text:
## Monday, 31 October 2022

### On This Day in Math - October 31

*The image is a pumpkin carved a few Halloweens ago by Sonja L. One of my Stats/Calc students (and a really good Bassoonist, Bassooner, Bassoon-enough)

I don’t like reciprocals.
They’re always trying to get one over on you.

The 304th day of the year; 304 is the sum of six consecutive primes starting with 41, and also the sum of eight consecutive primes starting with 23. (and for those who keep up with such things, it is also the record number of wickets taken in an English cricket season by Tich Freeman in 1928.)

There are 304 semi-primes less than 2^10, but 304 is NOT one of them. *Derek Orr

304 = 77^2 - 75^2 = 23^2 - 15^2 = 40^2 - 36^2

Math Joke for Halloween: Why do mathematicians confuse Halloween and Christmas? Because Oct 31 = Dec 25 (31 in base 8 (Octal) is the same quantity as 25 in Decimal)

EVENTS

In 1815, English chemist, Sir Humphrey Davy of London (Davy was actually from Penzance) patented the miner's safety lamp. Miners at work constantly met firedamp, an explosive mix of methane gas and air, during the working of coal. This was an almost insurmountable obstacle to the working of many of the collieries until the discovery of the safety lamp. The flame of the safety lamp is surrounded by a copper or iron gauze cylinder, with openings no more than 1/24-inch. Such a fine gauze prevents flame passing through, but fails if coarser. The wire absorbs or conducts away the heat of the flame contained inside the lamp so it does not explode gas outside the lamp. If firedamp is present, a pale blue flame appears around the central flame. This warns a miner to leave the area immediately! *TIS

1839 (Sometime in October) the ﬁrst teacher’s institute was held at Hartford, Connecticut, 26 men teachers attended a six week course sponsored by Henry Barnard and received the “opportunity of critically reviewing the studies which they will be called upon to teach, with a full explanation of all the principles involved.” The authority who gave instruction on higher mathematics was Charles Davies. *VFR

1903 At a New York meeting of the AMS F. N. Cole (1861-1927) presented a paper “On the factoring of large numbers.” He spoke not a word, but carefully raised 2 to the 67th power, then subtracted one. Moving over he computed 193,707,721 times 761,838,257,287. The calculations agreed, showing that 267 − 1 was not a Mersenne prime. E. T. Bell, in Mathematics—Queen and Servant of the Sciences, wrote, with his usual exageration, “For the ﬁrst and only time on record, an audience of the American Mathematical Society vigorously applauded the author or a paper delivered before it.” Later, in 1911, Bell asked Cole how long it had taken him to ﬁnd this factorization and he replied “Three years of Sundays.” It is instructive to check this arithmetic on your hand held calculator. [Eves, Adieu, 297◦; BAMS 10(1903), 134] *VFR

1915 Closing date for a prize consisting of a gold medal bearing the portrait of Weierstrass and 3000 Swedish crowns for the best essay on the theory of analytic functions. King Gustav V of Sweden founded the prize to commemorate the centenary of the birth of Weierstrass. *VFR

1918 The wife of the Russian mathematician Lyapunov died of tuberculosis. On the same day, Lyapunov shot himself. He died three days later, on 3 November 1918. *VFR

1933 Albert Einstein moved to the United States on 17 October 1933. Two weeks later, Halloween arrived. When a group of girls knocked on his door that evening and shouted 'Trick or Treat,' Einstein came to the front porch and played the violin for them: * @phalpern

In 1992, the Vatican admitted erring for over 359 years in formally condemning Galileo Galilei for entertaining scientific truths such as the Earth revolves around the sun it, which the Roman Catholic Church long denounced as anti-scriptural heresy. After 13 years of inquiry, the Pope's commission of historic, scientific and theological scholars brought the pope a "not guilty" finding for Galileo. *TIS In 1822 the church lifted the ban on the works of Galileo and in 1979 Pope John Paul II selected a commission to investigate. On Mar 31 of 1984 the Vatican newspaper, L’Observatore Romano, stated, “The so-called heresy of Galileo does not seem to have any foundation, neither theologically nor under canon law.” It still took until Oct 31, 1992, before Pope John Paul II declared that the church may have been mistaken in condemning Galileo. *Wik

BIRTHS

1711 Laura Maria Catarina Bassi (31 Oct 1711 in Bologna, Papal States, 20 Feb 1778 in Bologna, Papal States) was an Italian physicist and one of the earliest women to gain a position in an Italian university. *SAU She was the first woman in the world to earn a university chair in a scientific field of studies. She received a doctoral degree from the University of Bologna in May 1732, only the third academic qualification ever bestowed on a woman by a European university, and the first woman to earn a professorship in physics at a university in Europe. She was the first woman to be offered an official teaching position at a university in Europe.
In 1738, she married Giuseppe Veratti, a fellow academic with whom she had twelve children. After this, she was able to lecture from home on a regular basis and successfully petitioned the University for more responsibility and a higher salary to allow her to purchase her own equipment.
One of her principal patrons was Pope Benedict XIV. He supported less censorship of scholarly work, such as happened with Galileo, and he supported women figures in learning, including Agnesi.
She was mainly interested in Newtonian physics and taught courses on the subject for 28 years. She was one of the key figures in introducing Newton's ideas of physics and natural philosophy to Italy. She also carried out experiments of her own in all aspects of physics. In order to teach Newtonian physics and Franklinian electricity, topics that were not focused in the university curriculum, Bassi gave private lessons.[6] In her lifetime, she authored 28 papers, the vast majority of these on physics and hydraulics, though she did not write any books. She published only four of her papers.[2] Although only a limited number of her scientific works were left behind, much of her scientific impact is evident through her many correspondents including Voltaire, Francesco Algarotti, Roger Boscovich, Charles Bonnet, Jean Antoine Nollet, Giambattista Beccaria, Paolo Frisi, Alessandro Volta. Voltaire once wrote to her saying "There is no Bassi in London, and I would be much happier to be added to your Academy of Bologna than that of the English, even though it has produced a Newton". *Wik

1815 Karl (Theodor Wilhelm) Weierstrass (31 Oct 1815; 19 Feb 1897) was a German mathematician who is known as the "father of modern analysis" for his rigor in analysis led to the modern theory of functions, and considered one of the greatest mathematics teachers of all-time. He was doing mathematical research while a secondary school teacher, when in 1854, he published a paper on Abelian functions in the famous Crelle Journal. The paper so impressed the mathematical community that he shortly received an honorary doctorate and by 1856, he had a University appointment in Berlin. In 1871, he demonstrated that there exist continuous functions in an interval which have no derivatives nowhere in the interval. He also did outstanding work on complex variables. *TIS

1847 Galileo Ferraris (31 Oct 1847; 7 Feb 1897) Italian physicist who studied optics, acoustics and several fields of electrotechnics, but his most important discovery was the rotating magnetic field. He produced the field with two electromagnets in perpendicular planes, and each supplied with a current that was 90º out of phase. This could induce a current in a incorporated copper rotor, producing a motor powered by alternating current. He produced his first induction motor (with 4 poles) in May-Jun 1885. Its principles are now applied in the majority of today's a.c. motors, yet he refused to patent his invention, and preferred to place it at the service of everyone. *TIS

1890 Joseph Jean Camille Pérès (31 Oct 1890 in Clermont-Ferrand, France, 12 Feb 1962 in Paris, France) Pérès' work on analysis and mechanics was always influenced by Volterra, extending results of Volterra's on integral equations. His work in this area is now of relatively little importance since perhaps even for its day it was somewhat old fashioned.
A joint collaboration between Pérès and Volterra led to the first volume of Theorie generale des fonctionnelles published in 1936. Although the project was intended to lead to further volumes only this one was ever published. This work is discussed in where the author points out that the book belongs to an older tradition, being based on ideas introduced by Volterra himself from 1887 onwards. By the time the work was published the ideas it contained were no longer in the mainstream of development of functional analysis since topological and algebraic concepts introduced by Banach, von Neumann, Stone and others were determining the direction of the subject. However, the analysis which Pérès and Volterra studied proved important in developing ideas of mathematical physics rather than analysis and Pérès made good use of them in his applications. *SAU

1902 Abraham Wald (October 31, 1902 – December 13, 1950) was a mathematician born in Cluj, in the then Austria–Hungary (present-day Romania) who contributed to decision theory, geometry, and econometrics, and founded the field of statistical sequential analysis. He spent his researching years at Columbia University.*Wik

1919 Father Magnus J. Wenninger OSB (born Park Falls, Wisconsin, October 31, 1919) is a mathematician who works on constructing polyhedron models, and wrote the first book on their construction. *Wik

1925 John A. Pople (31 Oct 1925; 15 Mar 2004) British mathematician and chemist who, (with Walter Kohn), received the 1998 Nobel Prize in Chemistry for his work on computational methodology to study the quantum mechanics of molecules, their properties and how they act together in chemical reactions. Using Schrödinger's fundamental laws of quantum mechanics, he developed a computer program which, when provided with particulars of a molecule or a chemical reaction, outputs a description of the properties of that molecule or how a chemical reaction may take place - often used to illustrate or explain the results of different kinds of experiment. Pople provided his GAUSSIAN computer program to researchers (first published in 1970). Further developed, it is now used by thousands of chemists the world over. *TIS

1927 Narinder Singh Kapany (31 Oct 1927, )Indian-American physicist who is widely acknowledged as the father of fibre optics. He coined the term fibre optics for the technology transmitting light through fine glass strands in devices from endoscopy to high-capacity telephone lines that has changed the medical, communications and business worlds. While growing up in Dehradun in northern India, a teacher informed him that light only traveled in a straight line. He took this as a challenge and made the study of light his life work, initially at Imperial College, London. On 2 Jan 1954, Nature published his report of successfully transmitting images through fiber optical bundles. The following year he went to the U.S. to teach. In 1960, Optics Technology. He holds over 100 patents.*TIS

1935 Ronald Lewis Graham (born October 31, 1935- Jul 6 2020) is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. Graham was also featured in Ripley's Believe It or Not for being not only "one of the world's foremost mathematicians", but also "a highly skilled trampolinist and juggler", and past president of the International Jugglers' Association. He is currently the Chief Scientist at the California Institute for Telecommunications and Information Technology (also known as Cal-(IT)2) and the Irwin and Joan Jacobs Professor in Computer Science and Engineering at the University of California, San Diego. *Wik My current favorite Graham quote is, "An ideal math talk should contain one proof and one joke and they should not be the same."

