Math geeks rejoice: Facebook Messenger…

Math geeks rejoice: Facebook Messenger lets you write basic mathematical formulae in LaTex

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Math is AMAZING and we…

Math is AMAZING and we have to start treating it that way

Eugenia Cheng is the scientist in residence at the School of the Art Institute of Chicago, and author, whose latest book is “Beyond Infinity.”

Read her opinion on why math is in need of a makeover.

EUGENIA CHENG:

I would like you to meet a friend of mine. He’s really useful. Wait. That doesn’t make him sound very interesting, does it? Or fun.

Wouldn’t it be better to say, hi, I would like you to meet a friend of mine, she’s amazing, she’s brilliant?

We’d never introduce a friend by saying they’re useful. So, why are we doing that to math? Why do we keep going on about how important it is for everyone to learn math because it’s useful? Has that ever got a young person interested in anything?

I think math is fascinating and fun. Otherwise, I wouldn’t be a mathematician. Some math began life without any sign of practicality. Like, babies, they’re not exactly useful.

For example, Internet cryptography. It means we can do online shopping, online banking, and send e-mails. This comes from some number theory that existed just for its own sake 300 years ago. Most of engineering, medicine, lab science, weather forecasts and technology depends on calculus.

Calculus depends on irrational numbers that the Egyptians started wondering about thousands of years ago. The icosahedron is a satisfyingly symmetrical shape that was dreamt up by ancient Greek mathematicians.

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The Horrible Ways Mathematicians Have Died

A list of mathematicians who died under unfortunate or unfitting circumstances (note that these entries have been sorted by age at death):

Évariste Galois, 20 (1811-1832)
Presumably the youngest to qualify for inclusion on this list, Galois died at a meagre 20 years. He was shot in the stomach, and a full day later died, in hospital. The circumstances surrounding his death are not entirely known, only that he was killed in a duel. Speculation tends to indicate that the duel was motivated either by a matter related to his involvement with the radical Républicain movement, or by conflict arising from a romantic entanglement. The duel occurred only a month after his release from a six-month incarceration stemming from his disruptive political activities. Having predicted defeat, Galois jotted down what would become his mathematical legacy, on the eve of the duel. Sadly, his last words to his dear brother, Alfred:
“Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans!”
(Don’t cry, Alfred. I need all my courage to die at twenty.’)

Vladimir Markov, 25 (1871-1897)
He died of tuberculosis at only 25, despite having already established a reputation as a formidable mathematician. He was known for proving the Markov brothers’ inequality with his older brother Andrei Markov.

René Gâteaux, 25 (1889-1914)
A promising analyst, Gâteaux was killed during World War I. Gâteaux was born in the same town as Abraham de Moivre, and he was well regarded by his peers at the École Normale Supéerieure. Gâteaux was a fairly well-decorated soldier and was recalled for service in the war. He was killed during a retreat early in the conflict in France. Jacques Hadamard admired Gâteaux and worked to have the Prix Francoeur awarded to him posthumously.

Pavel Urysohn, 26 (1898-1924)
Urysohn is best known for his contributions to topology, especially in defining and expanding the definitions of dimension and compactness. He studied at Moscown University and became a professor there, but drowned while swimming in the sea, on vacation in France. He was accompanied by his colleague and friend, Pavel Alexandrov. It is believed that Alexandrov was incredibly distressed by this tragedy, and deeply regretted his inability to save his friend.

Niels Abel, 26 (1802-1829)
Plagued by poverty and a lack of renown, Abel and his work went unrecognized during his lifetime. He spent time in Paris hoping to gain recognition and publish his work, but was unable to afford adequate means to sustain his health. In addition to being underfed, Abel contracted pneumonia. His pneumonia worsened on a trip to visit his fiancée for Christmas. He soon died, only two days before a letter arrived indicating that a friend had managed to find secure him a place as a professor in Paris. He never saw his work take root, nor did he ever secure a paying job as a mathematician, nor did he have opportunity to marry his fiancée.

Frank Ramsey, 26 (1903-1930)
Ramsey is known for his work in mathematics, specifically combinatorics and logic/foundations, but is also remembered as a gifted philosopher and economist. Ramsey suffered from lifelong liver problems, and was often unable to focus on work for more than a few hours a day. In spite of this, he gained renown as a promising young philosopher and mathematician, until a severe attack of jaundice hospitalized him in 1930. He died during an operation meant to alleviate the problem.

