| 单词 | Riemann |
| 例句 | And it was in Germany—the land of Gauss and of Riemann—where infinity’s secrets were revealed. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z In the old days, before Riemann saw that the complex plane was really a sphere, functions like 1/x would stump mathematicians. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z The person who combined these two ideas was a student of Gauss’s: Georg Friedrich Bernhard Riemann. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Riemann made it perfectly acceptable to go off to infinity; since infinity is just a point on the sphere like any other point, it was no longer something to be feared. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Infinity and zero are simply opposite poles on the Riemann sphere, and they can switch places in a blink. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Riemann imagined a translucent ball sitting atop the complex plane, with the south pole of the ball touching zero. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z I should be dedicating my life to solving the unsolvable problems of mathematics, like the Riemann hypothesis or the Hodge conjecture. The Miscalculations of Lightning Girl 2018-05-01T00:00:00Z Riemann merged projective geometry with the complex numbers, and all of a sudden lines became circles, circles became lines, and zero and infinity became the poles on a globe full of numbers. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Ms. Riemann, who wrote a book about the experience with her husband, the journalist Torsten Sasse, said that the knowledge gained from the files was worth the pain. Piecing Together the History of Stasi Spying 2021-08-10T04:00:00Z Tell Facebook you have solved the elusive Riemann hypothesis, proving conclusively that the zeta function has its zeros only at the negative even integers and complex numbers. Perspective | Gene Weingarten: What to do with that half-hour before a nuclear missile strikes 2018-02-13T05:00:00Z Lutz Riemann was an East German actor known for playing a TV policeman. Piecing Together the History of Stasi Spying 2021-08-10T04:00:00Z Petra Riemann first heard about her father’s double life through a newspaper report. Piecing Together the History of Stasi Spying 2021-08-10T04:00:00Z Within minutes, expect a reference to the Riemann Zeta Function. Simon McBurney Discusses His ?Disappearing Number? 2010-07-02T08:19:00Z Let me show you how one can obtain this anomalous result using the Functional Equation of the Riemann Zeta Function. It?s Been a Gloomy Summer on New York Stages 2010-07-28T17:57:00Z Coward didn’t write about the Riemann Zeta Function; he wrote about stylish British swells quipping and bed hopping and drinking Champagne by the bucketful. It?s Been a Gloomy Summer on New York Stages 2010-07-28T17:57:00Z Indian authorities also found violations at Riemann Labs, whose cough syrup were linked to deaths of children in Cameroon. WHO flags India-made syrup in latest warning over contaminated drugs 2023-08-07T04:00:00Z India has found violations related to manufacturing and laboratory practices at drugmaker Riemann Labs, whose cough syrup was linked to the deaths of children in Cameroon, a government health official told Reuters on Wednesday. Advil maker Haleon raises annual revenue growth forecast on strong demand 2023-08-02T04:00:00Z This parallel was brought to light in the early 2000s by researcher Dieter Riemann and his colleagues at the University of Freiburg in Germany. How Dreams Reveal Brain Disorders 2023-06-05T04:00:00Z The Landau–Siegel zeros conjecture is similar to — and, some suspect, less challenging than — the Riemann hypothesis, another question on the randomness of primes and one of the biggest unsolved mysteries in mathematics. Mathematician Who Solved Prime-Number Riddle Claims New Breakthrough 2022-11-15T05:00:00Z Grifo scored at the third attempt in the 48th minute, when Riemann saved his penalty and the next shot from the rebound. Freiburg edges Bochum 1-0 for 3rd win in 4 Bundesliga games 2022-08-26T04:00:00Z Manuel Riemann saved Sané’s next effort in the 10th, and the Germany winger went close again 10 minutes later. Sadio Mané scores 2 as Bayern Munich routs Bochum 7-0 again 2022-08-21T04:00:00Z Program it to solve the Riemann hypothesis: It immediately converts the entire planet into a giant computer. Would "artificial superintelligence" lead to the end of life on Earth? It's not a stupid question 2022-08-06T04:00:00Z Estimate the volume of water in the swimming pool by using a Riemann sum with m = n = 2. Calculus, Volume 3 2016-03-30T00:00:00Z Solving either the Riemann or Landau-Siegel problems would mean that the distribution of prime numbers does not have huge statistical fluctuations. Mathematician Who Solved Prime-Number Riddle Claims New Breakthrough 2022-11-15T05:00:00Z The spot kick was awarded after Riemann dropped the ball and then grasped at Roland Sallai in a futile effort to hinder