (apices). This method, which has nearly all the advantages of positional arithmetic, was used in abacus calculations throughout the eleventh and twelfth centuries. *VFR More about Pope Sylvester's Abacus.

1792 U.S. Mint established. It was Jefferson who suggested decimal coinage. *VFR

1827 lead pencils were first manufactured by Joseph Dixon, who built his factory in Salem, Mass. Dixon was responsible for the development of the graphite industry in the U.S. In 1859 he patented graphite crucibles. When he died, the Joseph Dixon Crucible Company was the largest manufacturer of graphite products in the world. The first* pencil factory in the U.S. however, was started earlier by William Monroe of Concord, Mass., in Jun 1812. His first 30 pencils were bought by Benjamin Adams, a hardware dealer in Boston, Mass. The first pencils made in Great Britain (1584) used graphite from Borrowdale, Cumberland. *TIS
The Pencil, A History of Design and Circumstance

1845 Fizeau and Foucault take the first successful photograph of the sun. *VFR
"Taking advantage of a relatively new technology, the daguerreotype, French physicists Louis Fizeau and Leon Foucault made the first successful photographs of the sun on April 2, 1845. The original image, taken with an exposure of 1/60th of a second, was about 4.7 inches (12 centimeters) in diameter and captured several sunspots, visible in this reproduction. "( I find it interesting that the first photo of the sun was over five years after the first photo of the moon. Can you think why ?)

1933 Emmy Noether's right to teach at Gottingen was withdrawn because of her Jewish ancestry. The resulting infusion of scientists played a major role in transferring mathematical leadership from Germany to the United States. See AMM, 90(1983), 717. *VFR Thony Christie sent me a note assuring me that it was not her religion, but her politics. Seems she had Marxist leanings. Many others in her department were discharged at the same time. After the sweeping removal of "undesirables" Minister of Education Bernhard Rust supposedly had the following conversation with David Hilbert.
Rust: “I hear you have some problems in the mathematics department at Göttingen Herr Professor”.
Hilbert: “No, there are no problems; there is no mathematics department in Göttingen”.

1935 Sir Robert Watson-Watt received a patent on a radio device for detecting and locating an aircraft. He had submitted the idea to the Air Ministry in secret memo, Detection and location of aircraft by radio methods on Feb 12 of the same year. The method was tested on Feb 26 in a field just off the present day A5 in Northamptonshire near the village of Upper Stowe. *Wik

1948 Kurt Godel became a United States citizen. Being the diligent individual that he was, he studied the constitution carefully beforehand and felt that he had found a contradiction. On the way to the ceremony Einstein and Oskar Morgenstern tried to keep his mind on other issues, but when the judge called them into his chambers (so that he could meet Einstein) he asked Godel if he had anything to say. It was only with considerable effort that his friends were able to change the subject when Godel brought up the contradiction. *VFR
Paul O'Malley directed me to a site where a more complete version of this anecdote is spelled out by writer Jeffrey Kegler. I've included here a link to a pdf document that has a draft recollection of the story from Morgenstern.

1953 the journal Nature published a paper with this date from Francis Crick and James Watson, titled Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid, in which they described a double helix structure for DNA. The diagram published with the caper was captioned, "The figure is purely diagrammatic. The two ribbons symbolize the phosphate-sugar chains, and the horizontal rods the pairs of bases holding the chains together. The vertical line marks the fibre axis." *TIS

BIRTHS

1618 – Francesco Maria Grimaldi (2 April 1618 – 28 December 1663) was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna.
Between 1640 and 1650, working with Riccioli, he investigated the free fall of objects, confirming that the distance of fall was proportional to the square of the time taken. In astronomy, he built and used instruments to measure geological features on the Moon, and drew an accurate map or selenograph which was published by Riccioli. He was the first to make accurate observations on the diffraction of light (although by some accounts Leonardo da Vinci had earlier noted it), and coined the word 'diffraction'. Later physicists used his work as evidence that light was a wave, and Isaac Newton used it to arrive at his more comprehensive theory of light. *Wik
Thony Christie has a nice post about his influential work in the early investigation of refraction that is well worth reading.