Graham died of bronchiectasis on July 6, 2020, at the age of 84.

DEATHS

 *@willgater
1867 William Parsons, 3rd Earl of Rosse (17 Jun 1800, 31 Oct 1867) was an Irish astronomer who built the largest reflecting telescope of the 19th century. He learned to polish metal mirrors (1827) and spent the next few years building a 36-inch telescope. He later completed a giant 72-inch telescope (1845) which he named "Leviathan," It remained the largest ever built until decades after his death. He was the first to resolve the spiral shape of objects - previously seen as only clouds - which were much later identified as galaxies independent of our own Milky Way galaxy and millions of light-years away. His first such sighting was made in 1845, and by 1850 he had discovered 13 more. In 1848, he found and named the Crab Nebula (because he thought it resembled a crab), by which name it is still known. *TIS A reproduction of the Leviathan of Parsonstown (now Birr) Ireland at Birr Castle, Rosse’s ancestral home and is open to tourists.

1899 Juliusz Paweł Schauder (September 21, 1899, Lwów, Austria-Hungary – October, 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics.
He had to fight in World War I right after his graduation from school. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów.
Schauder was Jewish, and after the invasion of German troops in Lwów it was impossible for him to continue his work. In his letters to Swiss mathematicians, he wrote that he had important new results, but no paper to write them down. He was executed by the Gestapo, probably in October 1943.
Most of his mathematical work belongs to the field of functional analysis, being part of a large Polish group of mathematicians, i.e. Lwów School of Mathematics. They were pioneers in this area with wide applications in all parts of modern analysis. Schauder is best known for the Schauder fixed point theorem which is a major tool to prove the existence of solutions in various problems, the Schauder bases (a generalization of an orthonormal basis from Hilbert spaces to Banach spaces), and the Leray−Schauder principle, a way to establish solutions of partial differential equations from a priori estimates. *Wik

1988 George Eugene Uhlenbeck (6 Dec 1900, 31 Oct 1988) Dutch-American physicist who, with Samuel A. Goudsmit, proposed the concept of electron spin (Jan 1925) - a fourth quantum number which was a half integer. This provided Wolfgang Pauli's anticipated "fourth quantum number." In their experiment, a horizontal beam of silver atoms travelling through a vertical magnetic field was deflected in two directions according to the interaction of their spin (either "up" or "down") with the magnetic field. This was the first demonstration of this quantum effect, and an early confirmation of quantum theory. As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure, the kinetic theory of matter and extended Boltzmann's equation to dense gases.*TIS

Credits
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

## Sunday, 30 October 2022

### Soul Cakes, Halloween Began in Britain??

Well, you get old and eventually you learn stuff. I got a note from Charles Wells who said, " The *name* Halloween came from the British. It is the eve of All Saints Day (Hallowmas, November 1) which is celebrated all over Catholic Europe, not just in Britain. November 2 is All Souls Day, meaning the day for sinners as well as saints. That is the Day of the Dead in Mexico."
Later I added, "Thanks, Charles,but given that the Scottish, the English, the Welsh, and the Irish ALL seem to object to being included as "British", I will simply confirm what you have said with a quote from the Online Etymology Dictionary, "c.1745, Scottish shortening of Allhallow-even "Eve of All Saints, last night of October" (1556), the last night of the year in the old Celtic calendar, where it was Old Year's Night, a night for witches. Another pagan holiday given a cursory baptism and sent on its way. Hallowmas "All-saints" is first attested 1389."

I had just seen a BBC show that morning in 2009 and Sting had just released the Soul Cakes song, don't ask why, some things must be left unexplained.

Here is the story of soul cakes and halloween as told by Wikipedia: "A Soul cake is a small round cake which is traditionally made for All Souls' Day to celebrate the dead. The cakes, often simply referred to as souls, were given out to soulers (mainly consisting of children and the poor) who would go from door to door on Hallowmas (new word to me, obviously the eve of All Souls Day) singing and saying prayers for the dead. Each cake eaten would represent a soul being freed from Purgatory. The practice of giving and eating soul cakes is often seen as the origin of modern Trick or Treating." "The tradition of giving Soul Cakes originated in Britain during the Middle Ages, although similar practices for the souls of the dead were found as far south as Italy." "The cakes were usually filled with allspice, nutmeg, cinnamon, or other sweet spices, raisins or currants, and later were topped with the mark of a cross. They were traditionally set out with glasses of wine on All Hallows Eve, and on All Saints Day children would go "souling" by calling out: Soul, Soul, a soul cake!
I pray thee, good missus, a soul cake!
One for Peter, two for Paul,
three for Him what made us all!
Soul Cake, soul cake, please good missus, a soul cake.
An apple, a pear, a plum, or a cherry,
anything good thing to make us all merry.
One for Peter, one for Paul,
& three for Him who made us all. ...

lyrics from A Soalin', a holiday song written and performed by Peter, Paul and Mary (1963)."

See, nothing scary here. Have a Happy Halloween.

### On This Day in Math - October 30

'Mathematics is the science that uses easy words for hard ideas.'
~ Edward Kasner

The 303rd day of the year; there are 303 different bipartite graphs with 8 vertices. *What's Special About This Number

303 primes are below 2000. * Derek Orr

303 = 152^2 - 151^2 = 52^2 - 49^2

In the Gregorian calendar, 303 is the number of years that are not leap years in a period of 400 years.

EVENTS

1613 Kepler married his second wife (the ﬁrst died of typhus). She was ﬁfth on his slate of eleven candidates. The story that he used astrology in the choice is doubtful.*VFR Kepler married the 24-year-old Susanna Reuttinger. He wrote that she, "won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren.According to Kepler's biographers, this was a much happier marriage than his first. *Wik
The complete story from a letter of Kepler about his 2 year search is told here by Thony Christie.

1710 William Whiston, whom Newton had arranged to succeeded him as Lucasian Professor at Cambridge in 1701, was deprived of the chair and driven from Cambridge for his unorthodox religious views. Whiston was removed from his position at Cambridge, and denied membership in the Royal Society for his “heretical” views. He took the “wrong” side in the battle between Arianism (a unitarian view) and the Trinitarian view, but his brilliance still made the public attend to his proclamations. When he predicted the end of the world by a collision with a comet in October 16th of 1736 the Archbishop of Canterbury had to issue a denial to calm the panic (VFR put it this way, "it is not acceptable to be a unitarian at the College of the Whole and Undivided Trinity".
His translation of the works of Flavius Josephus may have contained a version of the famous Josephus Problem, and in 1702 Whiston's Euclid discusses the classic problem of the Rope Round the Earth, (if one foot of additional length is added, how high will the rope be). I am not sure of the dimensions in Whiston's problem, and would welcome input, I have searched the book and can not find the problem in it, but David Singmaster has said it is there, and he is not an easy source to reject. It is said that Ludwig Wittgenstein was fascinated by the problem and used to pose it to students regularly.

1735 Benjamin Franklin’s paper “On the Usefulness of Mathematics,” appeared in the Pennsylvania Gazette. [NCTM yearbook # 32(1970), p. 20]*VFR I have also seen the date given as October 30. Some historians also question whether or not this was actually written by Franklin.

1826 Abel presented a paper to the French Academy of Science that was ignored by Cauchy, who was to serve as referee. The paper was published some twenty years later.*VFR

In 1937, the closest approach to the earth by an asteroid, Hermes, was measured to be 485,000 miles, which, to an astronomer, is a mere hair's width (asteroid now lost).*TIS

1945 The first conference on Digital Computer Technique was held at MIT. The conference was sponsored by the National Research Council, Subcommittee Z on Calculating Machines and Computation. Attended by the Whirlwind team,(The Whirlwind computer was developed at the Massachusetts Institute of Technology. It is the first computer that operated in real time, used video displays for output, and the first that was not simply an electronic replacement of older mechanical systems) it influenced the direction of this computer. *CHM

1978 Laura Nickel and Curt Noll, eighteen year old students at California State at Hayward, show that 221,701 − 1 is prime. This was the largest prime known at that time. *VFR (By Feb of the next year, Noll had found another, 223209-1. By April, another larger Prime had been found.)

1992 The Vatican announced that a 13-year investigation into the Catholic Church’s condemnation of Galileo in 1633 will come to an end and that Galileo was right: The Copernican Theory, in which the Earth moves around the Sun, is correct and they erred in condemning Galileo. *New York Times for 31 October 1992.