Srinivasa Ramanujan, 32 (1887-1920)
The story of Ramanujan is well known among mathematicians, if not in general. Described as a prodigy, savant, genius, etc., Ramanujan taught himself mathematics as a youth and began to devise results in analytical number theory and other areas of mathematics in isolation. He was quite poor and unable to afford school, and his exclusive devotion to mathematics precluded him from scholarship funding. He spent much of his life seriously ill, and spent a fair amount of time unable to secure any position as a scholar or mathematician. Eventually, he came to England to work with G.H. Hardy. Sadly, his long-term illness continued, and he succumbed to a combination of malnutrition and a parasitic liver infection.

Maryam Mirzakhani, 40 (1977-2017)
Mirzakhani was born in Tehran, Iran, and – by her own estimation – was fortunate to come of age after the Iran-Iraq war when the political, social and economic environment had stabilized enough that she could focus on her studies. She dreamed of becoming a writer, but mathematics eventually swept her away. After earning her doctorate at Harvard, Mirzakhani accepted a position as assistant professor at Princeton University and as a research fellow at the Clay Mathematics Institute before joining the Stanford faculty. She was the first and to-date only female winner of the Fields Medal since its inception in 1936.
She had been battling breast cancer since 2013; the disease spread to her liver and bones in 2016. Mirzakhani was 40 years old. She died at Stanford Hospital. Mirzakhani is survived by her husband, Jan Vondrák, and a daughter, Anahita, as well as her parents, sister and two brothers.

Alan Turing, 41 (1912-1954)
Turing is famed for his work in code-breaking and in the foundations of computer science. Turing is also remembered as a victim of anti-gay persecution in Britain. In 1952, a sexual partner of his plotted to blackmail him (as homosexual relations were illegal), and Turing was eventually forced to go to the police. He was tried and convicted of “gross indecency.” Instead of prison, Turing opted for estrogen injections (intended to reduce libido). However, times were stressful due to anti-gay persecution and fear of Soviet espionage. Turing committed suicide in 1954, by eating a cyanide-laced apple, although the circumstances of his death were ambiguous enough (deliberately) so that his mother could maintain, for her own sake, that it was an accident.

Stanisław Saks, 44 (1897-1942)
Saks was an important figure in the development of the theory of integration and measure, and a frequent visitor of the ” Scottish Café.” He is remembered as a great mathematician, and as a an inspiring person who helped shape the follow generation of young mathematicians. Saks was a member of the Polish army (or undeground, it is unclear which he was in, or whether he was in both). The army retreated to Lvov, home of the famous Scottish Café, but in 1941, Lvov was also invaded. Saks fled Lvov, back to Warsaw, where he may have continued to work in Polish underground. There, in 1942, he was captured and executed by the Gestapo.

Dénes Kőnig, 60 (1884-1944)
The Nazi occupation of Hungary led to a series of anti-semitic atrocities. Kőnig was sympathetic to the plight of his persecuted colleagues, and worked to ease their suffering. The atrocities were their worst when the Hungarian National Socialist Party seized political power, and this led Kőnig to commit suicide days after the coup d’état.

Dmitri Egorov, 61 (1869-1931)
Egorov made important contributions in the areas of analysis, differential geometry, and integral equations, including a fundamental result named for him in real analysis. Luzin was Egorov’s first student, and was one member of a school that developed under Egorov to study real functions. Egorov became a leader and administrator in the Moscow Mathematical Society and at the Institute for Mechanics and Mathematics at Moscow Sate University. Egorov became a vocal opponent to the anti-religious persecution in the time following the Russian revolution, and was dismissed from the IMM. However, he remained active and well-respected in his position in the MMS, supported by his peers in the organization. Outside influences began to manipulate the society, and within a year, Egorov was dismissed from his position and arrested. He went on a hunger strike in prison and died in the prison hospital (or, as some reports state, at a colleague’s home).

Ludwig Boltzmann, 62 (1844-1906)
Famous for his contributions to statistical mechanics and thermodynamics, and for his push towards the “atomic model” of the universe, Boltzmann was remembered by those close to him as suffering from severe bouts of depression, due to what is retroactively presumed to be either bipolar disorder or atypical depression. While on vacation with his family, Boltzmann hanged himself. Although his work was accepted by some and refuted by others, it is believed that this was not necessarily the motivation for this suicide (instead owing at least in part to his mental condition as well).

