most recent 30 from http://mathoverflow.net 2013-05-21T13:36:50Z http://mathoverflow.net/feeds/user/498 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams Elisha Peterson 2009-10-28T00:10:17Z 2012-12-15T00:27:05Z

<p>Most math papers have few figures, if any, although sometimes a well-chosen figure can be a tremendous help in understanding mathematical concepts. Does anyone have any examples of notable uses of figures in mathematical writing and/or texts that make great use of figures/diagrams/illustrations?</p>

http://mathoverflow.net/questions/2939/tools-for-organizing-papers Elisha Peterson 2009-10-27T23:41:27Z 2010-08-15T11:45:39Z

<p>Much like a previous question on keeping research notes organized, my question is how people keep their pile of papers organized. I've got a stack of about 100 in my office, most of them classifying as "want to read", a couple "have read", and lots in between.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/858#858 Elisha Peterson 2009-10-17T11:37:50Z 2009-11-30T05:20:24Z

<p>Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.</p>

http://mathoverflow.net/questions/6139/how-can-i-learn-about-doing-linear-algebra-with-trace-diagrams/6203#6203 Elisha Peterson 2009-11-20T01:10:15Z 2009-11-20T01:10:15Z

<p>The best resource I can point a beginner to is the first few chapters of Stedman's book "Group Theory". He focuses on the specific example of 3-vector diagrams, and does a good job of including lots of sample calculations. Unfortunately, it's not available online. I have found Cvitanovic' book fascinating but tough to internalize. You might also try looking at some of the work of Jim Blinn (a compilation of his work is at <a href="http://research.microsoft.com/pubs/79791/UsingTensorDiagrams.pdf" rel="nofollow">http://research.microsoft.com/pubs/79791/UsingTensorDiagrams.pdf</a>), which has a lot of examples worked out. Another text that I commonly referred to is "The classical and quantum 6j-symbols" (<a href="http://books.google.com/books?id=mg8ISMd5mO0C" rel="nofollow">http://books.google.com/books?id=mg8ISMd5mO0C</a>), although this is limited to a special case of the diagrams.</p> <p>As for learning about the diagrams themselves, I think the only way to really get comfortable with it is to work out lots of examples. I filled endless chalkboards at the University of Maryland with the doodles... it's one of the fun parts of the subject. :)</p> <p>The paper you mentioned is focused on the applications of diagrams to ideas in traditional linear algebra. I have not found any other source that focuses exclusively on this use of diagrams, although Cvitanovic' book (for example) mentions without proof that one of his equations corresponds to the Cayley-Hamilton Theorem. This is probably because many mathematicians do not see much use in reproving old results (particularly if one must learn new notation to do so). I personally feel that there is sufficient beauty and elegance (once the notation is understood) in diagrammatic proofs of these "old proofs" to make them interesting. I also think that a deeper understanding of diagrammatic techniques is a worthy goal in itself. Others have mentioned some of the existing applications.</p> <p>The term "trace diagrams" originated in my thesis, so you won't find it in many published papers. I use it to mean the particular class of diagrams that are labeled by matrices. There are many other names. I first learned about them in the special case of "spin networks" (a special case), and Penrose has the strongest claim to historical priority, hence "Penrose tensor diagrams".</p>

http://mathoverflow.net/questions/873/short-introduction-to-planar-algebras Elisha Peterson 2009-10-17T13:40:09Z 2009-11-17T06:17:58Z

<p>Are there any good <strong>short</strong> expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.</p>

http://mathoverflow.net/questions/5558/math-vs-social-science/5594#5594 Elisha Peterson 2009-11-15T00:20:32Z 2009-11-15T00:20:32Z

<p>Social scientists have long looked at the connections between people and studied "social networks", e.g. the famous paper <em>The Strength of Weak Ties</em>. There is a huge push currently to infuse this area of social science with more mathematics, and there have been several recent articles in the Notices of the AMS about this "network science".</p>

http://mathoverflow.net/questions/5353/how-to-respond-to-i-was-never-much-good-at-maths-at-school/5593#5593 Elisha Peterson 2009-11-15T00:13:56Z 2009-11-15T00:13:56Z

<p>It seems to me that frequently people will say this because they find mathematics intimidating. I don't think they're trying to be rude.</p> <p>I try to get the idea across that mathematics is not always as complicated as it seems. For instance, the average adult may not know the term "integration", but a child can easily count the number of squares underneath a curve.</p>

http://mathoverflow.net/questions/5209/notions-of-matrix-differentiation Elisha Peterson 2009-11-12T17:41:10Z 2009-11-13T16:55:18Z