2012 After Hurricane Sandy came ashore in New Jersey on the 29th, the huge weather system was captured with an overlay to emphasize it's Fibonacci-like structure. *HT to Bob Mrotek for sending me this image

BIRTHS
1840 Joseph Jean Baptiste Neuberg (30 Oct 1840 in Luxembourg City, Luxembourg - 22 March 1926 in Liège, Belgium) Neuberg worked on the geometry of the triangle, discovering many interesting new details but no large new theory. Pelseneer writes, "The considerable body of his work is scattered among a large number of articles for journals; in it the influence of A Möbius is clear." *SAU

1844 George Henri Halphen (30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He did his studies at École Polytechnique (X 1862). He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry.*Wik

1863 Stanislaw Zaremba (3 Oct 1863 in Romanowka, Poland - 23 Nov 1942 in Kraków, Poland) From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. Zaremba played a crucial role in this transformation. Much of Zaremba's research work was in partial differential equations and potential theory. He also made major contributions to mathematical physics and to crystallography. He made important contributions to the study of viscoelastic materials around 1905. He showed how to make tensorial definitions of stress rate that were invariant to spin and thus were suitable for use in relations between the stress history and the deformation history of a material. He studied elliptic equations and in particular contributed to the Dirichlet principle.*SAU

1906 Andrei Nikolaevich Tikhonov (30 Oct 1906 in Gzhatska, Smolensk, Russia - November 8, 1993, Moscow) Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935. These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics.
The extremely deep investigations of Tikhonov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.
Tikhonov's work on mathematical physics continued throughout the 1940s and he was awarded the State Prize for this work in 1953. However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative. After a series of fundamental papers introducing the topic, the work was carried on by his students.
Another area in which Tikhonov made fundamental contributions was that of computational mathematics. Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ... and other branches of the natural sciences were evolved and put into practice. ... One of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes, which Tikhonov developed in collaboration with Samarskii.
In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems. He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems. Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin Prize for his work on ill-posed problems in 1966. In the same year he was elected to full membership of the USSR Academy of Sciences.*SAU

1907 Harold Davenport (30 Oct 1907 in Huncoat, Lancashire, England - 9 June 1969 in Cambridge, Cambridgeshire, England) Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He wrote a number of important textbooks and monographs including The higher arithmetic (1952)*SAU

1946 William Paul Thurston  (October 30, 1946 – August 21, 2012) American mathematician who was awarded the Fields Medal in 1983 for his work in topology. As early as his Ph.D. thesis entitled Foliations of 3-manifolds which are circle bundles (1972) that showed the existence of compact leaves in foliations of 3-manifolds, Thurston had been working in the field of topology. In the following years, Thurston's contributions to the field of foliations were recognized to be of considerable depth, set apart by their originality. This was also true of his subsequent work on Teichmüller space. *TIS

DEATHS

1626 Willebrord van Royen Snell (13 June 1580 in Leiden, Netherlands - 30 Oct 1626 in Leiden, Netherlands) Snell was a Dutch mathematician who is best known for the law of refraction, a basis of modern geometric optics; but this only become known after his death when Huygens published it. His father was Rudolph Snell (1546-1613), the professor of mathematics at Leiden. Snell also improved the classical method of calculating approximate values of π by polygons which he published in Cyclometricus (1621). Using his method 96 sided polygons gives π correct to 7 places while the classical method yields only 2 places. Van Ceulen's 35 places could be found with polygons of 230 sides rather than 262. In fact Van Ceulen's 35 places of π appear in print for the first time in this book by Snell. *SAU

1631 Michael Mästin (30 Sept 1550 in Göppingen, Baden-Würtemberg, Germany
- 30 Oct 1631 in Tübingen, Baden-Würtemberg, Germany) astronomer who was Kepler's teacher and who publicized the Copernican system. Michael Mästin was a German astronomer who was Kepler's teacher and who publicised the Copernican system. Perhaps his greatest achievement (other than being Kepler's teacher) is that he was the first to compute the orbit of a comet, although his method was not sound. He found, however, a sun centered orbit for the comet of 1577 which he claimed supported Copernicus's heliocentric system. He did show that the comet was further away than the moon, which contradicted the accepted teachings of Aristotle. Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system. However for the more advanced lectures he adopted the heliocentric approach - Kepler credited Mästlin with introducing him to Copernican ideas while he was a student at Tübingen (1589-94).*SAU

1739 Leonty Filippovich Magnitsky (June 9, 1669, Ostashkov – October 30, 1739, Moscow) was a Russian mathematician and educator. From 1701 and until his death, he taught arithmetic, geometry and trigonometry at the Moscow School of Mathematics and Navigation, becoming its director in 1716. In 1703, Magnitsky wrote his famous Arithmetic (Арифметика; 2,400 copies), which was used as the principal textbook on mathematics in Russia until the middle of the 18th century. This book was more an encyclopedia of mathematics than a textbook because most of its content was communicated for the first time in Russian literature. In 1703, Magnitsky also produced a Russian edition of Adriaan Vlacq's log tables called Таблицы логарифмов и синусов, тангенсов и секансов (Tables of Logarithms, Sines, Tangents, and Secants). Legend has it that Leonty Magnitsky was nicknamed Magnitsky by Peter the Great, who considered him a "people's magnet" *Wik

1805 Ormbsy MacKnight Mitchel (July 20, 1805 – October 30, 1862) American astronomer and major general in the American Civil War.
A multi-talented man, he was also an attorney, surveyor, and publisher. He is notable for publishing the first magazine in the United States devoted to astronomy. Known in the Union Army as "Old Stars", he is best known for ordering the raid that became famous as the Great Locomotive Chase during the Civil War. He was a classmate of Robert E. Lee and Joseph E. Johnston at West Point where he stayed as assistant professor of mathematics for three years after graduation.
The U.S. communities of Mitchell, Indiana, Mitchelville, South Carolina, and Fort Mitchell, Kentucky were named for him. A persistently bright region near the Mars south pole that was first observed by Mitchel in 1846 is also named in his honor. *TIA

1806 Alexander (Dallas) Bache (July 19, 1806 – February 17, 1867) was Ben Franklin's great grandson. A West Point trained physicist, Bache became the second Superintendent of the Coast Survey (1844-65). He made an ingenious estimate of ocean depth in 1856. He studied records of a tidal wave that had taken 12 hours to cross the Pacific. Knowing that wave speeds depend on depth, he calculated a 2 1/5-mile average depth for the Pacific (within 15% of the right value). Bache created the National Academy of Sciences, securing greater government involvement in science. Through the Franklin Institute he instituted boiler tests to promote safety for steamboats.*TIS

1975 Gustav Hertz (22 July 1887, 30 Oct 1975) German quantum physicist who, with James Franck, received the Nobel Prize for Physics in 1925 for the Franck-Hertz experiment, which confirmed the quantum theory that energy can be absorbed by an atom only in definite amounts and provided an important confirmation of the Bohr atomic model. He was a nephew of Heinrich Hertz. Although he fought on the German side in World War I, being of Jewish descent, he was forced to resign his professorship (1934) when Hitler took power. From 1945 he worked in the Soviet Union, and then in 1955 was a professor of physics in Leipzig, East Germany.*TIS

2007 Juha Heinonen, (23 July 1960 in Toivakka, Finland - 30 Oct 2007 in Ann Arbor, Michigan, USA) Professor of Mathematics passed away on October 30. He arrived in the Department in 1988 as a postdoctoral assistant professor, and became a professor in 2000. He was a leading researcher in geometric function theory, having published two books and numerous articles with many collaborators. Most recently, Juha served as Associate Chair for Graduate Studies in the Department, where he mentored many young mathematicians. *Math at U of M webpage memorial (Heinonen died at the age of 47 'after a brief but courageous battle with kidney cancer'. The Department of Mathematics at the University of Michigan established the Juha Heinonen Memorial Graduate Student Fellowship in his honour. An international conference in his memory Quasiconformal Mappings and Analysis on Metric Spaces was organised at the University of Michigan, Ann Arbor in May 2008.)

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

## Saturday, 29 October 2022

### #27 Barycenter ...from old Math Terms notes

Barycenter The word barycenter is another term for the center of gravity or centroid. The Greek root is barus which generally refers to weighty or heavy. The more ancient Indo-European root seems to have come from a word like "gwerus" and has relatives in our words for gravity and grave.
Another word derived from the same root is baryon, the name for a family of particles that are heavier (more massive) than mesons. The word barometer also comes from the same root and is so named because, in a sense, it measures how heavy the air is. Another related word still in current use is baritone, which literally means heavy voiced. The science names for the chemical barium and the ore from which we obtain it, barite, also called "heavy spar", are both from the same root.
The History of Math web site at St. Andrews University in Scotland credits the creation of barycenters to August Möbius (1790-1868):

In 1827 Möbius published Der barycentrische Calcul, a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle ABC then if weights a, b and c are placed at A, B and C respectively then a point P, the center of gravity, is determined. Möbius showed that every point P in the plane is determined by the homogeneous coordinates [a,b,c], the weights required to be placed at A, B and C to give the center of gravity at P. The importance here is that Möbius was considering directed quantities, an early appearance of vectors.

### On This Day in Math - October 29

Allez en avant, et la foi vous viendra
Push on and faith will catch up with you.

~Jean d'Alembert [advice to those who questioned the calculus](probably also great for students struggling with mathematics at any level)

The 302nd day of the year; There are 302 ways to play the first three moves in checkers.

