mm-3829 === Subject: : An exact 1-D integration challenge - 31 - we mean, a challenge for _Maple_ Hello the Maple fandom, None of the Maple versions since at least 1994 on can get this integral correctly. Is there a Maple soul who can show how to get to the exact value of the integral using Maple commands int(sin(tan(z/2))*sin(z), z= 0..Pi); ? Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director http://www.cybertester.com/ Cyber Tester, LLC http://maple.bug-list.org/ Maple Bugs Encyclopaedia http://www.CAS-testing.org/ CAS Testing === === Subject: : Re: An exact 1-D integration challenge - 31 - we mean, a challenge for _Maple_ value(student[changevar](tan(z/2)=y,Int(sin(tan(z/2))*sin(z), z= 0..Pi))); exp(-1) Pi Mate === === Subject: : ODE Problem Let the system dx/dt=x-y-x^3 dy/dt=x+-y^3 or a equivalent system dr/dt= r(1-r^2(1-1/2sin^2(2{theta}))) d{theta}/dt= 1-r^2sin(2{theta}) Prove the existence of a periodic trajectory G such that G are in the set {Xin R^2: 1<|X|<2 }. I prove the existence of a periodic trajectory for the system in the region {Xin R^2: 1<=|X|<=2 } using the Poincare-Bendixon theorem. How can I prove the last statment of the problem??? === === Subject: : Re: Proofs to programs So did you. You said, Church's Thesis, that these formal notions of computability exactly encompass our notion of an effective procedure. You just said that it was. No it doesn't. It means capable of being shown mathematically to be unprovable. You say that I am wrong but only repeat what I said. As far as Church's Thesis goes, 1. If the notion is informal, and thus not susceptible to proof or refutation, and by the same reasoning it is not susceptible to evaluation as to its being true or false. Thus it is meaningless to talk about it being true. 2. In particular, without a formal definition, it is subjective as to what effectively computable includes. One could say that if we know the exact rules as to the behavior of a Turing Machine at any point in time, then it is intuitively effectively computable as to whether or not it will ever terminate. C-B === === Subject: : Re: Proofs to programs The point of my post, once again, is that stringing them together as you originally had, while not strictly false, is very misleading, as it runs together a formal mathematical theorem with an informal conjuecture. But I've already made that clear more than once, and as you have purposely omitted the fact in your reply, you are clearly just trolling here. What a dishonest representation of my remarks. I never asserted the above. I said explicitly that formally unprovable is ambiguous between these two senses. You obviously are not interested in an honest discussion. So it is meaningless to speak of any proposition not susceptible to formal proof or refutation as true or false. This of course encompasses the vast majority of assertions we make all day long. Ridiculous. Well, I have no idea what you mean by subjective (you seem to want it to suggest arbitrariness and incommunicability), but fact is, effectively computable has no formal definition. Since you disagree, you obviously need to do some homework. You might start with the entry The Church-Turing Thesis in the online Stanford Encyclopedia of Philosophy. Meantime, back in the killfile you go. I'd forgotten how stupefyingly disingenuous you are. Bye. === === Subject: : Re: Proofs to programs You said the same thing: Church's Thesis, that these formal notions of computability exactly encompass our notion of an effective procedure. What's the difference? Did you write Your inference here is sound only if we take formally unprovable to mean capable of being shown mathematically to be unprovable when all it in fact means is simply not capable of being shown mathematically to be provable? Doesn't this include the above? Yes, if it is mathematical in nature. This is because truth and What did I say that contradicts that proposition? What did I say that is disingenuous? C-B === === Subject: : Re: Proofs to programs Everything? On second thought probably not; you seem sincere in some of your confusion and idiocy. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === === Subject: : Re: Proofs to programs What could be a better demonstration that ones Mathematical discoveries are flawless than his harshest critic being unable to come up with a single example of anything wrong? C-B CM said I was wrong then repeated what I said. When I quoted him, he said that was misrepresentation and that he never said it. Now who would be such a fool as to side with such silliness? Why, Aata and DU! CM: Your inference here is sound only if we take formally unprovable to mean capable of being shown mathematically to be unprovable when all it in fact means is simply not capable of being shown mathematically to be provable. C-B quoting CM: formally unprovable means not capable of being shown mathematically to be provable. CM: What a dishonest representation of my remarks. I never asserted the above. LOL : ) === === Subject: : Re: Proofs to programs Laughing at yourself, I hope. Menzel said that your inference is sound only IF we take 'formally unprovable' to mean a certain thing. Then you quoted him but cut off the first part so that you made it appeaer that Menzel had said that 'formally unprovable' means that certain thing. If you can't see these clear differences, then you really are beyond having a rational conversation with. MoeBlee === === Subject: : Re: Proofs to programs In case anyone wonders whether this matters (and/or whether the word dishonest is really appropriate), just for the record: I haven't been following this silly debate closely. Just now I noticed CB quoting CM as above, and I was suprprised to see that CM had said what CM said he'd said - I was about to post a reply saying that surely formally unprovable is ambiguous. Then I stopped to consider the fact that it was CB quoting CM here, thought about various things he'd accused me of saying in the past... Sho nuff. CM: Well, duh. This sounds like you really haven't been paying attention. ****** === === Subject: : Re: Proofs to programs Nothing that you have ever said in criticism of my posts (i.e., no comment of yours regarding my posts) has even been true. You said that my expressing a relation to synthesize a program amounts to programming, when in fact programming is equivalent to representing, and expressible and representable are very different. You said that R(P) to state that P is r.e. (some such syntax applied to the relation) is better than my use of input and output variables, when in fact it doesn't work because it isn't general enough and is a poor design because it unnecessarily goes outside of the syntax of Predicate Calculus, making Rules of Inference no longer apply. You said that it is trivial to formalize Turing's 1937 proof, but never provided such a formalization and nobody has even shown such a formal representation outside of my ARXIV paper. So why should anyone pay any attention to your reckless statements? === === Subject: : Re: Proofs to programs On Thu, 24 Aug 2006 03:59:41 -0500, David C Ullrich Not lately. As noted, CB had been in my killfile for quite some time, but a response I saw he'd made to someone else actually looked like part of a somewhat reasonable discussion, and since he said something that could benefit from a bit of clarification (it wasn't even that what he said was outright wrong) I unkilled him and replied. Silly me. Ah well, he is back where he belongs. === === Subject: : Re: More on Poincare conjecture I can't say I agree completely with the negative portrayal of Yau in involved/affected by this mess (the competition to become the leading Chinese mathematician), I would say that, if anything, this should not be thoughted of as a Chinese evil but more of a Personal Evil stemming from an individual's greed (for power, recognition, etc.) Ohm3 === === Subject: : Re: More on Poincare conjecture doesn't deserve, including the latest breakthrough on the Poincare conjecture by Grigori Perelman. -Kartik MANIFOLD DESTINY A legendary problem and the battle over who solved it. by SYLVIA NASAR AND DAVID GRUBER Issue of 2006-08-28 On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel in Beijing for a lecture by the Chinese mathematician Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal-the most coveted award in mathematics-a reputation in both disciplines as a thinker of unrivalled technical power. Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes in Beijing and Hong Kong, dividing his time between the United States and China. His lecture at the Friendship Hotel was part of an international conference on string theory, which he had organized with the support of the Chinese government, in part to promote the country's recent advances in theoretical physics. (More than six thousand students attended the keynote address, which was delivered by Yau's close friend Stephen Hawking, in the Great Hall of the People.) The subject of Yau's talk was something that few in his audience knew much about: the Poincar.8e conjecture, a century-old conundrum about the characteristics of three-dimensional spheres, which, because it has important implications for mathematics and cosmology and because it has eluded all attempts at solution, is regarded by mathematicians as a holy grail. Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves and black-rimmed glasses and, with his hands in his pockets, described how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a proof of the Poincar.8e conjecture a few weeks earlier. I'm very positive about Zhu and Cao's work, Yau said. Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle. He said that Zhu and Cao were indebted to his longtime American collaborator Richard Hamilton, who deserved most of the credit for solving the Poincar.8e. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. Nevertheless, Yau said, in Perelman's work, spectacular as it is, many key ideas of the proofs are sketched or outlined, and complete details are often missing. He added, We would like to get Perelman to make comments. But Perelman resides in St. Petersburg and refuses to communicate with other people. For ninety minutes, Yau discussed some of the technical details of his students' proof. When he was finished, no one asked any questions. That night, however, a Brazilian physicist posted a report of the lecture on his blog. Looks like China soon will take the lead also Grigory Perelman is indeed reclusive. He left his job as a researcher at the Steklov Institute of Mathematics, in St. Petersburg, last December; he has few friends; and he lives with his mother in an apartment on the outskirts of the city. Although he had never granted an interview before, he was cordial and frank when we visited him, in late June, shortly after Yau's conference in Beijing, taking us on a long walking tour of the city. I'm looking for some friends, and they don't have to be mathematicians, he said. The week before the conference, Perelman had spent hours discussing the Poincar.8e conjecture with Sir John M. Ball, the fifty-eight-year-old president of the International Mathematical Union, the discipline's influential professional association. The meeting, which took place at a conference center in a stately mansion overlooking the Neva River, was highly unusual. At the end of May, a committee of nine prominent mathematicians had voted to award Perelman a Fields Medal for his work on the Poincar.8e, and Ball had gone to St. Petersburg to persuade him to accept the prize in a public ceremony at the I.M.U.'s quadrennial congress, in Madrid, on August 22nd. The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.U. congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be as purely international and impersonal as possible. However, the Fields Medal, which is awarded every four years, to between two and four mathematicians, is supposed not only to reward past achievements but also to stimulate future research; for this reason, it is given only to mathematicians aged forty and younger. In recent decades, as the number of professional mathematicians has grown, the Fields Medal has become increasingly prestigious. Only forty-four medals have been awarded in nearly seventy years-including three for work closely related to the Poincar.8e conjecture-and no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. I refuse, he said simply. Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincar.8e on the Internet in three installments. Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive. Judgments about the accuracy of a proof are mediated by peer-reviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose pa-per is under consideration is kept secret. Publication implies that a proof is complete, correct, and original. By these standards, Perelman's proof was unorthodox. It was astonishingly brief for such an ambitious piece of work; logic sequences that could have been elaborated over many pages were often severely compressed. Moreover, the proof made no direct mention of the Poincar.8e and included many elegant results that were irrelevant to the central argument. But, four years later, at least two teams of experts had vetted the proof and had found no significant gaps or errors in it. A consensus was emerging in the math community: Perelman had solved the Poincar.8e. Even so, the proof's complexity-and Perelman's use of shorthand in making some of his most important claims-made it vulnerable to challenge. Few mathematicians had the expertise necessary to evaluate and defend it. After giving a series of lectures on the proof in the United States in continued to answer queries about it by e-mail, he had had minimal contact with colleagues and, for reasons no one understood, had not tried to publish it. Still, there was little doubt that Perelman, who turned forty on June 13th, deserved a Fields Medal. As Ball planned the I.M.U.'s 2006 congress, he began to conceive of it as a historic event. More than three thousand mathematicians would be attending, and King Juan Carlos of Spain had agreed to preside over the awards ceremony. The I.M.U.'s newsletter predicted that the congress would be remembered as the occasion when this conjecture became a theorem. Ball, determined to make sure that Perelman would be there, decided to go to St. Petersburg. Ball wanted to keep his visit a secret-the names of Fields Medal recipients are announced officially at the awards ceremony-and the conference center where he met with Perelman was deserted. For ten hours over two days, he tried to persuade Perelman to agree to accept the prize. Perelman, a slender, balding man with a curly beard, bushy eyebrows, and blue-green eyes, listened politely. He had not spoken English for three years, but he fluently parried Ball's entreaties, at one point taking Ball on a long walk-one of Perelman's favorite activities. As he summed up the conversation two weeks later: He proposed to me three alternatives: accept and come; accept and don't come, and we will send you the medal later; third, I don't accept the one. The Fields Medal held no interest for him, Perelman explained. It was completely irrelevant for me, he said. Everybody understood that if the proof is correct then no other recognition is needed. Proofs of the Poincar.8e have been announced nearly every year since the conjecture was formulated, by Henri Poincar.8e, more than a hundred years ago. Poincar.8e was a cousin of Raymond Poincar.8e, the President of France during the First World War, and one of the most creative mathematicians of the nineteenth century. Slight, myopic, and notoriously absent-minded, he conceived his famous problem in 1904, eight years before he died, and tucked it as an offhand question into the end of a sixty-five-page paper. Poincar.8e didn't make much progress on proving the conjecture. Cette question nous entra.94nerait trop loin (This question known as rubber-sheet geometry, for its focus on the intrinsic difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincar.8e used the term manifold to describe such an abstract topological space. The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere-even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is simply connected, meaning that no holes puncture it. Unlike a soccer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you cannot pull the slipknot closed without tearing the bagel. Two-dimensional manifolds were well understood by the mid-nineteenth century. But it remained unclear whether what was true for two dimensions was also true for three. Poincar.8e proposed that all closed, simply connected, three-dimensional manifolds-those which lack holes and are of finite extent-were spheres. The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. Proving it mathematically, however, was far from easy. Most attempts were merely embarrassing, but some led to important mathematical discoveries, including proofs of Dehn's Lemma, the Sphere Theorem, and the Loop Theorem, which are now fundamental concepts in topology. By the nineteen-sixties, topology had become one of the most productive areas of mathematics, and young topologists were launching regular attacks on the Poincar.8e. To the astonishment of most mathematicians, it turned out that manifolds of the fourth, fifth, and higher dimensions were more tractable than those of the third dimension. By 1982, Poincar.8e's conjecture had been proved in all dimensions except the third. In 2000, the Clay Mathematics Institute, a private foundation that promotes mathematical research, named the Poincar.8e one of the seven most important outstanding problems in mathematics and offered a million dollars to anyone who could prove it. My whole life as a mathematician has been dominated by the Poincar.8e conjecture, John Morgan, the head of the mathematics department at Columbia University, said. I never thought I'd see a solution. I thought nobody could touch it. Grigory Perelman did not plan to become a mathematician. There was never a decision point, he said when we met. We were outside the apartment building where he lives, in Kupchino, a neighborhood of drab high-rises. Perelman's father, who was an electrical engineer, encouraged his interest in math. He gave me logical and other math problems to think about, Perelman said. He got a lot of books for me to read. He taught me how to play chess. He was proud of me. Among the books his father gave him was a copy of Physics for Entertainment, which had been a best-seller in the Soviet Union in the nineteen-thirties. In the foreword, the book's author describes the contents as conundrums, brain-teasers, entertaining anecdotes, and unexpected comparisons, adding, I have quoted extensively from Jules Verne, H. G. Wells, Mark Twain and other writers, because, besides providing entertainment, the fantastic experiments these writers describe may well serve as instructive illustrations at physics classes. The book's topics included how to jump from a moving car, and why, according to the law of buoyancy, we would never drown in the Dead Sea. The notion that Russian society considered worthwhile what Perelman did for pleasure came as a surprise. By the time he was fourteen, he was the star performer of a local math club. In 1982, the year that Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and the gold medal at the International Mathematical Olympiad, in Budapest. He was friendly with his teammates but not close-I had no close friends, he said. He was one of two or three Jews in his grade, and he had a passion for opera, which also set him apart from his peers. His mother, a math teacher at a technical college, played the violin and began taking him to the opera when he was six. By the time Perelman was fifteen, he was spending his pocket money on records. He was thrilled to own a recording of a famous 1946 performance of La Traviata, featuring Licia Albanese as Violetta. Her voice was very good, he said. At Leningrad University, which Perelman entered in 1982, at the age of sixteen, he took advanced classes in geometry and solved a problem posed by Yuri Burago, a mathematician at the Steklov Institute, who later became his Ph.D. adviser. There are a lot of students of high ability who speak before thinking, Burago said. Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully. Burago added, He was not fast. Speed means nothing. Math doesn't depend on speed. It is about deep. At the Steklov in the early nineties, Perelman became an expert on the geometry of Riemannian and Alexandrov spaces-extensions of leading Russian and American mathematics journals. In 1992, Perelman was invited to spend a semester each at New York University and Stony Brook University. By the time he left for the United States, that fall, the Russian economy had collapsed. Dan Stroock, a mathematician at M.I.T., recalls smuggling wads of dollars into the country to deliver to a retired mathematician at the Steklov, who, like many of his colleagues, had become destitute. Perelman was pleased to be in the United States, the capital of the international mathematics community. He wore the same brown corduroy jacket every day and told friends at N.Y.U. that he lived on a diet of bread, cheese, and milk. He liked to walk to Brooklyn, where he had relatives and could buy traditional Russian brown bread. Some of his colleagues were taken aback by his fingernails, which were several inches long. If they grow, why wouldn't I let them grow? he would say when someone asked why he didn't cut them. Once a week, he and a young Chinese mathematician named Gang Tian drove to Princeton, to attend a seminar at the Institute for Advanced Study. For several decades, the institute and nearby Princeton University had been centers of topological research. In the late seventies, William Thurston, a Princeton mathematician who liked to test out his ideas using scissors and construction paper, proposed a taxonomy for classifying manifolds of three dimensions. He argued that, while the manifolds could be made to take on many different shapes, they nonetheless had a preferred geometry, just as a piece of silk draped over a dressmaker's mannequin takes on the mannequin's form. Thurston proposed that every three-dimensional manifold could be broken down into one or more of eight types of component, including a spherical type. Thurston's theory-which became known as the geometrization conjecture-describes all possible three-dimensional manifolds and is thus a powerful generalization of the Poincar.8e. If it was confirmed, then Poincar.8e's conjecture would be, too. Proving Thurston and Poincar.8e definitely swings open doors, Barry Mazur, a mathematician at Harvard, said. The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. This is a kind of twentieth-century Pythagorean theorem, Mazur added. It changes the landscape. In 1982, Thurston won a Fields Medal for his contributions to topology. That year, Richard Hamilton, a mathematician at Cornell, published a paper on an equation called the Ricci flow, which he suspected could be relevant for solving Thurston's conjecture and thus the Poincar.8e. Like a heat equation, which describes how heat distributes itself evenly through a substance-flowing from hotter to cooler parts of a metal sheet, for example-to create a more uniform temperature, the Ricci flow, by smoothing out irregularities, gives manifolds a more uniform geometry. Hamilton, the son of a Cincinnati doctor, defied the math profession's nerdy stereotype. Brash and irreverent, he rode horses, windsurfed, and had a succession of girlfriends. He treated math as merely one of life's pleasures. At forty-nine, he was considered a brilliant lecturer, but he had published relatively little beyond a students. Perelman had read Hamilton's papers and went to hear him give a talk at the Institute for Advanced Study. Afterward, Perelman shyly spoke to him. I really wanted to ask him something, Perelman recalled. He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton's openness and generosity-it really attracted me. I can't say that most mathematicians act like that. I was working on different things, though occasionally I would think about the Ricci flow, Perelman added. You didn't have to be a great mathematician to see that this would be useful for geometrization. I felt I didn't know very much. I kept asking questions. Shing-Tung Yau was also asking Hamilton questions about the Ricci flow. Yau and Hamilton had met in the seventies, and had become close, despite considerable differences in temperament and background. A mathematician at the University of California at San Diego who knows both men called them the mathematical loves of each other's lives. Yau's family moved to Hong Kong from mainland China in 1949, when he was five months old, along with hundreds of thousands of other refugees fleeing Mao's armies. The previous year, his father, a relief worker for the United Nations, had lost most of the family's savings in a series of failed ventures. In Hong Kong, to support his wife and eight children, he tutored college students in classical Chinese literature and philosophy. When Yau was fourteen, his father died of kidney cancer, leaving his mother dependent on handouts from Christian missionaries and whatever small sums she earned from selling handicrafts. Until then, Yau had been an indifferent student. But he began to devote himself to schoolwork, tutoring other students in math to make money. Part of the thing that drives Yau is that he sees his own life as being his father's revenge, said Dan Stroock, the M.I.T. mathematician, who has known Yau for twenty years. Yau's father was like the Talmudist whose children are starving. Yau studied math at the Chinese University of Hong Kong, where he attracted the attention of Shiing-Shen Chern, the pre.91minent Chinese mathematician, who helped him win a scholarship to the University of California at Berkeley. Chern was the author of a famous theorem combining topology and geometry. He spent most of his career in the United States, at Berkeley. He made frequent visits to Hong Kong, Taiwan, and, later, China, where he was a revered symbol of Chinese intellectual achievement, to promote the study of math and science. In 1969, Yau started graduate school at Berkeley, enrolling in seven graduate courses each term and auditing several others. He sent half of his scholarship money back to his mother in China and impressed his professors with his tenacity. He was obliged to share credit for his first major result when he learned that two other mathematicians were working on the same problem. In 1976, he proved a twenty-year-old conjecture pertaining to a type of manifold that is now crucial to string theory. A French mathematician had formulated a proof of the problem, which is known as Calabi's conjecture, but Yau's, because it was more general, was more powerful. (Physicists now refer to Calabi-Yau manifolds.) He was not so much thinking up some original way of looking at a subject but solving extremely hard technical problems that at the time only he could solve, by sheer intellect and force of will, Phillip Griffiths, a geometer and a former director of the Institute for Advanced Study, said. In 1980, when Yau was thirty, he became one of the youngest mathematicians ever to be appointed to the permanent faculty of the Institute for Advanced Study, and he began to attract talented students. He won a Fields Medal two years later, the first Chinese ever to do so. By this time, Chern was seventy years old and on the verge of retirement. According to a relative of Chern's, Yau decided that he was going to be the next famous Chinese mathematician and that it was time for Chern to step down. Harvard had been trying to recruit Yau, and when, in 1983, it was about to make him a second offer Phillip Griffiths told the dean of faculty a version of a story from The Romance of the Three Kingdoms, a Chinese classic. In the third century A.D., a Chinese warlord dreamed of creating an empire, but the most brilliant general in China was working for a rival. Three times, the warlord went to his enemy's kingdom to seek out the general. Impressed, the general agreed to join him, and together they succeeded in founding a dynasty. Taking the hint, the dean flew to Philadelphia, where Yau lived at the time, to make him an offer. Even so, Yau turned down the job. Finally, in 1987, he agreed to go to Harvard. Yau's entrepreneurial drive extended to collaborations with colleagues and students, and, in addition to conducting his own research, he began organizing seminars. He frequently allied himself with brilliantly inventive mathematicians, including Richard Schoen and William Meeks. But Yau was especially impressed by Hamilton, as much for his swagger as for his imagination. I can have fun with Hamilton, Yau told us during the string-theory conference in Beijing. I can go swimming with him. I go out with him and his girlfriends and all that. Yau was convinced that Hamilton could use the Ricci-flow equation to solve the Poincar.8e and Thurston conjectures, and he urged him to focus on the problems. Meeting Yau changed his mathematical life, a friend of both mathematicians said of Hamilton. This was the first time he had been on to something extremely big. Talking to Yau gave him courage and direction. Yau believed that if he could help solve the Poincar.8e it would be a victory not just for him but also for China. In the mid-nineties, Yau and several other Chinese scholars began meeting with President Jiang Zemin to discuss how to rebuild the country's scientific institutions, which had been largely destroyed during the Cultural Revolution. Chinese universities were in dire condition. According to Steve Smale, who won a Fields for proving the Poincar.8e in higher dimensions, and who, after retiring from Berkeley, taught in Hong Kong, Peking University had halls filled with the smell of urine, one common room, one office for all the assistant professors, and paid its faculty wretchedly low salaries. Yau persuaded a Hong Kong real-estate mogul to help finance a mathematics institute at the Chinese Academy of Sciences, in Beijing, and to endow a Fields-style medal for Chinese mathematicians under the age of forty-five. On his trips to China, Yau touted Hamilton and their joint work on the Ricci flow and the Poincar.8e as a model for young Chinese mathematicians. As he put it in Beijing, They always say that the whole country should learn from Mao or some big heroes. So I made a joke to them, but I was half serious. I said the whole country should learn from Hamilton. Grigory Perelman was learning from Hamilton already. In 1993, he began a two-year fellowship at Berkeley. While he was there, Hamilton gave several talks on campus, and in one he mentioned that he was working on the Poincar.8e. Hamilton's Ricci-flow strategy was extremely technical and tricky to execute. After one of his talks at Berkeley, he told Perelman about his biggest obstacle. As a space is smoothed under the Ricci flow, some regions deform into what mathematicians refer to as singularities. Some regions, called necks, become attenuated areas of infinite density. More troubling to Hamilton was a kind of singularity he called the cigar. If cigars formed, Hamilton worried, it might be impossible to achieve uniform geometry. Perelman realized that a paper he had written on Alexandrov spaces might help Hamilton prove Thurston's conjecture-and the Poincar.8e-once Hamilton solved the cigar problem. At some point, I asked Hamilton if he knew a certain collapsing result that I had proved but not published-which turned out to be very useful, Perelman said. Later, I realized that he didn't understand what I was talking about. Dan Stroock, of M.I.T., said, Perelman may have learned stuff from Yau and Hamilton, but, at the time, they were not learning from him. By the end of his first year at Berkeley, Perelman had written several strikingly original papers. He was asked to give a lecture at the 1994 I.M.U. congress, in Zurich, and invited to apply for jobs at Stanford, Princeton, the Institute for Advanced Study, and the University of Tel Aviv. Like Yau, Perelman was a formidable problem solver. Instead of spending years constructing an intricate theoretical framework, or defining new areas of research, he focussed on obtaining particular results. According to Mikhail Gromov, a renowned Russian geometer who has collaborated with Perelman, he had been trying to overcome a technical difficulty relating to Alexandrov spaces and had apparently been stumped. He couldn't do it, Gromov said. It was hopeless. Perelman told us that he liked to work on several problems at once. At Berkeley, however, he found himself returning again and again to Hamilton's Ricci-flow equation and the problem that Hamilton thought he could solve with it. Some of Perelman's friends noticed that he was becoming more and more ascetic. Visitors from St. Petersburg who stayed in his apartment were struck by how sparsely furnished it was. Others worried that he seemed to want to reduce life to a set of rigid axioms. When a member of a hiring committee at Stanford asked him for a C.V. to include with requests for letters of recommendation, Perelman balked. If they know my work, they don't need my C.V., he said. If they need my C.V., they don't know my work. Ultimately, he received several job offers. But he declined them all, and in the summer of 1995 returned to St. Petersburg, to his old job at the Steklov Institute, where he was paid less than a hundred dollars a month. (He told a friend that he had saved enough money in the United States to live on for the rest of his life.) His father had moved to Israel two years earlier, and his younger sister was planning to join him there after she finished college. His mother, however, had decided to remain in St. Petersburg, and Perelman moved in with her. I realize that in Russia I work better, he told colleagues at the Steklov. At twenty-nine, Perelman was firmly established as a mathematician and yet largely unburdened by professional responsibilities. He was free to pursue whatever problems he wanted to, and he knew that his work, should he choose to publish it, would be shown serious consideration. Yakov Eliashberg, a mathematician at Stanford who knew Perelman at Berkeley, thinks that Perelman returned to Russia in order to work on the Poincar.8e. Why not? Perelman said when we asked whether Eliashberg's hunch was correct. The Internet made it possible for Perelman to work alone while continuing to tap a common pool of knowledge. Perelman searched Hamilton's papers for clues to his thinking and gave several seminars on his work. He didn't need any help, Gromov said. He likes to be alone. He reminds me of Newton-this obsession with an idea, working by yourself, the disregard for other people's opinion. Newton was more obnoxious. Perelman is nicer, but very obsessed. In 1995, Hamilton published a paper in which he discussed a few of his ideas for completing a proof of the Poincar.8e. Reading the paper, Perelman realized that Hamilton had made no progress on overcoming his obstacles-the necks and the cigars. I hadn't seen any evidence of progress after early 1992, Perelman told us. Maybe he got stuck even earlier. However, Perelman thought he saw a way around