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Ars Mathematica #page { background: url("http://www.arsmathematica.net/wp-content/themes/default/images/kubrickbgwide.jpg") repeat-y top; border: none; } .recentcomments a{display:inline !important;padding: 0 !important;margin: 0 !important;} Ars Mathematica Dedicated to the mathematical arts. The Art of Mumford December 20th, 2008 by Walt The language of schemes relies on a dramatic extension of the notion of points. David Mumford’s Red Book on Varieties and Schemes is full of drawings that try to communicate these exotic new sorts of points. Lieven Le Bruyn explains. Posted in Uncategorized | 4 Comments » Your Vocabulary Lesson December 9th, 2008 by Walt Aimless websurfing has taught me that mathematics is apodictic. Now you know. Posted in Uncategorized | 6 Comments » The Appeal of Mathematics December 8th, 2008 by Walt I was musing on the fact that I have never heard a psychologically plausible account of the appeal of pure mathematics. (I say “pure” mathematics because I suspect pure and applied mathematics have different sources of appeal). By “psychologically plausible”, I mean one grounded on the psychology of individual mathematicians. Lots of mathematicians have written explanations of the appeal, but most of these are either of the form “Because mathematics is awesome.”, or “Because I’m awesome” While mathematics is awesome, and while I’m willing to grant the premise that I’m awesome pretty much any time it comes up, these explanations lack the kind of specificity I have in mind. One common explanation, for example, is that math is like music, which relies on the presupposition that music is intrinsically valuable, and that math has value by analogy. But why do we like music? What in the psychology of mathematicians makes math seem like music to them? These are harder questions than the original one. Another explanation is that math is challenging, which is a subspecies of the “I’m awesome”. But in what way is mathematics challenging to mathematicians? Mathematicians, as a group, do not strive to be Nietzschean superman endlessly trying to overcome their limitations, so why this particular challenge, rather than the Nathan’s hot dog eating contest, or climbing Everest?There are psychological explanations floating around as stereotypes, most of which are immensely unflattering, but are least examples of the kind of explanation I have in mind. One example is that mathematicians are like the Rain Man in that they just like repetitive tasks like counting or adding. Another example is that mathematicians can’t handle the real world, and so retreat to the safety of the world of numbers. These are both wrong and insulting, but they are at least grounded in the psychology of individual mathematicians. If anyone has a non-wrong explanation, I’d be curious to hear it. Posted in Uncategorized | 35 Comments » Thanks for Spin-Statistics Theorem November 29th, 2008 by Walt Cosmic Variance has a cute little tradition where for each Thanksgiving Day they pick a physics result to be thankful for. This year they pick the spin-statistics theorem, which explains why elementary particles with half-integer spins satisfy the Pauli exclusion principle. Posted in Physics, Uncategorized | No Comments » Physics Books for a Math Ph.D. Student November 18th, 2008 by Walt Slashdot has a thread discussing physics books recommendations for a math Ph.D. student who would like more physical intution into partial differential equations. There are some good suggestions, and many comments that come dangerously close to “You’re a math Ph.D. student and you don’t know what a PDE is?” Posted in Uncategorized | 23 Comments » Afterlife of Particle Accelerators November 13th, 2008 by Walt New Scientist has an interesting article, Where do science supermachines go when they die? that talks about what happens to the pieces of particle accelerators after they are decommissioned (or in the case of the Superconducting Supercollider, never turned on). Posted in Uncategorized | 1 Comment » Elsevier’s Chaos, Solitons, and Fractals November 11th, 2008 by Walt In the comments at n-category cafe, Zoran Skoda presents evidence that the journal Chaos, Solitons, and Fractals, a peer-reviewed journal published by Elsevier, is publishing pseudo-science. John Baez collects more evidence here. The journal is included in some of Elsevier’s journal bundles, so if you are at a school with a big library, you probably have access to the journal in electronic form, and can check it out yourself. Posted in Uncategorized | 20 Comments » What is a Statistic? November 1st, 2008 by Walt From the request thread, I was hoping for a nice easy softball, maybe from an undergraduate or mathematical amateur. Apparently, though, I have finally scared off anyone other than procrastinating professional mathematicians, who want me to actually write the posts I promised.In the comments here I promised a post explaining why most statistics satisfy the Central Limit Theorem. I thought I’d