1888 Antonio Signorini (2 April 1888 – 23 February 1963) was an influential Italian mathematical physicist and civil engineer of the 20th century. He is known for his work in finite elasticity, thermoelasticity and for formulating the Signorini problem.
The Signorini problem is the first variational inequality problem, : it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces. The name was coined by Gaetano Fichera to honour his teacher, Antonio Signorini: the original name coined by him is problem with ambiguous boundary conditions. The problem was posed by Antonio Signorini during a course taught at the Istituto Nazionale di Alta Matematica in 1959. The problem was taken up, in particular, by one of his students, Gaetano Fichera.
On the first days of January 1963, Fichera was able to give a complete proof of the existence and uniqueness of a solution for the problem with ambiguous boundary condition, which he called "Signorini problem" to honour his teacher. The preliminary note later published as Fichera 1963 was written up and submitted to Signorini exactly a week before his death: He was very satisfied to see a positive result.

1906 Shokichi Iyanaga (April 2, 1906 – June 1, 2006) was a Japanese mathematician. Iyanaga published many papers which arose through several courses such as algebraic topology, functional analysis, and geometry, which he taught. He became Professor at the University of Tokyo in 1942. It was during World War II. Towards the end of the war, many Japanese cities were bombarded and he had to find refuge in the countryside. He was busy in editing textbooks from primary and secondary schools and he continued to give courses and organise seminars.*Wik

1923 – George Spencer-Brown (born April 2, 1923, Grimsby, Lincolnshire, England, ) is a polymath best known as the author of Laws of Form. He describes himself as a "mathematician, consulting engineer, psychologist, educational consultant and practitioner, consulting psychotherapist, author, and poet.",*Wik
In a 1976 letter to the Editor of Nature, Spencer-Brown claimed a proof of the four-color theorem, which is not computer-assisted. The preface of the 1979 edition of Laws of Form repeats that claim, and further states that the generally accepted computational proof by Appel, Haken, and Koch has 'failed' (page xii). Spencer-Brown's claimed proof of the four-color theorem has yet to find any defenders; Kauffman provides a detailed review of parts of that work. *VFR

1934 Paul Joseph Cohen (2 Apr 1934, )American mathematician who received the Fields Medal (1966) for his fundamental work on the foundations of set theory. Cohen invented a technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalised continuum hypothesis. The continuum hypothesis problem was the first of Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900. Hilbert's famous speech The Problems of Mathematics challenged (then and now) mathematicians to solve these fundamental questions and Cohen has the distinction of solving Problem 1. He also worked on differential equations and harmonic analysis. *TIS

DEATHS

1952 Bernard(-Ferdinand) Lyot (27 Feb 1897; 2 Apr 1952 at age 55) French astronomer who invented the coronagraph (1930), an instrument which allows the observation of the solar corona when the Sun is not in eclipse. Earlier, using his expertise in optics, Lyot made a very sensitive polariscope to study polarization of light reflected from planets. Observing from the Pic du Midi Observatory, he determined that the lunar surface behaves like volcanic dust, that Mars has sandstorms, and other results on the atmospheres of the other planets. Modifications to his polarimeter created the coronagraph, with which he photographed the Sun's corona and its analyzed its spectrum. He found new spectral lines in the corona, and he made (1939) the first motion pictures of solar pro

Published on

minences.*TIS

1995 Hannes Olof Gösta Alfvén (30 May 1908, 2 Apr 1995 at age 86) was a Swedish astrophysicist who was one of the founders of the field of plasma physics (the study of ionized gases). He shared the 1970 Nobel Prize in Physics (with Frenchman Louis Néel). Alfvén was recognized “for fundamental work in magnetohydrodynamics with fruitful applications in different parts of plasma physics.” He conceived plasma cosmology as an alternative to the Big Bang theory of the origin of the universe. In the concept of plasma cosmology, the universe has no specific beginning nor has any forseeable end. Instead of a dominance by gravitational forces, the theory maintains that it is the electromagnetic forces of plasma throughout the universe that organizes the matter of the universe into its observed structure of stars. *TIS

Pat'sBlog

Saturday 6 April 2019

On This Day in Math - April 6

Friday 5 April 2019

On This Day in Math - April 5

Thursday 4 April 2019

On This Day in Math - April 4

Wednesday 3 April 2019

On This Day in Math - April 3

Tuesday 2 April 2019

On This Day in Math - April 2

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