Friday, 22 July 2011

On This Day in Math - July 22

The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.

~David van Dantzig

This is the 203rd day of the year; 203 is the 6th Bell number, i.e. it is the number of partitions of a set of size 6.

1694 Johann Bernoulli sent “L’Hospital’s rule” to L’Hospital under the terms of their agreement of 17 March 1694. *VFR  The agreement between them led to the first real calculus text in 1696.

1925 After Norbert Wiener suggested to his friend Phillip Franklin in a letter that they hang a sign outside their office at MIT reading “Wiener and Franklin. Wholesale and Retail Mathematicians and Exporters,” he wrote: “As to the state of the market: differential geometry seems rather quiet, and some of the principal operators have deserted it for other securities. Real and complex variables continue firm, without much change. Analysis situs has a bull market. Bull operators have been very active in differential equations, also. Quantum theory continues speculative, with chances of a very sharp rise, but the market contains a lot of wildcat stock. Hilbert, Brouwer, and Co. are doing well with mathematical logic.” From Science in America, ed. Nathan Reingold, p. 384.*VFR

1933 Wiley Post startled the world by completing the first solo airplane flight around the world. The 15,400 mile flight lasted seven days, 18 hours, 49 and 1/2 minutes. Two years later he was killed in an airplane crash with humorist, Will Rogers. [Scientific American, November 1933]*VFR   He had made an accompanied flight around the world in 1931. Born 22 Nov 1898, Wiley Post made his first solo flight in 1926, the year he got his flying license, signed by Orville Wright, despite wearing a patch over his left eye, lost in an oilfield accident. Post invented the first pressurized suit to wear when he flew around the world. Another credit was his research into the jet streams. He died with his passenger, humorist Will Rogers, 15 Aug 1935 in a plane crash in Alaska.*TIS

1966 Philipp Frank (20 Mar 1884; 22 Jul 1966 at age 82) Austrian-American physicist and mathematician whose theoretical work covered a broad range of mathematics, including variational calculus, Hamiltonian geometrical optics, Schrödinger wave mechanics, and relativity. Frank had a deep and lasting interest in the philosophy of science. In a number of writings, he strove to reconcile science and philosophy and “bring about the closest rapprochement between” them. The 1907 paper he wrote analyzing the law of causality caught Einstein's attention, who in 1912 recommended Frank as his successor as professor of theoretical physics at the German University of Prague. He held that position until 1938, when he moved to Harvard University in the U.S., first as visiting lecturer, but remaining there until retirement in 1954. He wrote on misinterpretations of the Theory of Relativity.*TIS

1976 “researchers from Univ of Illinois announced they had found an unavoidable set containing 1936 reducible configurations effectively proving the four color theorem.*VFR

1983 Science reported that Gerd Faltings of Wuppertal University in Germany proved the sixty-year­old Mordell conjecture: most equations of degree higher than three have only a finite number of rational solutions. In particular, this applies to Fermat’s Last Theorem. [Mathematics Magazine 57 (1984), p. 52].*VFR  In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension. It was proved by Gerd Faltings (1983), and is now known as Faltings' theorem.

1997  Apple Announces OS 8-Apple Computer Inc. announces a new operating system for its Macintosh computers, OS 8. An important move at a time when Apple's upper-level management and profits were experiencing significant problems, the new operating system offered new features such as easier integration of the Internet and a three-dimensional look. Immediately after the announcement, the software earned positive reviews from users, although it was not expected to end Apple's financial troubles as it faced growing competition from improvements in the Microsoft Windows operating system used on IBM-compatible PCs. *CHM

1784 Friedrich Wilhelm Bessel born. He is noted for the special class of functions that have become an indispensable tool in applied mathematics. This, like all of his mathematical work, was motivated by his work in astronomy. *VFR    In 1809, at the age of 26, Bessel was appointed director of Frederick William III of Prussia's new Königsberg Observatory and professor of astronomy, where he spent the rest of his career. His monumental task was determining the positions and proper motions for about 50,000 stars, which allowed the first accurate determination of interstellar distances. Bessel's work in determining the constants of precession, nutation and aberration won him further honors. Other than the sun, he was the first to measure the distance of a star, by parallax, of 61 Cygni (1838). In mathematical analysis, he is known for his Bessel function. *TIS

