Wednesday 2 November 2022

On This Dayin Math - November 2

The Boole Window, Lincoln Cathedral

[Arithmetic] is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.
~Henry Bell

The 306th day of the year; 306 is the sum of four consecutive primes starting with 71.

There are 306 triangular numbers with five digits. (students, how many triangular numbers have 3 digits... Can you calculate without listing them all?)

306 = 92 + 92 + 122; (not impressed?) you can write the same numerals with exponents and 306 =92 + 92 + 122 Really? Still not impressed, how about 306 =82 + 112 + 112 = 82 + 112 + 112


1749 Leonhard Euler wrote (June 28) to the Rev. Mr. Caspar Wetstein, Chaplain to His Royal Highness the Prince of Wales, concerning the Gradual Approach of the Earth to the Sun. Read before the Royal Society Nov 2, 1749. *Philosophical Transactions

1917 The poet Alfred Noyes was present for the "first light" of the Hooker telescope on November 2, 1917. Noyes used this night as the setting in the opening of Watchers of the Sky, the first volume in his trilogy The Torchbearers, an epic poem about the history of science. According to his account of the night, the first object viewed in the telescope was Jupiter and Noyes himself was the first to see one of the planet's moons through the telescope. *Wik The 100 inch telescope on Mount Wilson was the largest telescope in the world until the 200 inch in 1948 at Palomar Observatory. It is the location where Edwin Hubble made the observation that some of the fuzzy blobs in space were actually galaxies like our own. The campus on Mount Wilson is also the place where Michelson made his measurements of the speed of light. *Frederick Pohl, Chasing Science, pg 54

1944 One of the greatest chemists of the century, and one who inadvertently killed more people than any dictator, died on this day of one of his own inventions. Thomas Midgley Jr was instrumental in the discovery and introduction of the gasoline additive tetraethyl lead. He also developed a method to extract large quantities of bromine from seawater when he learned that bromine was needed to prevent tetraethyl lead from corroding engine valves and spark plugs. He also discovered that dichlorodifluoromethane, also known as Freon, could be used as a nontoxic and nonflammable refrigerant.
His work led to the release of large quantities of lead into the atmosphere as a result of the large-scale combustion of leaded gasoline all over the world. High atmospheric lead levels have been linked with serious long-term health problems from childhood, including neurological impairment.
Midgley died three decades before the ozone-depleting and greenhouse gas effects of CFCs in the atmosphere became widely known. Bill Bryson remarks that Midgley possessed "an instinct for the regrettable that was almost uncanny."
In 1940, at the age of 51, Midgley contracted poliomyelitis, which left him severely disabled. This led him to devise an elaborate system of strings and pulleys to help others lift him from bed. This system was the eventual cause of his own death when he was entangled in the ropes of this device and died of strangulation at the age of 55. Tetraethyl lead and CFCs have both become illegal, but I have no evidence that the device that killed him has been banned. *wik, *Priestly Medal bio

1979 Krachian’s ellipse algorithm for linear programming announced in Science. *VFR

1988 Computers in America went mad, thanks to a virus spread by graduate student Robert T. Morris Jr. *VFR Robert Morris, Jr., a graduate student in Computer Science at Cornell, wrote an experimental, self-replicating, self-propagating program called a worm and injected it into the Internet. He chose to release it from MIT, to disguise the fact that the worm came from Cornell. Morris soon discovered that the program was replicating and reinfecting machines at a much faster rate than he had anticipated---there was a bug. Ultimately, many machines at locations around the country either crashed or became ``catatonic.'' When Morris realized what was happening, he contacted a friend at Harvard to discuss a solution. Eventually, they sent an anonymous message from Harvard over the network, instructing programmers how to kill the worm and prevent reinfection. However, because the network route was clogged, this message did not get through until it was too late. Computers were affected at many sites, including universities, military sites, and medical research facilities. *MIT Edu

In 2000, an American astronaut and two Russian cosmonauts became the first permanent residents of the international space station, at the start of their four-month mission. After their Soyuz spacecraft linked up at 11:00am GMT, William Shepherd, Sergei Krikalev and Yuri Gidzenko entered the station, turned on the lights and life support systems, and proceeded to set up a live television link with the Russian mission control to confirm that the move-in was going well. They were confined to two of the space station’s three rooms until space shuttle Endeavor arrived in early Dec. with giant solar panels that would provide all the necessary power. *TIS

2011 The 2011 Rolf Schock Prize in Mathematics is being awarded to Michael Aschbacher for his "fundamental contributions to one of the largest mathematical projects ever, the classification of finite simple groups, notably his contribution to the quasi-thin case." The $75,000 Rolf Schock Prize in Mathematics is awarded by the Royal Swedish Academy of Sciences. An award ceremony will take place in Stockholm on November 2, 2011.
Aschbacher, the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology, received the AMS's Cole Prize in 1980 and became a member of the National Academy of Sciences in 1990.*MAA