302 is the sum of three consecutive squares 92+102+112

302 is a semiprime, 2 x 151.  Its reversal 203 = 7 x 29, is also a semiprime

EVENTS

1669 Newton, aged twenty-six, appointed Lucasian Professor at Cambridge. This post required Newton to lecture once each week on “some part of Geometry, Astronomy, Geography, Optics, Statics, or some other Mathematical discipline,” and to deposit ten of those lectures in the library each year. The students were required to attend, but like all other requirements they ignored this one too. We know of only three people who attended a lecture at Cambridge by Newton. [Westfall 208–210; Works, 3, xv] *VFR

1675 Leibniz ﬁrst used the integral sign. Also ﬁrst used “d”. He also constructed what he calls the “triangulum characteristicum,” which had been used before him by Pascal and Barrow. [Cajori, History of Mathematical Notations, vol. 2, p. 2; Struik’s Source Book mistakenly has 26 October]

VFR Historical notes for the calculus classroom ,
In these same pages he will write examples of the integrals of x2 and x3,and then illustrate that a constant multiple may be taken outside the integral as shown in the image below.

On the left is Liebniz integral sign with a vincula in place of todays parentheses to show that he is integrating the quantity (a/b) l   Then the open bottomed box is Liebniz symbol for equality,then he shows the constant (a/b) multiplied by the integral of l .

At this point, Leibniz does not include the dx, as in $\int x^2 = \frac{x^3}{3}$  even though it seems his definition of an integral as a summation would seem to require it.  By 1686 he will adopt it, as he wrote $\int \rho dx$

1856 William Rowan Hamilton submits a paper on "New Roots of Unity" which will be the foundation of his Icosian Calculus, and the Icosagon game he used as a simplification of the operations of the group. The symbols of the icosian calculus can be equated to moves between vertices on a dodecahedron. Hamilton’s work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. *Wik
The game set shown below included numbered pegs that could track your path around the twenty vertices of the dodecahedron

1878 Patent issued for Odhner calculating machine. *VFR Willigot T. Odhner was granted a patent for a calculating machine that performed multiplications by repeated additions. The patent, a modified and compact version of Gottfried von Leibniz stepped wheel, was acquired and embodied in Brunsviga calculators that sold into 1950s.*CHM

1929 "Black Tuesday", the great USA stock market crash. About 16 million shares were traded, and the Dow lost an additional 30 points, or 12%.. "Anyone who bought stocks in mid-1929 and held onto them saw most of his or her adult life pass by before getting back to even." Richard M. Salsman *Wik

1964 Asteroid "Lucifer" is discovered by astronomer Elizabeth Roemer. amhistorymuseum @amhistorymuseum Roemer was the winner of the 1946 Science Talent Search and is now Professor Emerita, Lunar and Planetary Laboratory, University of Arizona. *Smithsonian Institution Archives (Ok, it's pure trivia, but is she somehow related to Ole, who first measured the speed of light???)

1985 On October 29th, 1985, the 329th birthday of Edmond Halley, the British threw a big party in honor of the return of Halley's Comet. The Halley's Comet Royal Gala was held at Wembley Conference Centre, London. It was a combination Variety Show and "Who's Who" in British Society, hosted by Princess Anne of the British Royal Family. *Joseph M. Laufer, Halley's Comet Society, USA

In 1991, space probe Galileo become the first human object to fly past an asteroid, Gaspra, making its closest approach at a distance of 1,604 km, passing at a speed of 8 km/sec (5 mi/sec). The encounter provided much data, including 150 images, which showed Gaspra has numerous craters indicating it has suffered numerous collisions since its formation. Gaspra is about 20-km long and orbits the Sun in the main asteroid belt between Mars and Jupiter. Gaspra, asteroid 951, was discovered by Ukrainian astronomer Grigoriy N. Neujamin (1916) who named it after a Black Sea retreat. In the photograph (left), subtle color variations have been exaggerated by NASA to highlight changes in reflectivity, surface structure and composition. *TIS

1998, Nearly four decades after he became the first American to orbit Earth, John Glenn is relaunched into space. *@HISTORYmag

BIRTHS

1897 Edwin James George Pitman was born in Melbourne on 29 October 1897 and died at Kingston near Hobart on 21 July 1993.  In 1920 he completed the degree course and graduated B.A. (1921), B.Sc. (1922) and M.A. (1923). In the meantime he was appointed Acting Professor of Mathematics at Canterbury College, University of New Zealand (1922-23). He returned to Australia when appointed Tutor in Mathematics and Physics at Trinity and Ormond Colleges and Part-time Lecturer in Physics at the University of Melbourne (1924-25). In 1926 Pitman was appointed Professor of Mathematics at the University of Tasmania, a position he held until his retirement in 1962.
Pitman described himself as 'a mathematician who strayed into Statistics'; nevertheless, his contributions to statistical and probability theory were substantial.
Pitman was active in the formation of the Australian Mathematical Society in 1956. He also took an active part in the Summer Research Institutes organized by the Mathematical Society, and used them as a sounding board for his research on statistical inference.
He was a renowned member of the Statistical Society of Australia, attending its biennial conferences. In 1978 the Statistical society established the Pitman Medal.
Pitman presented the first systematic account of non-parametric inference and lectured extensively on the subject, both in Australia and in the United States. The kernel of the subject, as described by him, is 'Suppose that the sum of two samples A, B is the sample C. Then A, B are discordant if A is an unlikely sample from C.' Again, he writes, 'The approach to the subject, starting from the sample and working towards the population instead of the reverse, may be a bit of a novelty'; and later, 'the essential point of the method is that we do not have to worry about the populations which we do not know, but only about the sample values which we do know'.
The notes of the 'Lectures on Non-parametric Inference' given in the United States, though never published, have been widely circulated and have had a major impact on the development of the subject. Among the new concepts introduced in these Lectures are asymptotic power, efficacy, and asymptotic relative efficiency.
A major contribution to probability theory is his elegant treatment of the behavior of the characteristic function in the neighborhood of the origin, in three papers. This governs such properties as the existence of moments. There are also interesting properties of the Cauchy distribution, and of subexponential distributions.
On his death, on 21 July 1993, Edwin was buried at the Hobart Regional Cemetery in Kingston. He lives on in the memory of many of us who are grateful for his life and legacy.
*Evan J. Williams, Australian Academy of Science

1925 Klaus Friedrich Roth (29 Oct 1925, )German-born British mathematician who was awarded the Fields Medal in 1958. His major work has been in number theory, particularly the analytic theory of numbers. He solved in the famous Thue-Siegel problem (1955) concerning the approximation to algebraic numbers by rational numbers (for which he won the medal). Roth also proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turán of 1935).*TIS

DEATHS

1783 Jean le Rond D'Alembert (16 Nov 1717, 29 Oct 1783) was abandoned by his parents on the steps of Saint Jean le Rond, which was the baptistery of Notre-Dame, qv in Section 7-A-1. Foster parents were found and he was christened with the name of the saint. [Eves, vol. II, pp. 32 33. Okey, p. 297.] When he became famous, his mother attempted to reclaim him, but he rejected her. *VFR Known for his work in various fields of applied mathematics, in particular dynamics. In 1743 he published his Traité de dynamique (Treatise on Dynamics). The d'Alembert principle extends Newton's third law of motion, that Newton's law holds not only for fixed bodies but also for free moving bodies. D'Alembert also wrote on fluid dynamics, the theory of winds, the properties of vibrating strings and conducted experiments on the properties of sound . His most significant purely mathematical innovation was his invention and development of the theory of partial differential equations. He published eight volumes of mathematical studies (1761-80). He was editor of the mathematical and scientific articles for Denis Diderot's Encyclopédie.*TIS

1917 Giovanni Battista Guccia (21 Oct 1855 in Palermo, Italy - 29 Oct 1914 in Palermo, Italy) Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves. His results published in volume one of the Rendiconti del Circolo Matematico di Palermo were extended by Corrado Segre in 1888 and Castelnuovo in 1897. *SAU

1921 Konstantin Alekseevich Andreev (26 March 1848 in Moscow, Russia - 29 Oct 1921 Near Sevastopol, Crimea) Andreev is best known for his work on geometry, although he also made contributions to analysis. In the area of geometry he did major pieces of work on projective geometry. Let us note one particular piece of work for which he has not received the credit he deserves. Gram determinants were introduced by J P Gram in 1879 but Andreev invented them independently in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions. *SAU

1931 Gabriel Xavier Paul Koenigs (17 January 1858 Toulouse, France – 29 October 1931 Paris, France) was a French mathematician who worked on analysis and geometry. He was elected as Secretary General of the Executive Committee of the International Mathematical Union after the first world war, and used his position to exclude countries with whom France had been at war from the mathematical congresses.
He was awarded the Poncelet Prize for 1913.*Wik

1933 Paul Painlevé worked on differential equations. He served twice as prime-minister of France. *SAU

1951 Robert Aitken (31 Dec 1864, 29 Oct 1951) American astronomer who specialized in the study of double stars, of which he discovered more than 3,000. He worked at the Lick Observatory from 1895 to 1935, becoming director from 1930. Aitken made systematic surveys of binary stars, measuring their positions visually. His massive New General Catalogue of Double Stars within 120 degrees of the North Pole allowed orbit determinations which increased astronomers' knowledge of stellar masses. He also measured positions of comets and planetary satellites and computed orbits. He wrote an important book on binary stars, and he lectured and wrote widely for the public. *TIS

1993 Lipman Bers (May 22, 1914 – October 29, 1993) was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.*Wik

1993 Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem (Dilworth 1950) relating chains and antichains in partial orders; he was also the first to study antimatroids (Dilworth 1940). Dilworth advised 17 Ph.D. students and as of 2010 has 373 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist.*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, 28 October 2022

### On This Day in Math - October 28

"Big Fleas have little fleas upon their backs to bite 'em,

and little fleas have lesser fleas,

~Augustus De Morgan

The 301st day of the year; 301 is the sum of three consecutive primes starting at 97

$301 \equiv 1 Mod _b$ for every base,b, from 2 through 6   (Sixth grade version, if you divide 301 by any number 2 through 6, you get a remainder of 1)

301, like every odd number, is the difference of two consecutive squares, 151^2 - 150^2 .  It is also 25^2 - 18^2  (students should expand (x+7)^2 - x^2 to see why, and when this type of relation will next be useful.