1795 Gabriel Lam´e was born in Tours, in today's département of Indre-et-Loire.
He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves, and defined by the equation:
 \left|\,{x\over a}\,\right|^n + \left|\,{y\over b}\,\right|^n =1
where n is any positive real number.
He is also known for his running time analysis of the Euclidean algorithm. Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers a and b, the algorithm runs in no more than 5k steps, where k is the number of (decimal) digits of b. He also proved a special case of Fermat's last theorem. He actually thought that he found a complete proof for the theorem, but his proof was flawed. The Lamé functions are part of the theory of ellipsoidal harmonics. *Wik
Piet Hein's Super Ellipse is a Lame Curve

1822  Gregor Mendel (Original name (until 1843) Johann Mendel). Austrian pioneer in the study of heredity. He spent his adult life with the Augustinian monastery in Brunn, where as a geneticist, botanist and plant experimenter, he was the first to lay the mathematical foundation of the science of genetics, in what came to be called Mendelism. Over the period 1856-63, Mendel grew and analyzed over 28,000 pea plants. He carefully studied for each their plant height, pod shape, pod color, flower position, seed color, seed shape and flower color. He made two very important generalizations from his pea experiments, known today as the Laws of Heredity. Mendel coined the present day terms in genetics: recessiveness and dominance.

1826Giuseppe Piazzi (July 7, 1746 - July 22, 1826) an Italian mathematician and astronomer. He discovered the asteroid Ceres and established an observatory at Palermo, now the Osservatorio Astronomico di Palermo – Giuseppe S. Vaiana. (for more detail see the source article) *Today in Astronomy

1882 Konrad Knopp born. He is best known for comprehensive book on infinite series.*VFR

1887 - Gustav Hertz was born. Hertz was a German physicist who shares the 1925 Nobel Prize in Physics with James Franck for their Frank-Hertz experiment. The Frank-Hertz experiment shows that an atom absorbs energy in discrete amounts, confirming the quantum theory of atoms. This experiment was an important step confirming the Bohr model of the atom. *Science History

1902 Reinhold Baer - Baer's mathematical work was wide ranging; topology, abelian groups and geometry. His most important work, however, was in group theory, on the extension problem for groups, finiteness conditions, soluble and nilpotent groups.*SAU

1914  Edward (Rolke) Farber was an American who invented a portable, battery-operated stroboscopic flash unit for still cameras (1937) that effectively "stopped action." He began his career as a photojournalist on the staff of the Milwaukee Journal. After studying electrical engineering at Northwestern University, Farber went on to design flash equipment for the U.S. Army during World War II, and then established his own electronic-flash manufacturing firm. He was a good friend and collaborator of Harold Edgerton and developed the first practical portable strobe flash for news photographers. In 1942, the Milwaukee Journal became the first newspaper to furnish all of its photographers with the portable flash. Weighing only 13.5 pounds, it was a considerable improvement over the 90-pound units photographers used prior to Farber's invention. He sold his Strobe Research firm in 1954. He was a photographic adviser to the U.S. Government during its intercontinental ballistic missile testing program in the late 1950's.*TIS

1935 John Robert Stallings  In 1968 Stallings published his most famous paper On torsion-free groups with infinitely many ends in the Annals of Mathematics. L Neuwirth explains what is contained in the paper:-
In this remarkable paper, the author, using very little besides his bare hands, proves the following theorem:
1. If G is a torsion-free, finitely presented group, with infinitely many ends, then G is a non-trivial free product.
This simple sounding theorem proves to be very powerful, implying
(with a little work) the following two theorems:
2. A torsion-free, finitely generated group, containing a free subgroup of finite index, is itself free.
3. A finitely generated group of cohomological dimension 1 is free.
This last theorem answers a question which had been unanswered for over ten years and which had received considerable attention over that period of time. Theorem
2 answers a question of J-P Serre, who proved an analogue of Theorem 2 for pro-p groups. The proof of Theorem 1 is both combinatorial and geometric in nature and, as suggested, is self-contained.
For this truly outstanding paper the American Mathematical Society awarded Stallings their Frank Nelson Cole Prize in Algebra in 1970. Also in 1970 he was invited to address the International Congress of Mathematicians in Nice, France. He gave a talk on Group theory and 3-manifolds. He had been honoured in the previous year when invited to give the James K Whittemore Lecture in Mathematics at Yale University in 1969. His topic was Group theory and three-dimensional manifolds. This lecture and his Nice address were both published in 1971.
Among the 50 or so papers Stalling published, we should highlight another two which have proved particularly important: Topology on finite graphs (1983) and Non-positively curved triangles of groups (1991). The first of these introduced the 'Stallings subgroup graph' as a method to describe subgroups of free groups. It also introduced a foldings technique now known as 'Stallings' foldings method' which has been the basis for much later work. The second of these two papers introduced the notion of a triangle of groups which became the basis for later work on the theory of complexes of groups.*SAU