1815 George Boole (2 Nov 1815; 8 Dec 1864) English mathematician who helped establish modern symbolic logic and an algebra of logic, now called Boolean algebra. By replacing logical operations by symbols, Boole showed that the operations could be manipulated to give logically consistent results. Boole's logical algebra is essentially an algebra of classes, being based on such concepts as complement and union of classes.The study of mathematical or symbolic logic developed mainly from his ideas, and is basic to the design of digital computer circuits. Boolean also algebras find important applications in such diverse fields as topology, measure theory, probability and statistics.Boole also wrote important works on differential equations and other branches of mathematics. *TIS

1826 Henry John Stephen Smith (2 Nov 1826 in Dublin, Ireland, 9 Feb 1883 in Oxford, England) was an Irish mathematician whose most important contributions are in number theory where he worked on elementary divisors. He proved that any integer can be expressed as the sum of 5 squares and as the sum of 7 squares, showing in how many ways this could occur. In addition to solving these cases explicitly, he gave a method which would yield the number of ways that an integer can be expressed as the sum of k squares for any fixed k. He published his results in The orders and genera of quadratic forms containing more than three indeterminates published in the Proceedings of the Royal Society in 1867. Eisenstein had earlier proved the result for 3 squares and Jacobi for 2, 4 and 6 squares. Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms. *SAU

1871 Poul Heegaard (2 Nov 1871 in Copenhagen, Denmark - 7 Feb 1948 in Oslo, Norway) was a Danish mathematician who (with Max Dehn) was the first to classify compact surfaces.*SAU His 1898 thesis introduced a concept now called the Heegaard splitting of a 3-manifold. Heegaard's ideas allowed him to make a careful critique of work of Henri Poincaré. Poincaré had overlooked the possibility of the appearance of torsion in the homology groups of a space.
He later co-authored, with Max Dehn, a foundational article on combinatorial topology, in the form of an encyclopedia entry.
Heegaard studied mathematics at the University of Copenhagen, from 1889 to 1893 and following years of traveling, and teaching mathematics, he was appointed professor at University of Copenhagen in 1910.
Following a dispute with the faculty over, among other things, the hiring of Harald Bohr (The Brother of Niels Bohr, and Olmpic Soccer medalist) as professor at the University (Heegaard was against it); Heegaard accepted a professorship at Oslo in Norway, where he worked till his retirement in 1941.*Wik

1885 Harlow Shapley (2 Nov 1885; 20 Oct 1972) Astronomer, known as "The Modern Copernicus," who discovered the Sun's position in the galaxy. From 1914 to 1921 he was at Mt. Wilson Observatory, where he calibrated Henrietta S. Leavitt's period vs. luminosity relation for Cepheid variable stars and used it to determine the distances of globular clusters. He boldly and correctly proclaimed that the globulars outline the Galaxy, and that the Galaxy is far larger than was generally believed and centered thousands of light years away in the direction of Sagittarius. In the early 1920's, Shapley entered a "Great Debate" with Heber D. Curtis. They truly argued over the "Scale of the Universe." *TIS

1911 Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an American mathematician. He worked on mathematical logic, set theory, geometry, number theory, and combinatorics. Robinson (1937) set out a simpler and more conventional version of John Von Neumann's 1923 axiomatic set theory. Soon after Alfred Tarski joined Berkeley's mathematics department in 1942, Robinson began to do major work on the foundations of mathematics, building on Tarski's concept of "essential undecidability," by proving a number of mathematical theories undecidable. Robinson (1950) proved that an essentially undecidable theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic Q. Q is finitely axiomatizable because it lacks Peano arithmetic's axiom schema of induction; nevertheless Q, like Peano arithmetic, is incomplete and undecidable in the sense of Gödel. Robinson's work on undecidability culminated in his coauthoring Tarski et al. (1953), which established, among other things, the undecidability of group theory, lattice theory, abstract projective geometry, and closure algebras.
Robinson worked in number theory, even employing very early computers to obtain results. For example, he coded the Lucas-Lehmer primality test to determine whether 2n − 1 was prime for all prime n<2304 on a SWAC. In 1952, he showed that these Mersenne numbers were all composite except for 17 values of n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. He discovered the last 5 of these Mersenne primes, the largest ones known at the time. *Wik

1929 Richard E. Taylor (2 Nov 1929, ) Canadian physicist who in 1990 shared the Nobel Prize for Physics with Jerome Friedman and Henry Kendall for his collaboration in pioneering investigations concerning deep inelastic scattering of electrons on protons and bound neutrons, which have been of essential importance for the development of the quark model in particle physics. The team performed a series of experiments that confirmed the hypothesis that protons and neutrons are made up of quarks. This discovery was crucial to the formulation of the currently accepted theoretical description of matter and its interactions, known as the standard model. *TIS