Issai Schur, 66 (1875-1941)
Issai Schur is well known for his contributions to algebra, and to me personally for his theorem on monochromatic solutions to x+y=z. However, Schur was one of many Jewish mathematicians eventually displaced from Germany during the rise of Hitler and the beginnings of the second World War. Schur believed himself as much a German as his non-Jewish colleagues, but he was dismissed from his position in Berlin and was eventually so humiliated as to have borrowed money to pay the fee that Jewish refugees were charged to flee Germany. Having fled to Palestine, Schur lobbied unsuccessfully for a position elsewhere, particularly in the USA. Unable to find any such position, Schur died on his 66th birthday, in Palestine, of a heart attack. Schiffer can be credited for recounting an even that may best characterise the painfully unfortunate circumstances of Schur’s life after that:
Schur told me that the only person at the Mathematical Institute in Berlin who was kind to him was Grunsky, then a young lecturer. Long after the war, I talked to Grunsky about that remark and he literally started to cry: “You know what I did? I sent him a postcard to congratulate him on his sixtieth birthday. I admired him so much and was very respectful in that card. How lonely he must have been to remember such a small thing.”
This entry was inspired by the comments of Prof. V. Retakh, and some of the information was also found in Schur’s MacTutor biography.

Kurt Gödel, 71 (1906-1978)
The story of Gödel’s death is a particularly tragic one. Most famous for his (in)completeness theorems and other breakthroughs in mathematical logic, Gödel was well-known for bouts of paranoia. Later in life, he was convinced that his food was being poisoned. He refused to eat food unless his wife tasted it first, and when she fell ill, Gödel refused to eat. He then slowly died of starvation, refusing all food. Already weakened by a restricted diet (due to ulcers), Gödel wasted away to only 65 pounds at the time of his death.

Georg Cantor, 71 (1845-1918)
Cantor is credited with revolutionising set theory by introducing the arithmetics of (infinite) ordinal and cardinal numbers. His work was of great mathematical and philosophical importance, but it was widely criticized (quite harshly and viciously), by the likes of Poincaré, Wittgenstein, and most notably Kronecker.
Kronecker was head of Cantor’s department, and devoted a fair amount of energy in opposition to Cantor’s work. The circumstances of not just his death, but his life, caused Cantor a great deal of misery. Although he lived to be quite old, he died impoverished, underfed, confined to a sanatorium due to depression, of a heart attack.

Archimedes, 75 (287-212BC)
Often considered to be the greatest mathematician of Antiquity, Archimedes was prized as a philosopher and mathematician. He is well known for his contributions to mathematics, philosophy, science, and engineering. He is famed for use of the word “Eureka!” During the seige of Syracuse, it was ordered that Archimedes not be harmed. However, in a misunderstanding that is not well-documented, a Roman soldier cut Archimedes down by sword, in spite of the standing order to the contrary. It is sometimes said that Archimedes’ compass and/or straightedge was mistaken for a weapon; however this is apocryphal.

Isaac Newton, 84 (1643-1727)
Despite living a long life, Newton’s health is famously poor. As a child, Newton was poor, unathletic, and often ill. As an elderly man, Newton was eccentric and engrossed in contentious philosophical and religious debates. Upon his death, it was discovered that his body had a high concentration of mercury, which likely contributed to his poor health (and eccentricity).

Abraham de Moivre, 87 (1667-1754)
Despite being a gifted and renowned mathematician in France, de Moivre spent much of his life in poverty. He was a Calvinist, and when the Edict of Nantes was revoked in 1685 (a decision that is unequivocally considered to have damaged France), de Moivre left France for England. He remained virtually destitute, de Moivre was unable to secure employment and was often known to play chess for money in order to afford sustenance. Eventually succumbing to the ravages of poverty and old age, de Moivre predicted the day of his own death using a simple arithmetic progression in the number of hours he slept per day. The day he predicted 24 hours of sleep was the day he died.

Alexander Grothendieck, 86 (1928-2014)
Grothendieck is considered to be one of the most important mathematicians of the 20th century. Born to a mixed Russian-German anarchist family in 1925 Berlin, his parents fought in the Spanish Civil War and his father was ultimately killed by Nazis during the second World War. Grothendeick was a staunch and vocal pacifist and this lead him to abandon most of his mathematical work in 1972, due to the ties between research mathematics and defence science and other governmental/military influences. He won the Fields medal in 1966, but declined the Craford prize in 1988, the year of his complete retirement. Grothendieck entered seclusion in 1991 and his precise location is unknown.