notion, in the hope of collaborating. He did not answer, Perelman said. So I decided to work alone. Yau had no idea that Hamilton's work on the Poincar.8e had stalled. He was increasingly anxious about his own standing in the mathematics profession, particularly in China, where, he worried, a younger scholar could try to supplant him as Chern's heir. More than a decade had passed since Yau had proved his last major result, though he continued to publish prolifically. Yau wants to be the king of geometry, Michael Anderson, a geometer at Stony Brook, said. He believes that everything should issue from him, that he should have oversight. He doesn't like people encroaching on his territory. Determined to retain control over his field, Yau pushed his students to tackle big problems. At Harvard, he ran a notoriously tough seminar on differential geometry, which met for three hours at a time three times a week. Each student was assigned a recently published proof and asked to reconstruct it, fixing any errors and filling in gaps. Yau believed that a mathematician has an obligation to be explicit, and impressed on his students the importance of step-by-step rigor. There are two ways to get credit for an original contribution in mathematics. The first is to produce an original proof. The second is to identify a significant gap in someone else's proof and supply the missing chunk. However, only true mathematical gaps-missing or mistaken arguments-can be the basis for a claim of originality. Filling in gaps in exposition-shortcuts and abbreviations used to make a proof more efficient-does not count. When, in 1993, Andrew Wiles revealed that a gap had been found in his proof of Fermat's last theorem, the problem became fair game for anyone, until, the following year, Wiles fixed the error. Most mathematicians would agree that, by contrast, if a proof's implicit steps can be made explicit by an expert, then the gap is merely one of exposition, and the proof should be considered complete and correct. Occasionally, the difference between a mathematical gap and a gap in exposition can be hard to discern. On at least one occasion, Yau and his students have seemed to confuse the two, making claims of originality that other mathematicians believe are unwarranted. In 1996, a young geometer at Berkeley named Alexander Givental had proved a mathematical conjecture about mirror symmetry, a concept that is fundamental to string theory. Though other mathematicians found Givental's proof hard to follow, they were optimistic that he had solved the problem. As one geometer put it, Nobody at the time said it was incomplete and incorrect. In the fall of 1997, Kefeng Liu, a former student of Yau's who taught at Stanford, gave a talk at Harvard on mirror symmetry. According to two geometers in the audience, Liu proceeded to present a proof strikingly similar to Givental's, describing it as a paper that he had co-authored with Yau and another student of Yau's. Liu mentioned Givental but only as one of a long list of people who had contributed to the field, one of the geometers said. (Liu maintains that his proof was significantly different from Givental's.) Around the same time, Givental received an e-mail signed by Yau and his collaborators, explaining that they had found his arguments impossible to follow and his notation baffling, and had come up with a proof of their own. They praised Givental for his brilliant idea and will be acknowledged. A few weeks later, the paper, Mirror Principle I, appeared in the Asian Journal of Mathematics, which is co-edited by Yau. In it, Yau and his coauthors describe their result as the first complete proof of the mirror conjecture. They mention Givental's work only in passing. Unfortunately, they write, his proof, which has been read by many prominent experts, is incomplete. However, they did not identify a specific mathematical gap. Givental was taken aback. I wanted to know what their objection was, he told us. Not to expose them or defend myself. In March, 1998, he published a paper that included a three-page footnote in which he pointed out a number of similarities between Yau's proof and his own. Several months later, a young mathematician at the University of Chicago who was asked by senior colleagues to investigate the dispute concluded that Givental's proof was complete. Yau says that he had been working on the proof for years with his students and that they achieved their result independently of Givental. We had Around this time, Yau had his first serious conflict with Chern and the Chinese mathematical establishment. For years, Chern had been hoping to bring the I.M.U.'s congress to Beijing. According to several mathematicians who were active in the I.M.U. at the time, Yau made an eleventh-hour effort to have the congress take place in Hong Kong instead. But he failed to persuade a sufficient number of colleagues to go along with his proposal, and the I.M.U. ultimately decided to hold the 2002 congress in Beijing. (Yau denies that he tried to bring the congress to Hong Kong.) Among the delegates the I.M.U. appointed to a group that would be choosing speakers for the congress was Yau's most successful student, Gang Tian, who had been at N.Y.U. with Perelman and was now a professor at M.I.T. The host committee in Beijing also asked Tian to give a plenary address. Yau was caught by surprise. In March, 2000, he had published a survey of recent research in his field studded with glowing references to Tian and to their joint projects. He retaliated by organizing his first conference on string theory, which opened in Beijing a few days before the math congress began, in late August, 2002. He persuaded Stephen Hawking and several Nobel laureates to attend, and for days the Chinese newspapers were full of pictures of famous scientists. Yau even managed to arrange for his group to have an audience with Jiang Zemin. A mathematician who helped organize the math congress recalls that along the highway between Beijing and the airport there were billboards with pictures of Stephen Hawking plastered everywhere. That summer, Yau wasn't thinking much about the Poincar.8e. He had confidence in Hamilton, despite his slow pace. Hamilton is a very good friend, Yau told us in Beijing. He is more than a friend. He is a hero. He is so original. We were working to finish our proof. Hamilton worked on it for twenty-five years. You work, you get tired. He probably got a little tired-and you want to take a rest. Then, on November 12, 2002, Yau received an e-mail message from a Russian mathematician whose name didn't immediately register. May I bring to your attention my paper, the e-mail said. On November 11th, Perelman had posted a thirty-nine-page paper entitled The Entropy Formula for the Ricci Flow and Its Geometric Applications, on arXiv.org, a Web site used by mathematicians to then e-mailed an abstract of his paper to a dozen mathematicians in the United States-including Hamilton, Tian, and Yau-none of whom had heard from him for years. In the abstract, he explained that he had written a sketch of an eclectic proof of the geometrization conjecture. Perelman had not mentioned the proof or shown it to anyone. I didn't have any friends with whom I could discuss this, he said in St. Petersburg. I didn't want to discuss my work with someone I didn't trust. Andrew Wiles had also kept the fact that he was working on Fermat's last theorem a secret, but he had had a colleague vet the proof before making it public. Perelman, by casually posting a proof on the Internet of one of the most famous problems in mathematics, was not just flouting academic convention but taking a considerable risk. If the proof was flawed, he would be publicly humiliated, and there would be no way to prevent another mathematician from fixing any errors and claiming victory. But Perelman said he was not particularly concerned. My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased, he said. I never set out to be the sole solver of the Poincar.8e. Gang Tian was in his office at M.I.T. when he received Perelman's e-mail. He and Perelman had been friendly in 1992, when they were both at N.Y.U. and had attended the same weekly math seminar in Princeton. I immediately realized its importance, Tian said of Perelman's paper. Tian began to read the paper and discuss it with colleagues, who were equally enthusiastic. On November 19th, Vitali Kapovitch, a geometer, sent Perelman an e-mail: Hi Grisha, Sorry to bother you but a lot of people are asking me about your preprint The entropy formula for the Ricci . . . Do I understand it correctly that while you cannot yet do all the steps in the Hamilton program you can do enough so that using some collapsing results you can prove geometrization? Vitali. Perelman's response, the next day, was terse: That's correct. Grisha. In fact, what Perelman had posted on the Internet was only the first installment of his proof. But it was sufficient for mathematicians to see that he had figured out how to solve the Poincar.8e. Barry Mazur, the Harvard mathematician, uses the image of a dented fender to describe Perelman's achievement: Suppose your car has a dented fender and you call a mechanic to ask how to smooth it out. The mechanic would have a hard time telling you what to do over the phone. You would have to bring the car into the garage for him to examine. Then he could tell you where to give it a few knocks. What Hamilton introduced and Perelman completed is a procedure that is independent of the particularities of the blemish. If you apply the Ricci flow to a 3-D space, it will begin to undent it and smooth it out. The mechanic would not need to even see the car-just apply the equation. Perelman proved that the cigars that had troubled Hamilton could not actually occur, and he showed that the neck problem could be solved by performing an intricate sequence of mathematical surgeries: cutting out singularities and patching up the raw edges. Now we have a procedure to smooth things and, at crucial points, control the breaks, Mazur said. Colleagues at Princeton and Stony Brook extended similar invitations. Perelman accepted them all and was booked for a month of lectures Speaking of mathematicians generally, Fedor Nazarov, a mathematician at Michigan State University, said, After you've solved a problem, you have a great urge to talk about it. Hamilton and Yau were stunned by Perelman's announcement. We felt that nobody else would be able to discover the solution, Yau told us in Beijing. But then, in 2002, Perelman said that he published something. He basically did a shortcut without doing all the detailed estimates that we did. Moreover, Yau complained, Perelman's proof was written in such a messy way that we didn't understand. Perelman's April lecture tour was treated by mathematicians and by the press as a major event. Among the audience at his talk at Princeton were John Ball, Andrew Wiles, John Forbes Nash, Jr., who had proved the Riemannian embedding theorem, and John Conway, the inventor of the cellular automaton game Life. To the astonishment of many in the audience, Perelman said nothing about the Poincar.8e. Here is a guy who proved a world-famous theorem and didn't even mention it, Frank Quinn, a mathematician at Virginia Tech, said. He stated some key points and special properties, and then answered questions. He was establishing credibility. If he had beaten his chest and said, 'I solved it,' he would have got a huge amount of resistance. He added, People were expecting a strange sight. Perelman was much more normal than they expected. To Perelman's disappointment, Hamilton did not attend that lecture or the next ones, at Stony Brook. I'm a disciple of Hamilton's, though I haven't received his authorization, Perelman told us. But John Morgan, at Columbia, where Hamilton now taught, was in the audience at Stony Brook, and after a lecture he invited Perelman to speak at Columbia. Perelman, hoping to see Hamilton, agreed. The lecture took place on a Saturday morning. Hamilton showed up late and asked no questions during either the long discussion session that followed the talk or the lunch after that. I had the impression he had read only the first part of my paper, Perelman said. about Perelman's proof: Many experts, although not all, seem convinced that Perelman has stubbed out the cigars and tamed the narrow necks. But they are less confident that he can control the number of surgeries. That could prove a fatal flaw, Yau warns, noting that many other attempted proofs of the Poincar.8e conjecture have stumbled over similar missing steps. Proofs should be treated with skepticism until mathematicians have had a chance to review them thoroughly, Yau told us. Until then, he said, it's not math-it's religion. By mid-July, Perelman had posted the final two installments of his proof on the Internet, and mathematicians had begun the work of formal explication, painstakingly retracing his steps. In the United States, at least two teams of experts had assigned themselves this task: Gang Tian (Yau's rival) and John Morgan; and a pair of researchers at the University of Michigan. Both projects were supported by the Clay Institute, which planned to publish Tian and Morgan's work as a book. The book, in addition to providing other mathematicians with a guide to Perelman's logic, would allow him to be considered for the Clay Institute's million-dollar prize for solving the Poincar.8e. (To be eligible, a proof must be published in a peer-reviewed venue and withstand two years of scrutiny by the mathematical community.) Petersburg, he received a long e-mail from Tian, who said that he had just attended a two-week workshop at Princeton devoted to Perelman's proof. I think that we have understood the whole paper, Tian Perelman did not write back. As he explained to us, I didn't worry too much myself. This was a famous problem. Some people needed time to get accustomed to the fact that this is no longer a conjecture. I personally decided for myself that it was right for me to stay away from verification and not to participate in all these meetings. It is important for me that I don't influence this process. In July of that year, the National Science Foundation had given nearly a million dollars in grants to Yau, Hamilton, and several students of Yau's to study and apply Perelman's breakthrough. An entire branch of mathematics had grown up around efforts to solve the Poincar.8e, and now that branch appeared at risk of becoming obsolete. Michael Freedman, who won a Fields for proving the Poincar.8e conjecture for the fourth dimension, told the Times that Perelman's proof was a small sorrow for this particular branch of topology. Yuri Burago said, It kills the field. After this is done, many mathematicians will move to other branches of mathematics. Five months later, Chern died, and Yau's efforts to insure that he--not Tian-was recognized as his successor turned vicious. It's all about their primacy in China and their leadership among the expatriate Chinese, Joseph Kohn, a former chairman of the Prince-ton mathematics department, said. Yau's not jealous of Tian's mathematics, but he's jealous of his power back in China. Though Yau had not spent more than a few months at a time on mainland China since he was an infant, he was convinced that his status as the only Chinese Fields Medal winner should make him Chern's successor. In a speech he gave at Zhejiang University, in Hangzhou, during the When I stepped out from the airplane, I touched the soil of Beijing and felt great joy to be in my mother country, he said. I am proud to say that when I was awarded the Fields Medal in mathematics, I held no passport of any country and should certainly be considered Chinese. The following summer, Yau returned to China and, in a series of interviews with Chinese reporters, attacked Tian and the mathematicians ACADEMIC CORRUPTION IN CHINA, Yau called Tian a complete mess. He accused him of holding multiple professorships and of collecting a hundred and twenty-five thousand dollars for a few months' work at a Chinese university, while students were living on a hundred dollars a month. He also charged Tian with shoddy scholarship and plagiarism, and with intimidating his graduate students into letting him add his name to their papers. Since I promoted him all the way to his academic fame today, I should also take responsibility for his improper behavior, Yau was quoted as saying to a reporter, explaining why he felt obliged to speak out. In another interview, Yau described how the Fields committee had passed Tian over in 1988 and how he had lobbied on Tian's behalf with various prize committees, including one at the National Science Foundation, which awarded Tian five hundred thousand dollars in 1994. Tian was appalled by Yau's attacks, but he felt that, as Yau's former student, there was little he could do about them. His accusations were baseless, Tian told us. But, he added, I have deep roots in Chinese culture. A teacher is a teacher. There is respect. It is very hard for me to think of anything to do. While Yau was in China, he visited Xi-Ping Zhu, a prot.8eg.8e of his who was now chairman of the mathematics department at Sun Yat-sen tour in the United States, Yau had recruited Zhu and another student, Huai-Dong Cao, a professor at Lehigh University, to undertake an explication of Perelman's proof. Zhu and Cao had studied the Ricci flow under Yau, who considered Zhu, in particular, to be a mathematician of exceptional promise. We have to figure out whether Perelman's paper holds together, Yau told them. Yau arranged for Zhu to spend the 2005-06 academic year at Harvard, where he gave a