EVENTS

1386 Opening of the University of Heidelberg. It is the oldest university in Germany and was the third university established in the Holy Roman Empire. *Wik

1462 Archbishop Adolph of Nassau captured the city of Maintz and allowed his soldiers to plunder the city. This forced Gutenberg and his printers to ﬂee, but rather than nipping printing in the bud, it forced its spread to Strasburg, Cologne, Basel, Augsburg, Ulm, Nuremberg, Subiaco, and by 1470, Paris. [G. H. Putnam, Books and Their Makers During the Middle Ages (1896),
p. 372]. *VFR

1636 Harvard College founded. The only mathematical master’s thesis in the U.S. before 1700 was at Harvard. This was in 1693 when the candidate took the affirmative position on “Is the quadrature of a circle possible?”. *VFR

1752  Euler writes to Mersenne to say that he only knows of seven perfect numbers, those of the form $(2^p -1)(2^{p-1})$ with p = 2, 3, 5, 7, 13, 17, and 19.  He also says he is uncertain whether $2^{31} - 1$ is prime (it is), and adds that if it has a factor , it will be of the form 64n+1. Numbers of the form $2^p -1|$ are called Mersenne primes, and as of Jan 2020, there were only 51 known.  *L E Dickson, History of the Theory of Numbers  *GIMPS

1752 Euler publishes a paper listing the 161 numbers less than 15,000 for which $n^2+1$ is a prime. He also listed eight numbers for which $n^4 + 1$ is a prime; {1, 2, 4, 6, 16, 20, 24, and 34}.
He had described to Goldbach as early as July 9, 1743 a manor by which numbers of this form might be divisible.   *L. E. Dickson, History of the Theory of Numbers

1886 The Statue of Liberty was dedicated on Bedloe’s Island in New York Harbor. The sculptur Bartholin was present. The statue had almost been moved to another city when there was not enough interest in New York to pay the cost of building the pedestal.  Joseph Pulitzer, publisher of the World, a New York newspaper, announced a drive to raise $100,000 (the equivalent of$2.3 million today). Pulitzer pledged to print the name of every contributor, no matter how small the amount given.The drive captured the imagination of New Yorkers, especially when Pulitzer began publishing the notes he received from contributors. "A young girl alone in the world" donated "60 cents, the result of self denial."  As the donations flooded in, the committee resumed work on the pedestal. After five months of daily calls to donate to the statue fund, on August 11, 1885, the World announced that $102,000 had been raised from 120,000 donors, and that 80 percent of the total had been received in sums of less than one dollar. *Wik 1899 Robert Goddard has a "Cherry Tree Dream" of space flight. He will forever remember this as his "anniversary day.: He became interested in space when he read H. G. Wells' science fiction classic The War of the Worlds when he was 16 years old. His dedication to pursuing space flight became fixed on October 19, 1899. The 17-year-old Goddard climbed a cherry tree to cut off dead limbs. He was transfixed by the sky, and his imagination grew. He later wrote: On this day I climbed a tall cherry tree at the back of the barn … and as I looked toward the fields at the east, I imagined how wonderful it would be to make some device which had even the possibility of ascending to Mars, and how it would look on a small scale, if sent up from the meadow at my feet. I have several photographs of the tree, taken since, with the little ladder I made to climb it, leaning against it. It seemed to me then that a weight whirling around a horizontal shaft, moving more rapidly above than below, could furnish lift by virtue of the greater centrifugal force at the top of the path. I was a different boy when I descended the tree from when I ascended. Existence at last seemed very purposive. For the rest of his life he observed October 19 as "Anniversary Day", a private commemoration of the day of his greatest inspiration. *Wik 1938 The Indianapolis Star newspaper carried a story of a new proof of the Pythagorean Theorem by Ann Condit, a Junior at Central High School in South Bend, Ind. Her proof is unique in that it used the midpoint of the hypotenuse is the origin of all auxiliary lines and triangles. In less than two years her proof would appear in Elisha Scott Loomis' 2nd edition of his The Pythagorean Proposition, expanded to 344 proofs from the 230 proofs in his first edition. *David Acheson, The Wonder Book of Geometry ----------------------------------------------------------------------------------------------------------------------------------- 1957 Only three weeks after Sputnik went into space, young Denis Cox in Victoria, Australia sent a design for a spaceship addressed, "TO A TOP SCIENTIST AT Woomera ROCKET RANGE South Australia." His design included locations for Australian Insignia, four Rolls Royce Engines, guided missiles, etc, but advised the scientists, "YOU PUT IN OTHER DETAILS". The letter can be seen here at the Letters of Note web site Edited by Shaun Usher. On September 24, 2009, an article on ABC Australia's web page indicated that "The Defence Science Technology Organisation is now finally organising a letter from rocket scientists in response to the letter." In 1965, the Gateway Arch (630' (190m) high) was completed in St. Louis, Missouri. This graceful sweeping tapered curve of stainless steel is the tallest memorial in the U.S. The architect of the catenary curve arch (correct children, it is NOT a parabola) was Eero Saarinen who won the design competition in 1947. It was constructed 1961-66 in the Jefferson National Expansion Memorial Park, established on the banks of the Mississippi River, on 21 Dec 1935, to commemorate the westward growth of the United States between 1803 and 1890. Cost for the$30 million national monument was shared by the federal government and the City of St. Louis. The memorial arch has an observation room at the top for visitors reached by trams running inside the legs of the arch.*TIS

BIRTHS
1703 Antoine Deparcieux (28 Oct 1703 in Clotet-de-Cessous, France - 2 Sept 1768 in Paris, France) was a French mathematician who is best known for an early work on annuities and mortality.*SAU

1804 Pierre François Verhulst (28 October 1804, Brussels, Belgium – 15 February 1849, Brussels, Belgium) was a mathematician and a doctor in number theory from the University of Ghent in 1825. Verhulst published in 1838 the equation:

$\frac{dN}{dt} = r N (1-\frac{N}{k})$

when N(t) represents number of individuals at time t, r the intrinsic growth rate and k is the carrying capacity, or the maximum number of individuals that the environment can support. In a paper published in 1845 he called the solution to this the logistic function, and the equation is now called the logistic equation. This model was rediscovered in 1920 by Raymond Pearl and Lowell Reed, who promoted its wide and indiscriminate use.*Wik

1845 Ulisse Dini (14 Nov 1845 in Pisa, Italy - 28 Oct 1918 in Pisa, Italy) Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet. He discovered a condition, now known as the Dini condition, ensuring the convergence of a Fourier series in terms of the convergence of a definite integral. As well as trigonometric series, Dini studied results on potential theory. *SAU

1880 Michele Cipolla (born 28 October 1880 in Palermo; died 7 September 1947 in Palermo) was an Italian mathematician, mainly specializing in number theory.
He was a professor of Algebraic Analysis at the University of Catania and, later, the University of Palermo. He developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the problem of binomial congruence.*Wik

1937 Dr. Marcian Edward (Ted) Hoff, Jr. was born October 28, 1937 at Rochester, New York. He received a BEE (1958) from Rensselear Polytechnic Institute in Troy, NY. During the summers away from college he worked for General Railway Signal Company in Rochester where he made developments that produced his first two patents. He attended Stanford as a National Science Foundation Fellow and received a MS (1959) and Ph.D. (1962) in electrical engineering. He joined Intel in 1962. In 1980, he was named the first Intel Fellow, the highest technical position in the company. He spent a brief time as VP for Technology with Atari in the early 1980s and is currently VP and Chief Technical Officer with Teklicon, Inc. Other honors include the Stuart Ballantine Medal from the Franklin Institute.*CHM

1955 Bill Gates, cofounder and CEO of Microsoft Corporation, was born. Gates developed a version of BASIC for the Altair 8800 while being a student at Harvard. With the success of BASIC, he and co-developer Paul Allen founded Microsoft, which delivered an operating system for the IBM PC, the Microsoft Word word processing program, the Window system software, and other programs. *CHM

DEATHS
1703 John Wallis (23 Nov 1616, 28 Oct 1703) British mathematician who introduced the infinity math symbol. Wallis was skilled in cryptography and decoded Royalist messages for the Parliamentarians during the Civil War. Subsequently, he was appointed to the Savilian Chair of geometry at Oxford in 1649, a position he held until his death more than 50 years later. Wallis was part of a group interested in natural and experimental science which became the Royal Society, so Wallis is a founder member of the Royal Society and one of its first Fellows. Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton. *TIS