Note: Grothendieck was discovered to have died in 2014. He did, indeed, live in seclusion for many years, having withdrawn from mathematics until his death.

Do you want to speak English fluently?

What 5 things do you need to do in order to achieve fluency?

#1 practice every day!
If you don’t use it, you lose it. That means if you don’t practice in a regular basis, your brain will forget the information. So make sure you are practicing English on a daily basis. If even it’s singing an English song, or thinking of a phrase of word while you’re brushing your tooth. So don’t make excuses, just do it!

#2 don’t be afraid to fail
Making mistakes is all part of process. If you don’t fail then how can you be ever succeed? How will you ever learn? Speak, even if you don’t know how to pronounce the words. Write even when you don’t know how to spell the words. Have a conversation even you can’t fully understand what other person is saying! When children learn to speak the make mistake after mistake after mistake, but by trying the people around them are happy to help them and courage them to improve. So go out and make mistakes, be courageous and break out of your comfort zone.

#3 all things equally
Native language users can speak, listen, read and write! So if you want to be fluent at English you have to practice all these four skills equally. Start watching films, English series, following English podcasts or listen to the radio. Connect to natives or other English learners to practice your speaking in a regular basis, or just talk to yourself. Write a diary, write a story. Or even your shopping list in English. On the daily basis you can read an English newspaper, or following English blogs. So don’t stock just doing one thing!

#4 stop worrying about the rules
English is a weird language. It doesn’t always make sense, but us native don’t worry about it, neither should you. Don’t ask why that word is pronounced like that, why this rules has some exceptions, we haven’t made the language and its rules. Instead of why ask how: “why that word is pronounced?”

#5 learn like a native
If you want to speak English like a native, then learn the English like a native. Acquire the language, experience it. Listen to phrases and learn how to use it. Do not stock in rules.

Sofia Kovalevskaya: the woman who covered the walls of her room in theorems

When she was eleven years old, Sofia (sometimes called Sonja) Kovalevskaya covered the walls of her room in note sheets about differential and integral calculus by the Russian mathematician Mikhail Ostrogradski. These notes were from her father’s university years. This was how Sofia became familiar with calculus. Her first introduction to calculus was by the hand of her uncle Pyotr Krukovsky. He taught her the basics until she developed such a fascination with mathematics that she described it as “a mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.”

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Noncommutative Algebra

Noncommutative Algebra (Graduate Texts in Mathematics) 1993rd Edition

on March 7, 2004
Format: Hardcover
This is a very nice introduction to the theory of semisimple modules and rings, central simple algebras, and the Brauer group. It starts off with some review of basic concepts such as modules, tensor products, and field extensions – much of which is in the exercises. Indeed, a lot of the material is developed in the exercises so it is very useful to go through them. Most of the exercises aren’t exceedingly difficult either, and hints are provided for the more tricky ones.
Chapter one gets the ball rolling with the theory of simple modules and rings. The Wedderburn structure theorem is proved, as well as the structure theorem for simple artinian rings. These ideas are examined further in chapter two by analysing the Jacobson radical, thereby giving another characterization of semisimple rings. Chapter three goes on to study central simple algebras. Here they prove the famous Wedderburn theorem that every finite division ring is commutative, as well as the famous Frobenius theorem on real division algebras. Chapter four finally gives the definition of the Brauer group with a discussion of group cohomology. It proves that for a Galois extension K/k the second Galois cohomology group H^2(Gal(K/k),K*) is isomorphic to the Brauer group Br(K/k). The naturality of this isomorphism is discussed later on and developed through the exercises, although they never actually define what natuarality really is.
One way I think the book could be improved would be to introduce some more abstract homological algebra. The definition of group cohomology given is in terms of n-cochains. I think it would be useful to include the notion of derived functors here, and the general definition of the cohomolgy homology of a derived functor through the use of injective and projective resolutions. Then the group cohomology could be defined using the Ext functors, and the cocycle complexes obtained by examined the particular projective resolution called the Eilenberg-MacLane bar resolution. This method would help introduce the reader to homological algebra, and at the same time give some concrete use of such abstract constructions. The authors generally shy away from categorical language though, which might appeal to some readers.
The later chapters discuss primitive rings, some representation theory of finite groups, dimension theory of rings, and the Brauer group of a commutative ring. I have not gone through this later material in detail (yet), so I cannot give comment on it.
Overall, if you want to learn some non-commutative algebra, by all means buy this book and work out as many exercises as you can. I found the exposition to be quite illuminating and overall very well written.
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