1932 Melvin Schwartz (2 Nov 1932, ) American physicist and entrepreneur who, along with Leon M. Lederman and Jack Steinberger, received the Nobel Prize for Physics in 1988 for their research concerning neutrinos (subatomic particles that have no electric charge and virtually no mass). Using a beam of neutrinos, the team discovered a new kind of neutrino called a muon, and new information about the structure of particles called leptons. Neutrinos are produced when unstable atomic nuclei or subatomic particles disintegrate. Schwartz and his team wanted to study the "weak" nuclear force that creates certain kinds of radioactivity. The team used a particle accelerator to create a high-intensity beam of neutrinos. They studied the reactions produced when this beam hit other matter.*TIS


1914 Heinrich Friedrich Karl Ludwig Burkhardt (10 Oct 1861 in Schweinfurt, Germany
- 2 Nov 1914 in Munich, Germany) His main work was in analysis, particularly the theory of trigonometric series, and on the history of mathematics. Other topics on which Burkhardt published papers included groups, differential equations, differential geometry and mathematical physics. One of his papers, published in 1888, was Hyperelliptic sigma functions. Other papers include: Theorie der Cremonatransformationen (1892), Über Vectoranalysis (1896), Mathematisches und naturwissenschaftliches Denken (1902), Über Reihenentwicklungen nach oszillierenden Funktionen (1903), Zu den Funktionen des elliptischen Zylinders (1906), and Mathematische Miszellen aus der Vorlesungspraxis (1913). *SAU

1966 Petrus (Peter) Josephus Wilhelmus Debye (24 Mar 1884, 2 Nov 1966) was a Dutch physical chemist whose investigations of dipole moments, X rays, and light scattering in gases brought him the 1936 Nobel Prize for Chemistry. Most of his work was in chemical-physics with special interest in electrolytes and dipolar momentum analysis. He established a theory of specific heat with some improvements on that proposed by Einstein. Debye performed important work in the analysis of crystalline powders using X-ray diffraction techniques. He also determined the dimensions of gaseous molecules and the interatomic distances using X-rays.*TIS

1970 Abram Samoilovitch Besicovitch (24 Jan 1891 in Berdyansk, Russia -2 Nov 1970 in Cambridge, Cambridgeshire, England) Besicovitch left Petrograd for Copenhagen in 1924 and there worked with Harald Bohr. He had been awarded a Rockefeller Fellowship but his applications for permission to work abroad had been refused. He escaped across the border with a colleague J D Tamarkin under the cover of darkness. He managed to reach Copenhagen where he was supported financially for a year with the Rockefeller Fellowship. His interest in almost periodic functions came about through this year spent working with Harald Bohr. After he visited Oxford in 1925 Hardy, who quickly saw the mathematical genius in Besicovitch, found a post for him in Liverpool. At Cambridge Besicovitch lectured on analysis in most years but he also gave an advanced course on a topic which was directly connected with his research interests such as almost periodic functions, Hausdorff measure, or the geometry of plane sets. Besicovitch was famous for his work on almost periodic functions, his interest in which, as we mentioned above, came from his time in Copenhagen with Harald Bohr. In 1932 he wrote an influential text Almost periodic functions covering his work in this area.
One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas. The problem had been posed in 1917 by a Japanese mathematician S Kakeya and asked what was the smallest area in which a line segment of unit length could be rotated through 2p. Besicovitch proved in 1925 that given any e, an area of less than e could be found in which the rotation was possible. The figures that resulted from Besicovitch's construction were highly complicated, unbounded figures.
Other areas on which Besicovitch worked included geometric measure theory, Hausdorff measure, real function theory, and complex function theory. In addition to this work on deep mathematical theories, Besicovitch loved problems, particularly those which could be stated in elementary terms but which proved resistant to attack. Often he showed that the "obvious solution" to certain problems is false. An example of such a problem is the Lion and the Man problem posed by Richard Rado in the mid 1920s. *SAU

1993 Dura Kurepa (16 Aug 1907 in Majske Poljane near Glina, Croatia - 2 Nov 1993)
The topics which Kurepa investigated are very varied but lie mostly within topology, set theory and number theory. He published over 200 papers but this number rises to over 700 items if we include books, articles and reviews. He was fascinated by the continuum hypothesis and the axiom of choice. Perhaps best known is his work on trees and partitions, especially Aronszajn and Suslin trees. His book The Theory of Sets written in Serbo-Croatian and published in 1951 illustrates his interests in that particular area. After introducing the fundamental concepts and elementary operations in Chapter 1, he looks at cardinal numbers in the second chapter, then partially ordered sets and ordinal numbers in the third. Chapter 4 is on topological and metric spaces, with the fifth and final chapter on limiting processes in analysis, measure theory, Borel and Souslin sets.
In number theory he made many contributions, but perhaps his most famous is his open problem on the left factorial function. In 1971 he published his definition of !n, the left factorial function, defined by

!n = 0! + 1! + 2! + 3! + ... + (n-1)!.

Kurepa conjectured that the greatest common divisor of !n and n! was 2 for all n>1. There are many equivalent forms of the conjecture, but one of the most natural was given by Kurepa in the same 1971 paper, namely that !n is not divisible by n for any n>2. If the left factorial conjecture is false we certainly know that it will fail for n>106. *SAU

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

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