seminar on Perelman's proof and continued to work on his paper with Cao. On April 13th of this year, the thirty-one mathematicians on the editorial board of the Asian Journal of Mathematics received a brief e-mail from Yau and the journal's co-editor informing them that they had three days to comment on a paper by Xi-Ping Zhu and Huai-Dong Cao titled The Hamilton-Perelman Theory of Ricci Flow: The Poincar.8e and Geometrization Conjectures, which Yau planned to publish in the journal. The e-mail did not include a copy of the paper, reports from referees, or an abstract. At least one board member asked to see the paper but was told that it was not available. On April 16th, Cao received a message from Yau telling him that the paper had been accepted by the A.J.M., and an abstract was posted on the journal's Web site. A month later, Yau had lunch in Cambridge with Jim Carlson, the president of the Clay Institute. He told Carlson that he wanted to trade a copy of Zhu and Cao's paper for a copy of Tian and Morgan's book manuscript. Yau told us he was worried that Tian would try to steal from Zhu and Cao's work, and he wanted to give each party simultaneous access to what the other had written. I had a lunch with Carlson to request to exchange both manuscripts to make sure that nobody can copy the other, Yau said. Carlson demurred, explaining that the Clay Institute had not yet received Tian and Morgan's complete manuscript. By the end of the following week, the title of Zhu and Cao's paper on the A.J.M.'s Web site had changed, to A Complete Proof of the Poincar.8e and Geometrization Conjectures: Application of the Hamilton-Perelman Theory of the Ricci Flow. The abstract had also been revised. A new sentence explained, This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. Zhu and Cao's paper was more than three hundred pages long and filled the A.J.M.'s entire June issue. The bulk of the paper is devoted to reconstructing many of Hamilton's Ricci-flow results-including results that Perelman had made use of in his proof-and much of Perelman's proof of the Poincar.8e. In their introduction, Zhu and Cao credit Perelman with having brought in fresh new ideas to figure out important steps to overcome the main obstacles that remained in the program of Hamilton. However, they write, they were obliged to substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program. Mathematicians familiar with Perelman's proof disputed the idea that Zhu and Cao had contributed significant new approaches to the Poincar.8e. Perelman already did it and what he did was complete and correct, John Morgan said. I don't see that they did anything different. By early June, Yau had begun to promote the proof publicly. On June 3rd, at his mathematics institute in Beijing, he held a press conference. The acting director of the mathematics institute, attempting to explain the relative contributions of the different mathematicians who had worked on the Poincar.8e, said, Hamilton contributed over fifty per cent; the Russian, Perelman, about twenty-five per cent; and the Chinese, Yau, Zhu, and Cao et al., about thirty per cent. (Evidently, simple addition can sometimes trip up even a mathematician.) Yau added, Given the significance of the Poincar.8e, that Chinese mathematicians played a thirty-per-cent role is by no means easy. It is a very important contribution. On June 12th, the week before Yau's conference on string theory opened in Beijing, the South China Morning Post reported, Mainland mathematicians who helped crack a 'millennium math problem' will present the methodology and findings to physicist Stephen Hawking. . . . Yau Shing-Tung, who organized Professor Hawking's visit and is also Professor Cao's teacher, said yesterday he would present the findings to Professor Hawking because he believed the knowledge would help his research into the formation of black holes. On the morning of his lecture in Beijing, Yau told us, We want our contribution understood. And this is also a strategy to encourage Zhu, who is in China and who has done really spectacular work. I mean, important work with a century-long problem, which will probably have another few century-long implications. If you can attach your name in any way, it is a contribution. E. T. Bell, the author of Men of Mathematics, a witty history of the discipline published in 1937, once lamented the squabbles over priority which disfigure scientific history. But in the days before e-mail, blogs, and Web sites, a certain decorum usually prevailed. In 1881, Poincar.8e, who was then at the University of Caen, had an altercation with a German mathematician in Leipzig named Felix Klein. Poincar.8e had published several papers in which he labelled certain Poincar.8e, pointing out that he and others had done significant work on these functions, too. An exchange of polite letters between Leipzig and Caen ensued. Poincar.8e's last word on the subject was a quote from Goethe's Faust: Name ist Schall und Rauch. Loosely translated, that corresponds to Shakespeare's What's in a name? This, essentially, is what Yau's friends are asking themselves. I find myself getting annoyed with Yau that he seems to feel the need for more kudos, Dan Stroock, of M.I.T., said. This is a guy who did magnificent things, for which he was magnificently rewarded. He won every prize to be won. I find it a little mean of him to seem to be trying to get a share of this as well. Stroock pointed out that, twenty-five years ago, Yau was in a situation very similar to the one Perelman is in today. His most famous result, on Calabi-Yau manifolds, was hugely important for theoretical physics. Calabi outlined a program, Stroock said. In a real sense, Yau was Calabi's Perelman. Now he's on the other side. He's had no compunction at all in taking the lion's share of credit for Calabi-Yau. And now he seems to be resenting Perelman getting credit for completing Hamilton's program. I don't know if the analogy has ever occurred to him. Mathematics, more than many other fields, depends on collaboration. Most problems require the insights of several mathematicians in order to be solved, and the profession has evolved a standard for crediting individual contributions that is as stringent as the rules governing math itself. As Perelman put it, If everyone is honest, it is natural to share ideas. Many mathematicians view Yau's conduct over the Poincar.8e as a violation of this basic ethic, and worry about the damage it has caused the profession. Politics, power, and control have no legitimate role in our community, and they threaten the integrity of our field, Phillip Griffiths said. Perelman likes to attend opera performances at the Mariinsky Theatre, in St. Petersburg. Sitting high up in the back of the house, he can't make out the singers' expressions or see the details of their costumes. But he cares only about the sound of their voices, and he says that the acoustics are better where he sits than anywhere else in the theatre. Perelman views the mathematics community-and much of the larger world-from a similar remove. Before we arrived in St. Petersburg, on June 23rd, we had sent several messages to his e-mail address at the Steklov Institute, hoping to arrange a meeting, but he had not replied. We took a taxi to his apartment building and, reluctant to intrude on his privacy, left a book-a collection of John Nash's papers-in his mailbox, along with a card saying that we would be sitting on a bench in a nearby playground the following afternoon. The next day, after Perelman failed to appear, we left a box of pearl tea and a note describing some of the questions we hoped to discuss with him. We repeated this ritual a third time. Finally, believing that Perelman was out of town, we pressed the buzzer for his apartment, hoping at least to speak with his mother. A woman answered and let us inside. Perelman met us in the dimly lit hallway of the apartment. It turned out that he had not checked his Steklov e-mail address for months, and had not looked in his mailbox all week. He had no idea who we were. We arranged to meet at ten the following morning on Nevsky Prospekt. a four-hour walking tour of the city, commenting on every building and vista. After that, we all went to a vocal competition at the St. Petersburg Conservatory, which lasted for five hours. Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. He mentioned a dispute that he had had years earlier with a collaborator over how to credit the author of a particular proof, and said that he was dismayed by the discipline's lax ethics. It is not people who break ethical standards who are regarded as aliens, he said. It is people like me who are isolated. We asked him whether he had read Cao and Zhu's paper. It is not clear to me what new contribution did they make, he said. Apparently, Zhu did not quite understand the argument and reworked it. As for Yau, Perelman said, I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest. The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. As long as I was not conspicuous, I had a choice, Perelman explained. Either to make some ugly thing-a fuss about the math community's lack of integrity-or, if I didn't do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit. We asked Perelman whether, by refusing the Fields and withdrawing from his profession, he was eliminating any possibility of influencing the discipline. I am not a politician! he replied, angrily. Perelman would not say whether his objection to awards extended to the Clay Institute's million-dollar prize. I'm not going to decide whether to accept the prize until it is offered, he said. Mikhail Gromov, the Russian geometer, said that he understood Perelman's logic: To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness. Others might view Perelman's refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. The ideal scientist does science and cares about nothing else, he said. He wants to live this ideal. Now, I don't think he really lives on this ideal plane. But he wants to. === === Subject: : Re: More on Poincare conjecture mistakes, the proof is authentic. http://en.rian.ru/analysis/20060823/53055987.html === === Subject: : Re: More on Poincare conjecture Those people took NSF grant of about one million dollars just to refine and extend the concise proof by Perelman and make it a book of 400 pages. Now they want to take a larger part of credit for solving the problem... Meanwhile, one can find the following notice in the footnote to the first paper posted by Grisha (http://arxiv.org/abs/math.DG/0211159): I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I'd like to thank everyone who worked to make those opportunities available to me. How is that for fairness ? === === Subject: : Re: lagrange multipliers applied on payoff matrix Sorry, it was a typo | (1/w1) (1/w2) 0 | A = | (1/w1) 0 (1/w3) | | 0 (1/w2) (1/w3) | science student and have not studies such algorothms. Pls suggest some gud references and more hint to the procedure of solution. I am really struck. Regads, shiva. === === Subject: : Re: lagrange multipliers applied on payoff matrix hi, I have been able to solve for the above mentioned conditions. But I am struck for following condition, Here, I have been able to get values of Q as | 0 | | w1/(w1+w2) | | w2/(w1+w2) | but I am not able to solve for P. Actual solution of P is | w1/(w1+w2) | | w2/(w1+w2) | | 0 | Can some help me out. shiva === === Subject: : Re: lagrange multipliers applied on payoff matrix constraints, writing non-strict inequalities as strict, is fairly standard among beginning students. However, strict vs. non-strict is EXTREMELY IMPORTANT, and it is vital to write *exactly* what you mean. The theoretical properties of the two types of problems are very different. Again, the distinction is vital: one type of restriction gives a closed set, leading to the existence of a finite optimal solution, while the other type leads to an open set in which many optimization problems have no solutions at all. R.G. Vickson === === Subject: : Re: An exact 1-D integration challenge - 29 - we mean, a challenge for _Maple_ My hope is that mathematics is, in a sense, infinite. === === Subject: : Conformal Mapping I'm having trouble with the following problem and it's really bothering me: Find a conformal map f from C/[-1, 1] onto C. Suggestions and hints only please. Kyle Czarnecki === === Subject: : Re: Conformal Mapping On Thu, 24 Aug 2006 00:04:17 EDT, Narcoleptic Insomniac First we need to clarify something: When people say conformal mapping they often mean conformal equivalence, although the two are not the same thing. A conformal equivalence is obviously impossible here, right? Assuming you really meant conformal mapping: If g(z) = (z + 1/z)/2 then g almost works; it gives a conformal mapping from C[-1,1] onto C{1}. So you just need a conformal mapping from C{1} onto C. A hint for that: The Little Picard Theorem shows that if f is a nonconstant entire function and f is odd then f(C) = C. So you just need to prove that and then find such an f such that f' never vanishes. For example, ****** === === Subject: : Re: Conformal Mapping On Thu, 24 Aug 2006 04:54:57 -0500, That should have been C{-1,1}. Then the hint below needs to be modified: And the Big Picard Theorem shows that if f is a non-constant odd entire function then f(C) = C and also that f attains every non-zero value infinitely many times. ****** === === Subject: : Re: Conformal Mapping === === Subject: : Re: Conformal Mapping Huh? That's impossible, since C[-1,1] is obviously not simply connected. No, it doesn't quite do that. ****** === === Subject: : Re: Conformal Mapping At first I thought that was false as well (thought it should be an annulus). But what you say is true, in fact if K is any connected compact subset of C containing more than one point then CK is conformally equivalent to a punctured disk. (Proof, just for my own benefit so I'll remember this: to V{0}, where the complement of V in the extended plane is a connected set containing more than one point. Hence V is equivalent to the unit disk.) ****** === === Subject: : Re: Conformal Mapping A conformal mapping is not necessarily a covering map. For example, say g is entire, g(0) = 0 and g'(z) = exp(z^2). The Little Picard Theorem says that g(C) = C, since g is odd, Any non-zero value is attained infintely many times by g, so if f is the restriction of g to C{a} for a suitable a then f is a conformal map from C{a} onto C. ****** === === Subject: : Re: Conformal Mapping This is one of the standard methods for computing the transfinite diameter of a compact set; for example the corresponding quadratic map for the plane with an arc removed, something like f(z)=(1-Rz)/(1-R/z), was used by Widom in his generalisation of Szego's strong limit theorem to circular arcs, which is how I came across these mappings. Incidentally I think this shows there is no conformal mapping (=holomorphic with nowhere vanishing derivative) onto C, because it would give a non simply connected covering of a simply connected space. -- rusty === === Subject: : Re: Conformal Mapping Oops, that should have been from the exterior of the unit circle The domain C[-1,1] is thus conformally equivalent to the punctured unit disc. -- rusty === === Subject: : Re: Conformal Mapping This domain is conformally equivalent to the unit disc or upper half plane, so such a mapping cannot exist. The standard map f(z)=(z+ 1/z)/2 (described for example on page 269 of Nehari's book on Conformal Mapping) gives a 1-1 conformal mapping of the upper/lower half plane onto C[-1,1]. Standard texts on conformal mappings were prepared during the second world war, because conformal mapping was used in aerodynamics. In german there is von Koppenfels and Stallmann's Praxis der Konformen Abbildung; and in english Kober's Dictionary of Conformal Representations prepared for the British Admiralty between 1944 and 1948 and later published by Dover. (von Koppenfels did not survive the war.) -- rusty === === Subject: : Re: Latin Square I don't think your original statement is true without the assumption of associativity. Basically, (G,*) defines a magma/groupoid. Giving it a multiplication table that is a Latin square makes it a quasigroup (you basically define division). Endowing a quasigroup with a two-sided inverse makes it a loop. A loop (or a quasigroup) need not have the left inverse equal to the right inverse. For example, the Latin square 12345 24153 35421 41532 53214 defines a loop where x_1 is the identity element. But we have that x_2 * x_3 = x_3 * x_5 = x_4 * x_2 = x_5 * x_4 = x_1 = e (If you assume associativity, it is just a standard exercise to show that monoids becomes groups when endowed with a good multiplication table.) If A is symmetric, without associativity: The if direction: A is a Latin square, so it defines a quasi-group structure on (G,*), in particular we can solve a * x = b and y * a = b for given a, b. Since A is symmetric, x = y in above. Letting b = e and we are done. The only if direction doesn't seem to work. Suppose the multiplication tables is 123 212 321 It satisfies the assumptions that x_1 is an identity, and that every element has a two-sided inverse (themselves). Without associativity, the existence of inverses doesn't guarantee a division. Ohm3 === === Subject: : Re: Latin Square would have been symmetric in any case, but your example with the 5x5 matrix shows that it is not so. This is an exercise from Rotman's An Introduction to the Theory of Groups, 4th edition, page 18, exercise 1.42 ii). Apparently, associativity is not mentioned... === === Subject: : Re: Latin Square That's interesting, I just checked the