1916 Cleveland Abbe (3 Dec 1838, 28 Oct 1916) U.S. astronomer and first meteorologist, born in New York City, the "father of the U.S. Weather Bureau," which was later renamed the National Weather Service. Abbe inaugurated a private weather reporting and warning service at Cincinnati. His weather reports or bulletins began to be issued on Sept. 1, 1869. The Weather Service of the United States was authorized by Congress on 9 Feb 1870, and placed under the direction of the Signal Service. Abbe was the only person in the country who was already experienced in drawing weather maps from telegraphic reports and forecasting from them. Naturally, he was offered an important position in this new service which he accepted, beginning 3 Jan 1871, and was often the official forecaster of the weather.*TIS

1918 Edward Bouchet (15 Sept 1852, New Haven, Conn – 28 Oct 1918, New Haven, Conn) was the first African-American to earn a Ph.D. in Physics from an American university and the first African-American to graduate from Yale University in 1874. He completed his dissertation in Yale's Ph.D. program in 1876 becoming the first African-American to receive a Ph.D. (in any subject). His area of study was Physics. Bouchet was also the first African-American to be elected to Phi Beta Kappa.
Bouchet was also among 20 Americans (of any race) to receive a Ph.D. in physics and was the sixth to earn a Ph. D. in physics from Yale.
When Bouchet was born there were only three schools in New Haven open to black children. Bouchet was enrolled in the Artisan Street Colored School with only one teacher, who nurtured Bouchet's academic abilities. He attended the New Haven High School from 1866–1868 and then Hopkins School from 1868-1870 where he was named valedictorian (after graduating first in his class).
Bouchet was unable to find a university teaching position after college, most likely due racial discrimination. Bouchet moved to Philadelphia in 1876 and took a position at the Institute for Colored Youth (ICY). He taught physics and chemistry at the ICY for 26 years. The ICY was later renamed Cheyney University. He resigned in 1902 at the height of the W. E. B. Du Bois-Booker T. Washington controversy over the need for an industrial vs. collegiate education for blacks.
Bouchet spent the next 14 years holding a variety of jobs around the country. Between 1905 and 1908, Bouchet was director of academics at St. Paul's Normal and Industrial School in Lawrenceville, Virginia (presently, St. Paul's College). He was then principal and teacher at Lincoln High School in Gallipolis, Ohio from 1908 to 1913. He joined the faculty of Bishop College in Marshall, Texas in 1913. Illness finally forced him to retire in 1916 and he moved back to New Haven. He died there, in his childhood home, in 1918, at age of 66. He had never married and had no children.*Wik

1924 John Backus (3 Dec 1924, 28 Oct 1988) American computer scientist who invented the FORTRAN (FORmula TRANslation) programming language in the mid 1950s. He had previously developed an assembly language for IBM's 701 computer when he suggested the development of a compiler and higher level language for the IBM 704. As the first high-level computer programming language, FORTRAN was able to convert standard mathematical formulas and expressions into the binary code used by computers. Thus a non-specialist could write a program in familiar words and symbols, and different computers could use programs generated in the same language. This paved the way for other computer languages such as COBOL, ALGOL and BASIC. *TIS

1965 Luther Pfahler Eisenhart (13 January 1876 – 28 October 1965) was an American mathematician, best known today for his contributions to semi-Riemannian geometry.*Wik

1986 Irving Reiner (February 8, 1924, Brooklyn, New York – October 28, 1986) mathematician at the University of Illinois who worked on representation theory. He solved the problem of finding which abelian groups have a finite number of indecomposable modules. His book with Charles W. Curtis, (Curtis & Reiner 1962), was for many years the standard text on representation theory.*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

## Thursday, 27 October 2022

### Problems From the Land Down Under, and some more

Looking through the Gazette of the Australian Mathematical Society, and found their puzzle corner (July 2009 , so the exponents in the first problem are explained) ... really nice problems. I think I have this one, but I didn't prove it....

Digital deduction The numbers 2^2009 and 5^2009 are written on a piece of paper in decimal notation. How many digits are on this piece of paper? And this one has me puzzled (which is why they call them puzzles, I guess)..

Piles of stones There are 25 stones sitting in a pile next to a blackboard. You are allowed to take a pile and divide it into two smaller piles of size a and b, but then you must write the number a×b on the blackboard. You continue to do this until you are left with 25 piles, each with one stone. What is the maximum possible sum of the numbers written on the blackboard? Anyone know how to a) prove the first, or b) solve the second...

Do let me know....mostly down to chewing my pencil tips now....

Spoiler (I think) x x x x x x x
OK, for number one I went back to that old Polya-ism, "If there is a problem you can't solve, find a smaller problem you can solve."  Instead of 2010 I put in 1.  Well 2^1 has one digit and 5^1 has 1 digit so the answer is 2.  Repeating this with more numbers it seemed the solution was always n+1 digits for any exponent n.
Sue VanHattum gave a nice approach using base ten logarithms,
digits in 2^n = ceiling(log(2^n))
digits in 5^n = ceiling(log(5^n))
adding gives n+1, so we have n+1 digits.

OK, I think the total for the 25 stones will always be 300... I tried it about three different ways and they all came out the same... hmmmm... In fact, if we look at some smaller numbers for a guide, it seems that for any n, the sum of the products by this process will lead to $\dbinom{n}{2}$... now why is that? Anyone, Anyone??? Bueller?
Well I was right on that one, it seems, but the real understanding came when master problem solver, Joshua Zucker, explained, "The second problem I have seen many times in books as a strong induction exercise, but ... WHY does it come out the triangular numbers?
Well, the triangular numbers are the solution to the handshake problem.
When all the pebbles are in one pile, let them all shake hands.
At each splitting step, the number of points you score is equal to the number of handshakes you destroy.
At the end, you have all the pebbles in their own individual pile, so there are no more handshakes possible - they have all been destroyed.
Hence the score is equal to the initial number of handshakes.

Just for a kick, I picked out a couple of newer ones for you to try.  Enjoy and share your solutions:

For the geometry lovers, try this one.

In a regular nonagon, prove that the length difference between the longest diagonal and the shortest diagonal is equal to the side length. In other words, prove c−b = a in the diagram below.
 *Australian Mathematical Soc. Gazette
And here is one for that I think is an excellent problem for younger students to intuit a wonderful mix of problems solving ideas.  "Let S be a set of 10 distinct positive integers no more than 100. Prove that S contains two disjoint non-empty subsets which have the same sum."

I will come back in awhile and address possible approaches to each, (If I can solve them).


text:
# Custom implementation to provide an immutable range of a vector without copying it

Edit: A new and improved version can be found in this question, based on the feedback!

I needed to pass a const reference to part of a std::vector without copy.

Some research, particularly this SO question, suggested that it is not something supported by default, so I created my own code for it:

#include <iterator>
#include <vector>

template <typename T>
class ConstVectorSubrange{
public:
ConstVectorSubrange(typename std::vector<T>::const_iterator start_, uint32 size_)
: start(start_)
, range_size(size_)
{ }

ConstVectorSubrange(std::initializer_list<typename std::vector<T>::const_iterator> list)
: start(*list.begin())
, range_size(std::distance(*list.begin(), *(list.end()-1)))
{ }

const T& operator[](std::size_t index) const{
assert(index < range_size);
return *(start + index);
}
const T& back(void) const{
return *(cend()-1);
}
std::size_t size(void) const{
return range_size;
}
const typename std::vector<T>::const_iterator cbegin(void) const{
return start;
}
const typename std::vector<T>::const_iterator cend(void) const{
return (start + range_size);
}

private:
const typename std::vector<T>::const_iterator start;
const uint32 range_size;
};

An example of using this would be a member function where not the whole array is visible, although copies are to be avoided:

#include <vector>

class BigData{
std::vector<int> big_array = vector<int>(10000);
public:

ConstVectorSubrange get_last_five_thousand_elements(){
return{(big_array.cend()-5000), big_array.cend()};
}

};

How can this code be improved?

Especially considering the iterators might be invalidated after the creation of this object, is there a way to make it safer?

• Some C++ programmers will find the void arguments offensive. Nov 22 '21 at 5:24
• Yes, you are right, and I think I'll remove them in the next version! Do you know why that is by the way? Nov 22 '21 at 6:12
• They are not needed in C++. If they appeared in a header file that may be read by a C compiler, I think it would be fine. But they appear in the middle of templated code. No C compiler is going to read that. Nov 22 '21 at 6:18

# You are implementing std::span

What you are trying to implement is basically C++20's std::span. It might be worthwhile to look at how it is implemented, and copy its interface if possible, so if you ever want to change the codebase using your class to use std::span, it will be easy.

I'll also use std::span as a reference and point out differences between it and your class below.

# Use std::size_t for sizes consistently

Use std::size_t instead of uint32 for range_size. It is the proper type to store sizes. Another possible alternative is to use std::vector<T>::size_type, so you match exactly what std::vector uses to store sizes.

# Don't use an initializer list for begin/end pairs

The constructor that takes an initializer list does not make sense. If you want a constructor that takes a begin and end iterator pair, just make that explicit:

ConstVectorSubrange(typename std::vector<T>::const_iterator start, typename std::vector<T>::const_iterator end)
: start(start)
, range_size(end - start)
{ }

With an initializer list, you can pass any number of iterators to the constructor, including an empty initializer list.

# Add begin() and end() member functions

You only implemented cbegin() and cend(), but this is not enough to make range-for work. Add begin() and end() member functions. Note that these should also be const functions of course.