1575  Francisco Maurolico was an Italian Benedictine who wrote important books on Greek mathematics. He also worked on geometry, the theory of numbers, optics, conics and mechanics.*SAU

1826 Giuseppe Piazzi Italian astronomer and author, born in Valtellina, discovered the first asteroid - Ceres. He established an observatory at Palermo and mapped the positions of 7,646 stars. He also discovered that the star 61 Cygni had a large Proper Motion , which led Bessel to chose it as the object of his parallax studies. He discovered Ceres in 1801, but was able to make only three observations. Fortuitously, Gauss had recently developed mathematical techniques that allowed the orbit to be calculated. This was the first asteroid discovered. The thousandth Asteroid discovered was named Piazzia in his honor.*TIS  (His dates of birth and death are six days apart)

1869 John A. Roebling  German-American engineer who pioneered the design and construction of suspension bridges. In 1831 he immigrated to Saxonburg, near Pittsburgh, Pa., and shortly thereafter was employed by the Pennsylvania Railroad Corp. to survey its route across the Allegheny Mountains. He then demonstrated the practicability of steel cables in bridge construction and in 1841 established at Saxonburg the first U.S. factory to manufacture steel-wire rope. Roebling utilized steel cables in the construction of numerous suspension bridges including a railroad suspension bridge over the Niagara River at Niagara Falls (1851-55). He designed the Brooklyn Bridge. He died from injuries while supervising preliminary construction operations.*TIS

1915 Sir Sandford Fleming Scottish surveyor and leading railway engineer who divided world into time zones. He emigrated at age 17 years to Quebec, Canada, on April 24, 1845, as a surveyor. Later became one of the foremost railway engineers of his time. While in charge of the initial survey for the Canadian Pacific Railway, the first Canadian railway to span the continent, he realized the problems of coordinating such a long railway. This lead him to the idea of time zones, which contribution to the adoption of the present system of time zones earned him the title of "Father of Standard Time." Fleming also designed the first Canadian postage stamp. Issued in 1851, it cost three pennies and depicted the beaver, now the national animal of Canada.*TIS

1932 Reginald Aubrey Fessenden was a Canadian inventor and engineer with 300 patents. He broadcast the first program of voice and music. In 1893, Fessenden moved to Pittsburgh as the head of electrical engineering at the university, Fessenden read of Marconi's work and began experimenting himself. Marconi could only transmit Morse code. But Fessenden's goal was to transmit the human voice and music. He invented the "continuous wave": sound superimposed onto a radio wave for transmission. A radio receiver extracts the signal so the listener with the original sound. Fessenden made the first long-range transmissions of voice on Christmas Eve 1906 from a station at Brant Rock, Massachusetts, heard hundreds of miles out in the Atlantic.*TIS

1938  Ernest (William) Brown was a British astronomer who devoted his career to the theory of the Moon's motion and constructing accurate lunar tables. His theory took account of "the gravitational action of every particle of matter which can have a sensible effect on the Moon's motion," some 1500 terms. He then determined the numerical values of the constants by analyzing 150 years of Greenwich observations, and computed tables accurate to 0.01 arcsec. After 30 years of work, Brown published his lunar tables Tables of the Motion of the Moon in 1919. In 1926 Brown published a paper in which he ascribed fluctuations in the Moon's motion to irregular changes in the Earth's period of rotation, which has subsequently proved correct. *TIS