<p>There are a few standard notions of matrix derivatives, e.g.</p> <ul> <li>If <em>f</em> is a function defined on the entries of a matrix <em>A</em>, then one can talk about the matrix of partial derivatives of <em>f</em>.</li> <li>If the entries of a matrix are all functions of a scalar <em>x</em>, then it makes sense to talk about the derivative of the matrix as the matrix of derivatives of the entries.</li> </ul> <p>In the second case, it makes sense to talk about higher order derivatives, but in the first example the derivative provides a matrix from a scalar function, so you have to massage it a little bit to define a higher order derivative (e.g. take the trace of the resulting matrix).</p> <p>I was wondering what other notions of matrix differentiation might exist out there, particularly any notions that allow for higher-order differentiation. I am also interested in any connections between various forms of matrix derivatives. As this is related to an undergraduate research project, I am mostly looking for answers that include a minimum of advanced terminology, but a discussion of how more general concepts (e.g. differential forms, matrix exponentials, etc.) relate to matrix derivatives would also be helpful.</p>

http://mathoverflow.net/questions/3997/are-there-any-interesting-connections-between-game-theory-and-algebraic-topology/4101#4101 Elisha Peterson 2009-11-04T15:38:36Z 2009-11-04T15:38:36Z

<p>Sperner's Lemma comes to mind. It can be used to prove various results about fair division (being strongly related to the Brouwer Fixed Point Theorem). Ideas from homology can be used to simplify proofs of its generalized form. See Francis Su's papers (<a href="http://www.math.hmc.edu/~su/papers.html" rel="nofollow">http://www.math.hmc.edu/~su/papers.html</a>). </p>

http://mathoverflow.net/questions/4023/text-for-an-introductory-real-analysis-course/4100#4100 Elisha Peterson 2009-11-04T15:27:19Z 2009-11-04T15:27:19Z

<p>I recommend Frank Morgan's <em>Real Analysis</em> for its clarity, the concise chapters, and good exercises. It's much more accessible than Rudin... while I loved learning with Rudin, I don't think it's for everyone.</p>

http://mathoverflow.net/questions/4069/bivectors-in-3-and-4-dimensions/4099#4099 Elisha Peterson 2009-11-04T15:17:47Z 2009-11-04T15:17:47Z

<p>I believe the answer to your question on simplicity is no, e.g. (1,0,0,0) ^ (0,0,1,0) + (0,1,0,0) ^ (0,0,0,1) cannot be written in the form f ^ g.</p>

http://mathoverflow.net/questions/3739/your-experience-of-computer-science-programming-in-mathematics-education/3802#3802 Elisha Peterson 2009-11-02T15:26:09Z 2009-11-02T15:26:09Z

<p>I'm like a few previous responses in having enough CS training to handle basic programming skills. My difficulties have primarily been in mundane/trivial things like compiling, or relearning syntax after years of not using a language.</p> <p>In general, I think sometimes it's easy to overgeneralize and say that all younger students are good at computers. This may be partially true, but there are many younger students that are not at all comfortable with computers, and for which programming does not make sense. Although there is a lot of overlap, programming skill doesn't completely correlate with mathematical skill.</p> <p>Probably the difficulty is mostly in the different point-of-view. CS solutions/programs are built using techniques that differ from math solutions. One must be extremely precise, one must be able to extract a template out of a solution process, one must be able to adapt that template to the programming language. I think that is where the main difficulty (and much of the educational benefit) lies.</p>

http://mathoverflow.net/questions/3366/is-there-a-version-of-temperley-lieb-using-sl3-rather-than-sl2/3367#3367 Elisha Peterson 2009-10-30T01:55:41Z 2009-10-30T02:04:07Z

<p>Yes... I saw it first in Stedman's work (Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990), but it may also exist elsewhere. The basic idea is to combine symmetrization and anti-symmetrization across different sets of strands, roughly in correspondence with the Young tableaux.</p> <p>The other place to look for more general diagrams for general Lie algebras is Cvitanovic (Group Theory: Birdtracks, Lie's, and Exceptional Groups, Predrag Cvitanović, Princeton University Press, 2008, <a href="http://birdtracks.eu/" rel="nofollow">http://birdtracks.eu/</a>). The text is available online and it is extremely impressive.</p>

http://mathoverflow.net/questions/1785/how-do-you-keep-your-research-notes-organized/2954#2954 Elisha Peterson 2009-10-28T00:17:46Z 2009-10-28T00:17:46Z

<p>A completely different approach I have (hence a different answer) is to maintain a private wiki... I keep lots of partially written notes on the subjects I'm learning about or on partial approaches in new directions. The advantage is being able to access them from anywhere, without worrying about having a hard copy or the right computer. I organize them in lots of ways (tags, categories, etc.), can search through them easily, and have a history of any changes made. (The site I use is www.wikidot.com)</p>

http://mathoverflow.net/questions/1785/how-do-you-keep-your-research-notes-organized/2938#2938 Elisha Peterson 2009-10-27T23:38:29Z 2009-10-27T23:38:29Z

<p>I'm not always great with organization, but one cool thing I have done is scan my handwritten notes and add them to evernote, which has some really cool handwriting recognition software so I can search through them.</p> <p>The other thing I do is lots of notebooks with tab dividers, grouped by project.</p>

http://mathoverflow.net/questions/2487/plateaus-to-watch-out-for/2932#2932 Elisha Peterson 2009-10-27T23:14:33Z 2009-10-27T23:14:33Z