You can even consider removing cbegin() and cend(); they don't offer any advantage over begin() and end() for your class. Note that std::span also doesn't implement cbegin() and cend().

# Be consistent manipulating iterators

You used std::distance() to calculate the distance between the start and end iterator passed to the second constructor, but then used arithmetic operators to advance the iterators to a given index or to the end. I would either use arithmetic operators everywhere, or use std::distance and std::next everywhere to manipulate the iterators. The latter is preferred if you want to make your class more generic and have it work for other container types as well.

# Missing member functions

Intertestingly there is a back() function, but no front()? Also consider implementing all the member functions of std::vector where possible, like at(), data(), empty() and rbegin()/rend(). This ensures your subrange can be used as a drop-in replacement for a std::vector as much as possible.

It might also be interesting to add the extra member functions std::span adds, like first(), last() and subspan().

# Making T deducible

A big problem with your class is that the compiler isn't able to deduce T from the arguments passed to the constructors, due to the dependent type names being used. There are some ways to fix this. One is to provide deduction guides, but this requires C++17. Another way is not to make your class templated on the value type T, but rather on the iterator type. This will also greatly simplify your code, and at the same time make it work for other containers besides std::vector:

template<typename Iterator>
class ConstSubrange {
public:
using T = std::remove_reference_t<std::iter_reference_t<Iterator>>;

ConstSubrange(Iterator start, std::size_t size)
: start(start), range_size(size) {}

ConstSubrange(Iterator start, Iterator end)
: start(start), size(std::distance(start, end)) {}

const T& operator[](std::size_t index) const {
return *std::next(start, index);
}

const Iterator begin(void) const {
return start;
}

...
private:
const Iterator start;
const std::size_t range_size;
};

# Safety

Especially considering the iterators might be invalidated after the creation of this object, is there a way to make it safer?

Unfortunately, it's not possible to make a class like yours that provides a "view" on another container and have it protect you against invalidation; C++ just doesn't keep track of dependent lifetimes. Note that even a regular iterator doesn't provide this safety. Either you live with it, or make expensive copies of subranges, or use a programming language that uses garbage collection or has a borrow checker.

In a way your class is also too strict; it's perfectly fine to have it store regular iterators instead of const_iterators; this will allow you to modify a subrange of a vector.

• Initializer list is there or a kind of syntactic sugar, aiming to cover {vec.begin(),vec.end()}; Is there anything else I can do it with that? I get that it doesn't make any sense, maybe I am missing something. Nov 21 '21 at 17:20
• should I implement .begin() and .end() despite the class is supposed to have read-only access to the vector span it was constructed with? IN my case the strictness is for a reason, but for general usage, I'd understand that implementing begin() and end() would be better.. Nov 21 '21 at 17:22
• instead of std::advance, I think std::next is appropriate, in case you were referring to the back() function. I checked this question on SO. Nov 21 '21 at 19:04
• My suggestion to just use two iterator parameters also allows you to use {vec.begin(), vec.end()}. You should always implement begin() and end(), and make those functions const to ensure they also work with a const span, and having them return const iterators is also perfectly fine. The problem with deduction is that vector<T>::const_iterator is not a type but a type alias; the actual type of the iterator you pass in is something different (for libstdc++, it's a __gnu_cxx::__normal_iterator<T*, std::vector<T, std::allocator<T>>>). Nov 21 '21 at 22:30
• Template argument deduction is not bothered by aliases; it will deduce what you call a std::string just fine even though it's really a std::basic_string< ... >. The issue with not being able to deduce is because of a dependent type and it doesn't know the relationship between the Iterator type and T which is used as a template argument for the container and iterator. That's what deduction guides will tell it; e.g. "look at the Iterator's element type in this case". Nov 24 '21 at 17:52

You are asking how you can guard against invalid iterators; I assume you are most concerned about iterators that have become invalid after the vector had to re-allocate when it outgrew its capacity.

One solution is to store an index instead of an iterator; as long as the vector doesn't shrink the range will be valid, and its validity can be trivially checked by comparing to the vector's size.

This is only doable as long as your range class targets exclusively "indexable" containers like vectors (and not maps etc.). Downside: Because you only have an index you'll then also need an additional constructor parameter, a reference of the vector.

format: plain


text:
## three alarm clocks ring at intrevals of6 , 9, 15, minutes respectively .if they start ringing together,after what time will they next ring

Question

three alarm clocks ring at intrevals of6 , 9, 15, minutes respectively .if they start ringing together,after what time will they next ring together.. correct answer will be given thanks and made brainliest

in progress 0
2 months 2021-11-06T17:12:59+00:00 2 Answers 0 views 0

1. Step-by-step explanation:

LCM of 6,9,15 is 90

then we should divide 90 by 60 then we get 1.5 hours

So,by adding 1.5 to given time


text:
# 7.5 Strengths of ionic and covalent bonds  (Page 5/8)

 Page 5 / 8

## Key concepts and summary

The strength of a covalent bond is measured by its bond dissociation energy, that is, the amount of energy required to break that particular bond in a mole of molecules. Multiple bonds are stronger than single bonds between the same atoms. The enthalpy of a reaction can be estimated based on the energy input required to break bonds and the energy released when new bonds are formed. For ionic bonds, the lattice energy is the energy required to separate one mole of a compound into its gas phase ions. Lattice energy increases for ions with higher charges and shorter distances between ions. Lattice energies are often calculated using the Born-Haber cycle, a thermochemical cycle including all of the energetic steps involved in converting elements into an ionic compound.

## Key equations

• Bond energy for a diatomic molecule: $\text{XY}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{X}\left(g\right)+\text{Y}\left(g\right)\phantom{\rule{3em}{0ex}}{\text{D}}_{\text{X–Y}}=\text{Δ}H\text{°}$
• Enthalpy change: Δ H = ƩD bonds broken – ƩD bonds formed
• Lattice energy for a solid MX: $\text{MX}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{M}}^{n\text{+}}\left(g\right)+{\text{X}}^{n\text{−}}\left(g\right)\phantom{\rule{3em}{0ex}}\text{Δ}{H}_{\text{lattice}}$
• Lattice energy for an ionic crystal: $\text{Δ}{H}_{\text{lattice}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{C}\left({\text{Z}}^{\text{+}}\right)\left({\text{Z}}^{\text{−}}\right)}{{\text{R}}_{\text{o}}}$

## Chemistry end of chapter exercises

Which bond in each of the following pairs of bonds is the strongest?

(a) C–C or $\text{C}=\text{C}$

(b) C–N or $\text{C}\equiv \text{N}$

(c) $\text{C}\equiv \text{O}$ or $\text{C}=\text{O}$

(d) H–F or H–Cl

(e) C–H or O–H

(f) C–N or C–O

Using the bond energies in [link] , determine the approximate enthalpy change for each of the following reactions:

(a) ${\text{H}}_{2}\left(g\right)+{\text{Br}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}2\text{HBr}\left(g\right)$

(b) ${\text{CH}}_{4}\left(g\right)+{\text{I}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{CH}}_{3}\text{I}\left(g\right)+\text{HI}\left(g\right)$

(c) ${\text{C}}_{2}{\text{H}}_{4}\left(g\right)+3{\text{O}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}2{\text{CO}}_{2}\left(g\right)+2{\text{H}}_{2}\text{O}\left(g\right)$

(a) −114 kJ;
(b) 30 kJ;
(c) −1055 kJ

Using the bond energies in [link] , determine the approximate enthalpy change for each of the following reactions:

(a) ${\text{Cl}}_{2}\left(g\right)+3{\text{F}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}2{\text{ClF}}_{3}\left(g\right)$

(b) ${\text{H}}_{2}\text{C}={\text{CH}}_{2}\left(g\right)+{\text{H}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{H}}_{3}{\text{CCH}}_{3}\left(g\right)$

(c) $2{\text{C}}_{2}{\text{H}}_{6}\left(g\right)+7{\text{O}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}4{\text{CO}}_{2}\left(g\right)+6{\text{H}}_{2}\text{O}\left(g\right)$

When a molecule can form two different structures, the structure with the stronger bonds is usually the more stable form. Use bond energies to predict the correct structure of the hydroxylamine molecule:

The greater bond energy is in the figure on the left. It is the more stable form.

How does the bond energy of HCl( g ) differ from the standard enthalpy of formation of HCl( g )?

Using the standard enthalpy of formation data in Appendix G , show how the standard enthalpy of formation of HCl( g ) can be used to determine the bond energy.