book, and associativity is indeed not-assumed. I think Rotman made a mistake there. Since you discovered this issue, you should try writing him an e-mail to let him know of the counterexample. (Actually, the mistake can be easily made: I was almost tempted to respond to your original post with a proof until I realized that I used associativity implicitly in one of the steps that I skipped over.) Ohm3 === === Subject: : Re: Latin Square Are you saying you believe every element has a two-sided inverse if A is symmetric, even if a isn't a Latin square? Like, e a a a for example? -- === === Subject: : Re: Latin Square No, always mantaining the hypotesis of a matrix that is a LS. === === Subject: : How big is infinity? What if there is a largest natural number, Z? Is there any way we could determine the value of Z? What sort of axioms would be required for a set theory that assumed a largest natural and could such a set theory determine the value of Z? Obviously such a system wouldn't have a powerset axiom. If Z is the largest natural then floor(log2(Z)) is the largest natural that could have a complete powerset. Such a system might have different universes based on which element of the powerset of Log2(Z) are missing. For example, if Z=7 then the powerset of 3 only has 7 elements. Since we know 2^3 really equals 8, a universe where 2^3=7 must be missing some set of 3 or less elements. There might be 8 different universes each missing a different element of 3's powerset. Such a system couldn't assume every natural number has a successor. Such a system would have a largest prime. Even the pairing axiom might have to go because of the problem of the missing subsets. I am sure I am missing some obvious axioms. All axioms suggestions are welcomed and most will be equally ignored. Russell - Integers are an illusion === === Subject: : Re: How big is infinity? In sci.logic, Russell Easterly on Wed, 23 Aug 2006 22:58:54 -0700 Then what is Z+1? This gets silly. [rest snipped] -- #191, ewill3@earthlink.net Windows Vista. Because it's time to refresh your hardware. Trust us. === === Subject: : Re: How big is infinity? I know it's hard to understand Russell Easterly. Suppose we have the pseudocode: int main() { natural n; n = Z; { n = n - 1; } printf(DONE!n); return(0); } Then, is DONE! ever printed? David Bernier === === Subject: : Re: How big is infinity? Then it doesn't exist, obviously. === === Subject: : Re: How big is infinity? Then, equally obviously, Z is not a natural number. Which puts the kibosh on it being the largest natural number, yes? === === Subject: : Re: How big is infinity? Sorry that part is not obvious to me. === === Subject: : Re: How big is infinity? linux) *Of course* that is obvious, if you take any reasonably standard view of the natural numbers. But you provided an axiomatization which might be interesting but is certainly not about what folks mean when they say N. Whatever our formalization of N is, it shouldn't be modeled by {0} and {0,1} and so on! It's pretty non-controversial that, if n is in N, then so is n + 1. Deny this and you're not talking about the natural numbers any more. Whatever they're good for, there are surely enough to count any finite set of stuff. What else could we mean by N than that? -- === === Subject: : Re: How big is infinity? Well, I guess I'm not! I don't see why not. The assertion of the formalization is that the axioms capture various truths concerning the natural numbers, not that it captures all truths. I think you need to be very careful about saying that such and such is included in the meaning of a term. There was for a long time a belief in the ontological argument, that the mere fact of having the concept of God implied that God exists. Nowadays it's pretty much rejected. So while I agree it's pretty non-controversial - this is the stuff of psychology, and we are taught this stuff pretty young - I don't think that it is immediate that what we *mean* by natural numbers ensures that there *exist* natural numbers on and on forever. This would be the stuff of magic. === === Subject: : Re: How big is infinity? Regurgitating set theory and jumping through formalist hoops is not going to help here. The largest number? Here is the answer. A number occurs in an application in which the number is generated. So the largest number is the largest number in the application. Numbers are not transferable between applications. Therefore, the largest number in the count 1, 2, 3, 4, 5, 6. is 6. The largest number is six. If you think that that answer is a joke it is because it does not flatter your inflated conception of a conceptually impoverished mathematics. You will not be able to disprove this. Let your set theorists try. They will not succeed. === === Subject: : Re: How big is infinity? What is the largest number in the count 0, 1, 2, 3, ... then? Or how about the largest number in i+1? === === Subject: : Re: How big is infinity? You actually finished applying a count at three. You then hoped that we would assume that the count continued beyond it - out of sight. Either a) we must unfortunately conclude that a hidden assumed case, or count, is equivalent in truth value to a presented actual case or count, or b) assume that numbers can be split from their generative application and yet still be adequately presented. Neither of these are options. Of the square root of -1, 1, and their 'product', 1 is the only presented (numerically ascertainable) number. In other words, there is nothing for which an application for generating a largest number can be made. === === Subject: : Re: How big is infinity? Ethymologically, in-finite already says not-finite. === === Subject: : Re: How big is infinity? Yes well as noted many times over the course of various threads mathematikers prefer to maintain the term means bigger than any big big . . . when all it really means is undefined, unspecified, indeterminate or not finite. Of course mathematikers are modern mystics in any event and prefer their faith based mathematics to the facts. ~v~~ === === Subject: : Re: How big is infinity? How evil of them! -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === === Subject: : Re: How big is infinity? Check out: http://www.canoo.net/services/Controller?dispatch=wordformation&input=infini t&features=%28Cat+A%29&country=D&lookup=caseInSensitive === === Subject: : Re: How big is infinity? To answer the question posed in the === Subject: line: Pretty Big! - MO === === Subject: : Re: How big is infinity? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) They would hardly be the natural numbers then. -- Richard === === Subject: : Re: How big is infinity? If you can find it, I can add 1 to it. What sort of system would allow adding 1 to all but one number? All mathematics is equally illusional, but those illusions build very real bridges. === === Subject: : Re: How big is infinity? For instance, a sub-system of second-order PA with these axioms: 1/ N0 5/ Induction You can prove a lot with this system, e.g. the Euclidean Algorithm, existence and uniqueness of prime factorization, Quadratic Reciprocity, and perhaps Fermat's Last Theorem. The system can also prove its own consistency. === === Subject: : Re: How big is infinity? to have an member to which a 1 cannot be added. At least if I read Sn,m' correctly as equivalent to Sn= m' . === === Subject: : Re: How big is infinity? linux) No one asked for such a strong result. You asked what would *allow* adding 1 to all but one number. Evidently, this system would allow that. Interpret Sn,m as the *proposition* m is a successor to n. We don't have here an axiom that for every n, there is an m such that Sn,m. Seems to me that a model of this trivial theory is given by Sn,m as the empty relation (i.e., Sn,m is false for every n, m). So this theory doesn't commit one to much. It doesn't commit one to the existence of any successors at all, apparently. Nonetheless, abo says that you can prove rather a lot in the theory. I wouldn't know. -- Jesse F. Hughes Well, you know as soon as you have a new number I will be happy to add it to the list. Don't try those childish tit-for-tat games with me. -- Ross Finlayson on Cantor's theorem. === === Subject: : Re: How big is infinity? Hughes, the number goes on the end of the list. That's what happens when you well-order the reals. Do you propose that there's an end to the list? There is no universe in ZF. There's always a universe so ZF, where everything is a set yet everything is not a set, is inconsistent. There are various considerations of a maximal ordinal of a sort that have properties similar to zero. Where zero is a limit ordinal, in that it has no predecessor, but has a successor as does each of its, there's a possible consideration of a maximal ordinal that has no successor but does have a predecessor as do each of its. Some people use notions of extrapolations of what an infinity is and what its implications of existence are to reason around Goedelian incompleteness. There are lots of ways to approach infinity as a numerical concept. There are even a wide variety of useful ways, most or all of which are not transfinite cardinals. Skolemize, it's countable. Nobody in physics uses transfinite cardinals for anything, is that not so? Measure theory does not count, no pun intended, there are reasonable non-standard measure-theoretic principles available. Instead, various offshoots of real analysis lead to some considerations of what points are in the real ultimate hyperspace, and from that may be derived predictions. When people talk about non-standard real numbers, they're talking about the real numbers. As to what the natural numbers are, non-standard natural numbers are still only the natural numbers. ZF is inconsistent. Ross === === Subject: : Re: How big is infinity? linux) Oh, at the *end* of the list! I get it now! Well-order the reals, people! Quit bellyaching! Ross is waiting! -- No feeling sympathy for mathematicians who start marching with signs like 'Will work for food' in the future... I will not show mercy going forward. I was trained as a soldier in the United States Army after all... We play to win. --James Harris, feel his wrath! === === Subject: : Re: How big is infinity? (axiom 2/ has been changed so that it is corrrect...) That is indeed how I interpretted his comment. You are correct. Correct. In fact, the system has all initial segments {0,1,2,...,n} as models, as well as the standard model. === === Subject: : Re: How big is infinity? I take it that the notation Nx means x is in N. I'm not sure what the notation Sn,m' is supposed to mean. It appears that Sn is the successor of n; apparently, Sn is always defined for n in N. If that's so, which number doesn't have a successor? Dale. === === Subject: : Re: How big is infinity? linux) I think Sn,m means m is a successor of n. -- ...[W]hatever gifts I have, they are mine. And I do fully intend on NOT doing more research, NOT teaching, and NOT doing any number of things that other people may feel they have a right to tell me I should do, as when you had the chance with me, you crapped out. --James S. Harris === === Subject: : Re: How big is infinity? But then in this axiom I read this (and maybe it's just reading comprehension that's messing with me): For all n, for all m, for all m' If (n is in N) & (m is in N) & (m' is in N) & (m' is a successor of n) then m = m'. Or, more succinctly, For all m,m',n in N: and I can't make sense out of it. It seems to state that whenever any number ( m' ) is a successor of another number ( n ), then all numbers m are equal to m'. u As I said, maybe it's just poor reading on my part. Dale === === Subject: : Re: How big is infinity? Sorry, I left out a part: My apologies. === === Subject: : Re: How big is infinity? I wouldn't want to catalog all my omissions, that much that (An)(Ex)(Sn,x) is not assumed, so I understand where this is going. Dale === === Subject: : Re: How big is infinity? Or it could assume that zero is the successor of the largest number. If zero is allowed as a suucessor, a cyclic group. === === Subject: : Re: How big is infinity? But such a group cannot be ordered in any way consistent with the group operation. === === Subject: : Re: How big is infinity? I could take my domain of discourse to be {0,1,2,...,10}, and have sxy if y is the successor of x, +xyz if x+y=z, *xyz if x*y=z, for all x, y, z in my domain of discourse. I then get a theory in which there is a largest element. But most would regard it as pretty obvious that this isn't a theory about the natural numbers. Most would take it as essential to our concept of natural numbers that every natural number has a successor. You can investigate theories in which there is an object with no successors if you want. But what do you think will be the interest in doing so? === === Subject: : Re: How big is infinity? I'd take issue with this. It's not essential to our concept of cow that there are cows. The concept of a cow includes, essentially, only properties of cows, not that there exist cows. Down your path lies the ontological argument - does the existence of the concept of God imply the existence of God? Now sometimes the existence of a thing includes, essentially, the existence of another thing, but only when the second thing is included in the first. For example, the existence of a person implies the existence of that person's brain or mind. But I don't see any way that you can say that the concept of 11 is included in the concept of 3. === === Subject: : Re: How big is infinity? For me, that misses the point that natural numbers are understood as a system as much as they are understood to be individuals. I think that reflects a structuralist point of view. What is key in the concept of 'natural number' is not just that there is 0 as an individual, and 1 as an individual, and so on, but rather that, whatever it is we take natural numbers to be, each one is a part of a Peano system, of which successorship is crucial. For me, at least, key to the concept of natural numbers is that they are, indeed, counting numbers. And I think successorhip is crucial to counting. If you have 0 onto itself, and 1 onto itself, and so on, then you still don't have a system of counting numbers since counting also must permit always counting PAST any given point in a count. MoeBlee === === Subject: : Re: How big is infinity? Then let's see you define the set of natural numbers without reference to successors. They are as much a property of the set of naturals as multiple stomachs are of cowness === === Subject: : Re: How big is infinity? Well, I can define it without the assumption that every number has a successor - which I thought was the point under discussion. I do need to assume the existence of a successor relationship., in which case I can use the ancestral: In this assumption there is no assumption that the successor relation is total, i.e. that every number has a successor. (Note that normally I wouldn't define the natural numbers.) === === Subject: : Re: How big is infinity? What part of this represents the set of natural numbers? All I can see is, at most, the definition of a predicate. === === Subject: : Re: How big is infinity? I think you're working on a different meaning of definition than I. {x : Nx} is (obviously) the set of the natural numbers. === === Subject: : Re: How big is infinity? Read, In this definition... === === Subject: : Re: How big is infinity? But 11 is included (or at least implied) by the combination of the concept of 3 and the concept of successor of 3. === === Subject: : Re: How big is infinity? The fact that every natural number has a successor does not, in itself, imply that any natural numbers exist, any more than the fact that every cow has four legs implies that cows exist. You are reading more into the statement than is actually there. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. === === Subject: : Re: How big is infinity? ... I spend a lot of my time thinking about systems in which there is a largest element whose successor is either itself or the smallest element. I don't confuse them with the natural numbers, or the reals, but they are useful and interesting. Indeed, far, far more arithmetic is done in systems like that than in the actual natural numbers. Patricia === === Subject: : Re: How big is infinity? Just some random thoughts. Assume Z is the largest un-natural number. Z = Z+1 PowerSet(Z) = Z 1/Z = 0? I came up with a way to get Z=Z+1. We usually define successor with addition. Succ(x) = x+1 We can also define successor with multiplication: Succ(x) = x * (1 + 1/x) For example: Succ(1) = 1 * (1 + 1/1) = 2 Succ(2) = 2 * (1+1/2) = 3 ... If Z = Z+1 we can define successor as: Succ(x) = x * (1 + 1/x - 1/Z) Now: Succ(Z) = Z * (1+1/Z-1/Z) = Z Of course, this means: Succ(1) = 1 * (1 + 1/1 - 1/Z) = 1.9999...999 In fact: 1 = 0.999...999 2 = 1.999...998 ... I have no idea how to come up with a system where 1/Z = 0. Russell - 2 many 2 count === === Subject: : Re: How big is infinity? Does Russell ever have any other kind? === === Subject: : Re: How big is infinity? That's not mathematical. That's just vapid and low, mean. There is no universe in ZF. Classes are to sets as models are to theories. I wonder how the translations of the recently recovered reputed Archimedes manuscript proceed. There's only one theory with no axioms. Ross === === Subject: : Re: How big is infinity? There are, of course, all sorts of finite rings whose arithmetic is that of computers, which can do more with them in one day that all humans have done with infinite rings since day 1. === === Subject: : Re: How big is infinity? It's clear enough that an unsigned integer in any of various computer languages is a ring. What is a good way of thinking about signed integer types? Marshall === === Subject: : Re: How big is infinity? Actually it takes a whole set of integers, together with some operations, to form a ring according to the usual meaning of ring. === === Subject: : Re: How big is infinity? Pardon me for omitting details. I meant the signed two's compliment integers (the whole set) along with their 0, 1, + and *, and unary -. Marshall === === Subject: : Re: How big is infinity? But given, for every x, there is a y such that x+y=0, does it REALLY matter whether we conventionally call y -x or 0-x? Patricia === === Subject: : Re: How big is infinity? Although we don't usually call it -x when talking about unsigned numbers, in both 2's complement signed binary and unsigned binary, for any element x there is another element y such that x+y = 0. It is the number that would be -x if we were interpreting the numbers as 2's complement signed. I don't think a ring has to be infinite, or that there is any requirement that -x is not equal to x? What's missing? Patricia === === Subject: : Re: How big is infinity? Well, son of a gun. How about that. I would have sworn than *something* would have broken. If not associativity, then distributivity. But it doesn't appear to. Yes, it's a bit odd that the - element for MIN_VALUE is also MIN_VALUE, but as you say, that doesn't seem to contradict the definition. Wow. Marshall PS. Hi Patricia. Remember me? === === Subject: : Re: How big is infinity? I think the common ones are also rings. For example, consider 2's complement signed binary, such as Java int. Isn't it a commutative ring? Addition is associative and commutative, and has an identity element 0. Multiplication is associative and commutative, and has identity element 1. Multiplication distributes over addition. Indeed, the bit pattern operations for addition and multiplication are identical between unsigned binary and 2's complement signed binary. The differences are in division, remainder, and comparisons. Floating point types are not rings, because rounding error breaks associativity of addition. Patricia === === Subject: : Re: Eigenfunction expansion Here is a course by Anton Zettl on Sturm-Liouville theory that was incorporated into a book in 2005: http://www.math.niu.edu/SL2/papers/slp.pdf The classic books for ODEs are Titchmarsh's Eigenfunction Expansions, Courant & Hilbert, Vol 2 of Dunford and Schwartz, and of course Ince, although his approach via Sturm's theory of zeros might be considered a little old-fashioned these days. There are also many partial treatments in introductory functional analysis texts, such as Dieudonne's Foundations of Modern Analysis, or similar books by Reed & Simon, R.Zimmer or A.Friedman. For PDEs (Laplacian, Dirac operator) standard treatments using Sobolev spaces can be found in M.Taylor's PDE series or F.Warner's book on differentiable manifolds. Another excellent resource is Chapter XVII in Vol III of L.Hormander's treatise on PDEs. -- rusty === === Subject: : Re: Eigenfunction expansion Dave === === Subject: : Re: Eigenfunction expansion I guess you maybe looking for Spectral Theory. Any introductory PDE book should have something on it (if that's where you are coming from). In general, your probably want Fredholm theory and Functional Analysis, I recommend Reed and Simon, /Methods of Modern Mathematical Physics vol 1: Functional Analysis/. If you want *free*, perhaps this one would work: http://www.math.umn.edu/~garrett/m/fun/ (there are plenty of other ones on the internet.) Ohm3 === === Subject: : Re: Eigenfunction expansion Dave === === Subject: : elementary series problem here is a problem from rudin..quite elementary i guess...but somehow couldnt solve it.. prove that if summation of a sequence converges,then summation of the sequence defined such that n th term of this sequence is square root of nth term of above given sequence divided by n. === === Subject: : Re: elementary series problem Hint: Schwarz's inequality for finite sums. Best wishes Torsten. === === Subject: : Re: analysis with difficulty... Torsten Hennig ??: Why so difficult ? By induction on n, you get |f_n(x)|= |int_{0}^{x} f_(n-1)(t) dt| <= int_{0}^{x} |f_(n-1)(t)| dt <= (induction hypothesis) int_{0}^{x} t^(n-1)/(n-1)! * max_{y in [0;1]} |f_0(y)| dt = x^n/n! * max_{y in [0;1]} |f_0(y)|. You should use that sup{x in [0,1]} |f_n(x) - 0| <= 1/n! * max_{y in [0;1]} |f_0(y)| because the Weierstrass test for uniform convergence needs a majorant that does not depend on x. Best wishes Torsten. === === Subject: : Re: analysis with difficulty... yes, thank you very much for your advice. === === Subject: : Re: analysis with difficulty... Another proof is to note that T(f)(x) = int_0^x f(t) dt is a compact linear operator from C[0,1] into C[0,1], and to verify directly that every c ne 0 is in the resolvent of T, by directly solving Tf - c f = g. Thus the spectral radius of T is 0, and this is lim ||T^n||^(1/n). -- Ron Bruck ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === === Subject: : singular value decomposition Hi I have serious problems with finding accurate nullspaces and singular valued decompositions for large dense matrices (e.g. 108 X 108). The results I get from my software package are not consistent, since checking e.g. the nullspaces by a.nullspace(a) I do not get zeromatrices, also not approximately. Is this because the size of the matrix (I can not imagine that but...), or should it be possible to compute such problems with an accuracy of let's say at least 10^(-8)? friends from all countries of the world. Meanwhile, if you like music like me, i suggest this link: www.undergates.zoomshare.com/files/index.htm Write me and Kisses, Cassy You are aware that this group is intended for discussion and dissemination of math-based stuf are you not? You are aware that this post was likely sent to most usenet groups and the OP is not likely to come back and check for replies? And Cassy is probably not an 18 year-old woman from New Zealand? Initiating server query ...ooking up the domain name for IP: 87.28.107.12 The domain name for the IP address is: host12-107-static.28-87-b.business.telecomitalia.it She out of a server in Italy, probably not with hushmail That's easy - just integrate by parts! ****** === === Subject: : Re: The list of all natural numbers don't exist I think the concept completeness is useful to consider about the possibility of the existence of infinite sets. A set must be complete. In any other case, the concept set is not applicable. It has something to do with sets. I will think about a definition. In the given examples the finite case contains there whole structure as subnstructure in the infinite case it doesn't. I can't see a limit other than the structure which contains its structure as substructure. But since the structure doesn't converge I don't see the possibility of applying the concept of limits of the analysis. I would say if there is an antidiagonal in the diagonal proof of Georg Cantor, we can consider the list of reals as complete in that sense, that there are countable infinite many reals in the list. In this case, the antidiagonal would be a defined number. My claim is to say, that there is no defined antidiagonal since there are no complete lists of infinite many objects in any sense. The antidiagonal doesn't exist since a complete list of countable infinite many real numbers doesn't exist. Since every inifinite list is incomplete, it makes no sense to talk about a number, which is pretended not to be in the list. Albrecht S. Storz === === Subject: : Re: The list of all natural numbers don't exist For a set, S, the only issue is whether one can always answer the question of whether an given object is, or is not, a member. If the answer to that question is in the affirmative then then S is as complete as any set needs to be. Where in any axiom system for set theory is complete defined, or even used? What structure? But then that substructure, to becomplete,whatever that may mean, must also contain it structure as a subsubtructure, and so ad infinitum. The only Cantor proof of the uncountability of the reals, is the one called the first proof, which does not use the diagonal method at all. Cantor's diagonal proof, at least as published by Cantor, is a proof that the set of all infinite sequences of two symbols ( like, for example, endless binary strings) is uncountable. You are not talking about Cantor's diagonal proof then. === === Subject: : Re: The list of all natural numbers don't exist You have still not defined what you mean by complete. And you have given no justification whatsover for the statement a set must be complete. Define complete. You are the only one who claims this. You do not need convergence (in the sense of a metric) to define a limit. In general limits of sets do not involve convergence. The obvious limit is a stucture that contains any substructure found in any of the elemets of the sequence and only those substructures. Since no element of the sequence contains an infinite substructure, nor does the obvious limit. So the real numbers would be complete if a subset were complete but since it is not complete they are not complete. You have still not defined complete. Start by listing a few example of things that you think are complete and things that think are not complete. -William Hughes === === Subject: : Re: complex analysis with integral. Hmm. Not recognizing the name John Schutkeker I have a hard time deciding whether this is a silly joke. In case you're serious: No, this has nothing to do with RH. I don't see that it even has _that_ much to do with RH. Because (i) there's nothing subtle about the current question, the answer is very simple from the definitions, and (ii) there's nothing obvious about RH. Regarding (i): By _definition_ integral{0 to 00} e^(-zt) dt You can easily figure out what the integral from 0 to A is, and then it's easy to see for what values of z that converges. ****** === === Subject: : Re: complex analysis with integral. Professionally, I'm still in stealth mode. I hope to publish at least once in '07. Grant deadlines start on 9/27, so wish me luck at finally getting a paycheck by then. If you'd really like to check me out, search for my name in the sci.math archives, and that should bring a list of the topics that interest me. 'Til last month, Riemann hadn't been one of them. There's a very obvious mistake you can make at step one, but I won't bother to bore you with. In the interpretation if the formulae. In This Week's Finds in Mathematical Physics #124, 10/23/98, Baez defines Riemann's zeta function as zeta(s)=sum{n=1,inf}(n^-s) If s=1, the first term in this is 1/n, which is the same as 1/z, if n=z. Now, before you rip me up one side and down the other for confusing an integer with a complex number, I'd just like to point out that the *form* of that term is the same, although the input number, 'n' vs. 'z', is obviously quite disparate. And from what I've been reading lately, the most important thing in all mathematics is the form. But if you're saying that, by picking the first term from an infinite sum and saying Hey this is just like that, I have reached just a little too far, I wouldn't disagree. Perhaps I threw the baby out with the bath water. ;-P But hell, I'm a novice, am just getting my feet wet on Riemann, and boy is it fascinating! I hope the breakthrough comes before 2025, when I will be 65, because I fully expect it to change society. If I understand the issues correctly, when Riemann comes, public key crypto will no longer be a reliable technology. The all the doors will suddenly swing wide open to the hackers again, just like in the old days. === === Subject: : World-wide pi^2/6 mania http://tinylink.com/?wyynmSHXxq -- Clive Tooth www.clivetooth.dk Stock photos: http://submit.shutterstock.com/?ref=61771 === === Subject: : Re: World-wide pi^2/6 mania On Thu, 24 Aug 2006 10:21:24 +0100, The Last Danish Pastry A subtle in-joke about two numbers having or not having a common prime factor no doubt. See also: http://tedlab.mit.edu/~dr/Papers/Rohde-MRCA-two.pdf === === Subject: : Re: World-wide pi^2/6 mania Well, Copenhagen-wide pi^2/6 mania, anyway. === === Subject: : Re: World-wide pi^2/6 mania Oh boy, I'd like to do some complex integration stuff with the lower-right girl, she's cute :-) Maybe we can discuss our favourite numbers! Jeroen === === Subject: : Re: SAT is in P, CLIQUE is in P Proginoskes a .8ecrit : The objective is to prove that NP=P, if you can gives me graphs difficult to find cliques it maximum, or a 3-SAT which is well on satisfiable. If I give the answer, the algorithm deserves a particular intention. === === Subject: : Re: SAT is in P, CLIQUE is in P No, it does not work like that. The algorithm deserves attention only if it is correct. You may have an heuristic, which ---if it indeed tends to work---would be interesting (I guess) but would most certainly not prove that P=NP. In any case, even if you were able to solve challenge instances, that would not prove anything, as an algorithm may well be, say, O(e^n) but with the constant implied by the O extremely small so as to make small instances (where by small I mean not arbitrarily large) approachable. -- m === === Subject: : Re: SAT is in P, CLIQUE is in P Any paper that claims to have a clique algorithm must have a proof that it is correct. Even if the object is to show that the algorithm works for graphs which arise from 3-SAT graphs, a proof needs to be supplied, and Mimouni has not produced anything even remotely resembling a proof. (I'm not singling out Mimouni here, of course; I don't believe that Ibrahim Cahit's Spiral Colorings can be proven to work, even though it appears to be a useful heuristic which works in many contexts.) However, if it can be shown that there is a poly-time algorithm (A) for finding the maximum clique in a 3-SAT graph (a graph which arises from the CLIQUE/3-SAT reduction), then there is a poly-time algorithm for finding the maximum clique in any graph. This is true because of the fact that if it can be shown that any NP-complete problem is in P, then P=NP immediately follows: There is a poly-time reduction for the CLIQUES problem to 3-SAT, for an arbitrary graph G; the question about whether there is a clique of size k in a graph G can be converted into a 3-SAT formula F which is satisfiable iff G has a clique of size k. (This is because CLIQUES is in NP.) This 3-SAT formula F can be converted into a graph H (which is actually a 3-SAT graph)(since CLIQUES has been proven to be NP-complete), and then Algorithm A will supply values of the Boolean variables satisfying F (if some set of values exists), and this set of values will determine whether there is a clique of size k in G. (Once you have a poly-time algorithm for the decision problem as to whether there is a clique of size k in G, it can be extended to a poly-time algorithm to find the size of the largest clique in G, simply by letting k = 1, 2, 3, ..., |V(G)| and seeing how large k can be before you get a NO answer.) --- Christopher Heckman === === Subject: : Re: SAT is in P, CLIQUE is in P ok, I have a question: in is this algorithm where the nonpolynomial phase? For the first version of this algorithm, had given you against example, and i made a second version. Maintaining the algorithm by two part is devised, in the first each time one at a vertex and a finished number of clique and one please know if one can add the vertex in this clique or not (it is polynomial). And if the vertex can be in none of these clique, it (the vertex) will be in new cliques. When one finished one will find a number finished of clique, but not all clique them, then one passes to the second part, here one removes each edges which is in one cliques to find in the first part, therefore one obtains a new graph of small size. And one repeats the algorithm for this new graph, to find the others cliques. === === Subject: : Re: What is the MOST INTERESTING integer you know? I looked at these joys of 239 - here my comments: 239/169 is an approximant of sqrt(2) and what follows (the double equality with 5 arctan-values) are not really specific to 239. Of course the first *is* specific with the part the 7th that I suppressed above, but to add interest to number 239 by this, one must find 7 special - which indeed many people do ... ;-) and at least 7 occurs in the sequence (p_n) defined by p_1=2 and p_(n+1) = 2^p_n - 1 in which p_k is prime for k=1,2,3,4,5 (proven for k=5 by Lucas without computer - in 19th century ! - with his version of the primality test for Mersenne numbers) and p_6 is already so huge that it's much larger than the current largest primes known ;-) In the equality with 5 arctan-values, one can replace 239 by the numerator of any approximant of sqrt(2) of *odd* order (this resctriction comes from the fact that the n-th approximant a_n/b_n is such that a_n^2 - 2 * b_n^2 = (-1)^n and the arctan- formula works only when this is -1) and replace the other 4 arguments of arctan by the numerator and denominator of the two surrounding approximants to get a valid double equality. The thing with 4 squares needed follows from 239 = 7 mod 8 and is valid for any number 8*n+7 (n non-negat. integer) 239 remains a good candidate for a very interesting number, though. === === Subject: : Re: Primitive Pythagorean triangles: An infinite number of them or not? There appeared in the discussion the name of Euclides. I do not know, if the following is his proof, but the answer on the primary question is following: For prime number p = 2, we have the triangle (2,1). For prime number p = 3, we have the triangles (3,1) and (3,2). If n(p) is the number of prime numbers 1 till p, than n(p) is also the number of different triangles, and their sum is (n(p) over 2). The binomial grows faster than n(p). This formula can be easily adjusted, adding the triangle (1,1), or if triangles with the side 1 should not be counted. kunzmilan === === Subject: : Confused about Least Squares WAS: Least squares fitting A dumb least squares formulation for the best fit conic A.x^2 + B.x.y + C.y^2 + D.x + E.y + F = 0 to a set of 2D points gives rise to the following matrix [ xxxx xxxy xxyy xxx xxy xx] [ xxxy xxyy xyyy xxy xyy xy] [ xxyy xyyy yyyy xyy yyy yy] [ xxx xxy xyy xx xy x] [ xxy xyy yyy xy yy y] [ xx xy yy x y 1] Where abcd = sum_i (a_i.b_i.c_i.d_i), and so on: momenta. Solving for the values (A,B,C,D,E,F) results in a zero vector then. Therefore I have chosen F to be constant = - 1. Then the bottom row of the above matrix doesn't play a role then and I find some sensible solutions. But ... I'm not satisfied, because I don't quite understand what's happening. I cannot prove that it's always possible to assume that F = constant and non-zero. Even though I have imposed the _restriction_ that the origin of the coordinate system shall be the midpoint of the points, i.e. x = y = 0 in the above matrix. Worse, I think that F = nonzero constant can only be assumed if one is certain that the above matrix is singular. But it turns out that such is not the case. Please help. Han de Bruijn === === Subject: : Re: Confused about Least Squares I think, that You should simply realize Your problem as follows this eq. in the plane: Where for example for B,D,E =0 A=C we'll have eg. of circle: x^2 + y^2 = -F/A and only if F<0 then for every specyfic values A,B,C,D,E,F such circle will be degenerated to ellipse or moved out from center x;y=0 or this eq.