<p>My gut response is to say there are a limitless supply of these plateaus. There is so much out there that even the best mathematicians are limited in what they can understand well. So in terms of specific concept plateaus, well if you're like most of us you'll probably have lots of them, and that's a good thing.</p> <p>In terms of concepts, I think what I found tough was often much clearer after I lost an early misconception. E.g. for a long time I thought the Killing form in Lie algebra was using "killing" as a synonym for "erasing"... I tried to build my understanding around that conception and it didn't work very well (Killing is a name). A lot of "simple" mathematical ideas are known by proper names rather than descriptive terms, so as more of these accumulate you have to rely more on memorization than intuition.</p> <p>Outside of concepts, here's what I found tough:</p> <ul> <li><strong>Transition from coursework to research.</strong> Some people are very good at getting the A when the material is put in front of them, and most textbooks are good at giving you the necessary tools to solve the problems they present. I found the transition to more open-ended problems a significant challenge.</li> <li><strong>Understanding the frontier of a field.</strong> As stated in a previous response, it's tough to get to the frontier of a field. It takes a lot of work, and a lot of time. So graduate school requires a lot of perseverance.</li> </ul>

http://mathoverflow.net/questions/854/closest-grid-square-to-a-point-in-spherical-coordinates/1233#1233 Elisha Peterson 2009-10-19T16:20:03Z 2009-10-19T16:20:03Z

<p>So the question essentially is: given a point A on the sphere, and a curve, what is the closest point on the curve. Standard optimization guarantees this occurs at a point D where the curve is perpendicular to the geodesic AD. (And geodesics are pieces of great circles.)</p> <p>In your case, the curves are limited to theta=constant and phi=constant (parts of the spherical rectangles).</p> <p>If phi=constant, the perpendicular geodesics are all great circles through the north pole, i.e. theta=constant, so the closest point has the same value of theta as A.</p> <p>If the curve is theta=constant, there are a couple of ways to go. You might try Lagrange multipliers or some spherical trigonometry.</p>

http://mathoverflow.net/questions/854/closest-grid-square-to-a-point-in-spherical-coordinates/862#862 Elisha Peterson 2009-10-17T11:55:22Z 2009-10-17T11:55:22Z

<p>A couple of questions to clarify:</p> <ol> <li>Is the point A on the sphere or not?</li> <li>What do you mean by "spherical coordinate distance"? Is it the Euclidean distance when the points are expressed in spherical coordinates, or something else?</li> </ol>

http://mathoverflow.net/questions/812/teaching-statements-for-math-jobs/861#861 Elisha Peterson 2009-10-17T11:48:03Z 2009-10-17T11:48:03Z

<p>FYI, there are lots of examples of teaching statements out there on the web. Do a search for "mathematics teaching statement" and you should get plenty of hits. That said, it's probably better to write your own statement before taking a look at what else is out there... in my mind, it's too easy for teaching statements to sound the same.</p> <p>I would also suggest adding in specifics wherever possible to illustrate your point. Talk about how your "philosophy" translates in the classroom.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/859#859 Elisha Peterson 2009-10-17T11:41:27Z 2009-10-17T11:41:27Z

<p>Real Analysis, by Frank Morgan. The chapters are short and very directed. The proofs are written well. The exercises are well-selected. The book is written at a level accessible for most students.</p>

http://mathoverflow.net/questions/6139/how-can-i-learn-about-doing-linear-algebra-with-trace-diagrams/6203#6203 Elisha Peterson 2009-11-20T02:02:19Z 2009-11-20T02:02:19Z

Yes, I agree that &quot;tensor diagrams&quot; would be better. There is actually a wikipedia page for that, &quot;Penrose's tensor notation&quot; I think. But the flavor of that wikipedia page is a lot different than the one on trace diagrams.

http://mathoverflow.net/questions/5450/cocktail-party-math/5551#5551 Elisha Peterson 2009-11-14T23:58:08Z 2009-11-14T23:58:08Z

I love the pizza illustration!

http://mathoverflow.net/questions/5209/notions-of-matrix-differentiation/5229#5229 Elisha Peterson 2009-11-14T15:53:32Z 2009-11-14T15:53:32Z

This is very useful. Thank you!

http://mathoverflow.net/questions/3559/colloquial-catchy-statements-encoding-serious-mathematics/3615#3615 Elisha Peterson 2009-11-01T20:11:12Z 2009-11-01T20:11:12Z

I've always had issues with this one since I'm not convinced the &quot;stirring&quot; function would be continuous...

http://mathoverflow.net/questions/873/short-introduction-to-planar-algebras/874#874 Elisha Peterson 2009-10-17T16:01:57Z 2009-10-17T16:01:57Z

This looks great. Thanks!