him. Freiburg edges Bochum 1-0 for 3rd win in 4 Bundesliga games 2022-08-26T04:00:00Z Figure 1.17 For a function that is partly negative, the Riemann sum is the area of the rectangles above the x-axis less the area of the rectangles below the x-axis. Calculus, Vol 2 2016-03-30T00:00:00Z A sum of this form is called a Riemann sum, named for the 19th-century mathematician Bernhard Riemann, who developed the idea. Calculus, Volume 1 2016-03-30T00:00:00Z We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. Calculus, Volume 3 2016-03-30T00:00:00Z The conjecture is a cousin of the Riemann hypothesis — a way to predict the probability that numbers in a certain range are prime that was devised by German mathematician Bernhard Riemann in 1859. Mathematician Who Solved Prime-Number Riddle Claims New Breakthrough 2022-11-15T05:00:00Z Yet Ono says it would be unfounded to describe his work as “anything that suggests that we’re about to prove the Riemann hypothesis.” The Top Unsolved Questions in Mathematics Remain Mostly Mysterious 2021-05-28T04:00:00Z Recall that the proof of the Fundamental Theorem of Calculus used the concept of a Riemann sum to approximate the area under a curve by using rectangles. Calculus, Vol 2 2016-03-30T00:00:00Z Riemann sums give better approximations for larger values of n. Calculus, Volume 1 2016-03-30T00:00:00Z We compute triple integrals using Fubini’s Theorem rather than using the Riemann sum definition. Calculus, Volume 3 2016-03-30T00:00:00Z The Riemann hypothesis will probably remain at the top of mathematicians’ wishlists for years to come. Mathematician Who Solved Prime-Number Riddle Claims New Breakthrough 2022-11-15T05:00:00Z With interests in ‘wordplay and wordsmithing,’ Riemann said he considers himself a creative person. Five key counties to watch 2020-09-07T04:00:00Z For polar curves we use the Riemann sum again, but the rectangles are replaced by sectors of a circle. Calculus, Vol 2 2016-03-30T00:00:00Z We are now ready to define the area under a curve in terms of Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z For scalar line integrals, we chopped the domain curve into tiny pieces, chose a point in each piece, computed the function at that point, and took a limit of the corresponding Riemann sum. Calculus, Volume 3 2016-03-30T00:00:00Z The problem is similar to — but distinct from — the Riemann hypothesis, which is considered one of the most important problems in mathematics. Mathematician Who Solved Prime-Number Riddle Claims New Breakthrough 2022-11-15T05:00:00Z Clark spoke earlier Monday at a private service that was livestreamed on Facebook from the Riemann Family Funeral Home in Long Beach, where the couple lived. Mississippi couple's death from coronavirus came as they held each other's hands 2020-04-06T04:00:00Z This approach gives a Riemann sum approximation for the total area. Calculus, Vol 2 2016-03-30T00:00:00Z In this case, the associated Riemann sum is called a lower sum. Calculus, Volume 1 2016-03-30T00:00:00Z This March will mark one year on the job for Riemann, the Sheboygan Press reported. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z The eighth problem, which includes the Riemann hypothesis, is clearly still open. Hilbert Walked so the Clay Mathematics Institute Could Run 2019-10-20T04:00:00Z Two great enigmas form the center of this elegant novel, in which a brilliant mathematician attempts to solve the impenetrable Riemann hypothesis and learn the truth of her family history. Briefly Noted Book Reviews 2019-09-09T04:00:00Z Most mathematicians believe that the Riemann hypothesis is true and that P = NP is false. Proofs and Guarantees 2018-11-09T05:00:00Z Is the area under the curve on the given interval better approximated by the left Riemann sum or right Riemann sum? Calculus, Volume 1 2016-03-30T00:00:00Z Among other responsibilities, Riemann leads the charge on writing those freeway messages. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z It was on the Riemann hypothesis – an unsolved theorem about the distribution of prime numbers – and though we’ve just been discussing how exhausted she is after a 14-hour day, her eyes light up. ‘I spent seven years fighting to survive’: Chelsea Manning on whistleblowing and WikiLeaks 2018-10-07T04:00:00Z Mathematicians around the world were buzzing about the prospect of renowned theorist and geometer Michael Atiyah presenting a simple, new proof to the Riemann hypothesis, a challenging problem that has remained unsolved for 160 years. Top stories: a one-word Turing test, honey bees at risk, and Indigenous genome experts 2018-09-28T04:00:00Z Riemann found that the key to understanding their distribution lay within another set of numbers, the zeroes of a function called the Riemann zeta function that has both real and imaginary inputs. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8. Calculus, Volume 1 2016-03-30T00:00:00Z The messages are designed to impact driver behavior, Riemann said. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Which is easy, because you only need an expert’s knowledge of Riemann’s Hypothesis, the Fermi Paradox, and orbital mechanics in order to understand their qualifying system. Introducing the 'Ryder Cup Radicals' Email Exchange: The high stakes at the PGA Championship edition - Golf Digest 2018-08-08T04:00:00Z The related Riemann hypothesis is one of the great unsolved maths problems, and Holt helped me to understand why it’s important. Summer reads: Darwin’s dilemma on sexual selection, ecology versus empire, and the dark side of the greater good 2018-07-17T04:00:00Z By describing this constant using a relatively obscure relationship known as the Todd function, Atiyah claimed to be able to prove the Riemann hypothesis by contradiction. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z It is also worth noting here that we have retained the use of a regular partition in the Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z Sometimes, Riemann said, the messages don’t read fast enough or don’t look quite right when he’s driving by them in the morning, so they will edit them. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Voevodsky joins a line of great mathematicians, including Bernhard Riemann and Alexander Grothendieck, who built algebraic geometry into a deep and powerful science over the past two centuries. Vladimir Voevodsky (1966–2017) He couldn’t drop in a line about the Riemann hypothesis as an icebreaker. The science of spying: how the CIA secretly recruits academics 2017-10-10T04:00:00Z The Riemann hypothesis, one of the last great unsolved problems in math, was first proposed in 1859 by German mathematician Bernhard Riemann. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z Any partition can be used to form a Riemann sum. Calculus, Volume 1 2016-03-30T00:00:00Z Aside from the character count limitations from the software, Riemann said they want the messages to be short and easily comprehended by all ages. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z She made monumental contributions to the study of the dynamics and geometry of mathematical objects called Riemann surfaces. Maryam Mirzakhani (1977-2017) : Nature : Nature Research 2017-09-05T04:00:00Z Mirzakhani won the prize for her “outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”. Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes 2017-07-16T04:00:00Z However, Riemann’s formula only holds if one assumes that the real parts of these zeta function zeroes are all equal to one-half. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z This introduces a little more complex notation in our limits and makes the calculations more difficult without really gaining much additional insight, so we stick with regular partitions for the Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z Riemann’s brother was six years older than him and quickly got more involved in high school extracurricular activities, so while the route formally remained his brother’s, Riemann took over actually delivering the papers. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Computing such an average requires one to calculate the 'volume', or size, of the space of Riemann surfaces that contain a curve of a certain length. Maryam Mirzakhani (1977-2017) : Nature : Nature Research 2017-09-05T04:00:00Z “We want her to be just like other kids as much as possible, and really right now the only thing holding her back is being able to walk,” said Katie Riemann, Maddi’s mother. Kansas high-schoolers help toddlers with immobility 2017-06-17T04:00:00Z But despite a long and prolific career, the Riemann claim follows on the heels of more recent failed proofs. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z The route included the fire house and the Sheboygan Clinic and passed a grocery store that Riemann said he would occasionally pop into to spend his newly made coin. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z But this year’s event, held last night in New York, was more like the Hodge conjecture of frocks, or the Riemann hypothesis of style. Met Gala 2017: who nailed the trickiest dress code ever? 2017-05-02T04:00:00Z Without assuming unproven statements such as the k-tuple conjecture and the much-studied Riemann hypothesis, mathematicians’ understanding of the distribution of primes dries up. Peculiar pattern found in ‘random’ prime numbers 2016-03-13T05:00:00Z A famous mathematician today claimed he has solved the Riemann hypothesis, a problem relating to the distribution of prime numbers that has stood unsolved for nearly 160 years. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z Figure 5.17 For a function that is partly negative, the Riemann sum is the area of the rectangles above the x-axis less the area of the rectangles below the x-axis. Calculus, Volume 1 2016-03-30T00:00:00Z At that young age, Riemann said he already knew he was interested in the news. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Event organizer Veronica Romano of Riemann Family Funeral Homes says the raise money all year for the project. Volunteers needed to help place wreaths on veterans’ graves 2015-11-24T05:00:00Z Einstein emphasized that his general theory of relativity built on the work of Gauss and Riemann, giants of the mathematical world. History: Einstein was no lone genius 2015-11-15T05:00:00Z He says in an interview that, despite criticism, his work lays down a concrete basis for proving not only the Riemann hypothesis, but other unproven problems in mathematics. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10, 100. Calculus, Volume 1 2016-03-30T00:00:00Z At Sheboygan South High School, Riemann was involved with the student newspaper, Lake Breeze, and started really getting into photography. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z In 2004, de Branges released a purported solution to the Riemann hypothesis, which many consider the most important open problem in maths. Math Mystery: Shinichi Mochizuki and the Impenetrable Proof 2015-10-08T04:00:00Z In the first November paper, Einstein wrote that the theory was “a real triumph” of the mathematics of Carl Friedrich Gauss and Bernhard Riemann. History: Einstein was no lone genius 2015-11-15T05:00:00Z They all expressed concern about his desire to come out of retirement to present proofs based on shaky associations, and said it was unlikely that his proof of the Riemann hypothesis would be successful. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis 2018-09-24T04:00:00Z Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann sums, and both approaches are extremely cumbersome. Calculus, Volume 1 2016-03-30T00:00:00Z After graduating from South in 1999, Riemann went through Milwaukee Area Technical College’s television program and then went on to get a bachelor’s degree from Concordia University. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Riemann Funeral Home says services will be private. Former Long Beach mayor Mitchell dies at 59 2015-08-17T04:00:00Z It was during a talk in 1959 on what is seen to be one of the hardest problems in maths — the Riemann hypothesis — that the audience realized that there was something wrong with Nash. John Forbes Nash (1928-2015) : Nature : Nature Publishing Group 2015-06-23T04:00:00Z With OCD, Riemann says, people are so wrapped up in repetitive thoughts and behaviors that it interferes with their lives "either occupationally, academically, socially or in their relationships." Obsessive-Compulsive Disorder: Serious Business 2015-05-27T04:00:00Z In the following exercises, use a calculator to estimate the area under the curve by computing T10, the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Calculus, Volume 1 2016-03-30T00:00:00Z During his last semester there, Riemann won a scholarship from the Wisconsin Broadcasters Association. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Instead, Gauss encouraged a younger, brilliant pupil Bernhard Riemann to tackle the topic. What Einstein Should Have Known 2015-05-13T04:00:00Z Mirzakhani, according to the International Mathematical Union was recognized for “her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.” Maryam Mirzakhani becomes first woman awarded math’s “Nobel Prize” 2014-08-13T04:00:00Z "In our residential program, we probably refer one person a month, roughly, to be evaluated for some sort of neural procedure," Riemann says. Obsessive-Compulsive Disorder: Serious Business 2015-05-27T04:00:00Z The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z Some of Riemann’s other local favorites include the brat at Charcoal Inn on Eighth Street and a pastrami sandwich with no onion from Ella’s. Who cooks up those witty freeway messages? This guy 2020-03-07T05:00:00Z Although Riemann’s extension of geometry into many dimensions was the most important feature of his lecture, the most astonishing, with hindsight, was his suggestion that space might be curved into a closed ball. What Einstein Should Have Known 2015-05-13T04:00:00Z A text provided by the ICM further explains that she works on so-called Riemann surfaces and their deformations. A Huge First For Women: Female Mathematician Wins Fields Medal "It's a very structured and systematic way of exposing people to their feared situations and situations that will produce the unwanted obsessional thoughts," Riemann says. Obsessive-Compulsive Disorder: Serious Business 2015-05-27T04:00:00Z We already used the formal Riemann sum development of the volume formula when we developed the slicing method. Calculus, Volume 1 2016-03-30T00:00:00Z According to a citation released by the International Mathematical Union, Marzakhani won the prize for her "outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces". Fields Medal mathematics prize won by woman for first time in its history 2014-08-12T04:00:00Z Her thesis showed how to compute the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Maryam