1943 William Fogg Osgood died. Although his nickname was “Foggy,” this was not an apt description of him as a teacher. He instilled the habit of careful thought in Harvard students for 43 years. His A First Course in Differential and Integral Calculus (1907) was revised once and reprinted 17 times.*VFR

1959  David van Dantzig was at secondary school when he wrote his first mathematics paper. He was only thirteen years old at the time. However, his main interest in secondary school was not mathematics, rather it was chemistry. After leaving school he continued with his studies of chemistry, but this he did not enjoy and when he was forced to give up his academic studies to help support his family van Dantzig took on a number of jobs purely to make money.
By now van Dantzig knew that mathematics was the subject which he really wanted to study but he was not in a position to do so, both because he had to earn money and also because he did not have the necessary school qualifications. He put in hours of work on mathematics in the evenings after finishing his money earning tasks for the day. He took the state mathematics examinations in 1921, at a higher level the following year and again in 1923 he passed at a higher level still. Entering the University of Amsterdam to study mathematics he soon passed examinations which took him essentially to Master's Degree level.
Van Dantzig became an assistant to Schouten in 1927 at Delft Technical University. Then, for a short time, he taught at a teacher training institution, but he returned to Delft as a lecturer in 1932. This was the year in which he received his doctorate from Gröningen for a thesis which he submitted in 1931 Studiën over topologische Algebra. In this work he coined the now familiar term topological algebra but the thesis is memorable in other ways too. It -
... is a fine example of mathematical style: it consists of a concise string of definitions and theorems organised in such a way that in this context each theorem is obvious and none needs a proof.
He was promoted to extraordinary professor at Delft in 1938 and then an ordinary professor in 1940. The Dutch had tried to remain neutral when World War II broke out in 1939 but in the spring of 1940 German troops, in a strategic move on their way to attack France, entered Holland and the Dutch were defeated in a week. Van Dantzig was dismissed from his chair when the Germans occupied Holland and he was forced to move with his family from the Hague to Amsterdam.
After the war ended, he was appointed professor at the University of Amsterdam in 1946. In Amsterdam he was the cofounder of the research and service institution, the Mathematisch Centrum. He played a major role in both this Centre and in the University of Amsterdam where he continued to hold his chair until his death.
Van Dantzig studied differential geometry, electromagnetism and thermodynamics. His most important work was in topological algebra and in addition to his doctoral thesis which we mentioned above, he wrote a whole series of papers on topological algebra. He studied metrisation of groups, rings and fields. One paper classified fields with a locally compact topology.*SAU

1995  Otakar Boruvka  To many people Boruvka is best known for his solution of the Minimal Spanning Tree problem which he published in 1926 in two papers On a certain minimal problem (Czech) and Contribution to the solution of a problem of economical construction of electrical networks (Czech). Let us quote the problem as it appears in the second of these 1926 papers:-
There are n points in the plane whose mutual distances are different. The problem is to join them with a net in such a way that:
1. any two points are joined to each other either directly or by means of some other points;
2. the total length of the net will be minimal.
In modern graph theoretical terms this can be stated as: Given an undirected graph with weights assigned to its edges, find a spanning tree of minimal weight.
In fact the problem had been suggested to Boruvka before he became a university student. He had a friend, Jindrich Saxel, who worked for the firm West-Moravian Powerplants and he suggested the problem which he stated in terms of cities and the distances between them. At the time that Saxel suggested the problem to Boruvka, World War I was still happening and Czech universities were closed. Boruvka was offered a job with West-Moravian Powerplants at this time but declined. The authors of [8] write:-
The Minimal Spanning Tree problem is a cornerstone of Combinatorial Optimisation and in a sense its cradle. The problem is important both in its practical and theoretical applications. Moreover, recent development places Boruvka's pioneering work in a new and very contemporary context. One can even say that out of many available Minimal Spanning Tree algorithms, Boruvka's algorithm is presently the basis of the fastest known algorithms.  *SAU

*VFR = V Frederick Rickey, USMA
*TIS= Today in Science History
*Wik = Wikipedia
*SAU=St Andrews Univ. Math History
*CHM=Computer History Museum
Post a Comment