start slowly with an explanation of what a statistic is. A statistic is just something you compute from the data. This definition is so uninteresting that statistics books are a little apologetic about how contentless the definition sounds. (This usage of the term “statistic” was coined by Fisher. There is a cutting quote by Pearson on the terminology that is impossible to Google for, since all I remember is that it’s about the word statistic, and it involves Fisher and Pearson, who are probably the two most famous statisticians.)Probability distributions are mathematical abstractions, while statistics are numbers we compute from actual data. If we believe that we can model that data as if it is generated by a random variable, then we have to relate the statistic to some property of the probability distribution. Usually, we are interested in some property of the underlying distribution, and using statistics to estimate it. For example, we may be interested in the mean of the underlying random variable, which we can approximate by using the mean of data.Approximating the mean of the random variable in this way is a special case of a general technique to compute a property of a random variable. A random sample drawn from a probability distribution can be thought of as a (discrete) probability distribution in its own right. The property for the sample distribution can be used as an estimate of the property for the true distribution — this is known as the plug-in estimate for the property. An analog of the law of large numbers shows that this estimate converges to the true value.Next time: the analogue of the central limit theorem. Posted in Statistics | 9 Comments » Quantum Hyperion October 24th, 2008 by Walt Sean at Cosmic Variance has an interesting article on decoherence called Quantum Hyperion. It describes a paper by Zurek and Paz that calculates that if Saturn’s moon Hyperion were an isolated system, then within twenty years it would evolve into a non-localized quantum state. It is only the interaction with the outside world that keeps Hyperion looking like a moon. Posted in Uncategorized | 1 Comment » Lévy processes revisited October 22nd, 2008 by Walt I’ve been thinking about Lévy processes, a topic that I mentioned once before. A Lévy process is a generalization of both Brownian motion and a Poisson process. Brownian motion and Poisson processes are both continuous-time stochastic processes but have very different behavior. A Brownian motion follows a very jagged path that is almost always continuous. A Poisson process stays at one place for a long time, and then suddenly jumps to a new place. What they have in common is that changes over two disjoint time intervals are independent of each other, and if the two time intervals are the same the changes have the exact same distribution.Lévy processes include generalizations such as various combinations of Brownian motions and a Poisson process, but they also include more exotic possibilities. The sample path of a combination of a Brownian motion and a Poisson process will almost always have only finite number of discontinuities. In general, Lévy processes can generate sample paths with infinitely many jump discontinuities in almost every interval. Over a finite time horizon, this can give rise to fait-tailed distributions such as the Cauchy distribution. Posted in Uncategorized | 1 Comment » « Older Entries January 2009 M T W T F S S « Dec 1234 567891011 12131415161718 19202122232425 262728293031 The Art of Mumford Your Vocabulary Lesson The Appeal of Mathematics Thanks for Spin-Statistics Theorem Physics Books for a Math Ph.D. Student Recent Links Tagged With "analytic" - JabberTags on Rigid Analytic GeometryJonathan Vos Post on Your Vocabulary LessonBill Lee on Your Vocabulary LessonJonathan Vos Post on The Appeal of Mathematicsnotedscholar on The Art of Mumford December 2008 November 2008 October 2008 September 2008 August 2008 July 2008 June 2008 May 2008 April 2008 March 2008 February 2008 January 2008 December 2007 November 2007 October 2007 September 2007 August 2007 July 2007 June 2007 May 2007 April 2007 March 2007 February 2007 January 2007 December 2006 November 2006 October 2006 September 2006 August 2006 July 2006 June 2006 May 2006 April 2006 March 2006 February 2006 January 2006 December 2005 November 2005 October 2005 September 2005 August 2005 July 2005 June 2005 May 2005 Computer science Economics Mathematics Physics site Statistics Uncategorized It’s equal but it’s differentLambda the UltimateLieven Le Bruynn-Category CafeNeighborhood of InfinityNot Even WrongScott AaronsonUnapologetic MathematicianWhat’s new AMS BooksAMS BulletinAMS NoticesArXivCiteSeerCosma ShaliziJohn BaezMathWorldMSRI BooksPlanet MathStanford Encyclopedia of PhilosophyWikipedia Log in Entries RSS Comments RSS WordPress.org To contact us, email Walt or Robbie (or webmaster) at arsmath.net Ars Mathematica is proudly powered by WordPress Entries (RSS) and Comments (RSS). |
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