Mirzakhani: 'The more I spent time on maths, the more excited I got' 2014-08-12T04:00:00Z In particular, she has studied "moduli spaces" of these shapes, which map all of the possible geometries of a Riemann surface into their own, new space. First female winner for maths medal 2014-08-12T04:00:00Z On average, Riemann says, patients stay in his program about 60 days – with many people who arrive with severe symptoms making a dramatic improvement into the "mild" category by discharge. Obsessive-Compulsive Disorder: Serious Business 2015-05-27T04:00:00Z So, as we have done many times before, we form a partition, a Riemann sum, and, ultimately, a definite integral to calculate the force. Calculus, Volume 1 2016-03-30T00:00:00Z She won for her work on “the dynamics and geometry of Riemann surfaces and their moduli spaces.” Iranian is first woman to nab highest prize in maths 2014-08-11T04:00:00Z Critical mathematical applications such as Fourier transforms, Riemann zeta functions, Gaussian distributions, roots of unity, integrating over polar coordinates and pretty much anything involving trigonometry employs pi. Why Tau Trumps Pi 2014-06-25T04:00:00Z A hundred years later, Riemann used a more advanced and rigorous method, involving imaginary as well as real numbers, to calculate the zeta function and got the same answer: minus 1/12. Out There: In the End, It All Adds Up to – 2014-02-03T19:59:21Z "Obsession is an unwanted thought or image that generates a high level of anxiety," Riemann says. Obsessive-Compulsive Disorder: Serious Business 2015-05-27T04:00:00Z The arc length is first approximated using line segments, which generates a Riemann sum. Calculus, Volume 1 2016-03-30T00:00:00Z The Riemann zeta function is the analytic continuation of this function to the whole complex plane minus the point s=1. Does 1+2+3 Really Equal -1/12? 2014-01-20T15:05:02Z The first three were proved to be true in the 1960s, but the fourth and hardest — and the direct analogue of the Riemann hypothesis — was proved by Deligne in 1974. Belgian Mathematician Wins Abel Prize for Shaping Algebraic Geometry 2013-03-20T20:15:00.363Z No one expects Magic Johnson to offer proof of the Riemann hypothesis anytime soon, yet his wealth is estimated to be half a billion dollars. How Intelligence Relates To Wealth 2012-10-01T16:50:03Z I was like some two-dimensional being who might stand at the singular point of a Riemann's surface, and thus find himself in infinitesimal and inexplicable contact with a plane of existence not his own. Mysterious Psychic Forces An Account of the Author's Investigations in Psychical Research, Together with Those of Other European Savants 2012-03-28T02:00:29.747Z Even mathematics as an exact science is not immutable, and the geometries of Lebatchevsky and Riemann are as legitimate as Euclid's. Unicorns 2012-03-14T02:00:26.677Z The function in question is the Riemann zeta function, which is famous for its deep connections to questions about the distribution of prime numbers. Does 1+2+3 Really Equal -1/12? 2014-01-20T15:05:02Z The Riemann hypothesis itself remains “the most famous unsolved problems in mathematics”, says Gowers — which is in itself an indication of the significance of Deligne’s proof. Belgian Mathematician Wins Abel Prize for Shaping Algebraic Geometry 2013-03-20T20:15:00.363Z Riemann’s results are contained in the memoirs on “Abelian Integrals,” &c. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z So should you be studying Riemann sums every other day for two weeks and then take a month off before going back to them? Forget Ritalin and Cramming: Molecular Pathways in the Brain May Reveal the Best Learning Strategies 2012-01-05T22:45:00.250Z Professor Patterson knows Riemann and his "agogic accent," which, according to that editor of the Chopin Etudes, is a slight expansion in the value of the note; not a dynamic accent. Unicorns 2012-03-14T02:00:26.677Z The second hypothesis he succeeds in condemning, since, like all who preceded Bernhard Riemann, he is unable to conceive of the straight line as finite and closed. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z “Overall, the results show that fighting for market share limits profit growth at present,” Andreas Riemann, an analyst at Commerzbank, said by e-mail. L’Oreal Shares May Drop After First-Half Profit Misses Estimates 2011-08-30T22:23:02Z As the more general definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of aggregates, we shall, in what follows, adhere to Riemann’s definition. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z Graham Hoare, an editor at Mathematics Today, discusses what is perhaps the most celebrated case of mathematicians’ prescience: Georg Bernhard Riemann, who invented modern geometry and who is one of my personal heroes. When Math(s) Turns Out To Be Useful 2011-07-19T11:15:04.210Z The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Encyclopaedia Britannica, 11th Edition, Volume 10, Slice 6 "Foraminifera" to "Fox, Edward" 2011-04-22T02:00:08.637Z The projective method leads to a discrimination, first made by Klein,17 of two varieties of Riemann’s space; Klein calls these elliptic and spherical. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Carr� fils is the son of the well-known librettist of Faust and Romeo fame, and Choudens is connected with Choudens Fils, who publish this opera; but concerning the composer, Grove and Riemann are silent. Shakespeare and Music 2011-04-01T02:00:29.867Z Riemann; but the failure of an attempt made by Amp�re to prove that every continuous function must be differentiable may be regarded as the first step in the theory. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z Although to be fair to Einstein, the geometry of spacetime goes beyond Riemann’s: it is much more subtle and less easy to visualize, as I plan to discuss in a future post. When Math(s) Turns Out To Be Useful 2011-07-19T11:15:04.210Z It was shown by Riemann that the convergence or non-convergence of the series at a particular point x depends only upon the nature of the function in an arbitrarily small neighbourhood of the point x. Encyclopaedia Britannica, 11th Edition, Volume 10, Slice 6 "Foraminifera" to "Fox, Edward" 2011-04-22T02:00:08.637Z Riemann proceeds to specialize the manifold by considerations Measure of curvature. as to measurement. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z In 1824 the dissertation of Riemann, under Gauss, introduced the idea of an n-ply extended magnitude, or a study of n-dimensional manifolds and a new road was opened for mathematical intelligence. Creative Intelligence Essays in the Pragmatic Attitude For example, he has shown that Riemann’s totally discontinuous function, which is equal to 1 when x is rational and to 0 when x is irrational, can be represented by an analytical expression. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z Riemann, 93. rigaudon, 75; examples of, from Grieg, Rameau and MacDowell, 79, 81. Music: An Art and a Language Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multi-dimensional, non-Euclidean continuum. Relativity: The Special and General Theory Riemann’s work contains two fundamental conceptions, that of a manifold and that of the measure of curvature of a continuous manifold possessed of what he calls flatness in the smallest parts. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z These two cases are mentioned also by Riemann in “Goethe’s Romantechnik.” Laurence Sterne in Germany A Contribution to the Study of the Literary Relations of England and Germany in the Eighteenth Century As a song writer, Riemann considers him the legitimate successor of Schumann; and among the hundreds of songs which he produced are many of very great beauty. The Masters and their Music A series of illustrative programs with biographical, esthetical, and critical annotations In his studies for a history of musical notation, Dr. Hugo Riemann quotes an extract from an anonymous manuscript of the tenth century, in which the author gives directions for a set of organ pipes. A Popular History of the Art of Music From the Earliest Times Until the Present I don't care if he's Gauss and Riemann and Lorenz and Poincare and Minkowski and Whitehead and Einstein, all collapsed into one! The Mercenaries Its further discussion depends upon the measure of curvature, the second of Riemann’s fundamental conceptions. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Riemann also traces to Sterne, Fielding and their German followers, Goethe’s occasional use of the direct appeal to the reader. Laurence Sterne in Germany A Contribution to the Study of the Literary Relations of England and Germany in the Eighteenth Century The idealists in poetry, music, and philosophy gave place to great men of science, to figures such as those of Ludwig and Liebig, of Gauss, Riemann, and Helmholtz. Before the War His operas subsided into farce, the serious element being almost wholly lacking, and, according to Riemann, the last of them shows no improvement over the first. A Popular History of the Art of Music From the Earliest Times Until the Present So great an authority as the last edition of the Riemann Musik-Lexicon cannot forbear, even in 1915, to emphasize his "technic, absolutely developed in its every detail, and his fiery and poetic manner of interpretation." Violin Mastery Talks with Master Violinists and Teachers If α be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The clavier music, with the exception of a few bits, is of no great importance; still, I have played much of it with pleasure in Dr. Riemann's edition, and found many charming things. Haydn Not only the practical geometry of Euclid, but also its nearest generalisation, the practical geometry of Riemann, and therewith the general theory of relativity, rest upon this assumption. Sidelights on Relativity Riemann considers his greatest works for mixed chorus to be "Odysseus," "Arminius," "The Song of the Bell," and for male chorus "Frithjof," "Salamis" and "The Normans." A Popular History of the Art of Music From the Earliest Times Until the Present Riemann manages to get some new editorial comment upon it. Chopin : the Man and His Music The publication of Riemann’s dissertation was closely followed by two works of Hermann von Helmholtz,12 again undertaken in ignorance of the work of predecessors. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z It is Riemann's opportunity for pedagogic editing, and he does not miss it. Chopin : the Man and His Music Riemann repeats his trick of breaking a group, detaching a note for emphasis; although he is careful to retain the legato bow. Chopin : the Man and His Music Riemann calls Glinka "the Berlioz of Russia," in the originality of his invention and his clever technique; and something more, namely, that he created a national school of music for his country. A Popular History of the Art of Music From the Earliest Times Until the Present Riemann places half notes at the beginning of each measure, as a reminder of the necessary clinging of the thumbs. Chopin : the Man and His Music The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Riemann phrases the study with a multiplicity of legato bows and dynamic accents. Chopin : the Man and His Music Most of the others use 88 to the quarter, except Riemann, who espouses the more rapid gait of 96. Chopin : the Man and His Music Riemann exercises all his ingenuity to make the beginning clear to the eye. Chopin : the Man and His Music The heavy dynamic accents employed by Riemann are unmistakable. Chopin : the Man and His Music It begins with Riemann’s inaugural dissertation, which regards space as a particular case of a manifold; but the characteristic standpoint of the period is chiefly emphasized by Eugenio Beltrami. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z In the fourth bar, third group, third note of group, Klindworth and Riemann print E flat instead of D flat. Chopin : the Man and His Music Riemann and the others make the same addition. Chopin : the Man and His Music Here is Riemann's phrasing and grouping for the first few bars. Chopin : the Man and His Music Dr. Hugo Riemann, who has edited a few of the Preludes, phrases the first bars thus: Desperate and exasperating to the nerves is the second prelude in A minor. Chopin : the Man and His Music Klein, to whom the development of Cayley’s work is due, showed further that there are two forms of Riemann’s space, called by him the elliptic and the spherical. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z In Riemann the accentuation seems perverse, but there is no question as to its pedagogic value. Chopin : the Man and His Music It remained for Riemann to make some radical changes. Chopin : the Man and His Music But if this is a marked specimen of Riemann, examine his reading of the phrase wherein Chopin's triple rhythm is supplanted by duple. Chopin : the Man and His Music Kullak points out a variant in the fourteenth bar, G instead of B natural being used by Riemann. Chopin : the Man and His Music But −� of this quotient is defined by Riemann as the measure of curvature.15 Hence the measure of curvature is −1/R�, i.e. is constant and negative. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z But the illustration serves to show how Dr. Riemann is disposed to refine upon the gold of Chopin. Chopin : the Man and His Music Riemann has his own ideas of the phrasing of the following one, the fifth and familiar "Black Key" etude. Chopin : the Man and His Music All the other editions, except Riemann's, recommend the fifth finger on F, the fourth on C. Von Billow believes that small hands beginning with his system will achieve quicker results than by the Chopin fingering. Chopin : the Man and His Music Riemann detaches the eighth notes of the bass figures, as is his wont, for greater clearness. Chopin : the Man and His Music Another striking peculiarity of the Riemann phrasing is his heavy accent on the top E flat in the principal passage for the left hand. Chopin : the Man and His Music It may be noticed that Riemann even changes the arrangement of the bars. Chopin : the Man and His Music On this point Riemann must without question be considered right. Chopin : the Man and His Music As to phrasing and fingering I pin my faith to Riemann. Chopin : the Man and His Music Riemann breaks up the phrase in the bass figure, but I cannot see the gain on the musical side. Chopin : the Man and His Music Riemann accents the B, the E, A, B flat, C and F, at the close—perilous leaps for the left hand, but they bring into fine relief the exquisite harmonic web. Chopin : the Man and His Music Riemann has ideas of his own, both in the phrasing and figuration. Chopin : the Man and His Music In all the editions, Riemann's excepted, there is no doubt left as to the alternations of metres. Chopin : the Man and His Music |
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