$\begin{array}{}\\ \text{HCl}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}\left(g\right)+\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\phantom{\rule{0.2em}{0ex}}{\text{Cl}}_{2}\left(g\right)\phantom{\rule{3em}{0ex}}\text{Δ}{H}_{1}^{°}=\text{−Δ}{H}_{\text{f}\left[\text{HCl}\left(g\right)\right]}^{°}\\ \frac{1}{2}\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{H}\left(g\right)\phantom{\rule{8.5em}{0ex}}\text{Δ}{H}_{2}^{°}=\text{Δ}{H}_{\text{f}\left[\text{H}\left(g\right)\right]}^{°}\\ \underset{¯}{\frac{1}{2}\phantom{\rule{0.2em}{0ex}}{\text{Cl}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{Cl}\left(g\right)\phantom{\rule{8em}{0ex}}\text{Δ}{H}_{3}^{°}=\text{Δ}{H}_{\text{f}\left[\text{Cl}\left(g\right)\right]}^{°}}\\ \text{HCl}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{H}\left(g\right)+\text{Cl}\left(g\right)\phantom{\rule{5.5em}{0ex}}\text{Δ}{H}_{298}^{°}=\text{Δ}{H}_{1}^{°}+\text{Δ}{H}_{2}^{°}+\text{Δ}{H}_{3}^{°}\end{array}$
$\begin{array}{ll}{D}_{\text{HCl}}=\text{Δ}{H}_{298}^{°}& =\text{Δ}{H}_{\text{f}\left[\text{HCl}\left(g\right)\right]}^{°}+\text{Δ}{H}_{\text{f}\left[\text{H}\left(g\right)\right]}^{°}+\text{Δ}{H}_{\text{f}\left[\text{Cl}\left(g\right)\right]}^{°}\\ \\ \hfill & =-\left(-92.307\phantom{\rule{0.2em}{0ex}}\text{kJ}\right)+217.97\phantom{\rule{0.2em}{0ex}}\text{kJ}+121.3\phantom{\rule{0.2em}{0ex}}\text{kJ}\\ & =431.6\phantom{\rule{0.2em}{0ex}}\text{kJ}\end{array}$

Using the standard enthalpy of formation data in Appendix G , calculate the bond energy of the carbon-sulfur double bond in CS 2 .

Using the standard enthalpy of formation data in Appendix G , determine which bond is stronger: the S–F bond in SF 4 ( g ) or in SF 6 ( g )?

The S–F bond in SF 4 is stronger.

Using the standard enthalpy of formation data in Appendix G , determine which bond is stronger: the P–Cl bond in PCl 3 ( g ) or in PCl 5 ( g )?

Complete the following Lewis structure by adding bonds (not atoms), and then indicate the longest bond:

The C–C single bonds are longest.

Use the bond energy to calculate an approximate value of Δ H for the following reaction. Which is the more stable form of FNO 2 ?

Use principles of atomic structure to answer each of the following: This question is taken from the Chemistry Advanced Placement Examination and is used with the permission of the Educational Testing Service.

(a) The radius of the Ca atom is 197 pm; the radius of the Ca 2+ ion is 99 pm. Account for the difference.

(b) The lattice energy of CaO( s ) is –3460 kJ/mol; the lattice energy of K 2 O is –2240 kJ/mol. Account for the difference.

(c) Given these ionization values, explain the difference between Ca and K with regard to their first and second ionization energies.

Element First Ionization Energy (kJ/mol) Second Ionization Energy (kJ/mol)
K 419 3050
Ca 590 1140

(d) The first ionization energy of Mg is 738 kJ/mol and that of Al is 578 kJ/mol. Account for this difference.

(a) When two electrons are removed from the valence shell, the Ca radius loses the outermost energy level and reverts to the lower n = 3 level, which is much smaller in radius. (b) The +2 charge on calcium pulls the oxygen much closer compared with K, thereby increasing the lattice energy relative to a less charged ion. (c) Removal of the 4 s electron in Ca requires more energy than removal of the 4 s electron in K because of the stronger attraction of the nucleus and the extra energy required to break the pairing of the electrons. The second ionization energy for K requires that an electron be removed from a lower energy level, where the attraction is much stronger from the nucleus for the electron. In addition, energy is required to unpair two electrons in a full orbital. For Ca, the second ionization potential requires removing only a lone electron in the exposed outer energy level. (d) In Al, the removed electron is relatively unprotected and unpaired in a p orbital. The higher energy for Mg mainly reflects the unpairing of the 2 s electron.

The lattice energy of LiF is 1023 kJ/mol, and the Li–F distance is 200.8 pm. NaF crystallizes in the same structure as LiF but with a Na–F distance of 231 pm. Which of the following values most closely approximates the lattice energy of NaF: 510, 890, 1023, 1175, or 4090 kJ/mol? Explain your choice.

For which of the following substances is the least energy required to convert one mole of the solid into separate ions?

(a) MgO

(b) SrO

(c) KF

(d) CsF

(e) MgF 2

(d)

The reaction of a metal, M, with a halogen, X 2 , proceeds by an exothermic reaction as indicated by this equation: $\text{M}\left(s\right)+{\text{X}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{MX}}_{2}\left(s\right).$ For each of the following, indicate which option will make the reaction more exothermic. Explain your answers.

(b) a high ionization energy vs. a low ionization energy for M

(c) an increasing bond energy for the halogen

(d) a decreasing electron affinity for the halogen

(e) an increasing size of the anion formed by the halogen

The lattice energy of LiF is 1023 kJ/mol, and the Li–F distance is 201 pm. MgO crystallizes in the same structure as LiF but with a Mg–O distance of 205 pm. Which of the following values most closely approximates the lattice energy of MgO: 256 kJ/mol, 512 kJ/mol, 1023 kJ/mol, 2046 kJ/mol, or 4008 kJ/mol? Explain your choice.

4008 kJ/mol; both ions in MgO have twice the charge of the ions in LiF; the bond length is very similar and both have the same structure; a quadrupling of the energy is expected based on the equation for lattice energy

Which compound in each of the following pairs has the larger lattice energy? Note: Mg 2+ and Li + have similar radii; O 2– and F have similar radii. Explain your choices.

(a) MgO or MgSe

(b) LiF or MgO

(c) Li 2 O or LiCl

(d) Li 2 Se or MgO

Which compound in each of the following pairs has the larger lattice energy? Note: Ba 2+ and

(a) K 2 O or Na 2 O

(b) K 2 S or BaS

(c) KCl or BaS

(d) BaS or BaCl 2

(a) Na 2 O; Na + has a smaller radius than K + ; (b) BaS; Ba has a larger charge than K; (c) BaS; Ba and S have larger charges; (d) BaS; S has a larger charge

Which of the following compounds requires the most energy to convert one mole of the solid into separate ions?

(a) MgO

(b) SrO

(c) KF

(d) CsF

(e) MgF 2

Which of the following compounds requires the most energy to convert one mole of the solid into separate ions?

(a) K 2 S

(b) K 2 O

(c) CaS

(d) Cs 2 S

(e) CaO

(e)

The lattice energy of KF is 794 kJ/mol, and the interionic distance is 269 pm. The Na–F

distance in NaF, which has the same structure as KF, is 231 pm. Which of the following values is the closest approximation of the lattice energy of NaF: 682 kJ/mol, 794 kJ/mol, 924 kJ/mol, 1588 kJ/mol, or 3175 kJ/mol? Explain your answer.

what is matter
Matter is anything that occupied space
Rasidi
matter is anything that has mass and occupies space
Johnson
a plot of the atomic radius against atomic number shows a periodic variation with group 1elements occupying the peaks and group 4 the troughs.plot the graph
what is the periodic table all about
yeah
samuel
it's about predicting the different types of matter around us and organizing them in an elegant way
William
what is boyle law
the law states that at a constant absolute temperature the volume of a fixed of a gas is inversely proportional to the pressure of the gas...
Johnson
write short not on paper chromAtogphy
I am sorry, I don't understand the question. Are you referring to photospectometry, a machine that uses lightwaves to determine the chemical composition of a substance? Instead of the "coffee filter and solvent" method which is more crude, but simple enough to understand basic concepts.
Melissa
what is esterification?
Esterification is the general name for a chemical reaction in which two reactants (typically an alcohol and an acid) form an ester(-COOR) as the reaction product....
Johnson
pls i dont know chemistry wat shall i do?
Chancha
Please...what do you mean by you don't know chemistry..can you explain further
Johnson
I am afraid you may end up injuring yourself working with these chemical processes with limited lab experience. Please be extremely careful. Research chem analysis, precedure and equipment.
Melissa
Pls can I be enlighten about boys law on chemistry
samuel
how we Learn full periodic table
how
Imran
Hello guys am new here my name is Angel from Nigeria how are u guys doing?
Gud am from Nigeria too my name is sav
Michael
explain the atomic orbital
2nacl➡2na+cl2 2.58.5g 22.4l
what is neclear energy
nuclear energy is the energy released during nuclear fission or fusion.especially when used to generate electricity
Inusah
How many processes to break down CARBON DIOXIDE into carbon and oxygen
biology is my best subject
o really
Ajila
biology quick chemistry ass
You're in the wrong place lmao
Zeus
hi Guy
Charles
hello
daniel
hy
Kareem
How are you guys
daniel
fine and u
Kareem
were are you guys from?
l'm from Nigeria
Kareem
Brazil
Kareem
like what
Am from malawi
Hele
I'm from Nigeria, my name is Vincent. what is the relationship between volume an temperature?
Vincent
who is the father of chemistry?
Vincent
hey guys I'm new to the conservation, my name is Vincent from Nigeria, how are you guys doing?
Vincent
fine
Kareem
the relationship between volume and temperature is. volume is directly proportional to temperature
Inusah
Antoine lavoisier
Ujunwa
good evening everyone my name is Maryann
Ujunwa
an acid is any substance which when dissolved in water produces hydrogen ion as its only positive ion
its just me or you guys also hate chemistry?
I partially hate it
Oseni
there's some parts that I actually like to study
sometimes
Oseni
physics have this too
good evening everyone my name is Maryann
Ujunwa
Why acid is sour in taste. Please don't say naturally because there are a scientific reason in all natural thing
since it has a negatively charge nitrogen ions
Elvis
sorry, positively hydrogen ions, which has a pH less than 7
Elvis
Corrct, as the H+ ion concentration increases acidity increases.
Aakash
because it's pH value is smaller than 7.
Raju
what is molecules
samuel
what is acid
Faree