(*) will be false for all range of x;y or there will be few points or only one for to satisfied it. I think, that there are recent ready math. programs for to simulate such eq.(*) and so on You'll be able to understand the influance of A,B,C,D,E,F coefficients and their connection to described by You matrix. Have More Fun Ro-bin === === Subject: : Re: Confused about Least Squares Since you can scale all the coefficients (A,B,C,D,E,F) by a scalar and they still define the same conic, the problem of finding these coefficients for a given conic is not well-defined; there is a one dimensional family of solutions. It is no wonder that the matrix is singular unless you make some assumption. By scaling the coefficients, you can assume that either F = -1 or F = 0. If F = -1 is impossible, the matrix obtained in solving for the coefficients using this assumption will be singular and then F = 0 should be used. take out the trash before replying === === Subject: : Triangles Circumscribed about a Circle dou you know some elementary proof (that is an argument which doesn't use analysis) of the well-known result that the triangle of smallest area among all the triangles circumscribed about a given circle is the equilateral triangle? Maury === === Subject: : Re: Triangles Circumscribed about a Circle I'm not sure what you are looking for exists. However, here is a reference that contains answers to your questions: Chapters 4-6, Maxima and Minima Without Calculus, Ivan Nivan, Dolciani Mathematical Expositions No. 6, MAA, 1981. http://www.amazon.com/gp/product/088385306X/sr=8-1/qid=1156461129/ref=pd_bbs _1/002-1568955-0651239?ie=UTF8 === === Subject: : Partition number Hi! Is there exact form to describe the partition number? (Here, the partition number is such that, for integer N, P(N) is the number of method to construct N by summation several posive integers. For example, 4=1+1+1+1+1=1+1+2=1+3=2+2=4. So P(4)=4) If the partition number is not expressed of the function of N, is it possible to know whether the sum_n^ inf {p(n) / 2^n} converge or diverge? === === Subject: : Re: Partition number Close! If you put a little white space in your equation, like so: 4 = 1+1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4 you might have a better chance of counting up to 5 correctly. It would look even better as 4 = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 3 = 2 + 2 = 4 when you might even see you hadn't counted up to 4 correctly. -- === === Subject: : Re: Partition number :-) === === Subject: : Re: Partition number http://mathworld.wolfram.com/PartitionFunctionP.html -- DW === === Subject: : Re: Partition number Write partitions arranged as follows: 22 221 32 222 33 It is easy to find the rules, how the partitions in this list are formed. The column sums give binomial coefficients, and their sums are 2^(n-1). It seems that the sum_n^ inf {p(n) / 2^n} diverge. kunzmilan === === Subject: : Re: Partition number kunzmilan è.97.bc.bc: No, It converge. This problem is oriented from the other problem that whether the product m=1^inf {1 / {1-{1/2}^m} } converge or diverge. I have used the geometric series of each term, and get the above another problem. But, the sollution say that, the product m=1^inf {1 / {1-{1/2}^m} } converge since the sum 1^inf {1/2^m} converge. I don't understand this logic. Can anyone help me? === === Subject: : Re: Partition number Have you seen a theorem that says that (1 + a_1)(1 + a_2)(1 + a_3)... converges if and only if a_1 + a_2 + a_3 + ... converges? -- === === Subject: : Re: some Chinese books Don't worry. The are people much worse than you! I've met some native speakers of English (from the States) who tried to correct my use of invaluable, suggesting me to replace it with valuable, the reason being that 'invaluable' is the opposite of 'valuable'. I didn't really want to embarrass them, but I told them to look up a dictionary to learn that invaluable indeed means more valuable than valuable. I even taught them why this is so: too precisous to be valued. It's so ironic that a L2-learner is more accurate than the native speakers in this incidence. (And don't forget the Asian's general ability to spell English correctly -- a skill that many Americans can't master for mysterious reasons.) Words like infamous and inflammable are tricky, too! :) -- Lee Sau Dan .97.9b.8e.8d.93.85 ~{@nJX6X~} E-mail: danlee@informatik.uni-freiburg.de Home page: http://www.informatik.uni-freiburg.de/~danlee === === Subject: : Re: some Chinese books What do presently and momentarily mean to you? === === Subject: : Re: some Chinese books No, it does not. It means 'not valuable'. i.e. 'value cannot be assigned to it'. The problem is that the word valuable has strayed from its etymology, and invaluable has not -- because people have far less reason to talk about things being invaluable than valuable. Invaluable is quite a rare word -- there's no reason to use it in speech when priceless is far more readily available. (But someone who learned his English from books might not know that.) What an arrogant prick you must have seemed. Precious is highly emotive, far from a synonym for priceless. What's so ironic is that the L2-speaker _thinks_ he is more accurate than the native speakers. Asians -- you're quite a racist! -- cannot spell English any better than people from any other continent. Why don't you say what you really mean? Not to English-speakers. === === Subject: : Re: some Chinese books Is it really that rare? Personally, I feel that invaluable and priceless have different shades of meaning, and I don't think I'd often consider them interchangeable in actual usage. After all, something can have a value without having a price. I tend to use priceless to refer to the (usually, but not always, monetary) value of objects. An irreplaceable artifact is priceless. I feel much better about a priceless painting than I do about an invaluable painting. Invaluable, on the other hand, I use to refer to something like utility. I would say, His assistance was invaluable, not His assistance was priceless. Of course, that may just be my Texas dialect. I don't have a problem with That's priceless! in reference to a funny story, but it's not something a macho guy like me would actually say. I would have a problem with That's invaluable. :-p In my late twenties I learned a widely applicable spelling rule for the doubling of final consonants before suffixes from a Chinese lady (the same one in my Mrs. Cow story). She had been an English teacher in Taiwan and had taught the rule to her students. She pointed it out to me after noticing my frequent violations in a translation that she was checking. It was a major weakness in my otherwise fairly proficient spelling. I wonder if any of my English teachers were even aware of such a rule. I don't think I missed it in school. In fact, the only general spelling rule I recall from school is the incomplete 'i' before 'e', except after 'c', or when followed by 'g', (which, if followed slavishly, would lead to thier and wierd). My Asian granddaughter's spelling isn't so hot, but I'm sure that it's because she learned it in the States, rather than in Korea. -- Mike Wright http://www.raccoonbend.com === === Subject: : Re: some Chinese books Nu, what's the rule? And does it apply to British or American (e.g. travel(l)er, worship(p)er)? You mean, I before e, except after c, or when sounded like a, as in neighbor and weigh? === === Subject: : Re: some Chinese books Me too. I tend to associate priceless with physical objects that I can see. e.g. a gem, a genuine sculpture by da Vinci, etc. And I tend to use invaluable for abstract things. e.g. an invaluable experience. Long/short vowels? I've been using rote-memorization for that, until after reading about the big vowel shift in the history of English. Soon after that, I could explain why some consonants are doubled and others not when appending suffixes. I learnt this from an English teacher from Britain. I think this rule is quite well known among native speakers. I think that has to do with the norm. East-Asians value correctness very high. We aren't satified by something just good enoungh or works. -- Lee Sau Dan .97.9b.8e.8d.93.85 ~{@nJX6X~} E-mail: danlee@informatik.uni-freiburg.de Home page: http://www.informatik.uni-freiburg.de/~danlee === === Subject: : Re: some Chinese books [...] I'm not sure that it isn't more common than 'priceless'. Nope. That sounds about like the distinction that I make. I might. Yep. [...] Brian === === Subject: : Re: some Chinese books [...] Then you'll need tremendous willpower. Every elementary Thai book I've ever encountered uses some kind of (usually very bad) Romanization as well as the Thai script, at least for the first few chapters. Even though I can read the Thai, my eye always strays to the Roman. -- Richard Herring === === Subject: : Re: some Chinese books I don't think I'll bother with it. I'm quite sure I couldn't learn the Thai writing system without assigning sounds to the letters. My brain just doesn't work that way (when it works at all). -- Mike Wright http://www.raccoonbend.com === === Subject: : Re: some Chinese books One way to gain familiarity with an alphabet is to use PostScript or METAFONT to make a font for it, without worrying about what the characters sound like. I spent a little time doing that with some Sumerian characters before I gave up. When I typeset Japanese using TeX, I make up control sequences that have one of the pronunciations of the character as a mnemonic, but I'm not oriented to those mnemonics. Instead, after the endless routine of looking up characters in a character dictionary, I tend to start remembering the character numbers and the meanings of certain combinations of characters, but not necessarily the sound. Still, you've raised a very interesting question: what to do with oneself besides making or imagining sounds that these symbols are supposed to represent. One possibility is to attenuate the association with sound production. For example, bird calls are hard for humans to produce, so if you have a program that produces different bird calls in response to different letters, you might be taking a step in that direction. A few decades ago, they were predicting something called smellovision, which would enhance visual experiences by releasing certain scents. So, that might be another way to go. I saw something on TV recently about synesthesia. PC's and graphics software have made it possible for some synesthetics to create graphic images of how text appears to them, so that other individuals (including other synesthetics, who don't necessarily have the same responses to letters) can have some idea of their perceptions. Harvard U. Press published a book (I think it might be The mind of a mnemonist) about a famous mnemonist who was also completely synesthetic (the remark, What a crumbly yellow voice you have is attributed to him). When he remembered an arbitrary amount of fairly arbitrary text or information, it turned into a vivid visual experience, as was evident from his description of how it looked to him, almost like a story in which he was walking past varius landmarks. So, I do believe there is more that we can do besides imagine subvocal grunts while reading. The program mentioned the fact that almost all humans, no matter what language they speak, will agree when presented with two drawn figures, one kind out smooth and the other sharp and angular, that if one is to be called a takete and the other a malluma, that these would be the sharp one and the smooth one respectively. The program cited this as evidence that everyone has a certain amount of synesthetic capability. I don't know whether there is anything one can do to cultivate it. There were some experiments I learned about, maybe in Science News or maybe on NPR, in which they demonstrated that when a synesthetic person says that, say, 5 is a red character, that it really does look red to them and that this perception takes place before they can otherwise recognize the character. I think they did this by presenting a lot of 3's and one 5 in a cluster, so that it would have been hard to tell without looking carefully at it that there was a 5. People without this synesthetic perception had a lot of trouble, but the synesthete knew immediately from the red that there was a 5 there. -- Ignorantly, * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not reflect in any way on MIT. Also, I am nowhere near Boston. === === Subject: : Re: some Chinese books But why would I go to so much trouble to do things in what is, for me, an unnatural way? What's the payoff? I can already make lots of vocal sounds, but birdsong isn't included in the inventory. I've studied about ten languages, some more thoroughly than others, four of which don't use any version of the Roman alphabet--or five if you use Chinese characters to write Hokkien. I've always paid some degree of attention to sound, and it's worked pretty well so far. (Notice that I'm not talking about becoming a fluent speaker when going for reading knowledge, just about using pronunciation as part of the basis for learning.) Only with French and Hebrew did I limit my interest to reading knowledge right from the start. I'm sure no French speaker would have understood my French--and I probably couldn't have produced three sentences of conversational French, but my pronunciation was sufficient to give me a hook for translation using dictionaries and grammars. In the case of Hebrew, knowing the pronunciation of the letters lets me see Arabic cognates, which would surely be impossible otherwise. And, if I couldn't pronounce Arabic and Hokkien, how would I spot Arabic and Chinese loanwords in Malay and Indonesian? If I couldn't pronounce Hokkien, Vietnamese, and Japanese, how could I spot Chinese loanwords in Japanese and Vietnamese? Lee Sau Dan has mentioned Classical Chinese as a language which people often translate from, but do not speak. Actually, I don't move my lips when I read Chinese, but I do think in sound and may be subvocalizing. In particular, I don't think that Tang poetry would be quite the same without the sound. In fact, if I could hack it, I'd use Tang Min Hokkien to get closer to the original sounds than Mandarin can. I don't want to leave the impression that I've translated vast quantities of French and Hebrew, or that I spend a lot of time reading Tang poetry. I just did a little French (and Spanish) translation back around 1968-73 when no one else was available, and have only used Hebrew to check the details of Old Testament content. Banjo, guitar, programming, insect photography, and science reading take up just about all of my time these days. There's not much time left for poetry. I don't even paint anymore. :-( On September 2, I'll get to combine a couple of my interests when I go down to Dallas to see Abigail Washburn playing clawhammer banjo and singing in English and Mandarin: It's a good thing she learned to pronounce Mandarin, elst it would be pretty tough to learn those songs. -- Mike Wright http://www.raccoonbend.com === === Subject: : Re: some Chinese books That's why it is a job for a human rather than for a computer program. A human can use a certain amount of judgment in figuring out what what is going on. Sometimes even normal human intelligence is not enough. I have a book entitled (more or less, since I don't have it in front of me), Ambiguite dans japonais ecrit. In Sanskrit, ambiguity due to sandhi is deliberately exploited by poets. There is a poem which, if the sandhi is inverted one way, is a devotional religious poem, and if the sandhi is inverted another way, is a passionate love poem. I find it easier to read kanji than kana, even though I'm familiar with both of the kana syllabaries, because it is a lot easier to figure out where the boundaries are between the words. I would have a lot of trouble with text that was completely transliterated into kana unless the kana were neatly organized into separated words. Learning a language is one thing. Reading a language is something else. Reading is the act of extracting information from written material, no matter how one thinks about it. I am aware of the pronunciation of kana and kanji and I certainly have to use that information when I use a Japanese-English romaji dictionary to try to look up something written in kana or something in kanji whose pronunciation I have to guess at but which I can't find as an entry in a character dictionary. However, the mere fact that I imagine some kind of vocalizations, right or wrong, while reading is a far cry from actually being able to parse spoken language or to conduct a conversation or to read out loud or to improve one's accent. After one has a lot of practice reading, one might have some advantages in learning the spoken language, as well as some disadvantages, but reading is one thing and speaking and listening are something else. The activities are oriented to entirely different media. No one is saying that you can't think about what it sounds like or that such vocalizations don't serve a useful purpose. It's just not the main focus of the activity. Just find something you would move heaven and earth to be able to read and then move heaven and earth to read it. -- Ignorantly, * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not reflect in any way on MIT. Also, I am nowhere near Boston. === === Subject: : Re: some Chinese books The method you use to understand a text seems to be more akin to language decipherment than reading or translating. Reading a text implies that you already know to some extent the language in which the text is written. This is the beauty of being proficient in a second language you pick up a book and can understand it as you read it without grammars and without using dictionaries any more than a native speaker of the language would. Generally speaking , translation of a text from language A to language B involves more than simply transferring factual meanings from A to B. Of course, in scientific texts this is of paramount importance the form of language, apart from some formulaic, stylistic aspects is secondary. So basically you are using any form of clue available to you to unlock the meaning of the text you are interested in. How you may or may not vocalize the symbols on a page would be irrelevant. But you can't really say you are reading or translating in any meaningful sense. === === Subject: : Re: some Chinese books excellent question, mike. it is difficult to speak without actually hearing pronunciation. In fact, I am developing a talking chinese dictionary right now. I am also currently working on a Pinyin Inventory Project that records the voices of native speakers and replays it on demand for certain characters. The hope is to allow several speakers from different dialects to record. - http://www.corgilabs.com/blog/pinyin/ it will be available on this site early next month: - http://www.corgilabs.com/ -eo === === Subject: : Re: some Chinese books It's not easy to be certain without seeing the characters, but: Dictionary of Beijing Colloquialisms Standard Japanese for Sino-Japanese Exchange Modern Chinese Grammar First-Order Logic and First-Order Theory Compendium of Chinese Mental Calculation: Proportional(??) Calculation and Rapid Calculation === === Subject: : Re: some Chinese books That bi may be the character for pen. If that is the case, then bi suan means calculation using pencil & paper. -- Lee Sau Dan .97.9b.8e.8d.93.85 ~{@nJX6X~} E-mail: danlee@informatik.uni-freiburg.de Home page: http://www.informatik.uni-freiburg.de/~danlee === === Subject: : Re: some Chinese books There are only two bisuan entries in the l2-vol Hanyu Da Cidian: .95M.8eZ (pen + calculation) and .95@.8e_ (nose + acid). My money's on the first one. === === Subject: : Re: some Chinese books .94.8a.8eZ could be also be comparison calculation, the kind in which one is to determine which of two or more expression has the highest or lowest value. Tak -- ----------------------------------------------------------------+----- Tak To takto@alum.mit.eduxx --------------------------------------------------------------------^^ [taode takto ~{LU5B~}] NB: trim the xx to get my real email addr === === Subject: : Re: some Chinese books That's what I was thinking when I was looking up bisuan in the ABC Comprehensive Chinese-English Dictionary (John DeFrancis). === === Subject: : Re: some Chinese books -- Ignorantly, * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not reflect in any way on MIT. Also, I am nowhere near Boston. === === Subject: : Re: some Chinese books You mean you don't think it refers to a tingling sensation in the nose (due to sadness)? :-p I had too look up the second one, wondering what kind of acid the nose secretes. I guess .8e_ in this case is closer to sore than to sour. (I was very disappointed to learn that English sore and sour are not descended from the same root. It would have been a wonderful coincidence.) -- Mike Wright http://www.raccoonbend.com === === Subject: : Re: some Chinese books Then how would one translate .90S.8e_? It seem that .8e_ is more like twitching, which leads to soreness in the case of muscles. The sourness causes one to twitch. One's nose twitches when one is about to cry. Tak -- ----------------------------------------------------------------+----- Tak To takto@alum.mit.eduxx --------------------------------------------------------------------^^ [taode takto ~{LU5B~}] NB: trim the xx to get my real email addr === === Subject: : Re: some Chinese books It's hard to answer your concerns because you don't give any information about your own background in Japanese. But let's take your assertion about what is easy and what is difficult. In every language there are different reading levels (in a recent posting, I asked people if there were any objective or official standards for articulating different levels of literacy but got essentially no answers). I've met plenty of anglophone Americans who can't read an English translation of Braudel's Civilization and Capitalism and lots of people who can read advertisements but who can't read legal documents. So, it would not be surprising if I happened to read something I found easier rather than, say, a Japanese edition of Finnegans Wake. On the other hand, once one has read one thing, one is better prepared to read something else, even something more demanding. But to get back to your assertion: Let's test this assertion. Please suggest something written in Japanese that you believe can't be read by someone (me, for example) using dictionaries, grammars and other reference materials. Something short, please, since I don't have a lot of time to put into reading it. Maybe I can't read it. That will prove that I'm not yet at that level of reading Japanese. On the other hand, I've read other things in Japanese. I don't want to identify what I read at this time, but I don't mind saying that I chose my reading material because I found the content interesting, much in the same way that, even though I don't know Chinese, certain things attracted me to the Chinese books I mentioned in my original posting. I believe that motivation is a very important resource in studying any subject and languages are no exception. No stack of dictionaries and grammars will help one read something in an unfamiliar language if one doesn't feel like reading it or if one doesn't feel like doing the work required to actually use the reference materials. But if one is sufficiently motivated and doesn't mind doing a lot of work over a long period of time, one can accomplish a little. And all I wanted was to accomplish a little. I don't recommend this way of studying languages. It works for me because I don't have time to devote myself to doing it right and because this approach let's me work a little at a time over a long period of time without having to worry about whether I remember anything. The yardstick is that I was able to choose materials that I found interesting for some reason and was eventually able to read them and understand them and to write out English translations. Success is to be understood in the sense that I set a very specific task for myself to accomplish and I succeded in carrying it out. It's not a lot, but it is all I wanted to do. And it leaves me with some confidence that I can do it again, at least with some texts. Anyway, suggest something short to read, such that: (1) you think anyone with any claim to literacy, however meager, ought to be able to read it in Japanese; (2) you think a person who doesn't speak the language could not hope to read it by the brute force use of dictionaries and grammars; (3) it is easy (and hopefully free) for me to get a copy, e.g. either I can print it out from a library computer or every library with Japanese books has a copy and I can photocopy the page(s) I need to read. There is a very good chance that I won't be able to read it, particularly since I don't have the same kind of motivation to do so, and since I'll be under the kind of time pressure implicit in carrying on a conversation at the present pace about the merits of this way of doing things. But I don't mind giving it a try and I don't think the experience of trying will do me any harm. Let me mention another experiment of this kind that I did a couple of years ago. I was in Belgium and decided it would be fun to try to do a Flemish cross word puzzle, even though I never studied either the written or the spoken language. Fortunately, the apartment I was in had a crossword puzzle dictionary. It took me a couple of days, but I eventually filled in most of the puzzle. I then showed it to a friend who spoke Flemish and asked him to check it. There were a couple of mistakes but basically what I did was ok. I still don't speak or write Flemish, although I do know some German. -- Ignorantly, * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not reflect in any way on MIT. Also, I am nowhere near Boston. === === Subject: : Re: some Chinese books Of course. And even if you know a language way, you may not be able to comprehend every piece of text you can find. e.g. try to get a credit-card application form and _read the fine print_. Do you really understand the details of the terms written there without difficulty (and without prior legal training)? Even if you read it twice, you may still not understand it completely. Alternatively, try to go to the USPTO web site and read some patents. See how far you can get before getting lost. -- Lee Sau Dan .97.9b.8e.8d.93.85 ~{@nJX6X~} E-mail: danlee@informatik.uni-freiburg.de Home page: http://www.informatik.uni-freiburg.de/~danlee === === Subject: : Re: two real analysis questions I am sorry if this was posted more than once on some newsservers, but I posted it twice, and it never made it to Google, so I am replying to my post to see if that makes it. take out the trash before replying === === Subject: : Re: [CoT]: Homomorphism and Isomorphism Mea Culpa: I guess monoid-homomorphic monoids or group-homomorphic groups G aint what I was meaning.....rather it is the mapping f which Yes, thats right....he is targeting rings..... Ok....I guess I missed the nuance on the specific reference to onto homomorphism (Even then, I think it does sound weird to directly jump to surjective homomorphisms even before speaking about homomorphisms in general!!!). But then, I guess this hidden nuance does clear the cobwebs!!! gotcha!!!! Yes, I get it.....I missed the also inside the paranthesis in LeVeque's note on isomorphism the first time around!!! === === Subject: : Re: [CoT]: Homomorphism and Isomorphism days. My association with the Department is that of an alumnus. Definitely not. Why are you trying to adjectivise the word? The mapping is being characterized as a HOMOMORPHISM (noun, not adjective). [...] [...] Do drop the triple exclamation points. It makes you sound like a teenager at a rave. This is important for you to continue to keep in mind, though you keep ignoring it: LEVEQUE IS NOT TRYING TO TEACH YOU ABSTRACT abstract algebra merely as a convenient language for him to be able to speak about number theory. He has no interest whatsoever, in this book, on the general abstract algebra framework. He has no need of general ring morphisms; the only ring morphisms he will use are surjective. Since he is only interested in surjective ring homomorphisms, there is absolutely no use in him introducing the entire framework and taxonomy of morphisms (monomorphism, epimorphism, embedding, etc). It would only serve to confuse a reader who, if unfamiliar with them will see no use for them, and if familiar will not need to be reminded of them. SUGGESTION: If you are reading a math textbook, and something is not clear to you, you should read it again, carefully. A single missed word can completely change the meaning of the sentence. And if you still don't quite get it, you read it again. And again. And again. And again. If after half a dozen times of reading it over again ask. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === === Subject: : Re: [CoT]: Homomorphism and Isomorphism Have we all had the pleasure of asking on an exam whether G and H are isomorphic, and getting the reply, G is isomorphic, but H isn't.? -- Virtual Community for the Simulation and the Numerical Modeling - Finite Element Analysis FEA Computational Fluid Dynamics CFD Element Analysis FEA - Finite Element Method FEM and Computational Fluid Dynamics CFD for applications of civil and industrial engineering and Virtual Community for the Simulation and the Numerical Modeling http://it.groups.yahoo.com/group/fem-analysis/ Virtual Community for the Wind Engineering and the Aeroelasticity http://it.groups.yahoo.com/group/ingegneriadelvento/ The groups and' formed also from researchers and university teachers. And' very pleasant also your registration to the group! Gabriele Martufi (Wind Engineering Research and Structural Engineering Aeroelasticity) http://gabrielemartufi.altervista.org/ === === Subject: : Linear algebra question I have a question which has puzzled me for some time. Let A be an n*n positive definite symmetric matrix with real entries. By the spectral theorem, there exist matrices P and D such that A = PDP(-1),[where P(-1) stands for P inverse], and D is diagonal. Now look at the inner product (x,Ax), where x is a n-dimensional vector. I would like to prove that (x,Ax) = (x,Dx). Is this relation true? How does one prove it? === === Subject: : Re: Linear algebra question David Ullrich has said the answer is no. However, if you look at the set of all numbers of the form (x,Ax), and the set of all numbers of the form (x,Dx), these sets are equal. The proof goes like: (x,Ax) = (y,Dy) if y = Px. The result is also true in general, where A = PDP^(-1), and D is an upper-triangular matrix. Of course, you need to consider n-dimensional _complex_ vectors in this context. (If you restrict yourself to the complex vectors of norm 1, the set you end up with above is the field of values of A, a.k.a. the classical numerical range.) --- Christopher Heckman === === Subject: : Re: Linear algebra question Uh, yes. Primarily because the answer is no. ****** === === Subject: : Re: Linear algebra question Of course not - look at just about any non-trivial example. If you note that it's possible to take P to be an _orthogonal_ matrix above, so that the inverse of P is the same as the transpose, that shows you how to prove the _correct_ formula relating (x,Ax) to something about P. ****** === === Subject: : Nominate me for Abel Prize please Please nominate me for Abel Prize. http://www.mathematics21.org/binaries/funcoids-reloids.pdf (See also http://www.mathematics21.org/algebraic-general-topology.html) a. It introduces the most general theory in modern topology. b. It hides epsilon-delta notation behind a smart algebra. See all relevant details at http://www.mathematics21.org/abel-prize.html === === Subject: : Re: Nominate me for Abel Prize please ok ok... === === Subject: : Re: Nominate me for Abel Prize please ok ok... === === Subject: : Re: Nominate me for Abel Prize please ok ok... === === Subject: : Re: Nominate me for Abel Prize please Is the Abel prize not for lifetime achievement rather than mathematical bad taste? -- rusty === === Subject: : sampling distribution of the pearson's correlation between two binary variables let X,Y be random variables valued on {0,1}, p(X=1) = r p(Y=1) = s P(X=1,Y=1) = b The true pearson's correlation coefficient is rho = (b - rs) / sqrt(rs(1-r)(1-s)) My question is, what is the sampling distribution for rho after n paried samples of X and Y, given a certain triplet (r,s,b). Exact answers would be welcome, as would approximations valid for -0.3 < rho < 0.5. I believe this answer, or at least the variance of rho, can be found in The advanced theory of statistics by Kendall and Stuart Griffith press 1979 which I unfortunately don't have access to. -David Greenberg === === Subject: : Re: Solving this Equation For N Do your own homework. I already took and passed these courses. And learn to phrase your REQUESTS as such, not as orders. Arturo Magidin === === Subject: : Re: Solving this Equation For N This is starting(?) to smell link some pretty stinky bait. === === Subject: : Re: Solving this Equation For N Give me a break, use spaces in equations. Also learn how to cross post in stead of multi posting. Read about it at http://oakroadsystems.com/genl/unice.htm They're ease to read. For example a = b + xy < c usually gets read while a=b+xy